Subgroups — C₇×#61269 #13238485

Order 1792.

1750 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3495	C₂	1	2	yes	maximal
2	C₇×#3356	C₂	1	4	yes	maximal, complemented
3	C₇×#3352	C₂	1	4	yes	maximal, complemented
4	C₇×#3349	C₂	1	4	yes	maximal, complemented
5	C₇×#3344	C₂	1	4	yes	maximal, complemented
6	C₇×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3097	C₂	1	2	yes	maximal, complemented
8	C₇×#3095	C₂	1	2	yes	maximal, complemented
9	C₇×#3082	C₂	1	2	yes	maximal, complemented
10	C₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#470	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#2930	C₂	1	1	yes	maximal, characteristic
14	C₇×#2912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#2911	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₇×#131	C₂²	1	1	yes	characteristic
18	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
19	C₇×#556	C₂²	1	4	yes	—
20	C₇×#554	C₂²	1	4	yes	—
21	C₇×#552	—	2	4	no	—
22	C₇×#552	C₂²	1	4	yes	—
23	C₇×#552	C₂²	1	4	yes	—
24	C₇×#551	—	2	4	no	—
25	C₇×#544	C₂²	1	2	yes	—
26	C₇×#542	—	2	2	no	—
27	C₇×#541	—	2	2	no	—
28	C₇×#536	C₂²	1	2	yes	—
29	C₇×#533	—	2	2	no	—
30	C₇×#529	—	2	2	no	—
31	C₇×#525	C₂²	1	1	yes	characteristic
32	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
33	C₇×#520	C₂²	1	2	yes	—
34	C₇×#519	C₂²	1	2	yes	—
35	C₂×C₇×#117	C₂²	1	2	yes	—
36	C₇×#501	C₂²	1	4	yes	complemented
37	C₇×#499	C₂²	1	2	yes	complemented
38	C₇×#498	C₂²	1	2	yes	complemented
39	C₇×#485	C₂²	1	4	yes	—
40	C₇×#485	C₂²	1	4	yes	—
41	C₇×#485	C₂²	1	4	yes	complemented
42	C₇×#485	C₂²	1	4	yes	complemented
43	C₇×#484	C₂²	1	4	yes	—
44	C₇×#484	C₂²	1	4	yes	complemented
45	C₇×#483	C₂²	1	4	yes	—
46	C₇×#483	C₂²	1	4	yes	complemented
47	C₇×#479	C₂²	1	4	yes	complemented
48	C₇×#479	C₂²	1	4	yes	complemented
49	C₇×#477	C₂²	1	4	yes	complemented
50	C₇×#475	C₂²	1	4	yes	complemented
51	C₇×#470	C₂²	1	4	yes	—
52	C₇×#470	C₂²	1	4	yes	—
53	C₇×#470	C₂²	1	2	yes	complemented
54	C₇×#470	C₂²	1	2	yes	complemented
55	C₇×#467	C₂²	1	4	yes	complemented
56	C₇×#441	C₂²	1	2	yes	complemented
57	C₇×#441	C₂²	1	2	yes	complemented
58	C₄×C₇×Q₁₆	C₂²	1	4	yes	complemented
59	C₇×#436	C₂²	1	4	yes	—
60	C₇×#435	C₂²	1	4	yes	—
61	C₇×#428	C₂²	1	2	yes	—
62	C₇×#428	C₂²	1	2	yes	—
63	C₇×#428	C₂²	1	2	yes	—
64	C₇×#428	C₂²	1	2	yes	—
65	C₇×#428	C₂²	1	2	yes	—
66	C₇×#428	C₂²	1	2	yes	—
67	C₇×#428	C₂²	1	4	yes	complemented
68	C₇×#427	C₂²	1	1	yes	characteristic
69	C₇×#427	C₂²	1	1	yes	characteristic
70	C₇×#427	C₂²	1	2	yes	—
71	C₇×#427	C₂²	1	2	yes	complemented
72	C₂×C₇×#106	C₂²	1	1	yes	characteristic
73	C₂×C₇×#106	C₂²	1	2	yes	complemented
74	C₇×#424	C₂²	1	1	yes	characteristic
75	C₇×#419	C₂²	1	2	yes	—
76	C₇×#419	C₂²	1	2	yes	—
77	C₂×C₇×#103	C₂²	1	2	yes	—
78	C₇×#408	C₂²	1	1	yes	characteristic
79	C₇×#407	C₂²	1	1	yes	characteristic
80	C₇×#405	C₂²	1	1	yes	characteristic
81	#61269	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₇×#41	C₂³	1	1	yes	characteristic
83	C₂²×C₇×Q₈	C₂³	1	2	yes	—
84	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
85	C₂×C₇×Q₁₆	—	2	4	no	—
86	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
87	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
88	C₇×#131	—	2	4	no	—
89	C₂×C₇×M₁₆	—	2	4	no	—
90	C₂×C₇×M₁₆	C₂³	1	2	yes	—
91	C₂×C₇×M₁₆	C₂³	1	2	yes	—
92	C₂×C₇×M₁₆	C₂³	1	2	yes	—
93	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
94	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
95	C₂²×C₇×C₈	C₂³	1	2	yes	—
96	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
97	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
98	C₇×C₄⋊Q₈	—	2	2	no	—
99	C₇×C₄⋊Q₈	—	2	2	no	—
100	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
101	C₇×#126	—	2	4	no	—
102	C₇×#126	—	2	4	no	—
103	C₇×#126	—	2	4	no	—
104	C₇×#126	—	2	4	no	—
105	C₇×#125	—	2	2	no	—
106	C₇×#125	—	2	4	no	—
107	C₇×#125	—	2	4	no	—
108	C₇×#125	—	2	4	no	—
109	C₇×#125	C₂³	1	4	yes	—
110	C₇×#124	—	2	2	no	—
111	C₇×#124	—	2	4	no	—
112	C₇×#123	—	2	4	no	—
113	C₇×#123	—	2	4	no	—
114	C₇×#123	—	2	4	no	—
115	C₇×#123	D₄	1	2	yes	—
116	C₇×C₂²⋊Q₈	—	2	2	no	—
117	C₇×C₂²⋊Q₈	—	2	2	no	—
118	C₇×C₂²⋊Q₈	—	2	4	no	—
119	C₇×C₂²⋊Q₈	—	2	4	no	—
120	C₇×C₂²⋊Q₈	—	2	4	no	—
121	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
122	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
123	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
124	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
125	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
126	C₇×#121	D₄	1	2	yes	—
127	C₇×#120	D₄	1	2	yes	—
128	C₄×C₇×Q₈	—	2	2	no	—
129	C₄×C₇×Q₈	—	2	2	no	—
130	C₄×C₇×Q₈	—	2	2	no	—
131	C₄×C₇×Q₈	—	2	4	no	—
132	C₄×C₇×Q₈	C₂³	1	4	yes	—
133	C₄×C₇×Q₈	C₂³	1	4	yes	—
134	C₄×C₇×D₄	—	2	2	no	—
135	C₄×C₇×D₄	—	2	4	no	—
136	C₄×C₇×D₄	—	2	4	no	—
137	C₄×C₇×D₄	C₂³	1	4	yes	—
138	C₇×#117	—	2	2	no	—
139	C₇×#117	—	2	2	no	—
140	C₇×#117	—	2	4	no	—
141	C₇×#117	—	2	4	no	—
142	C₇×#117	—	2	4	no	—
143	C₇×#117	C₂³	1	2	yes	—
144	C₇×#117	C₂³	1	2	yes	—
145	C₇×#117	C₂³	1	4	yes	—
146	C₇×#117	C₂³	1	2	yes	—
147	C₇×#117	C₂³	1	2	yes	—
148	C₇×#117	C₂³	1	1	yes	characteristic
149	C₂×C₇×C₄⋊C₄	—	2	2	no	—
150	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
151	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
152	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
153	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
154	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
155	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
156	C₂×C₄²×C₇	—	2	2	no	—
157	C₇×#107	C₂³	1	2	yes	—
158	C₇×#107	C₂³	1	2	yes	—
159	C₇×#107	C₂³	1	4	yes	—
160	C₇×#107	C₂³	1	2	yes	—
161	C₇×#107	C₂³	1	4	yes	—
162	C₇×#107	C₂³	1	2	yes	—
163	C₇×#106	C₂³	1	2	yes	—
164	C₇×#106	C₂³	1	2	yes	—
165	C₇×#106	C₂³	1	1	yes	characteristic
166	C₇×#106	C₂³	1	1	yes	characteristic
167	C₇×#106	C₂³	1	2	yes	—
168	C₇×#106	C₂³	1	4	yes	—
169	C₇×#106	C₂³	1	2	yes	—
170	C₇×#106	C₂³	1	4	yes	—
171	C₇×#106	C₂³	1	1	yes	characteristic
172	C₇×#106	C₂³	1	1	yes	characteristic
173	C₇×C₄⋊C₈	C₂³	1	4	yes	—
174	C₇×#103	C₂³	1	4	yes	—
175	C₇×#103	C₂³	1	4	yes	—
176	C₇×#103	C₂³	1	4	yes	—
177	C₇×#103	C₂³	1	4	yes	—
178	C₇×#103	C₂³	1	4	yes	—
179	C₇×#103	C₂³	1	4	yes	—
180	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
181	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
182	C₇×#97	C₂³	1	2	yes	—
183	C₄×C₇×C₈	C₂³	1	2	yes	—
184	#3495	C₂×C₇	1	2	yes	—
185	#3356	C₂×C₇	1	4	yes	complemented
186	#3352	C₂×C₇	1	4	yes	complemented
187	#3349	C₂×C₇	1	4	yes	complemented
188	#3344	C₂×C₇	1	4	yes	complemented
189	#3163	C₂×C₇	1	1	yes	characteristic, complemented
190	#3097	C₂×C₇	1	2	yes	complemented
191	#3095	C₂×C₇	1	2	yes	complemented
192	#3082	C₂×C₇	1	2	yes	complemented
193	#3059	C₂×C₇	1	1	yes	characteristic, complemented
194	C₂×#470	C₂×C₇	1	1	yes	characteristic, complemented
195	#2950	C₂×C₇	1	1	yes	characteristic, complemented
196	#2930	C₂×C₇	1	1	yes	characteristic
197	#2912	C₂×C₇	1	1	yes	characteristic, complemented
198	#2911	C₂×C₇	1	1	yes	characteristic, complemented
199	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
200	C₇×#41	—	2	2	no	—
201	C₇×#41	—	2	2	no	—
202	C₂×C₇×Q₈	—	2	2	no	—
203	C₂×C₇×Q₈	—	2	4	no	—
204	C₂×C₇×Q₈	—	2	4	no	—
205	C₂×C₇×Q₈	—	2	4	no	—
206	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
207	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
208	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
209	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
210	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
211	C₂²×C₄×C₇	—	2	2	no	—
212	C₂²×C₄×C₇	—	2	2	no	—
213	C₂²×C₄×C₇	—	2	2	no	—
214	C₂²×C₄×C₇	—	2	2	no	—
215	C₂²×C₄×C₇	—	2	2	no	—
216	C₂²×C₄×C₇	—	2	2	no	—
217	C₂²×C₄×C₇	—	2	4	no	—
218	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
219	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
220	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
221	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
222	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
223	C₇×Q₁₆	—	2	4	no	—
224	C₇×Q₁₆	—	2	4	no	—
225	C₇×M₁₆	—	2	2	no	—
226	C₇×M₁₆	—	2	2	no	—
227	C₇×M₁₆	—	2	4	no	—
228	C₇×M₁₆	—	2	4	no	—
229	C₇×M₁₆	—	2	4	no	—
230	C₂×C₇×C₈	—	2	2	no	—
231	C₂×C₇×C₈	—	2	4	no	—
232	C₂×C₇×C₈	#41	1	4	yes	—
233	C₂×C₇×C₈	C₂⁴	1	2	yes	—
234	C₂×C₇×C₈	C₂⁴	1	2	yes	—
235	C₂×C₇×C₈	C₂⁴	1	2	yes	—
236	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
237	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
238	C₇×C₄⋊C₄	—	2	4	no	—
239	C₇×C₄⋊C₄	—	2	4	no	—
240	C₇×C₄⋊C₄	—	2	2	no	—
241	C₇×C₄⋊C₄	—	2	2	no	—
242	C₇×C₄⋊C₄	—	2	2	no	—
243	C₇×C₄⋊C₄	—	2	4	no	—
244	C₇×C₄⋊C₄	—	2	4	no	—
245	C₇×C₄⋊C₄	—	2	4	no	—
246	C₇×C₄⋊C₄	—	2	4	no	—
247	C₇×C₄⋊C₄	—	2	4	no	—
248	C₇×C₄⋊C₄	—	2	4	no	—
249	C₇×C₄⋊C₄	—	2	4	no	—
250	C₇×C₄⋊C₄	—	2	4	no	—
251	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
252	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
253	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
254	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
255	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
256	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
257	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
258	C₇×C₂²⋊C₄	—	2	2	no	—
259	C₇×C₂²⋊C₄	—	2	2	no	—
260	C₇×C₂²⋊C₄	—	2	2	no	—
261	C₇×C₂²⋊C₄	—	2	2	no	—
262	C₇×C₂²⋊C₄	—	2	4	no	—
263	C₇×C₂²⋊C₄	—	2	4	no	—
264	C₇×C₂²⋊C₄	—	2	4	no	—
265	C₇×C₂²⋊C₄	—	2	4	no	—
266	C₇×C₂²⋊C₄	—	2	4	no	—
267	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
268	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
269	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
270	C₄²×C₇	—	2	2	no	—
271	C₄²×C₇	—	2	2	no	—
272	C₄²×C₇	—	2	2	no	—
273	C₄²×C₇	—	2	2	no	—
274	C₄²×C₇	—	2	4	no	—
275	C₄²×C₇	—	2	4	no	—
276	C₄²×C₇	—	2	4	no	—
277	C₄²×C₇	C₂⁴	1	2	yes	—
278	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
279	C₂×#131	C₂²×C₇	1	1	yes	characteristic
280	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
281	#556	C₂²×C₇	1	4	yes	—
282	#554	C₂²×C₇	1	4	yes	—
283	#552	—	2	4	no	—
284	#552	C₂²×C₇	1	4	yes	—
285	#552	C₂²×C₇	1	4	yes	—
286	#551	—	2	4	no	—
287	#544	C₂²×C₇	1	2	yes	—
288	#542	—	2	2	no	—
289	#541	—	2	2	no	—
290	#536	C₂²×C₇	1	2	yes	—
291	#533	—	2	2	no	—
292	#529	—	2	2	no	—
293	#525	C₂²×C₇	1	1	yes	characteristic
294	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
295	#520	C₂²×C₇	1	2	yes	—
296	#519	C₂²×C₇	1	2	yes	—
297	C₂×#117	C₂²×C₇	1	2	yes	—
298	#501	C₂²×C₇	1	4	yes	complemented
299	#499	C₂²×C₇	1	2	yes	complemented
300	#498	C₂²×C₇	1	2	yes	complemented
301	#485	C₂²×C₇	1	4	yes	—
302	#485	C₂²×C₇	1	4	yes	—
303	#485	C₂²×C₇	1	4	yes	complemented
304	#485	C₂²×C₇	1	4	yes	complemented
305	#484	C₂²×C₇	1	4	yes	—
306	#484	C₂²×C₇	1	4	yes	complemented
307	#483	C₂²×C₇	1	4	yes	—
308	#483	C₂²×C₇	1	4	yes	complemented
309	#479	C₂²×C₇	1	4	yes	complemented
310	#479	C₂²×C₇	1	4	yes	complemented
311	#477	C₂²×C₇	1	4	yes	complemented
312	#475	C₂²×C₇	1	4	yes	complemented
313	#470	C₂²×C₇	1	4	yes	—
314	#470	C₂²×C₇	1	4	yes	—
315	#470	C₂²×C₇	1	2	yes	complemented
316	#470	C₂²×C₇	1	2	yes	complemented
317	#467	C₂²×C₇	1	4	yes	complemented
318	#441	C₂²×C₇	1	2	yes	complemented
319	#441	C₂²×C₇	1	2	yes	complemented
320	C₄×Q₁₆	C₂²×C₇	1	4	yes	complemented
321	#436	C₂²×C₇	1	4	yes	—
322	#435	C₂²×C₇	1	4	yes	—
323	#428	C₂²×C₇	1	2	yes	—
324	#428	C₂²×C₇	1	2	yes	—
325	#428	C₂²×C₇	1	2	yes	—
326	#428	C₂²×C₇	1	2	yes	—
327	#428	C₂²×C₇	1	2	yes	—
328	#428	C₂²×C₇	1	2	yes	—
329	#428	C₂²×C₇	1	4	yes	complemented
330	#427	C₂²×C₇	1	2	yes	—
331	#427	C₂²×C₇	1	1	yes	characteristic
332	#427	C₂²×C₇	1	1	yes	characteristic
333	#427	C₂²×C₇	1	2	yes	complemented
334	C₂×#106	C₂²×C₇	1	1	yes	characteristic
335	C₂×#106	C₂²×C₇	1	2	yes	complemented
336	#424	C₂²×C₇	1	1	yes	characteristic
337	#419	C₂²×C₇	1	2	yes	—
338	#419	C₂²×C₇	1	2	yes	—
339	C₂×#103	C₂²×C₇	1	2	yes	—
340	#408	C₂²×C₇	1	1	yes	characteristic
341	#407	C₂²×C₇	1	1	yes	characteristic
342	#405	C₂²×C₇	1	1	yes	characteristic
343	C₂³×C₇	—	2	2	no	—
344	C₂³×C₇	—	4	2	no	—
345	C₂³×C₇	C₂²×D₄	1	2	yes	—
346	C₂³×C₇	C₂²×D₄	1	2	yes	—
347	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
348	C₇×Q₈	—	2	2	no	—
349	C₇×Q₈	—	2	4	no	—
350	C₇×Q₈	—	2	4	no	—
351	C₇×Q₈	—	4	4	no	—
352	C₇×Q₈	—	8	4	no	—
353	C₇×D₄	—	2	4	no	—
354	C₇×D₄	—	2	2	no	—
355	C₇×D₄	—	4	4	no	—
356	C₂×C₄×C₇	—	2	2	no	—
357	C₂×C₄×C₇	—	2	4	no	—
358	C₂×C₄×C₇	—	2	2	no	—
359	C₂×C₄×C₇	—	2	2	no	—
360	C₂×C₄×C₇	—	2	2	no	—
361	C₂×C₄×C₇	—	2	2	no	—
362	C₂×C₄×C₇	—	2	2	no	—
363	C₂×C₄×C₇	—	2	2	no	—
364	C₂×C₄×C₇	—	2	2	no	—
365	C₂×C₄×C₇	—	2	4	no	—
366	C₂×C₄×C₇	—	2	2	no	—
367	C₂×C₄×C₇	—	2	4	no	—
368	C₂×C₄×C₇	—	4	4	no	—
369	C₂×C₄×C₇	—	4	4	no	—
370	C₂×C₄×C₇	—	8	4	no	—
371	C₂×C₄×C₇	—	8	2	no	—
372	C₂×C₄×C₇	—	8	2	no	—
373	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
374	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
375	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
376	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
377	C₂×C₄×C₇	C₂×#41	1	2	yes	—
378	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
379	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
380	C₇×C₈	—	2	2	no	—
381	C₇×C₈	—	2	2	no	—
382	C₇×C₈	—	2	2	no	—
383	C₇×C₈	C₂×#41	1	2	yes	—
384	C₇×C₈	C₂×#41	1	2	yes	—
385	C₂×#41	C₂³×C₇	1	1	yes	characteristic
386	C₂²×Q₈	C₂³×C₇	1	2	yes	—
387	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
388	C₂×Q₁₆	—	2	4	no	—
389	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
390	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
391	#131	—	2	4	no	—
392	C₂×M₁₆	—	2	4	no	—
393	C₂×M₁₆	C₂³×C₇	1	2	yes	—
394	C₂×M₁₆	C₂³×C₇	1	2	yes	—
395	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
396	C₂×M₁₆	C₂³×C₇	1	2	yes	—
397	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₇	1	2	yes	—
400	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
401	C₄⋊Q₈	—	2	2	no	—
402	C₄⋊Q₈	—	2	2	no	—
403	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
404	#126	—	2	4	no	—
405	#126	—	2	4	no	—
406	#126	—	2	4	no	—
407	#126	—	2	4	no	—
408	#125	—	2	2	no	—
409	#125	—	2	4	no	—
410	#125	—	2	4	no	—
411	#125	—	2	4	no	—
412	#125	C₂³×C₇	1	4	yes	—
413	#124	—	2	2	no	—
414	#124	—	2	4	no	—
415	#123	—	2	4	no	—
416	#123	—	2	4	no	—
417	#123	—	2	4	no	—
418	#123	C₇×D₄	1	2	yes	—
419	C₂²⋊Q₈	—	2	2	no	—
420	C₂²⋊Q₈	—	2	2	no	—
421	C₂²⋊Q₈	—	2	4	no	—
422	C₂²⋊Q₈	—	2	4	no	—
423	C₂²⋊Q₈	—	2	4	no	—
424	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
425	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
426	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
427	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
428	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
429	#121	C₇×D₄	1	2	yes	—
430	#120	C₇×D₄	1	2	yes	—
431	C₄×Q₈	—	2	2	no	—
432	C₄×Q₈	—	2	2	no	—
433	C₄×Q₈	—	2	2	no	—
434	C₄×Q₈	—	2	4	no	—
435	C₄×Q₈	C₂³×C₇	1	4	yes	—
436	C₄×Q₈	C₂³×C₇	1	4	yes	—
437	C₄×D₄	—	2	2	no	—
438	C₄×D₄	—	2	4	no	—
439	C₄×D₄	—	2	4	no	—
440	C₄×D₄	C₂³×C₇	1	4	yes	—
441	#117	—	2	2	no	—
442	#117	—	2	2	no	—
443	#117	—	2	4	no	—
444	#117	—	2	4	no	—
445	#117	—	2	4	no	—
446	#117	C₂³×C₇	1	2	yes	—
447	#117	C₂³×C₇	1	2	yes	—
448	#117	C₂³×C₇	1	2	yes	—
449	#117	C₂³×C₇	1	2	yes	—
450	#117	C₂³×C₇	1	4	yes	—
451	#117	C₂³×C₇	1	1	yes	characteristic
452	C₂×C₄⋊C₄	—	2	2	no	—
453	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
454	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
455	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
456	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
457	C₂×C₂²⋊C₄	—	2	2	no	—
458	C₂×C₂²⋊C₄	—	2	2	no	—
459	C₂×C₄²	—	2	2	no	—
460	#107	C₂³×C₇	1	2	yes	—
461	#107	C₂³×C₇	1	2	yes	—
462	#107	C₂³×C₇	1	2	yes	—
463	#107	C₂³×C₇	1	2	yes	—
464	#107	C₂³×C₇	1	4	yes	—
465	#107	C₂³×C₇	1	4	yes	—
466	#106	C₂³×C₇	1	2	yes	—
467	#106	C₂³×C₇	1	2	yes	—
468	#106	C₂³×C₇	1	1	yes	characteristic
469	#106	C₂³×C₇	1	4	yes	—
470	#106	C₂³×C₇	1	1	yes	characteristic
471	#106	C₂³×C₇	1	1	yes	characteristic
472	#106	C₂³×C₇	1	1	yes	characteristic
473	#106	C₂³×C₇	1	2	yes	—
474	#106	C₂³×C₇	1	2	yes	—
475	#106	C₂³×C₇	1	4	yes	—
476	C₄⋊C₈	C₂³×C₇	1	4	yes	—
477	#103	C₂³×C₇	1	4	yes	—
478	#103	C₂³×C₇	1	4	yes	—
479	#103	C₂³×C₇	1	4	yes	—
480	#103	C₂³×C₇	1	4	yes	—
481	#103	C₂³×C₇	1	4	yes	—
482	#103	C₂³×C₇	1	4	yes	—
483	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
484	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
485	#97	C₂³×C₇	1	2	yes	—
486	C₄×C₈	C₂³×C₇	1	2	yes	—
487	C₂²×C₇	—	2	2	no	—
488	C₂²×C₇	—	2	2	no	—
489	C₂²×C₇	—	2	4	no	—
490	C₂²×C₇	—	2	2	no	—
491	C₂²×C₇	—	2	2	no	—
492	C₂²×C₇	—	2	2	no	—
493	C₂²×C₇	—	8	2	no	—
494	C₂²×C₇	#547	1	2	yes	—
495	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
496	C₄×C₇	—	2	4	no	—
497	C₄×C₇	—	2	2	no	—
498	C₄×C₇	—	2	2	no	—
499	C₄×C₇	—	4	2	no	—
500	C₄×C₇	—	4	2	no	—
501	C₄×C₇	—	4	2	no	—
502	C₄×C₇	—	4	4	no	—
503	C₄×C₇	—	4	2	no	—
504	C₄×C₇	—	4	4	no	—
505	C₄×C₇	#547	1	2	yes	—
506	C₄×C₇	C₂²×#41	1	1	yes	characteristic
507	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
508	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
509	#41	—	2	2	no	—
510	#41	—	2	2	no	—
511	C₂×Q₈	—	2	2	no	—
512	C₂×Q₈	—	2	4	no	—
513	C₂×Q₈	—	2	4	no	—
514	C₂×Q₈	—	2	4	no	—
515	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
516	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
517	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
518	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
519	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
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	2	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
528	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
530	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
531	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
532	Q₁₆	—	2	4	no	—
533	Q₁₆	—	2	4	no	—
534	M₁₆	—	2	2	no	—
535	M₁₆	—	2	2	no	—
536	M₁₆	—	2	4	no	—
537	M₁₆	—	2	4	no	—
538	M₁₆	—	2	4	no	—
539	C₂×C₈	—	2	2	no	—
540	C₂×C₈	—	2	4	no	—
541	C₂×C₈	C₇×#41	1	4	yes	—
542	C₂×C₈	C₂⁴×C₇	1	2	yes	—
543	C₂×C₈	C₂⁴×C₇	1	2	yes	—
544	C₂×C₈	C₂⁴×C₇	1	2	yes	—
545	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
546	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	4	no	—
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	—	2	4	no	—
556	C₄⋊C₄	—	2	4	no	—
557	C₄⋊C₄	—	2	4	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
561	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
562	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
563	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
564	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
565	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
566	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
567	C₂²⋊C₄	—	2	2	no	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₂²⋊C₄	—	2	2	no	—
570	C₂²⋊C₄	—	2	2	no	—
571	C₂²⋊C₄	—	2	4	no	—
572	C₂²⋊C₄	—	2	4	no	—
573	C₂²⋊C₄	—	2	4	no	—
574	C₂²⋊C₄	—	2	4	no	—
575	C₂²⋊C₄	—	2	4	no	—
576	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
577	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
578	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
579	C₄²	—	2	2	no	—
580	C₄²	—	2	2	no	—
581	C₄²	—	2	2	no	—
582	C₄²	—	2	2	no	—
583	C₄²	—	2	4	no	—
584	C₄²	—	2	4	no	—
585	C₄²	—	2	4	no	—
586	C₄²	C₂⁴×C₇	1	2	yes	—
587	C₂×C₇	—	2	2	no	—
588	C₂×C₇	—	4	2	no	—
589	C₂×C₇	—	4	2	no	—
590	C₂×C₇	C₂×#547	1	1	yes	characteristic
591	C₂×C₇	#3586	1	2	yes	—
592	C₂³	—	2	2	no	—
593	C₂³	—	4	2	no	—
594	C₂³	C₂²×C₇×D₄	1	2	yes	—
595	C₂³	C₂²×C₇×D₄	1	2	yes	—
596	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
597	Q₈	—	2	2	no	—
598	Q₈	—	2	4	no	—
599	Q₈	—	2	4	no	—
600	Q₈	—	4	4	no	—
601	Q₈	—	8	4	no	—
602	D₄	—	2	4	no	—
603	D₄	—	2	2	no	—
604	D₄	—	4	4	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	4	4	no	—
618	C₂×C₄	—	4	4	no	—
619	C₂×C₄	—	8	4	no	—
620	C₂×C₄	—	8	2	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
623	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
624	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
625	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
626	C₂×C₄	C₂×C₇×#41	1	2	yes	—
627	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
628	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
629	C₈	—	2	2	no	—
630	C₈	—	2	2	no	—
631	C₈	—	2	2	no	—
632	C₈	C₂×C₇×#41	1	2	yes	—
633	C₈	C₂×C₇×#41	1	2	yes	—
634	C₇	#61269	1	1	yes	characteristic, direct product factor, Sylow
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	4	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	8	2	no	—
642	C₂²	C₇×#547	1	2	yes	—
643	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
644	C₄	—	2	4	no	—
645	C₄	—	2	2	no	—
646	C₄	—	2	2	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	2	no	—
650	C₄	—	4	4	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	4	no	—
653	C₄	C₇×#547	1	2	yes	—
654	C₄	C₂²×C₇×#41	1	1	yes	characteristic
655	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
656	C₂	—	2	2	no	—
657	C₂	—	4	2	no	—
658	C₂	—	4	2	no	—
659	C₂	C₂×C₇×#547	1	1	yes	characteristic
660	C₂	C₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#525 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3356	C₂	semidirect
3. C₇×#3352	C₂	semidirect
4. C₇×#3349	C₂	semidirect
5. C₇×#3344	C₂	semidirect
6. C₇×#3163	C₂	semidirect
7. C₇×#3097	C₂	semidirect
8. C₇×#3095	C₂	semidirect
9. C₇×#3082	C₂	semidirect
10. C₇×#3059	C₂	semidirect
11. C₂×C₇×#470	C₂	semidirect
12. C₇×#2950	C₂	semidirect
14. C₇×#2912	C₂	semidirect
15. C₇×#2911	C₂	semidirect
36. C₇×#501	C₂²	semidirect
37. C₇×#499	C₂²	semidirect
38. C₇×#498	C₂²	semidirect
41. C₇×#485	C₂²	semidirect
42. C₇×#485	C₂²	semidirect
44. C₇×#484	C₂²	semidirect
46. C₇×#483	C₂²	semidirect
47. C₇×#479	C₂²	semidirect
48. C₇×#479	C₂²	semidirect
49. C₇×#477	C₂²	semidirect
50. C₇×#475	C₂²	semidirect
53. C₇×#470	C₂²	semidirect
54. C₇×#470	C₂²	semidirect
55. C₇×#467	C₂²	semidirect
56. C₇×#441	C₂²	semidirect
57. C₇×#441	C₂²	semidirect
58. C₄×C₇×Q₁₆	C₂²	semidirect
67. C₇×#428	C₂²	semidirect
71. C₇×#427	C₂²	semidirect
73. C₂×C₇×#106	C₂²	semidirect
81. #61269	C₇	direct (N × H)
185. #3356	C₂×C₇	semidirect
186. #3352	C₂×C₇	semidirect
187. #3349	C₂×C₇	semidirect
188. #3344	C₂×C₇	semidirect
189. #3163	C₂×C₇	semidirect
190. #3097	C₂×C₇	semidirect
191. #3095	C₂×C₇	semidirect
192. #3082	C₂×C₇	semidirect
193. #3059	C₂×C₇	semidirect
194. C₂×#470	C₂×C₇	semidirect
195. #2950	C₂×C₇	semidirect
197. #2912	C₂×C₇	semidirect
198. #2911	C₂×C₇	semidirect
298. #501	C₂²×C₇	semidirect
299. #499	C₂²×C₇	semidirect
300. #498	C₂²×C₇	semidirect
303. #485	C₂²×C₇	semidirect
304. #485	C₂²×C₇	semidirect
306. #484	C₂²×C₇	semidirect
308. #483	C₂²×C₇	semidirect
309. #479	C₂²×C₇	semidirect
310. #479	C₂²×C₇	semidirect
311. #477	C₂²×C₇	semidirect
312. #475	C₂²×C₇	semidirect
315. #470	C₂²×C₇	semidirect
316. #470	C₂²×C₇	semidirect
317. #467	C₂²×C₇	semidirect
318. #441	C₂²×C₇	semidirect
319. #441	C₂²×C₇	semidirect
320. C₄×Q₁₆	C₂²×C₇	semidirect
329. #428	C₂²×C₇	semidirect
333. #427	C₂²×C₇	semidirect
335. C₂×#106	C₂²×C₇	semidirect
634. C₇	#61269	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×#3495	2
C₇×#3356	2
C₇×#3352	2
C₇×#3349	2
C₇×#3344	2
C₇×#3163	2
C₇×#3097	2
C₇×#3095	2
C₇×#3082	2
C₇×#3059	2
C₂×C₇×#470	2
C₇×#2950	2
C₇×#2930	2
C₇×#2912	2
C₇×#2911	2
#61269	7

Subgroups — C7×#61269 #13238485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61269 #13238485