Subgroups — C₇×#21899 #13199115

Order 1792.

1242 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3444	C₂	1	1	yes	maximal, characteristic
2	C₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1986	C₂	1	2	yes	maximal, complemented
5	C₇×#1980	C₂	1	2	yes	maximal, complemented
6	C₇×#1896	C₂	1	4	yes	maximal, complemented
7	C₇×#1806	C₂	1	4	yes	maximal, complemented
8	C₇×#533	C₄	1	1	yes	characteristic, complemented
9	C₇×#533	C₄	1	1	yes	characteristic, complemented
10	C₇×#533	C₂²	1	1	yes	characteristic
11	C₇×#532	C₄	1	1	yes	characteristic, complemented
12	C₇×#532	C₄	1	1	yes	characteristic, complemented
13	C₇×#532	C₂²	1	1	yes	characteristic
14	C₇×#529	—	2	1	no	—
15	C₇×#529	—	2	1	no	—
16	C₇×#529	—	2	1	no	—
17	C₇×#529	—	2	1	no	—
18	C₇×#529	C₄	1	2	yes	complemented
19	C₇×#529	C₄	1	2	yes	complemented
20	C₇×#525	—	2	1	no	—
21	C₇×#525	—	2	1	no	—
22	C₇×#525	C₂²	1	2	yes	—
23	C₄×C₇×#41	—	2	1	no	—
24	C₄×C₇×#41	—	2	1	no	—
25	C₄×C₇×#41	C₂²	1	2	yes	—
26	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
27	C₂²×C₄²×C₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
29	C₇×#432	—	2	1	no	—
30	C₇×#432	—	2	1	no	—
31	C₇×#431	—	2	1	no	—
32	C₇×#431	—	2	1	no	—
33	C₇×#423	C₂²	1	2	yes	—
34	C₂×C₇×#104	C₂²	1	8	yes	—
35	C₇×#407	C₂²	1	2	yes	—
36	C₄×C₇×M₁₆	C₂²	1	2	yes	—
37	C₄×C₇×C₄⋊C₄	—	2	1	no	—
38	C₄×C₇×C₄⋊C₄	—	2	1	no	—
39	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
40	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
41	C₂×C₇×#95	—	2	1	no	—
42	C₇×#339	C₂²	1	8	yes	—
43	#21899	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	—	2	1	no	—
45	C₂×C₇×#41	—	2	1	no	—
46	C₂×C₇×#41	C₂³	1	2	yes	—
47	C₂³×C₄×C₇	—	2	2	no	—
48	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₇×M₁₆	C₂³	1	4	yes	—
52	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
53	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
54	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
55	C₇×#127	C₂×C₄	1	1	yes	characteristic, complemented
56	C₇×#126	—	2	2	no	—
57	C₇×#126	—	2	2	no	—
58	C₇×#125	—	2	1	no	—
59	C₇×#125	—	2	1	no	—
60	C₇×#125	C₂×C₄	1	2	yes	complemented
61	C₇×#124	—	2	1	no	—
62	C₇×#124	—	2	1	no	—
63	C₇×#124	C₂×C₄	1	2	yes	complemented
64	C₇×#123	—	2	2	no	—
65	C₇×#123	—	2	2	no	—
66	C₇×#123	—	2	2	no	—
67	C₇×C₂²⋊Q₈	—	2	1	no	—
68	C₇×C₂²⋊Q₈	—	2	1	no	—
69	C₇×C₂²⋊Q₈	—	2	1	no	—
70	C₇×C₂²⋊Q₈	—	2	1	no	—
71	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
72	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
73	C₇×#121	—	2	1	no	—
74	C₇×#121	—	2	1	no	—
75	C₇×#121	—	2	1	no	—
76	C₇×#121	—	2	1	no	—
77	C₇×#121	C₂×C₄	1	2	yes	—
78	C₇×#121	C₂×C₄	1	2	yes	—
79	C₇×#120	—	2	2	no	—
80	C₄×C₇×Q₈	—	2	1	no	—
81	C₄×C₇×Q₈	—	2	1	no	—
82	C₄×C₇×Q₈	—	2	1	no	—
83	C₄×C₇×Q₈	—	2	1	no	—
84	C₄×C₇×Q₈	C₂×C₄	1	2	yes	complemented
85	C₄×C₇×Q₈	C₂×C₄	1	2	yes	complemented
86	C₄×C₇×D₄	—	2	2	no	—
87	C₄×C₇×D₄	—	2	2	no	—
88	C₄×C₇×D₄	—	2	2	no	—
89	C₄×C₇×D₄	—	2	2	no	—
90	C₄×C₇×D₄	—	2	2	no	—
91	C₄×C₇×D₄	—	2	2	no	—
92	C₄×C₇×D₄	—	2	1	no	—
93	C₄×C₇×D₄	—	2	1	no	—
94	C₄×C₇×D₄	—	2	1	no	—
95	C₄×C₇×D₄	—	2	1	no	—
96	C₄×C₇×D₄	C₂×C₄	1	2	yes	complemented
97	C₄×C₇×D₄	C₂×C₄	1	2	yes	complemented
98	C₇×#117	—	2	1	no	—
99	C₇×#117	—	2	1	no	—
100	C₇×#117	—	2	2	no	—
101	C₇×#117	—	2	2	no	—
102	C₇×#117	—	2	2	no	—
103	C₇×#117	C₂³	1	2	yes	—
104	C₂×C₇×C₄⋊C₄	—	2	8	no	—
105	C₂×C₇×C₄⋊C₄	—	2	2	no	—
106	C₂×C₇×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄²×C₇	—	2	2	no	—
108	C₂×C₄²×C₇	—	2	2	no	—
109	C₂×C₄²×C₇	—	4	4	no	—
110	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₇	C₂³	1	4	yes	—
121	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
123	C₇×C₄⋊C₈	—	2	2	no	—
124	C₇×C₄⋊C₈	—	2	2	no	—
125	C₇×#104	—	4	8	no	—
126	C₇×C₂²⋊C₈	—	2	2	no	—
127	C₇×C₂²⋊C₈	—	2	2	no	—
128	C₇×#97	—	2	2	no	—
129	C₄×C₇×C₈	—	2	2	no	—
130	C₇×#95	—	2	2	no	—
131	C₇×#95	—	2	2	no	—
132	#3444	C₂×C₇	1	1	yes	characteristic
133	#2920	C₂×C₇	1	1	yes	characteristic, complemented
134	C₂×C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
135	#1986	C₂×C₇	1	2	yes	complemented
136	#1980	C₂×C₇	1	2	yes	complemented
137	#1896	C₂×C₇	1	4	yes	complemented
138	#1806	C₂×C₇	1	4	yes	complemented
139	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
140	C₇×#41	—	2	4	no	—
141	C₇×#41	—	2	4	no	—
142	C₇×#41	—	4	4	no	—
143	C₇×#41	—	4	4	no	—
144	C₂×C₇×Q₈	—	2	1	no	—
145	C₂×C₇×Q₈	—	2	1	no	—
146	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
147	C₂×C₇×D₄	—	2	2	no	—
148	C₂×C₇×D₄	—	2	2	no	—
149	C₂×C₇×D₄	—	2	2	no	—
150	C₂×C₇×D₄	—	2	1	no	—
151	C₂×C₇×D₄	—	2	1	no	—
152	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
153	C₂²×C₄×C₇	—	2	4	no	—
154	C₂²×C₄×C₇	—	2	2	no	—
155	C₂²×C₄×C₇	—	2	2	no	—
156	C₂²×C₄×C₇	—	2	8	no	—
157	C₂²×C₄×C₇	—	2	4	no	—
158	C₂²×C₄×C₇	—	2	2	no	—
159	C₂²×C₄×C₇	—	2	2	no	—
160	C₂²×C₄×C₇	—	2	2	no	—
161	C₂²×C₄×C₇	—	4	2	no	—
162	C₂²×C₄×C₇	—	4	4	no	—
163	C₂²×C₄×C₇	—	4	4	no	—
164	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
173	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
182	C₇×M₁₆	—	2	4	no	—
183	C₇×M₁₆	—	2	4	no	—
184	C₂×C₇×C₈	—	2	4	no	—
185	C₇×C₄⋊C₄	—	2	8	no	—
186	C₇×C₄⋊C₄	—	2	8	no	—
187	C₇×C₄⋊C₄	—	2	1	no	—
188	C₇×C₄⋊C₄	—	2	1	no	—
189	C₇×C₄⋊C₄	—	2	1	no	—
190	C₇×C₄⋊C₄	—	2	1	no	—
191	C₇×C₄⋊C₄	—	2	2	no	—
192	C₇×C₄⋊C₄	—	2	2	no	—
193	C₇×C₄⋊C₄	—	2	2	no	—
194	C₇×C₄⋊C₄	—	2	2	no	—
195	C₇×C₄⋊C₄	—	2	2	no	—
196	C₇×C₄⋊C₄	—	2	2	no	—
197	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
198	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
199	C₇×C₂²⋊C₄	—	2	2	no	—
200	C₇×C₂²⋊C₄	—	2	2	no	—
201	C₇×C₂²⋊C₄	—	2	2	no	—
202	C₇×C₂²⋊C₄	—	2	2	no	—
203	C₇×C₂²⋊C₄	—	2	2	no	—
204	C₇×C₂²⋊C₄	—	2	2	no	—
205	C₇×C₂²⋊C₄	—	2	2	no	—
206	C₇×C₂²⋊C₄	—	2	2	no	—
207	C₇×C₂²⋊C₄	—	2	2	no	—
208	C₄²×C₇	—	2	2	no	—
209	C₄²×C₇	—	2	2	no	—
210	C₄²×C₇	—	2	2	no	—
211	C₄²×C₇	—	2	8	no	—
212	C₄²×C₇	—	4	4	no	—
213	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
214	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
215	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
216	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
217	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
218	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
219	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
220	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
221	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
222	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
223	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
224	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
225	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
229	#533	C₄×C₇	1	1	yes	characteristic, complemented
230	#533	C₄×C₇	1	1	yes	characteristic, complemented
231	#533	C₂²×C₇	1	1	yes	characteristic
232	#532	C₄×C₇	1	1	yes	characteristic, complemented
233	#532	C₄×C₇	1	1	yes	characteristic, complemented
234	#532	C₂²×C₇	1	1	yes	characteristic
235	#529	—	2	1	no	—
236	#529	—	2	1	no	—
237	#529	—	2	1	no	—
238	#529	—	2	1	no	—
239	#529	C₄×C₇	1	2	yes	complemented
240	#529	C₄×C₇	1	2	yes	complemented
241	#525	—	2	1	no	—
242	#525	—	2	1	no	—
243	#525	C₂²×C₇	1	2	yes	—
244	C₄×#41	—	2	1	no	—
245	C₄×#41	—	2	1	no	—
246	C₄×#41	C₂²×C₇	1	2	yes	—
247	C₂²×C₄⋊C₄	—	2	1	no	—
248	C₂²×C₄²	C₂²×C₇	1	2	yes	—
249	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
250	#432	—	2	1	no	—
251	#432	—	2	1	no	—
252	#431	—	2	1	no	—
253	#431	—	2	1	no	—
254	#423	C₂²×C₇	1	2	yes	—
255	C₂×#104	C₂²×C₇	1	8	yes	—
256	#407	C₂²×C₇	1	2	yes	—
257	C₄×M₁₆	C₂²×C₇	1	2	yes	—
258	C₄×C₄⋊C₄	—	2	1	no	—
259	C₄×C₄⋊C₄	—	2	1	no	—
260	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
261	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
262	C₂×#95	—	2	1	no	—
263	#339	C₂²×C₇	1	8	yes	—
264	C₂³×C₇	—	2	1	no	—
265	C₂³×C₇	—	2	1	no	—
266	C₂³×C₇	—	2	2	no	—
267	C₂³×C₇	—	2	2	no	—
268	C₂³×C₇	—	2	2	no	—
269	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
272	C₇×Q₈	—	2	4	no	—
273	C₇×Q₈	—	4	2	no	—
274	C₇×Q₈	—	4	2	no	—
275	C₇×D₄	—	2	4	no	—
276	C₇×D₄	—	4	2	no	—
277	C₇×D₄	—	4	2	no	—
278	C₇×D₄	—	4	2	no	—
279	C₇×D₄	—	4	4	no	—
280	C₇×D₄	—	4	4	no	—
281	C₇×D₄	—	4	2	no	—
282	C₂×C₄×C₇	—	2	1	no	—
283	C₂×C₄×C₇	—	2	1	no	—
284	C₂×C₄×C₇	—	2	2	no	—
285	C₂×C₄×C₇	—	2	2	no	—
286	C₂×C₄×C₇	—	2	2	no	—
287	C₂×C₄×C₇	—	2	2	no	—
288	C₂×C₄×C₇	—	2	2	no	—
289	C₂×C₄×C₇	—	2	2	no	—
290	C₂×C₄×C₇	—	2	2	no	—
291	C₂×C₄×C₇	—	2	2	no	—
292	C₂×C₄×C₇	—	2	2	no	—
293	C₂×C₄×C₇	—	2	2	no	—
294	C₂×C₄×C₇	—	2	2	no	—
295	C₂×C₄×C₇	—	2	8	no	—
296	C₂×C₄×C₇	—	2	2	no	—
297	C₂×C₄×C₇	—	2	2	no	—
298	C₂×C₄×C₇	—	2	2	no	—
299	C₂×C₄×C₇	—	4	1	no	—
300	C₂×C₄×C₇	—	4	1	no	—
301	C₂×C₄×C₇	—	4	1	no	—
302	C₂×C₄×C₇	—	4	1	no	—
303	C₂×C₄×C₇	—	4	4	no	—
304	C₂×C₄×C₇	—	4	16	no	—
305	C₂×C₄×C₇	—	4	4	no	—
306	C₂×C₄×C₇	—	4	4	no	—
307	C₂×C₄×C₇	—	4	4	no	—
308	C₂×C₄×C₇	—	4	4	no	—
309	C₂×C₄×C₇	#104	1	4	yes	complemented
310	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
311	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
312	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
313	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
315	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
316	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
317	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
318	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
319	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
320	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
321	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
323	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
324	C₇×C₈	—	4	4	no	—
325	C₂×#41	—	2	1	no	—
326	C₂×#41	—	2	1	no	—
327	C₂×#41	C₂³×C₇	1	2	yes	—
328	C₂³×C₄	—	2	2	no	—
329	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
330	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
331	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
332	C₂×M₁₆	C₂³×C₇	1	4	yes	—
333	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic, complemented
334	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic, complemented
335	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic, complemented
336	#127	C₂×C₄×C₇	1	1	yes	characteristic, complemented
337	#126	—	2	2	no	—
338	#126	—	2	2	no	—
339	#125	—	2	1	no	—
340	#125	—	2	1	no	—
341	#125	C₂×C₄×C₇	1	2	yes	complemented
342	#124	—	2	1	no	—
343	#124	—	2	1	no	—
344	#124	C₂×C₄×C₇	1	2	yes	complemented
345	#123	—	2	2	no	—
346	#123	—	2	2	no	—
347	#123	—	2	2	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	2	1	no	—
352	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
353	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
354	#121	—	2	1	no	—
355	#121	—	2	1	no	—
356	#121	—	2	1	no	—
357	#121	—	2	1	no	—
358	#121	C₂×C₄×C₇	1	2	yes	—
359	#121	C₂×C₄×C₇	1	2	yes	—
360	#120	—	2	2	no	—
361	C₄×Q₈	—	2	1	no	—
362	C₄×Q₈	—	2	1	no	—
363	C₄×Q₈	—	2	1	no	—
364	C₄×Q₈	—	2	1	no	—
365	C₄×Q₈	C₂×C₄×C₇	1	2	yes	complemented
366	C₄×Q₈	C₂×C₄×C₇	1	2	yes	complemented
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	—	2	1	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	C₂×C₄×C₇	1	2	yes	complemented
378	C₄×D₄	C₂×C₄×C₇	1	2	yes	complemented
379	#117	—	2	1	no	—
380	#117	—	2	1	no	—
381	#117	—	2	2	no	—
382	#117	—	2	2	no	—
383	#117	—	2	2	no	—
384	#117	C₂³×C₇	1	2	yes	—
385	C₂×C₄⋊C₄	—	2	8	no	—
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄²	—	2	2	no	—
389	C₂×C₄²	—	2	2	no	—
390	C₂×C₄²	—	4	4	no	—
391	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
392	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
393	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
394	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
395	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
396	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
397	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
398	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
399	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
400	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₇	1	4	yes	—
402	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
404	C₄⋊C₈	—	2	2	no	—
405	C₄⋊C₈	—	2	2	no	—
406	#104	—	4	8	no	—
407	C₂²⋊C₈	—	2	2	no	—
408	C₂²⋊C₈	—	2	2	no	—
409	#97	—	2	2	no	—
410	C₄×C₈	—	2	2	no	—
411	#95	—	2	2	no	—
412	#95	—	2	2	no	—
413	C₂²×C₇	—	2	4	no	—
414	C₂²×C₇	—	2	4	no	—
415	C₂²×C₇	—	2	1	no	—
416	C₂²×C₇	—	2	1	no	—
417	C₂²×C₇	—	2	1	no	—
418	C₂²×C₇	—	2	1	no	—
419	C₂²×C₇	—	2	1	no	—
420	C₂²×C₇	—	2	1	no	—
421	C₂²×C₇	—	4	2	no	—
422	C₂²×C₇	—	4	2	no	—
423	C₂²×C₇	—	4	2	no	—
424	C₂²×C₇	C₂×#104	1	2	yes	—
425	C₂²×C₇	#421	1	2	yes	—
426	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
427	C₂²×C₇	#535	1	1	yes	characteristic
428	C₂²×C₇	#536	1	1	yes	characteristic
429	C₄×C₇	—	2	1	no	—
430	C₄×C₇	—	2	1	no	—
431	C₄×C₇	—	2	1	no	—
432	C₄×C₇	—	2	1	no	—
433	C₄×C₇	—	2	1	no	—
434	C₄×C₇	—	2	1	no	—
435	C₄×C₇	—	2	1	no	—
436	C₄×C₇	—	2	1	no	—
437	C₄×C₇	—	4	8	no	—
438	C₄×C₇	—	4	2	no	—
439	C₄×C₇	—	4	2	no	—
440	C₄×C₇	—	4	2	no	—
441	C₄×C₇	C₂×#104	1	2	yes	—
442	C₄×C₇	C₂×#104	1	2	yes	—
443	C₄×C₇	#520	1	2	yes	—
444	C₄×C₇	#537	1	2	yes	—
445	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
446	#41	—	2	4	no	—
447	#41	—	2	4	no	—
448	#41	—	4	4	no	—
449	#41	—	4	4	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	8	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	4	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
471	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
472	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
473	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
474	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
475	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
476	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
477	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
480	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
481	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
482	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
483	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
484	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
485	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
486	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
488	M₁₆	—	2	4	no	—
489	M₁₆	—	2	4	no	—
490	C₂×C₈	—	2	4	no	—
491	C₄⋊C₄	—	2	8	no	—
492	C₄⋊C₄	—	2	8	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
504	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	—	2	8	no	—
518	C₄²	—	4	4	no	—
519	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
520	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
521	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
522	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
523	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
524	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
525	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
526	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
527	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
528	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
529	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
530	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
531	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
532	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
533	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
534	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
535	C₂×C₇	—	2	1	no	—
536	C₂×C₇	—	2	1	no	—
537	C₂×C₇	—	2	2	no	—
538	C₂×C₇	—	2	2	no	—
539	C₂×C₇	—	8	2	no	—
540	C₂×C₇	#2311	1	1	yes	characteristic
541	C₂×C₇	C₂²×#104	1	1	yes	characteristic
542	C₂×C₇	C₂×#421	1	1	yes	characteristic
543	C₂³	—	2	1	no	—
544	C₂³	—	2	1	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	2	2	no	—
548	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
549	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
550	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
551	Q₈	—	2	4	no	—
552	Q₈	—	4	2	no	—
553	Q₈	—	4	2	no	—
554	D₄	—	2	4	no	—
555	D₄	—	4	2	no	—
556	D₄	—	4	2	no	—
557	D₄	—	4	2	no	—
558	D₄	—	4	4	no	—
559	D₄	—	4	4	no	—
560	D₄	—	4	2	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	4	16	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	C₇×#104	1	4	yes	complemented
589	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
590	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
591	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
592	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
593	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
594	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
595	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
596	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
597	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
598	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
599	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
600	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
601	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
602	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
603	C₈	—	4	4	no	—
604	C₇	#21899	1	1	yes	characteristic, direct product factor, Sylow
605	C₂²	—	2	4	no	—
606	C₂²	—	2	4	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	1	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	2	1	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	1	no	—
613	C₂²	—	4	2	no	—
614	C₂²	—	4	2	no	—
615	C₂²	—	4	2	no	—
616	C₂²	C₂×C₇×#104	1	2	yes	—
617	C₂²	C₇×#421	1	2	yes	—
618	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
619	C₂²	C₇×#535	1	1	yes	characteristic
620	C₂²	C₇×#536	1	1	yes	characteristic
621	C₄	—	2	1	no	—
622	C₄	—	2	1	no	—
623	C₄	—	2	1	no	—
624	C₄	—	2	1	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	1	no	—
627	C₄	—	2	1	no	—
628	C₄	—	2	1	no	—
629	C₄	—	4	8	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	2	no	—
632	C₄	—	4	2	no	—
633	C₄	C₂×C₇×#104	1	2	yes	—
634	C₄	C₂×C₇×#104	1	2	yes	—
635	C₄	C₇×#520	1	2	yes	—
636	C₄	C₇×#537	1	2	yes	—
637	C₂	—	2	1	no	—
638	C₂	—	2	1	no	—
639	C₂	—	2	2	no	—
640	C₂	—	2	2	no	—
641	C₂	—	8	2	no	—
642	C₂	C₇×#2311	1	1	yes	characteristic
643	C₂	C₂²×C₇×#104	1	1	yes	characteristic
644	C₂	C₂×C₇×#421	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×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2920	C₂	semidirect
3. C₂×C₄×C₇×C₄⋊C₄	C₂	semidirect
4. C₇×#1986	C₂	semidirect
5. C₇×#1980	C₂	semidirect
6. C₇×#1896	C₂	semidirect
7. C₇×#1806	C₂	semidirect
8. C₇×#533	C₄	semidirect
9. C₇×#533	C₄	semidirect
11. C₇×#532	C₄	semidirect
12. C₇×#532	C₄	semidirect
18. C₇×#529	C₄	semidirect
19. C₇×#529	C₄	semidirect
43. #21899	C₇	direct (N × H)
52. C₇×C₄⋊Q₈	C₂×C₄	semidirect
53. C₇×C₄⋊Q₈	C₂×C₄	semidirect
54. C₇×C₄⋊Q₈	C₂×C₄	semidirect
55. C₇×#127	C₂×C₄	semidirect
60. C₇×#125	C₂×C₄	semidirect
63. C₇×#124	C₂×C₄	semidirect
84. C₄×C₇×Q₈	C₂×C₄	semidirect
85. C₄×C₇×Q₈	C₂×C₄	semidirect
96. C₄×C₇×D₄	C₂×C₄	semidirect
97. C₄×C₇×D₄	C₂×C₄	semidirect
133. #2920	C₂×C₇	semidirect
134. C₂×C₄×C₄⋊C₄	C₂×C₇	semidirect
135. #1986	C₂×C₇	semidirect
136. #1980	C₂×C₇	semidirect
137. #1896	C₂×C₇	semidirect
138. #1806	C₂×C₇	semidirect
229. #533	C₄×C₇	semidirect
230. #533	C₄×C₇	semidirect
232. #532	C₄×C₇	semidirect
233. #532	C₄×C₇	semidirect
239. #529	C₄×C₇	semidirect
240. #529	C₄×C₇	semidirect
309. C₂×C₄×C₇	#104	semidirect
333. C₄⋊Q₈	C₂×C₄×C₇	semidirect
334. C₄⋊Q₈	C₂×C₄×C₇	semidirect
335. C₄⋊Q₈	C₂×C₄×C₇	semidirect
336. #127	C₂×C₄×C₇	semidirect
341. #125	C₂×C₄×C₇	semidirect
344. #124	C₂×C₄×C₇	semidirect
365. C₄×Q₈	C₂×C₄×C₇	semidirect
366. C₄×Q₈	C₂×C₄×C₇	semidirect
377. C₄×D₄	C₂×C₄×C₇	semidirect
378. C₄×D₄	C₂×C₄×C₇	semidirect
588. C₂×C₄	C₇×#104	semidirect
604. C₇	#21899	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×#3444	2
C₇×#2920	2
C₂×C₄×C₇×C₄⋊C₄	2
C₇×#1986	2
C₇×#1980	2
C₇×#1896	2
C₇×#1806	2
#21899	7

Subgroups — C7×#21899 #13199115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21899 #13199115