Subgroups — C₇×#22505 #13199721

Order 1792.

1262 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3475	C₂	1	1	yes	maximal, characteristic
2	C₇×D₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1986	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1896	C₂	1	2	yes	maximal, complemented
7	C₇×#1871	C₂	1	2	yes	maximal, complemented
8	C₇×#1826	C₂	1	2	yes	maximal, complemented
9	C₇×#1806	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×#104	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#339	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
13	C₇×#556	C₄	1	1	yes	characteristic, complemented
14	C₇×#555	C₄	1	1	yes	characteristic, complemented
15	C₇×#554	C₄	1	1	yes	characteristic, complemented
16	C₇×#553	C₄	1	2	yes	complemented
17	C₇×#552	—	2	1	no	—
18	C₇×#552	C₄	1	2	yes	complemented
19	C₇×#551	—	2	2	no	—
20	C₇×#547	C₄	1	1	yes	characteristic, complemented
21	C₇×#546	—	2	1	no	—
22	C₇×#533	C₂²	1	1	yes	characteristic
23	C₇×#529	—	2	2	no	—
24	C₇×#529	—	2	1	no	—
25	C₇×#525	C₂²	1	2	yes	—
26	C₄×C₇×#41	—	2	1	no	—
27	C₄×C₇×#41	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₇×D₄	—	2	1	no	—
29	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
30	C₂×C₇×#117	C₂²	1	2	yes	—
31	C₂²×C₄²×C₇	C₂²	1	2	yes	—
32	C₇×#436	—	2	1	no	—
33	C₇×#435	—	2	2	no	—
34	C₇×C₈×D₄	—	2	1	no	—
35	C₇×#423	C₂²	1	1	yes	characteristic
36	C₂×C₇×#104	C₂²	1	1	yes	characteristic
37	C₂×C₇×#104	C₂²	1	1	yes	characteristic
38	C₂×C₇×#104	C₂²	1	2	yes	—
39	C₇×#407	C₂²	1	2	yes	—
40	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
41	C₄×C₇×C₄⋊C₄	—	2	1	no	—
42	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
43	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
44	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
45	C₄³×C₇	C₂²	1	1	yes	characteristic
46	C₇×#339	C₂²	1	2	yes	—
47	C₇×#339	C₂²	1	2	yes	—
48	C₇×#339	C₂²	1	1	yes	characteristic
49	C₇×#339	C₂²	1	2	yes	—
50	C₇×#339	C₂²	1	2	yes	—
51	C₇×#339	C₂²	1	2	yes	—
52	C₇×#339	C₂²	1	1	yes	characteristic
53	#22505	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₇×#41	—	2	1	no	—
55	C₂×C₇×#41	C₂³	1	1	yes	characteristic
56	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₇	—	2	2	no	—
58	C₂³×C₄×C₇	C₂³	1	2	yes	—
59	C₂×C₇×M₁₆	—	2	2	no	—
60	C₂×C₇×M₁₆	—	2	4	no	—
61	C₂×C₇×M₁₆	—	4	2	no	—
62	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₇×M₁₆	C₂³	1	2	yes	—
64	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
65	C₂²×C₇×C₈	—	2	2	no	—
66	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
67	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
68	C₇×#126	—	2	2	no	—
69	C₇×#126	—	2	2	no	—
70	C₇×#126	—	2	2	no	—
71	C₇×#125	—	2	2	no	—
72	C₇×#125	—	2	1	no	—
73	C₇×#125	C₂×C₄	1	2	yes	complemented
74	C₇×#124	—	2	1	no	—
75	C₇×#124	—	2	2	no	—
76	C₇×#123	—	2	2	no	—
77	C₇×#123	—	2	2	no	—
78	C₇×#123	—	2	2	no	—
79	C₇×#123	—	2	2	no	—
80	C₇×C₂²⋊Q₈	—	2	1	no	—
81	C₇×C₂²⋊Q₈	—	2	1	no	—
82	C₇×C₂²⋊Q₈	—	2	1	no	—
83	C₇×C₂²⋊Q₈	—	2	2	no	—
84	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
85	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
86	C₇×#121	—	2	2	no	—
87	C₇×#121	—	2	1	no	—
88	C₇×#121	C₂×C₄	1	2	yes	complemented
89	C₇×#121	C₂×C₄	1	2	yes	complemented
90	C₇×#120	—	2	2	no	—
91	C₄×C₇×Q₈	—	2	1	no	—
92	C₄×C₇×Q₈	—	2	1	no	—
93	C₄×C₇×Q₈	—	2	1	no	—
94	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
95	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
96	C₄×C₇×D₄	—	2	8	no	—
97	C₄×C₇×D₄	—	2	1	no	—
98	C₄×C₇×D₄	—	2	2	no	—
99	C₄×C₇×D₄	—	2	1	no	—
100	C₄×C₇×D₄	—	2	1	no	—
101	C₄×C₇×D₄	—	2	2	no	—
102	C₄×C₇×D₄	—	2	2	no	—
103	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
104	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
105	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
106	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic, complemented
107	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic, complemented
108	C₄×C₇×D₄	D₄	1	1	yes	characteristic
109	C₄×C₇×D₄	D₄	1	2	yes	—
110	C₄×C₇×D₄	D₄	1	1	yes	characteristic
111	C₇×#117	—	2	2	no	—
112	C₇×#117	—	2	2	no	—
113	C₇×#117	—	2	1	no	—
114	C₇×#117	—	2	2	no	—
115	C₇×#117	—	2	1	no	—
116	C₇×#117	C₂×C₄	1	2	yes	complemented
117	C₇×#117	C₂×C₄	1	2	yes	complemented
118	C₇×#117	C₂³	1	1	yes	characteristic
119	C₇×#117	C₂³	1	2	yes	—
120	C₂×C₇×C₄⋊C₄	—	2	1	no	—
121	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	complemented
122	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	complemented
123	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
124	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
125	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
126	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
127	C₂×C₄²×C₇	—	2	1	no	—
128	C₂×C₄²×C₇	—	2	2	no	—
129	C₂×C₄²×C₇	—	2	1	no	—
130	C₂×C₄²×C₇	—	2	4	no	—
131	C₂×C₄²×C₇	—	2	1	no	—
132	C₂×C₄²×C₇	—	2	2	no	—
133	C₂×C₄²×C₇	—	4	2	no	—
134	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₇	C₂³	1	2	yes	—
136	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
138	C₇×C₄⋊C₈	—	2	1	no	—
139	C₇×C₄⋊C₈	—	2	1	no	—
140	C₇×#104	—	2	2	no	—
141	C₇×#104	—	2	2	no	—
142	C₇×#104	—	4	2	no	—
143	C₇×C₂²⋊C₈	—	2	2	no	—
144	C₇×C₂²⋊C₈	—	2	2	no	—
145	C₇×#97	—	2	1	no	—
146	C₄×C₇×C₈	—	2	1	no	—
147	C₇×#95	—	2	1	no	—
148	C₇×#95	—	2	2	no	—
149	#3475	C₂×C₇	1	1	yes	characteristic
150	D₄×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
151	C₄²×D₄	C₂×C₇	1	1	yes	characteristic, complemented
152	#1994	C₂×C₇	1	1	yes	characteristic, complemented
153	#1986	C₂×C₇	1	1	yes	characteristic, complemented
154	#1896	C₂×C₇	1	2	yes	complemented
155	#1871	C₂×C₇	1	2	yes	complemented
156	#1826	C₂×C₇	1	2	yes	complemented
157	#1806	C₂×C₇	1	1	yes	characteristic, complemented
158	C₄×#104	C₂×C₇	1	1	yes	characteristic, complemented
159	C₂×#339	C₂×C₇	1	2	yes	complemented
160	C₂⁴×C₇	C₂×D₄	1	2	yes	—
161	C₇×#41	—	2	1	no	—
162	C₇×#41	—	2	1	no	—
163	C₇×#41	—	2	1	no	—
164	C₇×#41	—	2	1	no	—
165	C₇×#41	—	4	4	no	—
166	C₂×C₇×Q₈	—	2	1	no	—
167	C₂×C₇×Q₈	—	2	1	no	—
168	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₇×D₄	—	2	4	no	—
170	C₂×C₇×D₄	—	2	2	no	—
171	C₂×C₇×D₄	—	2	1	no	—
172	C₂×C₇×D₄	—	2	2	no	—
173	C₂×C₇×D₄	C₂²⋊C₄	1	2	yes	—
174	C₂×C₇×D₄	C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₇×D₄	C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	—	2	1	no	—
178	C₂²×C₄×C₇	—	2	2	no	—
179	C₂²×C₄×C₇	—	2	2	no	—
180	C₂²×C₄×C₇	—	2	2	no	—
181	C₂²×C₄×C₇	—	2	2	no	—
182	C₂²×C₄×C₇	—	2	8	no	—
183	C₂²×C₄×C₇	—	2	2	no	—
184	C₂²×C₄×C₇	—	2	1	no	—
185	C₂²×C₄×C₇	—	2	2	no	—
186	C₂²×C₄×C₇	—	2	2	no	—
187	C₂²×C₄×C₇	—	2	1	no	—
188	C₂²×C₄×C₇	—	2	1	no	—
189	C₂²×C₄×C₇	—	2	1	no	—
190	C₂²×C₄×C₇	—	4	4	no	—
191	C₂²×C₄×C₇	—	4	4	no	—
192	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
193	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
194	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
198	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
200	C₇×M₁₆	—	2	1	no	—
201	C₇×M₁₆	—	2	1	no	—
202	C₇×M₁₆	—	2	2	no	—
203	C₇×M₁₆	—	2	2	no	—
204	C₇×M₁₆	—	2	1	no	—
205	C₇×M₁₆	—	4	4	no	—
206	C₇×M₁₆	#41	1	2	yes	complemented
207	C₂×C₇×C₈	—	2	1	no	—
208	C₂×C₇×C₈	—	2	1	no	—
209	C₂×C₇×C₈	—	2	2	no	—
210	C₂×C₇×C₈	—	4	4	no	—
211	C₇×C₄⋊C₄	—	2	4	no	—
212	C₇×C₄⋊C₄	—	2	2	no	—
213	C₇×C₄⋊C₄	—	2	2	no	—
214	C₇×C₄⋊C₄	—	2	1	no	—
215	C₇×C₄⋊C₄	—	2	2	no	—
216	C₇×C₄⋊C₄	—	2	2	no	—
217	C₇×C₄⋊C₄	—	2	1	no	—
218	C₇×C₄⋊C₄	—	2	1	no	—
219	C₇×C₄⋊C₄	—	2	2	no	—
220	C₇×C₄⋊C₄	—	2	2	no	—
221	C₇×C₄⋊C₄	—	2	1	no	—
222	C₇×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
223	C₇×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
224	C₇×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
225	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
226	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
227	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
228	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
229	C₇×C₂²⋊C₄	—	2	8	no	—
230	C₇×C₂²⋊C₄	—	2	2	no	—
231	C₇×C₂²⋊C₄	—	2	2	no	—
232	C₇×C₂²⋊C₄	—	2	2	no	—
233	C₇×C₂²⋊C₄	—	2	2	no	—
234	C₇×C₂²⋊C₄	—	2	1	no	—
235	C₇×C₂²⋊C₄	—	2	2	no	—
236	C₇×C₂²⋊C₄	—	2	2	no	—
237	C₇×C₂²⋊C₄	—	2	1	no	—
238	C₇×C₂²⋊C₄	—	2	1	no	—
239	C₇×C₂²⋊C₄	—	2	2	no	—
240	C₇×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
241	C₇×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
242	C₇×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
243	C₇×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
244	C₄²×C₇	—	2	4	no	—
245	C₄²×C₇	—	2	2	no	—
246	C₄²×C₇	—	2	1	no	—
247	C₄²×C₇	—	2	2	no	—
248	C₄²×C₇	—	2	1	no	—
249	C₄²×C₇	—	2	1	no	—
250	C₄²×C₇	—	2	2	no	—
251	C₄²×C₇	—	2	1	no	—
252	C₄²×C₇	—	2	1	no	—
253	C₄²×C₇	—	2	2	no	—
254	C₄²×C₇	—	2	2	no	—
255	C₄²×C₇	—	2	1	no	—
256	C₄²×C₇	—	4	4	no	—
257	C₄²×C₇	—	4	2	no	—
258	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
259	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
260	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
261	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
262	C₄²×C₇	#41	1	2	yes	complemented
263	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
264	#556	C₄×C₇	1	1	yes	characteristic, complemented
265	#555	C₄×C₇	1	1	yes	characteristic, complemented
266	#554	C₄×C₇	1	1	yes	characteristic, complemented
267	#553	C₄×C₇	1	2	yes	complemented
268	#552	—	2	1	no	—
269	#552	C₄×C₇	1	2	yes	complemented
270	#551	—	2	2	no	—
271	#547	C₄×C₇	1	1	yes	characteristic, complemented
272	#546	—	2	1	no	—
273	#533	C₂²×C₇	1	1	yes	characteristic
274	#529	—	2	2	no	—
275	#529	—	2	1	no	—
276	#525	C₂²×C₇	1	2	yes	—
277	C₄×#41	—	2	1	no	—
278	C₄×#41	C₂²×C₇	1	1	yes	characteristic
279	C₂×C₄×D₄	—	2	1	no	—
280	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
281	C₂×#117	C₂²×C₇	1	2	yes	—
282	C₂²×C₄²	C₂²×C₇	1	2	yes	—
283	#436	—	2	1	no	—
284	#435	—	2	2	no	—
285	C₈×D₄	—	2	1	no	—
286	#423	C₂²×C₇	1	1	yes	characteristic
287	C₂×#104	C₂²×C₇	1	1	yes	characteristic
288	C₂×#104	C₂²×C₇	1	1	yes	characteristic
289	C₂×#104	C₂²×C₇	1	2	yes	—
290	#407	C₂²×C₇	1	2	yes	—
291	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
292	C₄×C₄⋊C₄	—	2	1	no	—
293	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
294	C₄×C₂²⋊C₄	—	2	2	no	—
295	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
296	C₄³	C₂²×C₇	1	1	yes	characteristic
297	#339	C₂²×C₇	1	2	yes	—
298	#339	C₂²×C₇	1	1	yes	characteristic
299	#339	C₂²×C₇	1	2	yes	—
300	#339	C₂²×C₇	1	2	yes	—
301	#339	C₂²×C₇	1	2	yes	—
302	#339	C₂²×C₇	1	2	yes	—
303	#339	C₂²×C₇	1	1	yes	characteristic
304	C₂³×C₇	—	2	2	no	—
305	C₂³×C₇	—	2	2	no	—
306	C₂³×C₇	—	2	4	no	—
307	C₂³×C₇	—	2	1	no	—
308	C₂³×C₇	—	4	2	no	—
309	C₂³×C₇	C₂×C₂²⋊C₄	1	2	yes	—
310	C₂³×C₇	C₂×C₂²⋊C₄	1	2	yes	—
311	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
312	C₇×Q₈	—	2	1	no	—
313	C₇×Q₈	—	2	1	no	—
314	C₇×Q₈	—	4	2	no	—
315	C₇×Q₈	—	8	1	no	—
316	C₇×D₄	—	2	4	no	—
317	C₇×D₄	—	2	1	no	—
318	C₇×D₄	—	2	1	no	—
319	C₇×D₄	—	2	2	no	—
320	C₇×D₄	—	4	1	no	—
321	C₇×D₄	—	4	1	no	—
322	C₇×D₄	—	4	4	no	—
323	C₇×D₄	—	8	2	no	—
324	C₇×D₄	#104	1	4	yes	complemented
325	C₂×C₄×C₇	—	2	2	no	—
326	C₂×C₄×C₇	—	2	2	no	—
327	C₂×C₄×C₇	—	2	2	no	—
328	C₂×C₄×C₇	—	2	2	no	—
329	C₂×C₄×C₇	—	2	2	no	—
330	C₂×C₄×C₇	—	2	2	no	—
331	C₂×C₄×C₇	—	2	2	no	—
332	C₂×C₄×C₇	—	2	2	no	—
333	C₂×C₄×C₇	—	2	2	no	—
334	C₂×C₄×C₇	—	2	2	no	—
335	C₂×C₄×C₇	—	2	2	no	—
336	C₂×C₄×C₇	—	2	2	no	—
337	C₂×C₄×C₇	—	2	2	no	—
338	C₂×C₄×C₇	—	2	2	no	—
339	C₂×C₄×C₇	—	2	2	no	—
340	C₂×C₄×C₇	—	2	4	no	—
341	C₂×C₄×C₇	—	2	1	no	—
342	C₂×C₄×C₇	—	2	1	no	—
343	C₂×C₄×C₇	—	2	1	no	—
344	C₂×C₄×C₇	—	2	2	no	—
345	C₂×C₄×C₇	—	2	2	no	—
346	C₂×C₄×C₇	—	4	1	no	—
347	C₂×C₄×C₇	—	4	16	no	—
348	C₂×C₄×C₇	—	4	1	no	—
349	C₂×C₄×C₇	—	4	2	no	—
350	C₂×C₄×C₇	—	4	2	no	—
351	C₂×C₄×C₇	—	4	1	no	—
352	C₂×C₄×C₇	—	4	4	no	—
353	C₂×C₄×C₇	—	4	1	no	—
354	C₂×C₄×C₇	—	4	1	no	—
355	C₂×C₄×C₇	—	4	2	no	—
356	C₂×C₄×C₇	—	8	1	no	—
357	C₂×C₄×C₇	—	8	2	no	—
358	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
359	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
360	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
361	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
362	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
363	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
364	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
365	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
366	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
367	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
368	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
369	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
370	C₇×C₈	—	2	2	no	—
371	C₇×C₈	—	4	2	no	—
372	C₇×C₈	—	4	1	no	—
373	C₂×#41	—	2	1	no	—
374	C₂×#41	C₂³×C₇	1	1	yes	characteristic
375	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
376	C₂³×C₄	—	2	2	no	—
377	C₂³×C₄	C₂³×C₇	1	2	yes	—
378	C₂×M₁₆	—	2	2	no	—
379	C₂×M₁₆	—	2	4	no	—
380	C₂×M₁₆	—	4	2	no	—
381	C₂×M₁₆	C₂³×C₇	1	2	yes	—
382	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
383	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
384	C₂²×C₈	—	2	2	no	—
385	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic, complemented
386	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic, complemented
387	#126	—	2	2	no	—
388	#126	—	2	2	no	—
389	#126	—	2	2	no	—
390	#125	—	2	2	no	—
391	#125	—	2	1	no	—
392	#125	C₂×C₄×C₇	1	2	yes	complemented
393	#124	—	2	1	no	—
394	#124	—	2	2	no	—
395	#123	—	2	2	no	—
396	#123	—	2	2	no	—
397	#123	—	2	2	no	—
398	#123	—	2	2	no	—
399	C₂²⋊Q₈	—	2	1	no	—
400	C₂²⋊Q₈	—	2	1	no	—
401	C₂²⋊Q₈	—	2	1	no	—
402	C₂²⋊Q₈	—	2	2	no	—
403	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	complemented
404	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	complemented
405	#121	—	2	2	no	—
406	#121	—	2	1	no	—
407	#121	C₂×C₄×C₇	1	2	yes	complemented
408	#121	C₂×C₄×C₇	1	2	yes	complemented
409	#120	—	2	2	no	—
410	C₄×Q₈	—	2	1	no	—
411	C₄×Q₈	—	2	1	no	—
412	C₄×Q₈	—	2	1	no	—
413	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic, complemented
414	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic, complemented
415	C₄×D₄	—	2	8	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	1	no	—
419	C₄×D₄	—	2	1	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
423	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
424	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
425	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic, complemented
426	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic, complemented
427	C₄×D₄	C₇×D₄	1	1	yes	characteristic
428	C₄×D₄	C₇×D₄	1	2	yes	—
429	C₄×D₄	C₇×D₄	1	1	yes	characteristic
430	#117	—	2	2	no	—
431	#117	—	2	2	no	—
432	#117	—	2	1	no	—
433	#117	—	2	2	no	—
434	#117	—	2	1	no	—
435	#117	C₂×C₄×C₇	1	2	yes	complemented
436	#117	C₂×C₄×C₇	1	2	yes	complemented
437	#117	C₂³×C₇	1	1	yes	characteristic
438	#117	C₂³×C₇	1	2	yes	—
439	C₂×C₄⋊C₄	—	2	1	no	—
440	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	complemented
441	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	complemented
442	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
443	C₂×C₂²⋊C₄	—	2	2	no	—
444	C₂×C₂²⋊C₄	—	2	2	no	—
445	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
446	C₂×C₄²	—	2	2	no	—
447	C₂×C₄²	—	2	1	no	—
448	C₂×C₄²	—	2	1	no	—
449	C₂×C₄²	—	2	4	no	—
450	C₂×C₄²	—	2	1	no	—
451	C₂×C₄²	—	2	2	no	—
452	C₂×C₄²	—	4	2	no	—
453	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
454	C₂×C₄²	C₂³×C₇	1	2	yes	—
455	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
456	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
457	C₄⋊C₈	—	2	1	no	—
458	C₄⋊C₈	—	2	1	no	—
459	#104	—	2	2	no	—
460	#104	—	2	2	no	—
461	#104	—	4	2	no	—
462	C₂²⋊C₈	—	2	2	no	—
463	C₂²⋊C₈	—	2	2	no	—
464	#97	—	2	1	no	—
465	C₄×C₈	—	2	1	no	—
466	#95	—	2	1	no	—
467	#95	—	2	2	no	—
468	C₂²×C₇	—	2	2	no	—
469	C₂²×C₇	—	2	1	no	—
470	C₂²×C₇	—	2	1	no	—
471	C₂²×C₇	—	2	2	no	—
472	C₂²×C₇	—	2	2	no	—
473	C₂²×C₇	—	2	2	no	—
474	C₂²×C₇	—	2	2	no	—
475	C₂²×C₇	—	4	4	no	—
476	C₂²×C₇	—	4	4	no	—
477	C₂²×C₇	—	4	1	no	—
478	C₂²×C₇	—	4	1	no	—
479	C₂²×C₇	—	4	1	no	—
480	C₂²×C₇	C₂×#104	1	4	yes	—
481	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
482	C₂²×C₇	#546	1	1	yes	characteristic
483	C₂²×C₇	#547	1	1	yes	characteristic
484	C₄×C₇	—	2	4	no	—
485	C₄×C₇	—	2	2	no	—
486	C₄×C₇	—	2	2	no	—
487	C₄×C₇	—	2	1	no	—
488	C₄×C₇	—	2	1	no	—
489	C₄×C₇	—	2	1	no	—
490	C₄×C₇	—	4	2	no	—
491	C₄×C₇	—	4	4	no	—
492	C₄×C₇	—	4	1	no	—
493	C₄×C₇	—	4	1	no	—
494	C₄×C₇	—	4	2	no	—
495	C₄×C₇	—	4	1	no	—
496	C₄×C₇	—	4	2	no	—
497	C₄×C₇	C₂×#104	1	2	yes	—
498	C₄×C₇	C₄×#41	1	1	yes	characteristic
499	C₄×C₇	#520	1	1	yes	characteristic
500	C₄×C₇	#546	1	1	yes	characteristic
501	C₄×C₇	#547	1	1	yes	characteristic
502	C₂⁴	C₂×C₇×D₄	1	2	yes	—
503	#41	—	2	1	no	—
504	#41	—	2	1	no	—
505	#41	—	2	1	no	—
506	#41	—	2	1	no	—
507	#41	—	4	4	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	2	no	—
515	C₂×D₄	C₇×C₂²⋊C₄	1	2	yes	—
516	C₂×D₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
517	C₂×D₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
518	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	8	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	4	4	no	—
533	C₂²×C₄	—	4	4	no	—
534	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
535	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
536	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
537	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
538	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
539	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
540	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
541	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
542	M₁₆	—	2	1	no	—
543	M₁₆	—	2	1	no	—
544	M₁₆	—	2	2	no	—
545	M₁₆	—	2	2	no	—
546	M₁₆	—	2	1	no	—
547	M₁₆	—	4	4	no	—
548	M₁₆	C₇×#41	1	2	yes	complemented
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	2	no	—
552	C₂×C₈	—	4	4	no	—
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	2	no	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	—	2	2	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	1	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
565	C₄⋊C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
566	C₄⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
567	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
568	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
569	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
570	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
571	C₂²⋊C₄	—	2	8	no	—
572	C₂²⋊C₄	—	2	2	no	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	2	no	—
575	C₂²⋊C₄	—	2	2	no	—
576	C₂²⋊C₄	—	2	1	no	—
577	C₂²⋊C₄	—	2	2	no	—
578	C₂²⋊C₄	—	2	2	no	—
579	C₂²⋊C₄	—	2	1	no	—
580	C₂²⋊C₄	—	2	1	no	—
581	C₂²⋊C₄	—	2	2	no	—
582	C₂²⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
583	C₂²⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
584	C₂²⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
585	C₂²⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
586	C₄²	—	2	4	no	—
587	C₄²	—	2	2	no	—
588	C₄²	—	2	1	no	—
589	C₄²	—	2	2	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	—	2	2	no	—
593	C₄²	—	2	1	no	—
594	C₄²	—	2	1	no	—
595	C₄²	—	2	2	no	—
596	C₄²	—	2	2	no	—
597	C₄²	—	2	1	no	—
598	C₄²	—	4	4	no	—
599	C₄²	—	4	2	no	—
600	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
601	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
602	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
603	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
604	C₄²	C₇×#41	1	2	yes	complemented
605	C₂×C₇	—	2	1	no	—
606	C₂×C₇	—	2	1	no	—
607	C₂×C₇	—	2	4	no	—
608	C₂×C₇	—	4	2	no	—
609	C₂×C₇	—	8	1	no	—
610	C₂×C₇	#2339	1	1	yes	characteristic
611	C₂×C₇	C₂²×#104	1	1	yes	characteristic
612	C₂×C₇	#2901	1	1	yes	characteristic
613	C₂³	—	2	2	no	—
614	C₂³	—	2	2	no	—
615	C₂³	—	2	4	no	—
616	C₂³	—	2	1	no	—
617	C₂³	—	4	2	no	—
618	C₂³	C₂×C₇×C₂²⋊C₄	1	2	yes	—
619	C₂³	C₂×C₇×C₂²⋊C₄	1	2	yes	—
620	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
621	Q₈	—	2	1	no	—
622	Q₈	—	2	1	no	—
623	Q₈	—	4	2	no	—
624	Q₈	—	8	1	no	—
625	D₄	—	2	4	no	—
626	D₄	—	2	1	no	—
627	D₄	—	2	1	no	—
628	D₄	—	2	2	no	—
629	D₄	—	4	1	no	—
630	D₄	—	4	1	no	—
631	D₄	—	4	4	no	—
632	D₄	—	8	2	no	—
633	D₄	C₇×#104	1	4	yes	complemented
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	4	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	16	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	2	no	—
659	C₂×C₄	—	4	2	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	4	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	4	2	no	—
665	C₂×C₄	—	8	1	no	—
666	C₂×C₄	—	8	2	no	—
667	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
668	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
669	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
670	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
671	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
672	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
673	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
674	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
675	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
676	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
677	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
678	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
679	C₈	—	2	2	no	—
680	C₈	—	4	2	no	—
681	C₈	—	4	1	no	—
682	C₇	#22505	1	1	yes	characteristic, direct product factor, Sylow
683	C₂²	—	2	1	no	—
684	C₂²	—	2	1	no	—
685	C₂²	—	2	2	no	—
686	C₂²	—	2	2	no	—
687	C₂²	—	2	2	no	—
688	C₂²	—	2	2	no	—
689	C₂²	—	2	2	no	—
690	C₂²	—	4	4	no	—
691	C₂²	—	4	4	no	—
692	C₂²	—	4	1	no	—
693	C₂²	—	4	1	no	—
694	C₂²	—	4	1	no	—
695	C₂²	C₂×C₇×#104	1	4	yes	—
696	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
697	C₂²	C₇×#546	1	1	yes	characteristic
698	C₂²	C₇×#547	1	1	yes	characteristic
699	C₄	—	2	4	no	—
700	C₄	—	2	1	no	—
701	C₄	—	2	1	no	—
702	C₄	—	2	1	no	—
703	C₄	—	2	2	no	—
704	C₄	—	2	2	no	—
705	C₄	—	4	4	no	—
706	C₄	—	4	2	no	—
707	C₄	—	4	1	no	—
708	C₄	—	4	2	no	—
709	C₄	—	4	1	no	—
710	C₄	—	4	2	no	—
711	C₄	—	4	1	no	—
712	C₄	C₂×C₇×#104	1	2	yes	—
713	C₄	C₄×C₇×#41	1	1	yes	characteristic
714	C₄	C₇×#520	1	1	yes	characteristic
715	C₄	C₇×#546	1	1	yes	characteristic
716	C₄	C₇×#547	1	1	yes	characteristic
717	C₂	—	2	1	no	—
718	C₂	—	2	1	no	—
719	C₂	—	2	4	no	—
720	C₂	—	4	2	no	—
721	C₂	—	8	1	no	—
722	C₂	C₇×#2339	1	1	yes	characteristic
723	C₂	C₂²×C₇×#104	1	1	yes	characteristic
724	C₂	C₇×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₂²⋊C₄
prime radical	C₇×#3475	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₇×D₄×M₁₆	C₂	semidirect
3. C₄²×C₇×D₄	C₂	semidirect
4. C₇×#1994	C₂	semidirect
5. C₇×#1986	C₂	semidirect
6. C₇×#1896	C₂	semidirect
7. C₇×#1871	C₂	semidirect
8. C₇×#1826	C₂	semidirect
9. C₇×#1806	C₂	semidirect
10. C₄×C₇×#104	C₂	semidirect
11. C₂×C₇×#339	C₂	semidirect
13. C₇×#556	C₄	semidirect
14. C₇×#555	C₄	semidirect
15. C₇×#554	C₄	semidirect
16. C₇×#553	C₄	semidirect
18. C₇×#552	C₄	semidirect
20. C₇×#547	C₄	semidirect
53. #22505	C₇	direct (N × H)
66. C₇×C₄⋊Q₈	C₂×C₄	semidirect
67. C₇×C₄⋊Q₈	C₂×C₄	semidirect
73. C₇×#125	C₂×C₄	semidirect
84. C₇×C₂²⋊Q₈	C₂×C₄	semidirect
85. C₇×C₂²⋊Q₈	C₂×C₄	semidirect
88. C₇×#121	C₂×C₄	semidirect
89. C₇×#121	C₂×C₄	semidirect
94. C₄×C₇×Q₈	C₂×C₄	semidirect
95. C₄×C₇×Q₈	C₂×C₄	semidirect
106. C₄×C₇×D₄	C₂×C₄	semidirect
107. C₄×C₇×D₄	C₂×C₄	semidirect
116. C₇×#117	C₂×C₄	semidirect
117. C₇×#117	C₂×C₄	semidirect
121. C₂×C₇×C₄⋊C₄	C₂×C₄	semidirect
122. C₂×C₇×C₄⋊C₄	C₂×C₄	semidirect
150. D₄×M₁₆	C₂×C₇	semidirect
151. C₄²×D₄	C₂×C₇	semidirect
152. #1994	C₂×C₇	semidirect
153. #1986	C₂×C₇	semidirect
154. #1896	C₂×C₇	semidirect
155. #1871	C₂×C₇	semidirect
156. #1826	C₂×C₇	semidirect
157. #1806	C₂×C₇	semidirect
158. C₄×#104	C₂×C₇	semidirect
159. C₂×#339	C₂×C₇	semidirect
206. C₇×M₁₆	#41	semidirect
262. C₄²×C₇	#41	semidirect
264. #556	C₄×C₇	semidirect
265. #555	C₄×C₇	semidirect
266. #554	C₄×C₇	semidirect
267. #553	C₄×C₇	semidirect
269. #552	C₄×C₇	semidirect
271. #547	C₄×C₇	semidirect
324. C₇×D₄	#104	semidirect
385. C₄⋊Q₈	C₂×C₄×C₇	semidirect
386. C₄⋊Q₈	C₂×C₄×C₇	semidirect
392. #125	C₂×C₄×C₇	semidirect
403. C₂²⋊Q₈	C₂×C₄×C₇	semidirect
404. C₂²⋊Q₈	C₂×C₄×C₇	semidirect
407. #121	C₂×C₄×C₇	semidirect
408. #121	C₂×C₄×C₇	semidirect
413. C₄×Q₈	C₂×C₄×C₇	semidirect
414. C₄×Q₈	C₂×C₄×C₇	semidirect
425. C₄×D₄	C₂×C₄×C₇	semidirect
426. C₄×D₄	C₂×C₄×C₇	semidirect
435. #117	C₂×C₄×C₇	semidirect
436. #117	C₂×C₄×C₇	semidirect
440. C₂×C₄⋊C₄	C₂×C₄×C₇	semidirect
441. C₂×C₄⋊C₄	C₂×C₄×C₇	semidirect
548. M₁₆	C₇×#41	semidirect
604. C₄²	C₇×#41	semidirect
633. D₄	C₇×#104	semidirect
682. C₇	#22505	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3475	2
C₇×D₄×M₁₆	2
C₄²×C₇×D₄	2
C₇×#1994	2
C₇×#1986	2
C₇×#1896	2
C₇×#1871	2
C₇×#1826	2
C₇×#1806	2
C₄×C₇×#104	2
C₂×C₇×#339	2
#22505	7

Subgroups — C7×#22505 #13199721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22505 #13199721