Subgroups — C₇×#60206 #13237422

Order 1792.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3455	C₂	1	1	yes	maximal, characteristic
2	C₇×#2967	C₂	1	8	yes	maximal, complemented
3	C₇×#2966	C₂	1	8	yes	maximal, complemented
4	C₇×#2961	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2932	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2930	C₂	1	2	yes	maximal, complemented
9	C₇×#2929	C₂	1	2	yes	maximal, complemented
10	C₇×#2923	C₂	1	2	yes	maximal, complemented
11	C₇×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₇×#131	C₂	1	2	yes	maximal, complemented
13	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₇×#131	C₂²	1	2	yes	—
15	C₂×C₇×#131	C₂²	1	2	yes	—
16	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
17	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#542	C₄	1	4	yes	—
19	C₇×#541	C₄	1	2	yes	—
20	C₇×#540	C₄	1	2	yes	—
21	C₇×#539	C₄	1	4	yes	—
22	C₇×#538	C₄	1	4	yes	—
23	C₇×#533	C₂²	1	1	yes	characteristic
24	C₇×#532	C₂²	1	1	yes	characteristic
25	C₇×#529	C₂²	1	8	yes	—
26	C₇×#525	C₂²	1	2	yes	—
27	C₄×C₇×#41	C₂²	1	2	yes	—
28	C₂×C₇×#117	C₂²	1	1	yes	characteristic
29	C₇×#446	C₂²	1	4	yes	complemented
30	C₇×#446	C₂²	1	8	yes	complemented
31	C₇×C₈×Q₈	C₂²	1	4	yes	complemented
32	C₇×#436	C₂²	1	8	yes	—
33	C₇×#436	C₂²	1	4	yes	complemented
34	C₇×#435	C₂²	1	8	yes	—
35	C₇×#435	C₂²	1	8	yes	—
36	C₇×#435	C₂²	1	8	yes	complemented
37	C₇×C₈×D₄	C₂²	1	8	yes	—
38	C₇×C₈×D₄	C₂²	1	4	yes	complemented
39	C₇×#433	C₂²	1	8	yes	—
40	C₇×#433	C₂²	1	8	yes	complemented
41	C₇×#432	C₂²	1	8	yes	complemented
42	C₇×#424	C₂²	1	2	yes	—
43	C₇×#424	C₂²	1	2	yes	—
44	C₇×#423	C₂²	1	1	yes	characteristic
45	C₇×#423	C₂²	1	1	yes	characteristic
46	C₇×#408	C₂²	1	2	yes	—
47	C₇×#408	C₂²	1	2	yes	—
48	C₇×#407	C₂²	1	2	yes	—
49	C₇×#405	C₂²	1	2	yes	—
50	C₇×#405	C₂²	1	2	yes	—
51	C₇×#405	C₂²	1	4	yes	—
52	C₇×#405	C₂²	1	4	yes	—
53	C₇×#405	C₂²	1	4	yes	complemented
54	C₇×#405	C₂²	1	4	yes	complemented
55	C₄×C₇×M₁₆	C₂²	1	2	yes	—
56	C₄×C₇×M₁₆	C₂²	1	4	yes	complemented
57	C₂×C₇×#97	C₂²	1	1	yes	characteristic
58	C₂×C₇×#97	C₂²	1	1	yes	characteristic
59	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
60	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
61	C₂×C₄×C₇×C₈	C₂²	1	4	yes	complemented
62	#60206	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₇×#41	C₂³	1	2	yes	—
64	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
65	C₇×#131	—	2	8	no	—
66	C₇×#131	—	2	8	no	—
67	C₂×C₇×M₁₆	—	2	4	no	—
68	C₂×C₇×M₁₆	C₂³	1	2	yes	—
69	C₂×C₇×M₁₆	C₂³	1	2	yes	—
70	C₂×C₇×M₁₆	C₂³	1	4	yes	—
71	C₂×C₇×M₁₆	C₂³	1	4	yes	—
72	C₂×C₇×M₁₆	C₂³	1	4	yes	—
73	C₂²×C₇×C₈	—	2	4	no	—
74	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₇×C₈	C₂³	1	4	yes	—
76	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₇×C₈	C₂³	1	2	yes	—
78	C₂²×C₇×C₈	C₂³	1	4	yes	—
79	C₂²×C₇×C₈	C₂³	1	2	yes	—
80	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
81	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
82	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
83	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
84	C₇×#127	C₂×C₄	1	1	yes	characteristic
85	C₇×#126	C₂×C₄	1	8	yes	—
86	C₇×#126	C₂×C₄	1	8	yes	—
87	C₇×#125	C₂×C₄	1	8	yes	—
88	C₇×#125	C₂×C₄	1	2	yes	—
89	C₇×#124	C₂×C₄	1	8	yes	—
90	C₇×#124	C₂×C₄	1	2	yes	—
91	C₇×#123	C₂×C₄	1	4	yes	—
92	C₇×#123	C₂×C₄	1	8	yes	—
93	C₇×#123	C₂×C₄	1	8	yes	—
94	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
95	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
96	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
97	C₇×#121	C₂×C₄	1	4	yes	—
98	C₇×#121	C₂×C₄	1	8	yes	—
99	C₇×#121	C₂×C₄	1	4	yes	—
100	C₇×#120	C₂×C₄	1	4	yes	—
101	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
102	C₄×C₇×Q₈	C₂³	1	4	yes	—
103	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
104	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
105	C₄×C₇×D₄	C₂³	1	8	yes	—
106	C₄×C₇×D₄	C₂³	1	4	yes	—
107	C₇×#117	C₂×C₄	1	2	yes	—
108	C₇×#117	C₂×C₄	1	2	yes	—
109	C₇×#117	C₂³	1	4	yes	—
110	C₇×#117	C₂³	1	2	yes	—
111	C₇×#117	C₂³	1	4	yes	—
112	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
113	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
114	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₇	C₂³	1	4	yes	—
117	C₇×C₄⋊C₈	C₂³	1	4	yes	—
118	C₇×C₄⋊C₈	C₂³	1	4	yes	—
119	C₇×C₄⋊C₈	C₂³	1	8	yes	—
120	C₇×C₄⋊C₈	C₂³	1	8	yes	—
121	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
122	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
123	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
124	C₇×#97	C₂³	1	4	yes	—
125	C₇×#97	C₂³	1	4	yes	—
126	C₇×#97	C₂³	1	4	yes	—
127	C₇×#97	C₂³	1	4	yes	—
128	C₇×#97	C₂³	1	4	yes	—
129	C₄×C₇×C₈	C₂³	1	2	yes	—
130	C₄×C₇×C₈	C₂³	1	2	yes	—
131	C₄×C₇×C₈	C₂³	1	4	yes	—
132	C₄×C₇×C₈	C₂³	1	4	yes	—
133	C₄×C₇×C₈	C₂³	1	2	yes	—
134	C₄×C₇×C₈	C₂³	1	2	yes	—
135	C₄×C₇×C₈	C₂³	1	4	yes	—
136	#3455	C₂×C₇	1	1	yes	characteristic
137	#2967	C₂×C₇	1	8	yes	complemented
138	#2966	C₂×C₇	1	8	yes	complemented
139	#2961	C₂×C₇	1	1	yes	characteristic, complemented
140	#2960	C₂×C₇	1	1	yes	characteristic, complemented
141	#2932	C₂×C₇	1	1	yes	characteristic, complemented
142	#2931	C₂×C₇	1	1	yes	characteristic, complemented
143	#2930	C₂×C₇	1	2	yes	complemented
144	#2929	C₂×C₇	1	2	yes	complemented
145	#2923	C₂×C₇	1	2	yes	complemented
146	#2921	C₂×C₇	1	1	yes	characteristic, complemented
147	C₄×#131	C₂×C₇	1	2	yes	complemented
148	C₂×#405	C₂×C₇	1	1	yes	characteristic, complemented
149	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
150	C₇×#41	—	2	4	no	—
151	C₇×#41	—	2	4	no	—
152	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
153	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
154	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
155	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
156	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
157	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₇	—	2	2	no	—
159	C₂²×C₄×C₇	—	2	2	no	—
160	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
162	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
163	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
164	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
165	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
168	C₇×M₁₆	—	2	4	no	—
169	C₇×M₁₆	—	2	4	no	—
170	C₇×M₁₆	—	2	8	no	—
171	C₇×M₁₆	—	2	8	no	—
172	C₇×M₁₆	—	2	8	no	—
173	C₂×C₇×C₈	—	2	2	no	—
174	C₂×C₇×C₈	—	2	2	no	—
175	C₂×C₇×C₈	—	2	4	no	—
176	C₂×C₇×C₈	—	2	4	no	—
177	C₂×C₇×C₈	—	2	8	no	—
178	C₂×C₇×C₈	—	2	8	no	—
179	C₂×C₇×C₈	#41	1	4	yes	—
180	C₂×C₇×C₈	#41	1	4	yes	—
181	C₂×C₇×C₈	C₂⁴	1	2	yes	—
182	C₂×C₇×C₈	C₂⁴	1	4	yes	—
183	C₂×C₇×C₈	C₂⁴	1	4	yes	—
184	C₂×C₇×C₈	C₂⁴	1	2	yes	—
185	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
186	C₂×C₇×C₈	C₂⁴	1	2	yes	—
187	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
188	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
189	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
190	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
191	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
192	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
193	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
194	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
195	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
196	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
197	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
198	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
199	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
200	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
201	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
202	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
203	C₄²×C₇	C₂²×C₄	1	2	yes	—
204	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
206	C₄²×C₇	C₂²×C₄	1	8	yes	—
207	C₄²×C₇	C₂⁴	1	2	yes	—
208	C₄²×C₇	C₂⁴	1	2	yes	—
209	C₄²×C₇	C₂⁴	1	4	yes	—
210	C₂×#131	C₂²×C₇	1	2	yes	—
211	C₂×#131	C₂²×C₇	1	2	yes	—
212	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
213	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
214	#542	C₄×C₇	1	4	yes	—
215	#541	C₄×C₇	1	2	yes	—
216	#540	C₄×C₇	1	2	yes	—
217	#539	C₄×C₇	1	4	yes	—
218	#538	C₄×C₇	1	4	yes	—
219	#533	C₂²×C₇	1	1	yes	characteristic
220	#532	C₂²×C₇	1	1	yes	characteristic
221	#529	C₂²×C₇	1	8	yes	—
222	#525	C₂²×C₇	1	2	yes	—
223	C₄×#41	C₂²×C₇	1	2	yes	—
224	C₂×#117	C₂²×C₇	1	1	yes	characteristic
225	#446	C₂²×C₇	1	4	yes	complemented
226	#446	C₂²×C₇	1	8	yes	complemented
227	C₈×Q₈	C₂²×C₇	1	4	yes	complemented
228	#436	C₂²×C₇	1	8	yes	—
229	#436	C₂²×C₇	1	4	yes	complemented
230	#435	C₂²×C₇	1	8	yes	—
231	#435	C₂²×C₇	1	8	yes	—
232	#435	C₂²×C₇	1	8	yes	complemented
233	C₈×D₄	C₂²×C₇	1	8	yes	—
234	C₈×D₄	C₂²×C₇	1	4	yes	complemented
235	#433	C₂²×C₇	1	8	yes	—
236	#433	C₂²×C₇	1	8	yes	complemented
237	#432	C₂²×C₇	1	8	yes	complemented
238	#424	C₂²×C₇	1	2	yes	—
239	#424	C₂²×C₇	1	2	yes	—
240	#423	C₂²×C₇	1	1	yes	characteristic
241	#423	C₂²×C₇	1	1	yes	characteristic
242	#408	C₂²×C₇	1	2	yes	—
243	#408	C₂²×C₇	1	2	yes	—
244	#407	C₂²×C₇	1	2	yes	—
245	#405	C₂²×C₇	1	4	yes	—
246	#405	C₂²×C₇	1	4	yes	—
247	#405	C₂²×C₇	1	2	yes	—
248	#405	C₂²×C₇	1	2	yes	—
249	#405	C₂²×C₇	1	4	yes	complemented
250	#405	C₂²×C₇	1	4	yes	complemented
251	C₄×M₁₆	C₂²×C₇	1	2	yes	—
252	C₄×M₁₆	C₂²×C₇	1	4	yes	complemented
253	C₂×#97	C₂²×C₇	1	1	yes	characteristic
254	C₂×#97	C₂²×C₇	1	1	yes	characteristic
255	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
256	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₇	1	4	yes	complemented
258	C₂³×C₇	—	2	2	no	—
259	C₂³×C₇	—	4	2	no	—
260	C₂³×C₇	C₂³×C₄	1	2	yes	—
261	C₂³×C₇	C₂³×C₄	1	4	yes	—
262	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
263	C₇×Q₈	—	2	4	no	—
264	C₇×Q₈	—	4	4	no	—
265	C₇×D₄	—	2	8	no	—
266	C₇×D₄	—	2	4	no	—
267	C₇×D₄	—	4	4	no	—
268	C₇×D₄	—	4	8	no	—
269	C₂×C₄×C₇	—	2	2	no	—
270	C₂×C₄×C₇	—	2	4	no	—
271	C₂×C₄×C₇	—	2	2	no	—
272	C₂×C₄×C₇	—	2	4	no	—
273	C₂×C₄×C₇	—	2	4	no	—
274	C₂×C₄×C₇	—	2	2	no	—
275	C₂×C₄×C₇	—	4	2	no	—
276	C₂×C₄×C₇	—	4	8	no	—
277	C₂×C₄×C₇	—	4	2	no	—
278	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
279	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
280	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
281	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
282	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
283	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
284	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
285	C₂×C₄×C₇	C₂×#41	1	2	yes	—
286	C₂×C₄×C₇	C₂×#41	1	2	yes	—
287	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
288	C₇×C₈	—	2	2	no	—
289	C₇×C₈	—	2	2	no	—
290	C₇×C₈	—	2	4	no	—
291	C₇×C₈	—	2	4	no	—
292	C₇×C₈	C₂×#41	1	2	yes	—
293	C₇×C₈	C₂×#41	1	2	yes	—
294	C₇×C₈	C₂×#41	1	4	yes	—
295	C₂×#41	C₂³×C₇	1	2	yes	—
296	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
297	#131	—	2	8	no	—
298	#131	—	2	8	no	—
299	C₂×M₁₆	—	2	4	no	—
300	C₂×M₁₆	C₂³×C₇	1	4	yes	—
301	C₂×M₁₆	C₂³×C₇	1	2	yes	—
302	C₂×M₁₆	C₂³×C₇	1	2	yes	—
303	C₂×M₁₆	C₂³×C₇	1	4	yes	—
304	C₂×M₁₆	C₂³×C₇	1	4	yes	—
305	C₂²×C₈	—	2	4	no	—
306	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₇	1	2	yes	—
308	C₂²×C₈	C₂³×C₇	1	2	yes	—
309	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₇	1	4	yes	—
311	C₂²×C₈	C₂³×C₇	1	4	yes	—
312	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
314	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
315	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
316	#127	C₂×C₄×C₇	1	1	yes	characteristic
317	#126	C₂×C₄×C₇	1	8	yes	—
318	#126	C₂×C₄×C₇	1	8	yes	—
319	#125	C₂×C₄×C₇	1	8	yes	—
320	#125	C₂×C₄×C₇	1	2	yes	—
321	#124	C₂×C₄×C₇	1	8	yes	—
322	#124	C₂×C₄×C₇	1	2	yes	—
323	#123	C₂×C₄×C₇	1	4	yes	—
324	#123	C₂×C₄×C₇	1	8	yes	—
325	#123	C₂×C₄×C₇	1	8	yes	—
326	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
327	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	—
328	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
329	#121	C₂×C₄×C₇	1	4	yes	—
330	#121	C₂×C₄×C₇	1	8	yes	—
331	#121	C₂×C₄×C₇	1	4	yes	—
332	#120	C₂×C₄×C₇	1	4	yes	—
333	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
334	C₄×Q₈	C₂³×C₇	1	4	yes	—
335	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
336	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
337	C₄×D₄	C₂³×C₇	1	8	yes	—
338	C₄×D₄	C₂³×C₇	1	4	yes	—
339	#117	C₂×C₄×C₇	1	2	yes	—
340	#117	C₂×C₄×C₇	1	2	yes	—
341	#117	C₂³×C₇	1	4	yes	—
342	#117	C₂³×C₇	1	2	yes	—
343	#117	C₂³×C₇	1	4	yes	—
344	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
345	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
346	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₇	1	4	yes	—
349	C₄⋊C₈	C₂³×C₇	1	4	yes	—
350	C₄⋊C₈	C₂³×C₇	1	4	yes	—
351	C₄⋊C₈	C₂³×C₇	1	8	yes	—
352	C₄⋊C₈	C₂³×C₇	1	8	yes	—
353	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
354	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
355	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
356	#97	C₂³×C₇	1	4	yes	—
357	#97	C₂³×C₇	1	4	yes	—
358	#97	C₂³×C₇	1	4	yes	—
359	#97	C₂³×C₇	1	4	yes	—
360	#97	C₂³×C₇	1	4	yes	—
361	C₄×C₈	C₂³×C₇	1	2	yes	—
362	C₄×C₈	C₂³×C₇	1	2	yes	—
363	C₄×C₈	C₂³×C₇	1	2	yes	—
364	C₄×C₈	C₂³×C₇	1	2	yes	—
365	C₄×C₈	C₂³×C₇	1	4	yes	—
366	C₄×C₈	C₂³×C₇	1	4	yes	—
367	C₄×C₈	C₂³×C₇	1	4	yes	—
368	C₂²×C₇	—	2	2	no	—
369	C₂²×C₇	—	2	2	no	—
370	C₂²×C₇	—	2	4	no	—
371	C₂²×C₇	—	2	8	no	—
372	C₂²×C₇	—	2	4	no	—
373	C₂²×C₇	—	4	4	no	—
374	C₂²×C₇	C₄×#41	1	2	yes	—
375	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
376	C₄×C₇	—	2	2	no	—
377	C₄×C₇	—	2	4	no	—
378	C₄×C₇	—	2	4	no	—
379	C₄×C₇	—	2	2	no	—
380	C₄×C₇	—	2	8	no	—
381	C₄×C₇	—	2	2	no	—
382	C₄×C₇	C₄×#41	1	2	yes	—
383	C₄×C₇	C₂²×#41	1	1	yes	characteristic
384	C₄×C₇	C₂²×#41	1	1	yes	characteristic
385	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
386	#41	—	2	4	no	—
387	#41	—	2	4	no	—
388	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
389	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
390	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
391	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
392	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
393	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
398	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
399	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
400	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
401	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
403	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
404	M₁₆	—	2	4	no	—
405	M₁₆	—	2	4	no	—
406	M₁₆	—	2	8	no	—
407	M₁₆	—	2	8	no	—
408	M₁₆	—	2	8	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	4	no	—
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	—	2	8	no	—
414	C₂×C₈	—	2	8	no	—
415	C₂×C₈	C₇×#41	1	4	yes	—
416	C₂×C₈	C₇×#41	1	4	yes	—
417	C₂×C₈	C₂⁴×C₇	1	2	yes	—
418	C₂×C₈	C₂⁴×C₇	1	4	yes	—
419	C₂×C₈	C₂⁴×C₇	1	4	yes	—
420	C₂×C₈	C₂⁴×C₇	1	2	yes	—
421	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
422	C₂×C₈	C₂⁴×C₇	1	2	yes	—
423	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
424	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
425	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
426	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
427	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
428	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
429	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
430	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
431	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
432	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
433	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
434	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
435	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
436	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
437	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
438	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
439	C₄²	C₂²×C₄×C₇	1	2	yes	—
440	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
441	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
442	C₄²	C₂²×C₄×C₇	1	8	yes	—
443	C₄²	C₂⁴×C₇	1	2	yes	—
444	C₄²	C₂⁴×C₇	1	2	yes	—
445	C₄²	C₂⁴×C₇	1	4	yes	—
446	C₂×C₇	—	2	2	no	—
447	C₂×C₇	—	4	4	no	—
448	C₂×C₇	—	4	2	no	—
449	C₂×C₇	C₂×C₄×#41	1	1	yes	characteristic
450	C₂×C₇	#3573	1	2	yes	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	C₂³×C₄×C₇	1	2	yes	—
454	C₂³	C₂³×C₄×C₇	1	4	yes	—
455	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
456	Q₈	—	2	4	no	—
457	Q₈	—	4	4	no	—
458	D₄	—	2	8	no	—
459	D₄	—	2	4	no	—
460	D₄	—	4	4	no	—
461	D₄	—	4	8	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
472	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
473	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
474	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
475	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
476	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
477	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
478	C₂×C₄	C₂×C₇×#41	1	2	yes	—
479	C₂×C₄	C₂×C₇×#41	1	2	yes	—
480	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
481	C₈	—	2	2	no	—
482	C₈	—	2	2	no	—
483	C₈	—	2	4	no	—
484	C₈	—	2	4	no	—
485	C₈	C₂×C₇×#41	1	2	yes	—
486	C₈	C₂×C₇×#41	1	2	yes	—
487	C₈	C₂×C₇×#41	1	4	yes	—
488	C₇	#60206	1	1	yes	characteristic, direct product factor, Sylow
489	C₂²	—	2	2	no	—
490	C₂²	—	2	8	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	C₄×C₇×#41	1	2	yes	—
496	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
497	C₄	—	2	4	no	—
498	C₄	—	2	2	no	—
499	C₄	—	2	8	no	—
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	4	no	—
503	C₄	C₄×C₇×#41	1	2	yes	—
504	C₄	C₂²×C₇×#41	1	1	yes	characteristic
505	C₄	C₂²×C₇×#41	1	1	yes	characteristic
506	C₂	—	2	2	no	—
507	C₂	—	4	2	no	—
508	C₂	—	4	4	no	—
509	C₂	C₂×C₄×C₇×#41	1	1	yes	characteristic
510	C₂	C₇×#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₇×C₈	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂⁴×C₄
prime radical	C₇×#3455	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2967	C₂	semidirect
3. C₇×#2966	C₂	semidirect
4. C₇×#2961	C₂	semidirect
5. C₇×#2960	C₂	semidirect
6. C₇×#2932	C₂	semidirect
7. C₇×#2931	C₂	semidirect
8. C₇×#2930	C₂	semidirect
9. C₇×#2929	C₂	semidirect
10. C₇×#2923	C₂	semidirect
11. C₇×#2921	C₂	semidirect
12. C₄×C₇×#131	C₂	semidirect
13. C₂×C₇×#405	C₂	semidirect
29. C₇×#446	C₂²	semidirect
30. C₇×#446	C₂²	semidirect
31. C₇×C₈×Q₈	C₂²	semidirect
33. C₇×#436	C₂²	semidirect
36. C₇×#435	C₂²	semidirect
38. C₇×C₈×D₄	C₂²	semidirect
40. C₇×#433	C₂²	semidirect
41. C₇×#432	C₂²	semidirect
53. C₇×#405	C₂²	semidirect
54. C₇×#405	C₂²	semidirect
56. C₄×C₇×M₁₆	C₂²	semidirect
61. C₂×C₄×C₇×C₈	C₂²	semidirect
62. #60206	C₇	direct (N × H)
137. #2967	C₂×C₇	semidirect
138. #2966	C₂×C₇	semidirect
139. #2961	C₂×C₇	semidirect
140. #2960	C₂×C₇	semidirect
141. #2932	C₂×C₇	semidirect
142. #2931	C₂×C₇	semidirect
143. #2930	C₂×C₇	semidirect
144. #2929	C₂×C₇	semidirect
145. #2923	C₂×C₇	semidirect
146. #2921	C₂×C₇	semidirect
147. C₄×#131	C₂×C₇	semidirect
148. C₂×#405	C₂×C₇	semidirect
225. #446	C₂²×C₇	semidirect
226. #446	C₂²×C₇	semidirect
227. C₈×Q₈	C₂²×C₇	semidirect
229. #436	C₂²×C₇	semidirect
232. #435	C₂²×C₇	semidirect
234. C₈×D₄	C₂²×C₇	semidirect
236. #433	C₂²×C₇	semidirect
237. #432	C₂²×C₇	semidirect
249. #405	C₂²×C₇	semidirect
250. #405	C₂²×C₇	semidirect
252. C₄×M₁₆	C₂²×C₇	semidirect
257. C₂×C₄×C₈	C₂²×C₇	semidirect
488. C₇	#60206	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×#3455	2
C₇×#2967	2
C₇×#2966	2
C₇×#2961	2
C₇×#2960	2
C₇×#2932	2
C₇×#2931	2
C₇×#2930	2
C₇×#2929	2
C₇×#2923	2
C₇×#2921	2
C₄×C₇×#131	2
C₂×C₇×#405	2
#60206	7

Subgroups — C7×#60206 #13237422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60206 #13237422