Subgroups — C₇×#21857 #13199073

Order 1792.

1108 conjugacy classes of subgroups.

The subgroup lattice has 634 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	C₂×C₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1995	C₂	1	2	yes	maximal, complemented
5	C₇×#1963	C₂	1	2	yes	maximal, complemented
6	C₇×#1871	C₂	1	4	yes	maximal, complemented
7	C₇×#1806	C₂	1	4	yes	maximal, complemented
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₇×#533	C₄	1	1	yes	characteristic, complemented
10	C₇×#533	C₄	1	1	yes	characteristic, complemented
11	C₇×#533	C₂²	1	1	yes	characteristic
12	C₇×#532	C₄	1	1	yes	characteristic, complemented
13	C₇×#532	C₄	1	1	yes	characteristic, complemented
14	C₇×#532	C₂²	1	1	yes	characteristic
15	C₇×#529	—	2	1	no	—
16	C₇×#529	—	2	1	no	—
17	C₇×#529	—	2	1	no	—
18	C₇×#529	—	2	1	no	—
19	C₇×#529	C₄	1	2	yes	complemented
20	C₇×#529	C₄	1	2	yes	complemented
21	C₇×#525	—	2	1	no	—
22	C₇×#525	—	2	1	no	—
23	C₇×#525	C₂²	1	2	yes	—
24	C₄×C₇×#41	—	2	1	no	—
25	C₄×C₇×#41	—	2	1	no	—
26	C₄×C₇×#41	C₂²	1	2	yes	—
27	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
28	C₇×#423	—	2	1	no	—
29	C₇×#423	—	2	1	no	—
30	C₇×#423	C₂²	1	2	yes	—
31	C₇×#423	C₂²	1	2	yes	—
32	C₂×C₇×C₄⋊C₈	—	2	1	no	—
33	C₂×C₇×C₄⋊C₈	—	2	1	no	—
34	C₂×C₇×#104	C₂²	1	8	yes	—
35	C₇×#383	—	2	1	no	—
36	C₇×#382	—	2	1	no	—
37	C₇×#381	—	2	1	no	—
38	C₇×#380	—	2	1	no	—
39	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
40	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₇×#376	C₂²	1	2	yes	—
42	C₇×#339	C₂²	1	8	yes	—
43	#21857	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	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₇×M₁₆	—	2	2	no	—
51	C₂×C₇×M₁₆	—	4	4	no	—
52	C₂×C₇×M₁₆	C₂³	1	4	yes	—
53	C₂²×C₇×C₈	—	2	2	no	—
54	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
55	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
56	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
57	C₇×#127	C₂×C₄	1	1	yes	characteristic
58	C₇×#126	—	2	2	no	—
59	C₇×#126	—	2	2	no	—
60	C₇×#125	—	2	1	no	—
61	C₇×#125	—	2	1	no	—
62	C₇×#125	C₂×C₄	1	2	yes	—
63	C₇×#124	—	2	1	no	—
64	C₇×#124	—	2	1	no	—
65	C₇×#124	C₂×C₄	1	2	yes	—
66	C₇×#123	—	2	2	no	—
67	C₇×#123	—	2	2	no	—
68	C₇×#123	—	2	2	no	—
69	C₇×C₂²⋊Q₈	—	2	1	no	—
70	C₇×C₂²⋊Q₈	—	2	1	no	—
71	C₇×C₂²⋊Q₈	—	2	1	no	—
72	C₇×C₂²⋊Q₈	—	2	1	no	—
73	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
74	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
75	C₇×#121	—	2	1	no	—
76	C₇×#121	—	2	1	no	—
77	C₇×#121	—	2	1	no	—
78	C₇×#121	—	2	1	no	—
79	C₇×#121	C₂×C₄	1	2	yes	—
80	C₇×#121	C₂×C₄	1	2	yes	—
81	C₇×#120	—	2	2	no	—
82	C₄×C₇×Q₈	—	2	1	no	—
83	C₄×C₇×Q₈	—	2	1	no	—
84	C₄×C₇×Q₈	—	2	1	no	—
85	C₄×C₇×Q₈	—	2	1	no	—
86	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
87	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
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	2	no	—
93	C₄×C₇×D₄	—	2	2	no	—
94	C₄×C₇×D₄	—	2	1	no	—
95	C₄×C₇×D₄	—	2	1	no	—
96	C₄×C₇×D₄	—	2	1	no	—
97	C₄×C₇×D₄	—	2	1	no	—
98	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
99	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
100	C₇×#117	—	2	1	no	—
101	C₇×#117	—	2	1	no	—
102	C₇×#117	—	2	2	no	—
103	C₇×#117	—	2	2	no	—
104	C₇×#117	—	2	2	no	—
105	C₇×#117	C₂³	1	2	yes	—
106	C₂×C₇×C₄⋊C₄	—	2	2	no	—
107	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
108	C₂×C₄²×C₇	—	2	2	no	—
109	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
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	D₄	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	C₂³	1	4	yes	—
120	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
122	C₇×C₄⋊C₈	—	2	2	no	—
123	C₇×C₄⋊C₈	—	2	2	no	—
124	C₇×C₄⋊C₈	—	2	2	no	—
125	C₇×C₄⋊C₈	—	2	2	no	—
126	C₇×#104	—	4	8	no	—
127	C₇×#95	—	2	2	no	—
128	C₇×#95	—	2	2	no	—
129	C₇×#95	—	2	2	no	—
130	C₇×#95	—	2	2	no	—
131	#3444	C₂×C₇	1	1	yes	characteristic
132	C₂×#423	C₂×C₇	1	1	yes	characteristic, complemented
133	#2283	C₂×C₇	1	1	yes	characteristic, complemented
134	#1995	C₂×C₇	1	2	yes	complemented
135	#1963	C₂×C₇	1	2	yes	complemented
136	#1871	C₂×C₇	1	4	yes	complemented
137	#1806	C₂×C₇	1	4	yes	complemented
138	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
139	C₇×#41	—	2	4	no	—
140	C₇×#41	—	2	4	no	—
141	C₇×#41	—	4	4	no	—
142	C₇×#41	—	4	4	no	—
143	C₂×C₇×Q₈	—	2	1	no	—
144	C₂×C₇×Q₈	—	2	1	no	—
145	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
146	C₂×C₇×D₄	—	2	2	no	—
147	C₂×C₇×D₄	—	2	2	no	—
148	C₂×C₇×D₄	—	2	2	no	—
149	C₂×C₇×D₄	—	2	1	no	—
150	C₂×C₇×D₄	—	2	1	no	—
151	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
152	C₂²×C₄×C₇	—	2	2	no	—
153	C₂²×C₄×C₇	—	2	2	no	—
154	C₂²×C₄×C₇	—	2	2	no	—
155	C₂²×C₄×C₇	—	2	2	no	—
156	C₂²×C₄×C₇	—	2	2	no	—
157	C₂²×C₄×C₇	—	2	4	no	—
158	C₂²×C₄×C₇	—	4	2	no	—
159	C₂²×C₄×C₇	—	4	2	no	—
160	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₇	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
178	C₇×M₁₆	—	2	8	no	—
179	C₇×M₁₆	—	4	4	no	—
180	C₂×C₇×C₈	—	2	4	no	—
181	C₂×C₇×C₈	—	4	4	no	—
182	C₇×C₄⋊C₄	—	2	1	no	—
183	C₇×C₄⋊C₄	—	2	1	no	—
184	C₇×C₄⋊C₄	—	2	1	no	—
185	C₇×C₄⋊C₄	—	2	1	no	—
186	C₇×C₄⋊C₄	—	2	2	no	—
187	C₇×C₄⋊C₄	—	2	2	no	—
188	C₇×C₄⋊C₄	—	2	2	no	—
189	C₇×C₄⋊C₄	—	2	2	no	—
190	C₇×C₄⋊C₄	—	2	2	no	—
191	C₇×C₄⋊C₄	—	2	2	no	—
192	C₇×C₄⋊C₄	—	4	4	no	—
193	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
194	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
195	C₇×C₂²⋊C₄	—	2	2	no	—
196	C₇×C₂²⋊C₄	—	2	2	no	—
197	C₇×C₂²⋊C₄	—	2	2	no	—
198	C₇×C₂²⋊C₄	—	2	2	no	—
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	—	4	4	no	—
205	C₄²×C₇	—	2	2	no	—
206	C₄²×C₇	—	2	2	no	—
207	C₄²×C₇	—	2	2	no	—
208	C₄²×C₇	—	2	4	no	—
209	C₄²×C₇	—	2	4	no	—
210	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
211	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
212	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
224	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
226	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
227	#533	C₄×C₇	1	1	yes	characteristic, complemented
228	#533	C₄×C₇	1	1	yes	characteristic, complemented
229	#533	C₂²×C₇	1	1	yes	characteristic
230	#532	C₄×C₇	1	1	yes	characteristic, complemented
231	#532	C₄×C₇	1	1	yes	characteristic, complemented
232	#532	C₂²×C₇	1	1	yes	characteristic
233	#529	—	2	1	no	—
234	#529	—	2	1	no	—
235	#529	—	2	1	no	—
236	#529	—	2	1	no	—
237	#529	C₄×C₇	1	2	yes	complemented
238	#529	C₄×C₇	1	2	yes	complemented
239	#525	—	2	1	no	—
240	#525	—	2	1	no	—
241	#525	C₂²×C₇	1	2	yes	—
242	C₄×#41	—	2	1	no	—
243	C₄×#41	—	2	1	no	—
244	C₄×#41	C₂²×C₇	1	2	yes	—
245	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
246	#423	—	2	1	no	—
247	#423	—	2	1	no	—
248	#423	C₂²×C₇	1	2	yes	—
249	#423	C₂²×C₇	1	2	yes	—
250	C₂×C₄⋊C₈	—	2	1	no	—
251	C₂×C₄⋊C₈	—	2	1	no	—
252	C₂×#104	C₂²×C₇	1	8	yes	—
253	#383	—	2	1	no	—
254	#382	—	2	1	no	—
255	#381	—	2	1	no	—
256	#380	—	2	1	no	—
257	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
258	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
259	#376	C₂²×C₇	1	2	yes	—
260	#339	C₂²×C₇	1	8	yes	—
261	C₂³×C₇	—	2	2	no	—
262	C₂³×C₇	—	2	2	no	—
263	C₂³×C₇	—	2	2	no	—
264	C₂³×C₇	—	4	1	no	—
265	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
268	C₇×Q₈	—	2	4	no	—
269	C₇×Q₈	—	4	2	no	—
270	C₇×Q₈	—	4	2	no	—
271	C₇×D₄	—	2	4	no	—
272	C₇×D₄	—	4	2	no	—
273	C₇×D₄	—	4	2	no	—
274	C₇×D₄	—	4	4	no	—
275	C₇×D₄	—	4	4	no	—
276	C₇×D₄	—	4	4	no	—
277	C₂×C₄×C₇	—	2	2	no	—
278	C₂×C₄×C₇	—	2	2	no	—
279	C₂×C₄×C₇	—	2	2	no	—
280	C₂×C₄×C₇	—	2	2	no	—
281	C₂×C₄×C₇	—	2	2	no	—
282	C₂×C₄×C₇	—	2	1	no	—
283	C₂×C₄×C₇	—	2	1	no	—
284	C₂×C₄×C₇	—	2	1	no	—
285	C₂×C₄×C₇	—	2	1	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	—	4	1	no	—
293	C₂×C₄×C₇	—	4	1	no	—
294	C₂×C₄×C₇	—	4	1	no	—
295	C₂×C₄×C₇	—	4	1	no	—
296	C₂×C₄×C₇	—	4	1	no	—
297	C₂×C₄×C₇	—	4	1	no	—
298	C₂×C₄×C₇	—	4	1	no	—
299	C₂×C₄×C₇	—	4	4	no	—
300	C₂×C₄×C₇	—	4	2	no	—
301	C₂×C₄×C₇	—	4	2	no	—
302	C₂×C₄×C₇	—	4	4	no	—
303	C₂×C₄×C₇	—	4	4	no	—
304	C₂×C₄×C₇	—	4	4	no	—
305	C₂×C₄×C₇	—	4	4	no	—
306	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
308	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
309	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
317	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
318	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
319	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
320	C₇×C₈	—	4	4	no	—
321	C₂×#41	—	2	1	no	—
322	C₂×#41	—	2	1	no	—
323	C₂×#41	C₂³×C₇	1	2	yes	—
324	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
325	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
326	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
327	C₂×M₁₆	—	2	2	no	—
328	C₂×M₁₆	—	4	4	no	—
329	C₂×M₁₆	C₂³×C₇	1	4	yes	—
330	C₂²×C₈	—	2	2	no	—
331	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
332	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
333	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
334	#127	C₂×C₄×C₇	1	1	yes	characteristic
335	#126	—	2	2	no	—
336	#126	—	2	2	no	—
337	#125	—	2	1	no	—
338	#125	—	2	1	no	—
339	#125	C₂×C₄×C₇	1	2	yes	—
340	#124	—	2	1	no	—
341	#124	—	2	1	no	—
342	#124	C₂×C₄×C₇	1	2	yes	—
343	#123	—	2	2	no	—
344	#123	—	2	2	no	—
345	#123	—	2	2	no	—
346	C₂²⋊Q₈	—	2	1	no	—
347	C₂²⋊Q₈	—	2	1	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
351	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
352	#121	—	2	1	no	—
353	#121	—	2	1	no	—
354	#121	—	2	1	no	—
355	#121	—	2	1	no	—
356	#121	C₂×C₄×C₇	1	2	yes	—
357	#121	C₂×C₄×C₇	1	2	yes	—
358	#120	—	2	2	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	—	2	1	no	—
361	C₄×Q₈	—	2	1	no	—
362	C₄×Q₈	—	2	1	no	—
363	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
364	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	2	no	—
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	1	no	—
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
376	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
377	#117	—	2	1	no	—
378	#117	—	2	1	no	—
379	#117	—	2	2	no	—
380	#117	—	2	2	no	—
381	#117	—	2	2	no	—
382	#117	C₂³×C₇	1	2	yes	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₄²	—	2	2	no	—
386	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
387	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
388	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
389	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
390	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
391	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
392	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
393	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
394	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
395	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₇	1	4	yes	—
397	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
399	C₄⋊C₈	—	2	2	no	—
400	C₄⋊C₈	—	2	2	no	—
401	C₄⋊C₈	—	2	2	no	—
402	C₄⋊C₈	—	2	2	no	—
403	#104	—	4	8	no	—
404	#95	—	2	2	no	—
405	#95	—	2	2	no	—
406	#95	—	2	2	no	—
407	#95	—	2	2	no	—
408	C₂²×C₇	—	2	1	no	—
409	C₂²×C₇	—	2	1	no	—
410	C₂²×C₇	—	2	1	no	—
411	C₂²×C₇	—	2	1	no	—
412	C₂²×C₇	—	2	1	no	—
413	C₂²×C₇	—	2	1	no	—
414	C₂²×C₇	—	2	1	no	—
415	C₂²×C₇	—	4	4	no	—
416	C₂²×C₇	—	4	2	no	—
417	C₂²×C₇	—	4	2	no	—
418	C₂²×C₇	—	4	2	no	—
419	C₂²×C₇	#421	1	2	yes	—
420	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
421	C₂²×C₇	#537	1	2	yes	—
422	C₄×C₇	—	2	1	no	—
423	C₄×C₇	—	2	1	no	—
424	C₄×C₇	—	2	1	no	—
425	C₄×C₇	—	2	1	no	—
426	C₄×C₇	—	2	1	no	—
427	C₄×C₇	—	2	1	no	—
428	C₄×C₇	—	2	1	no	—
429	C₄×C₇	—	2	1	no	—
430	C₄×C₇	—	2	1	no	—
431	C₄×C₇	—	4	2	no	—
432	C₄×C₇	—	4	2	no	—
433	C₄×C₇	—	4	2	no	—
434	C₄×C₇	—	8	4	no	—
435	C₄×C₇	#421	1	2	yes	—
436	C₄×C₇	#520	1	2	yes	—
437	C₄×C₇	#535	1	1	yes	characteristic
438	C₄×C₇	#536	1	1	yes	characteristic
439	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
440	#41	—	2	4	no	—
441	#41	—	2	4	no	—
442	#41	—	4	4	no	—
443	#41	—	4	4	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	4	2	no	—
460	C₂²×C₄	—	4	2	no	—
461	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
462	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
463	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
464	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
465	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
466	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
467	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
468	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
469	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
471	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
472	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
473	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
474	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
475	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
476	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
477	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
479	M₁₆	—	2	8	no	—
480	M₁₆	—	4	4	no	—
481	C₂×C₈	—	2	4	no	—
482	C₂×C₈	—	4	4	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	4	4	no	—
494	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
495	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₄²	—	2	2	no	—
507	C₄²	—	2	2	no	—
508	C₄²	—	2	2	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	2	4	no	—
511	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
512	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
513	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
514	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
515	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
516	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
517	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
518	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
519	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
520	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
521	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
522	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
523	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
524	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
525	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
526	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
527	C₂×C₇	—	2	2	no	—
528	C₂×C₇	—	2	2	no	—
529	C₂×C₇	—	4	1	no	—
530	C₂×C₇	—	8	2	no	—
531	C₂×C₇	#2311	1	1	yes	characteristic
532	C₂×C₇	C₂×#421	1	1	yes	characteristic
533	C₂×C₇	C₂×#421	1	1	yes	characteristic
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	4	1	no	—
538	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
539	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
540	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
541	Q₈	—	2	4	no	—
542	Q₈	—	4	2	no	—
543	Q₈	—	4	2	no	—
544	D₄	—	2	4	no	—
545	D₄	—	4	2	no	—
546	D₄	—	4	2	no	—
547	D₄	—	4	4	no	—
548	D₄	—	4	4	no	—
549	D₄	—	4	4	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	4	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
580	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
581	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
582	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
583	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
584	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
585	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
586	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
587	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
588	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
589	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
590	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
591	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
592	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
593	C₈	—	4	4	no	—
594	C₇	#21857	1	1	yes	characteristic, direct product factor, Sylow
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	1	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	4	4	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	C₇×#421	1	2	yes	—
607	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
608	C₂²	C₇×#537	1	2	yes	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	2	no	—
621	C₄	—	8	4	no	—
622	C₄	C₇×#421	1	2	yes	—
623	C₄	C₇×#520	1	2	yes	—
624	C₄	C₇×#535	1	1	yes	characteristic
625	C₄	C₇×#536	1	1	yes	characteristic
626	C₂	—	2	2	no	—
627	C₂	—	2	2	no	—
628	C₂	—	4	1	no	—
629	C₂	—	8	2	no	—
630	C₂	C₇×#2311	1	1	yes	characteristic
631	C₂	C₂×C₇×#421	1	1	yes	characteristic
632	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. C₂×C₇×#423	C₂	semidirect
3. C₇×#2283	C₂	semidirect
4. C₇×#1995	C₂	semidirect
5. C₇×#1963	C₂	semidirect
6. C₇×#1871	C₂	semidirect
7. C₇×#1806	C₂	semidirect
9. C₇×#533	C₄	semidirect
10. C₇×#533	C₄	semidirect
12. C₇×#532	C₄	semidirect
13. C₇×#532	C₄	semidirect
19. C₇×#529	C₄	semidirect
20. C₇×#529	C₄	semidirect
43. #21857	C₇	direct (N × H)
132. C₂×#423	C₂×C₇	semidirect
133. #2283	C₂×C₇	semidirect
134. #1995	C₂×C₇	semidirect
135. #1963	C₂×C₇	semidirect
136. #1871	C₂×C₇	semidirect
137. #1806	C₂×C₇	semidirect
227. #533	C₄×C₇	semidirect
228. #533	C₄×C₇	semidirect
230. #532	C₄×C₇	semidirect
231. #532	C₄×C₇	semidirect
237. #529	C₄×C₇	semidirect
238. #529	C₄×C₇	semidirect
594. C₇	#21857	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
C₂×C₇×#423	2
C₇×#2283	2
C₇×#1995	2
C₇×#1963	2
C₇×#1871	2
C₇×#1806	2
#21857	7

Subgroups — C7×#21857 #13199073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21857 #13199073