Subgroups — C₇×#60826 #13238042

Order 1792.

2026 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3500	C₂	1	1	yes	maximal, characteristic
3	C₇×#3223	C₂	1	4	yes	maximal, complemented
4	C₇×#3222	C₂	1	4	yes	maximal, complemented
5	C₇×#3216	C₂	1	4	yes	maximal, complemented
6	C₇×#3215	C₂	1	4	yes	maximal, complemented
7	C₇×#3152	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#497	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3135	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3044	C₂	1	2	yes	maximal, complemented
12	C₇×#3043	C₂	1	2	yes	maximal, complemented
13	C₇×#3011	C₂	1	2	yes	maximal, complemented
14	C₇×#3009	C₂	1	2	yes	maximal, complemented
15	C₇×#2923	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
17	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
18	C₂×C₇×#137	C₂²	1	2	yes	—
19	C₂×C₇×#137	C₂²	1	2	yes	complemented
20	C₂²×C₇×Q₁₆	C₂²	1	2	yes	complemented
21	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
22	C₇×#557	—	2	2	no	—
23	C₇×#556	—	2	2	no	—
24	C₇×#552	—	2	4	no	—
25	C₇×#548	—	2	2	no	—
26	C₇×#546	—	2	4	no	—
27	C₇×#546	—	2	2	no	—
28	C₇×#541	C₂²	1	4	yes	complemented
29	C₇×#536	C₂²	1	2	yes	—
30	C₇×#536	C₂²	1	1	yes	characteristic
31	C₇×#536	C₂²	1	2	yes	—
32	C₇×#535	C₂²	1	2	yes	—
33	C₇×#535	C₂²	1	1	yes	characteristic
34	C₇×#533	C₂²	1	4	yes	complemented
35	C₇×#533	C₂²	1	1	yes	characteristic, complemented
36	C₇×#532	C₂²	1	2	yes	—
37	C₇×#532	C₂²	1	1	yes	characteristic
38	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₇×#124	C₂²	1	2	yes	—
40	C₂×C₇×#124	C₂²	1	1	yes	characteristic
41	C₇×#519	C₂²	1	2	yes	complemented
42	C₄×C₇×#41	C₂²	1	1	yes	characteristic
43	C₇×#497	C₂²	1	2	yes	complemented
44	C₇×#497	C₂²	1	4	yes	complemented
45	C₇×#497	C₂²	1	2	yes	complemented
46	C₇×#497	C₂²	1	4	yes	complemented
47	C₇×#494	C₂²	1	2	yes	complemented
48	C₇×#492	C₂²	1	2	yes	complemented
49	C₇×#491	C₂²	1	2	yes	complemented
50	C₇×#490	C₂²	1	2	yes	complemented
51	C₇×#489	C₂²	1	2	yes	complemented
52	C₇×#489	C₂²	1	4	yes	complemented
53	C₇×#489	C₂²	1	2	yes	complemented
54	C₇×#488	C₂²	1	4	yes	complemented
55	C₇×#464	C₂²	1	8	yes	complemented
56	C₇×#464	C₂²	1	8	yes	complemented
57	C₇×#462	C₂²	1	8	yes	complemented
58	C₇×#460	C₂²	1	8	yes	complemented
59	C₇×#452	C₂²	1	8	yes	—
60	C₇×#452	C₂²	1	8	yes	complemented
61	C₇×#451	C₂²	1	8	yes	complemented
62	C₇×#450	C₂²	1	8	yes	—
63	C₇×#433	C₂²	1	4	yes	complemented
64	C₇×#432	C₂²	1	4	yes	complemented
65	C₇×#424	C₂²	1	2	yes	complemented
66	C₇×#419	C₂²	1	2	yes	complemented
67	C₇×#418	C₂²	1	2	yes	—
68	C₇×#416	C₂²	1	2	yes	—
69	C₇×#416	C₂²	1	2	yes	complemented
70	C₇×#408	C₂²	1	2	yes	—
71	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
72	C₂×C₇×#97	C₂²	1	1	yes	characteristic, complemented
73	#60826	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₇×2₋¹⁺⁴	—	4	4	no	—
75	C₇×H(5,𝔽₂)	—	4	4	no	—
76	C₂×C₇×#41	—	2	1	no	—
77	C₂×C₇×#41	—	2	2	no	—
78	C₂×C₇×#41	—	2	1	no	—
79	C₂×C₇×#41	C₂³	1	1	yes	characteristic
80	C₂×C₇×#41	C₂³	1	1	yes	characteristic
81	C₂×C₇×#41	C₂³	1	2	yes	—
82	C₂×C₇×#41	C₂³	1	2	yes	—
83	C₂×C₇×#41	C₂³	1	1	yes	characteristic
84	C₂²×C₇×Q₈	—	2	1	no	—
85	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
86	C₂²×C₇×Q₈	C₂³	1	2	yes	—
87	C₂²×C₇×Q₈	C₂³	1	2	yes	—
88	C₂²×C₇×D₄	—	2	2	no	—
89	C₂²×C₇×D₄	—	2	1	no	—
90	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
91	C₂²×C₇×D₄	C₂³	1	2	yes	—
92	C₇×#137	—	4	4	no	—
93	C₇×#137	—	4	4	no	—
94	C₂×C₇×Q₁₆	—	4	4	no	—
95	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
96	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
97	C₂×C₇×Q₁₆	C₂³	1	8	yes	—
98	C₂×C₇×SD₁₆	—	4	4	no	—
99	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
100	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
101	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
102	C₂×C₇×M₁₆	C₂³	1	2	yes	—
103	C₂²×C₇×C₈	C₂³	1	2	yes	—
104	C₇×C₄⋊Q₈	—	2	2	no	—
105	C₇×C₄⋊Q₈	—	2	2	no	—
106	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
107	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
108	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
109	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
110	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
111	C₇×#127	C₂³	1	1	yes	characteristic
112	C₇×#127	C₂³	1	2	yes	—
113	C₇×#126	—	2	8	no	—
114	C₇×#125	—	2	8	no	—
115	C₇×#125	—	2	4	no	—
116	C₇×#125	C₂³	1	2	yes	—
117	C₇×#124	—	2	2	no	—
118	C₇×#124	—	2	4	no	—
119	C₇×#124	—	2	2	no	—
120	C₇×#124	C₂³	1	2	yes	—
121	C₇×#124	C₂³	1	1	yes	characteristic
122	C₇×#124	C₂³	1	2	yes	—
123	C₇×#124	C₂³	1	1	yes	characteristic
124	C₇×#124	C₂³	1	2	yes	—
125	C₇×#124	C₂³	1	2	yes	—
126	C₇×#123	—	2	4	no	—
127	C₇×#123	—	2	2	no	—
128	C₇×#123	—	2	4	no	—
129	C₇×C₂²⋊Q₈	—	2	2	no	—
130	C₇×C₂²⋊Q₈	—	2	4	no	—
131	C₇×C₂²⋊Q₈	—	2	2	no	—
132	C₇×C₂²⋊Q₈	—	2	4	no	—
133	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
134	C₇×#121	—	2	2	no	—
135	C₇×#121	—	2	4	no	—
136	C₇×#121	—	2	2	no	—
137	C₇×#121	—	2	4	no	—
138	C₇×#120	—	2	2	no	—
139	C₇×#120	—	2	4	no	—
140	C₄×C₇×Q₈	—	2	4	no	—
141	C₄×C₇×Q₈	—	2	4	no	—
142	C₄×C₇×Q₈	—	2	2	no	—
143	C₄×C₇×Q₈	D₄	1	2	yes	—
144	C₄×C₇×Q₈	C₂³	1	8	yes	—
145	C₄×C₇×D₄	—	2	4	no	—
146	C₄×C₇×D₄	—	2	2	no	—
147	C₄×C₇×D₄	D₄	1	4	yes	—
148	C₄×C₇×D₄	D₄	1	2	yes	—
149	C₇×#117	—	2	4	no	—
150	C₇×#117	—	2	4	no	—
151	C₇×#117	C₂³	1	2	yes	—
152	C₇×#117	C₂³	1	1	yes	characteristic
153	C₇×#117	C₂³	1	2	yes	—
154	C₂×C₇×C₄⋊C₄	—	2	2	no	—
155	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
156	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
157	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
158	C₂×C₄²×C₇	C₂³	1	2	yes	—
159	C₇×C₄⋊C₈	C₂³	1	8	yes	—
160	C₇×#103	C₂³	1	8	yes	—
161	C₇×#103	C₂³	1	4	yes	—
162	C₇×#103	C₂³	1	4	yes	—
163	C₇×#102	C₂³	1	8	yes	—
164	C₇×#102	C₂³	1	4	yes	—
165	C₇×#102	C₂³	1	4	yes	—
166	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
167	C₇×#97	C₂³	1	2	yes	—
168	C₇×#97	C₂³	1	2	yes	—
169	C₇×#97	C₂³	1	2	yes	—
170	C₄×C₇×C₈	C₂³	1	2	yes	—
171	#3509	C₂×C₇	1	1	yes	characteristic, complemented
172	#3500	C₂×C₇	1	1	yes	characteristic
173	#3223	C₂×C₇	1	4	yes	complemented
174	#3222	C₂×C₇	1	4	yes	complemented
175	#3216	C₂×C₇	1	4	yes	complemented
176	#3215	C₂×C₇	1	4	yes	complemented
177	#3152	C₂×C₇	1	1	yes	characteristic, complemented
178	C₂×#497	C₂×C₇	1	1	yes	characteristic, complemented
179	#3140	C₂×C₇	1	1	yes	characteristic, complemented
180	#3135	C₂×C₇	1	1	yes	characteristic, complemented
181	#3044	C₂×C₇	1	2	yes	complemented
182	#3043	C₂×C₇	1	2	yes	complemented
183	#3011	C₂×C₇	1	2	yes	complemented
184	#3009	C₂×C₇	1	2	yes	complemented
185	#2923	C₂×C₇	1	1	yes	characteristic, complemented
186	C₂⁴×C₇	—	2	2	no	—
187	C₂⁴×C₇	—	2	1	no	—
188	C₇×#41	—	2	4	no	—
189	C₇×#41	—	2	4	no	—
190	C₇×#41	—	2	4	no	—
191	C₇×#41	—	4	4	no	—
192	C₇×#41	—	4	4	no	—
193	C₇×#41	—	4	8	no	—
194	C₇×#41	—	4	4	no	—
195	C₇×#41	—	4	4	no	—
196	C₂×C₇×Q₈	—	2	2	no	—
197	C₂×C₇×Q₈	—	2	2	no	—
198	C₂×C₇×Q₈	—	2	2	no	—
199	C₂×C₇×Q₈	—	2	1	no	—
200	C₂×C₇×Q₈	—	2	4	no	—
201	C₂×C₇×Q₈	—	2	4	no	—
202	C₂×C₇×Q₈	—	2	4	no	—
203	C₂×C₇×Q₈	—	4	4	no	—
204	C₂×C₇×Q₈	—	4	4	no	—
205	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
206	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
207	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
208	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
209	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
210	C₂×C₇×D₄	—	2	2	no	—
211	C₂×C₇×D₄	—	2	1	no	—
212	C₂×C₇×D₄	—	2	2	no	—
213	C₂×C₇×D₄	—	2	1	no	—
214	C₂×C₇×D₄	—	2	2	no	—
215	C₂×C₇×D₄	—	2	1	no	—
216	C₂×C₇×D₄	—	2	4	no	—
217	C₂×C₇×D₄	—	2	2	no	—
218	C₂×C₇×D₄	—	2	4	no	—
219	C₂×C₇×D₄	—	4	4	no	—
220	C₂×C₇×D₄	—	4	8	no	—
221	C₂×C₇×D₄	—	4	4	no	—
222	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
223	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
224	C₂×C₇×D₄	C₂⁴	1	2	yes	—
225	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
226	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
227	C₂²×C₄×C₇	—	2	2	no	—
228	C₂²×C₄×C₇	—	2	1	no	—
229	C₂²×C₄×C₇	—	2	2	no	—
230	C₂²×C₄×C₇	—	2	2	no	—
231	C₂²×C₄×C₇	—	2	1	no	—
232	C₂²×C₄×C₇	—	2	2	no	—
233	C₂²×C₄×C₇	—	2	1	no	—
234	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
235	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
236	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
237	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
238	C₇×Q₁₆	—	4	8	no	—
239	C₇×Q₁₆	—	4	4	no	—
240	C₇×Q₁₆	—	4	4	no	—
241	C₇×SD₁₆	—	4	8	no	—
242	C₇×SD₁₆	—	4	4	no	—
243	C₇×SD₁₆	—	4	4	no	—
244	C₇×M₁₆	—	2	4	no	—
245	C₂×C₇×C₈	—	2	4	no	—
246	C₂×C₇×C₈	C₂⁴	1	4	yes	—
247	C₂×C₇×C₈	C₂⁴	1	4	yes	—
248	C₇×C₄⋊C₄	—	2	4	no	—
249	C₇×C₄⋊C₄	—	2	4	no	—
250	C₇×C₄⋊C₄	—	2	2	no	—
251	C₇×C₄⋊C₄	—	2	8	no	—
252	C₇×C₄⋊C₄	—	2	4	no	—
253	C₇×C₄⋊C₄	—	2	2	no	—
254	C₇×C₄⋊C₄	—	2	4	no	—
255	C₇×C₄⋊C₄	—	2	8	no	—
256	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
257	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
258	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
259	C₇×C₂²⋊C₄	—	2	2	no	—
260	C₇×C₂²⋊C₄	—	2	8	no	—
261	C₇×C₂²⋊C₄	—	2	4	no	—
262	C₇×C₂²⋊C₄	—	2	2	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	2	no	—
267	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
268	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
269	C₄²×C₇	—	2	8	no	—
270	C₄²×C₇	—	2	2	no	—
271	C₄²×C₇	C₂×D₄	1	4	yes	—
272	C₄²×C₇	C₂×D₄	1	2	yes	—
273	C₄²×C₇	C₂⁴	1	1	yes	characteristic
274	C₄²×C₇	C₂⁴	1	1	yes	characteristic
275	C₄²×C₇	C₂⁴	1	2	yes	—
276	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
277	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic
278	C₂×#137	C₂²×C₇	1	2	yes	—
279	C₂×#137	C₂²×C₇	1	2	yes	complemented
280	C₂²×Q₁₆	C₂²×C₇	1	2	yes	complemented
281	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
282	#557	—	2	2	no	—
283	#556	—	2	2	no	—
284	#552	—	2	4	no	—
285	#548	—	2	2	no	—
286	#546	—	2	4	no	—
287	#546	—	2	2	no	—
288	#541	C₂²×C₇	1	4	yes	complemented
289	#536	C₂²×C₇	1	2	yes	—
290	#536	C₂²×C₇	1	1	yes	characteristic
291	#536	C₂²×C₇	1	2	yes	—
292	#535	C₂²×C₇	1	2	yes	—
293	#535	C₂²×C₇	1	1	yes	characteristic
294	#533	C₂²×C₇	1	4	yes	complemented
295	#533	C₂²×C₇	1	1	yes	characteristic, complemented
296	#532	C₂²×C₇	1	1	yes	characteristic
297	#532	C₂²×C₇	1	2	yes	—
298	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
299	C₂×#124	C₂²×C₇	1	1	yes	characteristic
300	C₂×#124	C₂²×C₇	1	2	yes	—
301	#519	C₂²×C₇	1	2	yes	complemented
302	C₄×#41	C₂²×C₇	1	1	yes	characteristic
303	#497	C₂²×C₇	1	4	yes	complemented
304	#497	C₂²×C₇	1	4	yes	complemented
305	#497	C₂²×C₇	1	2	yes	complemented
306	#497	C₂²×C₇	1	2	yes	complemented
307	#494	C₂²×C₇	1	2	yes	complemented
308	#492	C₂²×C₇	1	2	yes	complemented
309	#491	C₂²×C₇	1	2	yes	complemented
310	#490	C₂²×C₇	1	2	yes	complemented
311	#489	C₂²×C₇	1	2	yes	complemented
312	#489	C₂²×C₇	1	2	yes	complemented
313	#489	C₂²×C₇	1	4	yes	complemented
314	#488	C₂²×C₇	1	4	yes	complemented
315	#464	C₂²×C₇	1	8	yes	complemented
316	#464	C₂²×C₇	1	8	yes	complemented
317	#462	C₂²×C₇	1	8	yes	complemented
318	#460	C₂²×C₇	1	8	yes	complemented
319	#452	C₂²×C₇	1	8	yes	—
320	#452	C₂²×C₇	1	8	yes	complemented
321	#451	C₂²×C₇	1	8	yes	complemented
322	#450	C₂²×C₇	1	8	yes	—
323	#433	C₂²×C₇	1	4	yes	complemented
324	#432	C₂²×C₇	1	4	yes	complemented
325	#424	C₂²×C₇	1	2	yes	complemented
326	#419	C₂²×C₇	1	2	yes	complemented
327	#418	C₂²×C₇	1	2	yes	—
328	#416	C₂²×C₇	1	2	yes	—
329	#416	C₂²×C₇	1	2	yes	complemented
330	#408	C₂²×C₇	1	2	yes	—
331	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
332	C₂×#97	C₂²×C₇	1	1	yes	characteristic, complemented
333	C₂³×C₇	—	2	2	no	—
334	C₂³×C₇	—	2	1	no	—
335	C₂³×C₇	—	2	1	no	—
336	C₂³×C₇	—	4	4	no	—
337	C₂³×C₇	—	4	2	no	—
338	C₂³×C₇	—	8	2	no	—
339	C₂³×C₇	—	8	4	no	—
340	C₂³×C₇	C₂²×D₄	1	2	yes	—
341	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
342	C₇×Q₈	—	2	2	no	—
343	C₇×Q₈	—	2	4	no	—
344	C₇×Q₈	—	2	4	no	—
345	C₇×Q₈	—	2	4	no	—
346	C₇×Q₈	—	2	2	no	—
347	C₇×Q₈	—	2	2	no	—
348	C₇×Q₈	—	4	4	no	—
349	C₇×Q₈	—	4	4	no	—
350	C₇×Q₈	—	4	4	no	—
351	C₇×Q₈	—	4	2	no	—
352	C₇×D₄	—	2	4	no	—
353	C₇×D₄	—	2	2	no	—
354	C₇×D₄	—	2	4	no	—
355	C₇×D₄	—	2	2	no	—
356	C₇×D₄	—	2	2	no	—
357	C₇×D₄	—	4	4	no	—
358	C₇×D₄	—	4	4	no	—
359	C₇×D₄	—	4	4	no	—
360	C₇×D₄	—	4	2	no	—
361	C₇×D₄	—	4	2	no	—
362	C₇×D₄	—	4	2	no	—
363	C₇×D₄	—	8	4	no	—
364	C₇×D₄	—	8	2	no	—
365	C₂×C₄×C₇	—	2	4	no	—
366	C₂×C₄×C₇	—	2	4	no	—
367	C₂×C₄×C₇	—	2	4	no	—
368	C₂×C₄×C₇	—	2	2	no	—
369	C₂×C₄×C₇	—	2	1	no	—
370	C₂×C₄×C₇	—	2	1	no	—
371	C₂×C₄×C₇	—	4	2	no	—
372	C₂×C₄×C₇	—	4	4	no	—
373	C₂×C₄×C₇	—	4	4	no	—
374	C₂×C₄×C₇	—	4	4	no	—
375	C₂×C₄×C₇	—	4	4	no	—
376	C₂×C₄×C₇	—	4	4	no	—
377	C₂×C₄×C₇	—	4	2	no	—
378	C₂×C₄×C₇	—	4	2	no	—
379	C₂×C₄×C₇	—	4	2	no	—
380	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
381	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
382	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
383	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
384	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
385	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
386	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
387	C₇×C₈	—	2	4	no	—
388	C₇×C₈	—	2	4	no	—
389	2₋¹⁺⁴	—	4	4	no	—
390	H(5,𝔽₂)	—	4	4	no	—
391	C₂×#41	—	2	1	no	—
392	C₂×#41	—	2	2	no	—
393	C₂×#41	—	2	1	no	—
394	C₂×#41	C₂³×C₇	1	1	yes	characteristic
395	C₂×#41	C₂³×C₇	1	1	yes	characteristic
396	C₂×#41	C₂³×C₇	1	2	yes	—
397	C₂×#41	C₂³×C₇	1	2	yes	—
398	C₂×#41	C₂³×C₇	1	1	yes	characteristic
399	C₂²×Q₈	—	2	1	no	—
400	C₂²×Q₈	C₂³×C₇	1	2	yes	—
401	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
402	C₂²×Q₈	C₂³×C₇	1	2	yes	—
403	C₂²×D₄	—	2	2	no	—
404	C₂²×D₄	—	2	1	no	—
405	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
406	C₂²×D₄	C₂³×C₇	1	2	yes	—
407	#137	—	4	4	no	—
408	#137	—	4	4	no	—
409	C₂×Q₁₆	—	4	4	no	—
410	C₂×Q₁₆	C₂³×C₇	1	8	yes	—
411	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
412	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
413	C₂×SD₁₆	—	4	4	no	—
414	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
415	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
416	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
417	C₂×M₁₆	C₂³×C₇	1	2	yes	—
418	C₂²×C₈	C₂³×C₇	1	2	yes	—
419	C₄⋊Q₈	—	2	2	no	—
420	C₄⋊Q₈	—	2	2	no	—
421	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
422	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
423	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
424	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
425	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
426	#127	C₂³×C₇	1	2	yes	—
427	#127	C₂³×C₇	1	1	yes	characteristic
428	#126	—	2	8	no	—
429	#125	—	2	8	no	—
430	#125	—	2	4	no	—
431	#125	C₂³×C₇	1	2	yes	—
432	#124	—	2	2	no	—
433	#124	—	2	4	no	—
434	#124	—	2	2	no	—
435	#124	C₂³×C₇	1	2	yes	—
436	#124	C₂³×C₇	1	1	yes	characteristic
437	#124	C₂³×C₇	1	2	yes	—
438	#124	C₂³×C₇	1	2	yes	—
439	#124	C₂³×C₇	1	2	yes	—
440	#124	C₂³×C₇	1	1	yes	characteristic
441	#123	—	2	4	no	—
442	#123	—	2	2	no	—
443	#123	—	2	4	no	—
444	C₂²⋊Q₈	—	2	2	no	—
445	C₂²⋊Q₈	—	2	4	no	—
446	C₂²⋊Q₈	—	2	2	no	—
447	C₂²⋊Q₈	—	2	4	no	—
448	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
449	#121	—	2	2	no	—
450	#121	—	2	4	no	—
451	#121	—	2	2	no	—
452	#121	—	2	4	no	—
453	#120	—	2	2	no	—
454	#120	—	2	4	no	—
455	C₄×Q₈	—	2	4	no	—
456	C₄×Q₈	—	2	4	no	—
457	C₄×Q₈	—	2	2	no	—
458	C₄×Q₈	C₇×D₄	1	2	yes	—
459	C₄×Q₈	C₂³×C₇	1	8	yes	—
460	C₄×D₄	—	2	4	no	—
461	C₄×D₄	—	2	2	no	—
462	C₄×D₄	C₇×D₄	1	4	yes	—
463	C₄×D₄	C₇×D₄	1	2	yes	—
464	#117	—	2	4	no	—
465	#117	—	2	4	no	—
466	#117	C₂³×C₇	1	2	yes	—
467	#117	C₂³×C₇	1	1	yes	characteristic
468	#117	C₂³×C₇	1	2	yes	—
469	C₂×C₄⋊C₄	—	2	2	no	—
470	C₂×C₂²⋊C₄	—	2	2	no	—
471	C₂×C₂²⋊C₄	—	2	4	no	—
472	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
473	C₂×C₄²	C₂³×C₇	1	2	yes	—
474	C₄⋊C₈	C₂³×C₇	1	8	yes	—
475	#103	C₂³×C₇	1	4	yes	—
476	#103	C₂³×C₇	1	4	yes	—
477	#103	C₂³×C₇	1	8	yes	—
478	#102	C₂³×C₇	1	4	yes	—
479	#102	C₂³×C₇	1	4	yes	—
480	#102	C₂³×C₇	1	8	yes	—
481	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
482	#97	C₂³×C₇	1	2	yes	—
483	#97	C₂³×C₇	1	2	yes	—
484	#97	C₂³×C₇	1	2	yes	—
485	C₄×C₈	C₂³×C₇	1	2	yes	—
486	C₂²×C₇	—	2	2	no	—
487	C₂²×C₇	—	2	4	no	—
488	C₂²×C₇	—	2	2	no	—
489	C₂²×C₇	—	4	2	no	—
490	C₂²×C₇	—	4	4	no	—
491	C₂²×C₇	—	4	2	no	—
492	C₂²×C₇	—	4	1	no	—
493	C₂²×C₇	—	4	2	no	—
494	C₂²×C₇	—	4	1	no	—
495	C₂²×C₇	—	16	2	no	—
496	C₂²×C₇	—	16	4	no	—
497	C₂²×C₇	#535	1	2	yes	—
498	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
499	C₄×C₇	—	2	2	no	—
500	C₄×C₇	—	2	4	no	—
501	C₄×C₇	—	2	2	no	—
502	C₄×C₇	—	2	2	no	—
503	C₄×C₇	—	4	4	no	—
504	C₄×C₇	—	4	1	no	—
505	C₄×C₇	—	4	2	no	—
506	C₄×C₇	—	4	1	no	—
507	C₄×C₇	—	4	4	no	—
508	C₄×C₇	#535	1	2	yes	—
509	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
510	C₂⁴	—	2	2	no	—
511	C₂⁴	—	2	1	no	—
512	#41	—	2	4	no	—
513	#41	—	2	4	no	—
514	#41	—	2	4	no	—
515	#41	—	4	4	no	—
516	#41	—	4	4	no	—
517	#41	—	4	8	no	—
518	#41	—	4	4	no	—
519	#41	—	4	4	no	—
520	C₂×Q₈	—	2	2	no	—
521	C₂×Q₈	—	2	2	no	—
522	C₂×Q₈	—	2	2	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	4	no	—
525	C₂×Q₈	—	2	4	no	—
526	C₂×Q₈	—	2	4	no	—
527	C₂×Q₈	—	4	4	no	—
528	C₂×Q₈	—	4	4	no	—
529	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
530	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
531	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
532	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
533	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
534	C₂×D₄	—	2	2	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	—	2	2	no	—
537	C₂×D₄	—	2	1	no	—
538	C₂×D₄	—	2	2	no	—
539	C₂×D₄	—	2	1	no	—
540	C₂×D₄	—	2	4	no	—
541	C₂×D₄	—	2	2	no	—
542	C₂×D₄	—	2	4	no	—
543	C₂×D₄	—	4	4	no	—
544	C₂×D₄	—	4	8	no	—
545	C₂×D₄	—	4	4	no	—
546	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
547	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
548	C₂×D₄	C₂⁴×C₇	1	2	yes	—
549	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
550	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	2	2	no	—
554	C₂²×C₄	—	2	2	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	2	2	no	—
557	C₂²×C₄	—	2	1	no	—
558	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
559	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
560	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
561	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
562	Q₁₆	—	4	8	no	—
563	Q₁₆	—	4	4	no	—
564	Q₁₆	—	4	4	no	—
565	SD₁₆	—	4	8	no	—
566	SD₁₆	—	4	4	no	—
567	SD₁₆	—	4	4	no	—
568	M₁₆	—	2	4	no	—
569	C₂×C₈	—	2	4	no	—
570	C₂×C₈	C₂⁴×C₇	1	4	yes	—
571	C₂×C₈	C₂⁴×C₇	1	4	yes	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	4	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	8	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	4	no	—
579	C₄⋊C₄	—	2	8	no	—
580	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
581	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
582	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
583	C₂²⋊C₄	—	2	2	no	—
584	C₂²⋊C₄	—	2	8	no	—
585	C₂²⋊C₄	—	2	4	no	—
586	C₂²⋊C₄	—	2	2	no	—
587	C₂²⋊C₄	—	2	4	no	—
588	C₂²⋊C₄	—	2	4	no	—
589	C₂²⋊C₄	—	2	4	no	—
590	C₂²⋊C₄	—	2	2	no	—
591	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
592	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
593	C₄²	—	2	8	no	—
594	C₄²	—	2	2	no	—
595	C₄²	C₂×C₇×D₄	1	4	yes	—
596	C₄²	C₂×C₇×D₄	1	2	yes	—
597	C₄²	C₂⁴×C₇	1	1	yes	characteristic
598	C₄²	C₂⁴×C₇	1	1	yes	characteristic
599	C₄²	C₂⁴×C₇	1	2	yes	—
600	C₂×C₇	—	2	2	no	—
601	C₂×C₇	—	4	2	no	—
602	C₂×C₇	—	8	1	no	—
603	C₂×C₇	—	8	2	no	—
604	C₂×C₇	—	8	1	no	—
605	C₂×C₇	C₂×#535	1	1	yes	characteristic
606	C₂×C₇	#3586	1	2	yes	—
607	C₂³	—	2	2	no	—
608	C₂³	—	2	1	no	—
609	C₂³	—	2	1	no	—
610	C₂³	—	4	4	no	—
611	C₂³	—	4	2	no	—
612	C₂³	—	8	2	no	—
613	C₂³	—	8	4	no	—
614	C₂³	C₂²×C₇×D₄	1	2	yes	—
615	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
616	Q₈	—	2	2	no	—
617	Q₈	—	2	4	no	—
618	Q₈	—	2	4	no	—
619	Q₈	—	2	4	no	—
620	Q₈	—	2	2	no	—
621	Q₈	—	2	2	no	—
622	Q₈	—	4	4	no	—
623	Q₈	—	4	4	no	—
624	Q₈	—	4	4	no	—
625	Q₈	—	4	2	no	—
626	D₄	—	2	4	no	—
627	D₄	—	2	4	no	—
628	D₄	—	2	2	no	—
629	D₄	—	2	2	no	—
630	D₄	—	2	2	no	—
631	D₄	—	4	4	no	—
632	D₄	—	4	4	no	—
633	D₄	—	4	4	no	—
634	D₄	—	4	2	no	—
635	D₄	—	4	2	no	—
636	D₄	—	4	2	no	—
637	D₄	—	8	4	no	—
638	D₄	—	8	2	no	—
639	C₂×C₄	—	2	4	no	—
640	C₂×C₄	—	2	4	no	—
641	C₂×C₄	—	2	4	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	4	2	no	—
646	C₂×C₄	—	4	4	no	—
647	C₂×C₄	—	4	4	no	—
648	C₂×C₄	—	4	4	no	—
649	C₂×C₄	—	4	4	no	—
650	C₂×C₄	—	4	4	no	—
651	C₂×C₄	—	4	2	no	—
652	C₂×C₄	—	4	2	no	—
653	C₂×C₄	—	4	2	no	—
654	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
655	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
656	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
657	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
658	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
659	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
660	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
661	C₈	—	2	4	no	—
662	C₈	—	2	4	no	—
663	C₇	#60826	1	1	yes	characteristic, direct product factor, Sylow
664	C₂²	—	2	2	no	—
665	C₂²	—	2	4	no	—
666	C₂²	—	2	2	no	—
667	C₂²	—	4	2	no	—
668	C₂²	—	4	4	no	—
669	C₂²	—	4	2	no	—
670	C₂²	—	4	1	no	—
671	C₂²	—	4	2	no	—
672	C₂²	—	4	1	no	—
673	C₂²	—	16	2	no	—
674	C₂²	—	16	4	no	—
675	C₂²	C₇×#535	1	2	yes	—
676	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
677	C₄	—	2	2	no	—
678	C₄	—	2	4	no	—
679	C₄	—	2	2	no	—
680	C₄	—	2	2	no	—
681	C₄	—	4	4	no	—
682	C₄	—	4	1	no	—
683	C₄	—	4	2	no	—
684	C₄	—	4	1	no	—
685	C₄	—	4	4	no	—
686	C₄	C₇×#535	1	2	yes	—
687	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
688	C₂	—	2	2	no	—
689	C₂	—	4	2	no	—
690	C₂	—	8	2	no	—
691	C₂	—	8	1	no	—
692	C₂	—	8	1	no	—
693	C₂	C₂×C₇×#535	1	1	yes	characteristic
694	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×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3509	C₂	semidirect
3. C₇×#3223	C₂	semidirect
4. C₇×#3222	C₂	semidirect
5. C₇×#3216	C₂	semidirect
6. C₇×#3215	C₂	semidirect
7. C₇×#3152	C₂	semidirect
8. C₂×C₇×#497	C₂	semidirect
9. C₇×#3140	C₂	semidirect
10. C₇×#3135	C₂	semidirect
11. C₇×#3044	C₂	semidirect
12. C₇×#3043	C₂	semidirect
13. C₇×#3011	C₂	semidirect
14. C₇×#3009	C₂	semidirect
15. C₇×#2923	C₂	semidirect
19. C₂×C₇×#137	C₂²	semidirect
20. C₂²×C₇×Q₁₆	C₂²	semidirect
28. C₇×#541	C₂²	semidirect
34. C₇×#533	C₂²	semidirect
35. C₇×#533	C₂²	semidirect
38. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
41. C₇×#519	C₂²	semidirect
43. C₇×#497	C₂²	semidirect
44. C₇×#497	C₂²	semidirect
45. C₇×#497	C₂²	semidirect
46. C₇×#497	C₂²	semidirect
47. C₇×#494	C₂²	semidirect
48. C₇×#492	C₂²	semidirect
49. C₇×#491	C₂²	semidirect
50. C₇×#490	C₂²	semidirect
51. C₇×#489	C₂²	semidirect
52. C₇×#489	C₂²	semidirect
53. C₇×#489	C₂²	semidirect
54. C₇×#488	C₂²	semidirect
55. C₇×#464	C₂²	semidirect
56. C₇×#464	C₂²	semidirect
57. C₇×#462	C₂²	semidirect
58. C₇×#460	C₂²	semidirect
60. C₇×#452	C₂²	semidirect
61. C₇×#451	C₂²	semidirect
63. C₇×#433	C₂²	semidirect
64. C₇×#432	C₂²	semidirect
65. C₇×#424	C₂²	semidirect
66. C₇×#419	C₂²	semidirect
69. C₇×#416	C₂²	semidirect
71. C₄×C₇×M₁₆	C₂²	semidirect
72. C₂×C₇×#97	C₂²	semidirect
73. #60826	C₇	direct (N × H)
171. #3509	C₂×C₇	semidirect
173. #3223	C₂×C₇	semidirect
174. #3222	C₂×C₇	semidirect
175. #3216	C₂×C₇	semidirect
176. #3215	C₂×C₇	semidirect
177. #3152	C₂×C₇	semidirect
178. C₂×#497	C₂×C₇	semidirect
179. #3140	C₂×C₇	semidirect
180. #3135	C₂×C₇	semidirect
181. #3044	C₂×C₇	semidirect
182. #3043	C₂×C₇	semidirect
183. #3011	C₂×C₇	semidirect
184. #3009	C₂×C₇	semidirect
185. #2923	C₂×C₇	semidirect
279. C₂×#137	C₂²×C₇	semidirect
280. C₂²×Q₁₆	C₂²×C₇	semidirect
288. #541	C₂²×C₇	semidirect
294. #533	C₂²×C₇	semidirect
295. #533	C₂²×C₇	semidirect
298. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
301. #519	C₂²×C₇	semidirect
303. #497	C₂²×C₇	semidirect
304. #497	C₂²×C₇	semidirect
305. #497	C₂²×C₇	semidirect
306. #497	C₂²×C₇	semidirect
307. #494	C₂²×C₇	semidirect
308. #492	C₂²×C₇	semidirect
309. #491	C₂²×C₇	semidirect
310. #490	C₂²×C₇	semidirect
311. #489	C₂²×C₇	semidirect
312. #489	C₂²×C₇	semidirect
313. #489	C₂²×C₇	semidirect
314. #488	C₂²×C₇	semidirect
315. #464	C₂²×C₇	semidirect
316. #464	C₂²×C₇	semidirect
317. #462	C₂²×C₇	semidirect
318. #460	C₂²×C₇	semidirect
320. #452	C₂²×C₇	semidirect
321. #451	C₂²×C₇	semidirect
323. #433	C₂²×C₇	semidirect
324. #432	C₂²×C₇	semidirect
325. #424	C₂²×C₇	semidirect
326. #419	C₂²×C₇	semidirect
329. #416	C₂²×C₇	semidirect
331. C₄×M₁₆	C₂²×C₇	semidirect
332. C₂×#97	C₂²×C₇	semidirect
663. C₇	#60826	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×#3509	2
C₇×#3500	2
C₇×#3223	2
C₇×#3222	2
C₇×#3216	2
C₇×#3215	2
C₇×#3152	2
C₂×C₇×#497	2
C₇×#3140	2
C₇×#3135	2
C₇×#3044	2
C₇×#3043	2
C₇×#3011	2
C₇×#3009	2
C₇×#2923	2
#60826	7

Subgroups — C7×#60826 #13238042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60826 #13238042