Subgroups — C₇×#60318 #13237534

Order 1792.

1812 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×#3529	C₂	1	1	yes	maximal, characteristic
2	C₇×#2995	C₂	1	2	yes	maximal, complemented
3	C₇×#2993	C₂	1	4	yes	maximal, complemented
4	C₇×#2980	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2979	C₂	1	2	yes	maximal, complemented
6	C₇×#2977	C₂	1	2	yes	maximal, complemented
7	C₇×#2976	C₂	1	4	yes	maximal, complemented
8	C₇×#2968	C₂	1	4	yes	maximal, complemented
9	C₇×#2966	C₂	1	2	yes	maximal, complemented
10	C₇×#2933	C₂	1	4	yes	maximal, complemented
11	C₇×#2932	C₂	1	2	yes	maximal, complemented
12	C₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#2930	C₂	1	2	yes	maximal, complemented
14	C₂×C₇×#131	C₂²	1	1	yes	characteristic
15	C₂×C₇×#131	C₂²	1	2	yes	—
16	C₂×C₇×#131	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₇×M₁₆	C₂²	1	2	yes	complemented
18	C₇×#563	C₄	1	2	yes	—
19	C₇×#563	C₄	1	1	yes	characteristic
20	C₇×#562	C₄	1	4	yes	—
21	C₇×#562	C₄	1	2	yes	—
22	C₇×H(3,𝔽₄)	C₄	1	1	yes	characteristic
23	C₇×#560	C₄	1	4	yes	—
24	C₇×#560	C₄	1	2	yes	—
25	C₇×#532	C₂²	1	1	yes	characteristic
26	C₇×#532	C₂²	1	2	yes	—
27	C₇×#529	C₂²	1	2	yes	—
28	C₇×#529	C₂²	1	4	yes	—
29	C₇×#525	C₂²	1	2	yes	—
30	C₇×#525	C₂²	1	4	yes	—
31	C₇×#446	C₂²	1	2	yes	complemented
32	C₇×#446	C₂²	1	4	yes	complemented
33	C₇×C₈×Q₈	C₂²	1	2	yes	complemented
34	C₇×#436	C₂²	1	4	yes	—
35	C₇×#436	C₂²	1	2	yes	—
36	C₇×#436	C₂²	1	4	yes	complemented
37	C₇×#436	C₂²	1	4	yes	complemented
38	C₇×#436	C₂²	1	4	yes	complemented
39	C₇×#435	C₂²	1	4	yes	—
40	C₇×#435	C₂²	1	4	yes	—
41	C₇×#435	C₂²	1	4	yes	—
42	C₇×#435	C₂²	1	4	yes	complemented
43	C₇×#435	C₂²	1	4	yes	complemented
44	C₇×#435	C₂²	1	4	yes	complemented
45	C₇×#435	C₂²	1	4	yes	complemented
46	C₇×#435	C₂²	1	4	yes	complemented
47	C₇×#435	C₂²	1	4	yes	complemented
48	C₇×C₈×D₄	C₂²	1	4	yes	—
49	C₇×C₈×D₄	C₂²	1	2	yes	—
50	C₇×C₈×D₄	C₂²	1	4	yes	complemented
51	C₇×C₈×D₄	C₂²	1	4	yes	complemented
52	C₇×C₈×D₄	C₂²	1	4	yes	complemented
53	C₇×#433	C₂²	1	4	yes	complemented
54	C₇×#433	C₂²	1	4	yes	complemented
55	C₇×#433	C₂²	1	4	yes	complemented
56	C₇×#432	C₂²	1	2	yes	complemented
57	C₇×#431	C₂²	1	4	yes	complemented
58	C₇×#424	C₂²	1	2	yes	complemented
59	C₇×#424	C₂²	1	4	yes	complemented
60	C₇×#423	C₂²	1	1	yes	characteristic, complemented
61	C₇×#423	C₂²	1	2	yes	complemented
62	C₇×#408	C₂²	1	2	yes	—
63	C₇×#408	C₂²	1	1	yes	characteristic
64	C₇×#408	C₂²	1	2	yes	—
65	C₇×#408	C₂²	1	4	yes	—
66	C₇×#408	C₂²	1	1	yes	characteristic, complemented
67	C₇×#408	C₂²	1	2	yes	complemented
68	C₇×#407	C₂²	1	2	yes	complemented
69	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	complemented
70	C₇×#405	C₂²	1	2	yes	complemented
71	C₇×#405	C₂²	1	4	yes	complemented
72	C₂×C₇×#97	C₂²	1	2	yes	complemented
73	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
74	#60318	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₇×#41	C₂³	1	1	yes	characteristic
76	C₂×C₇×#41	C₂³	1	2	yes	—
77	C₂×C₇×#41	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₇	C₂³	1	2	yes	—
79	C₇×#131	—	4	2	no	—
80	C₇×#131	—	4	2	no	—
81	C₇×#131	—	4	4	no	—
82	C₂×C₇×M₁₆	—	4	4	no	—
83	C₂×C₇×M₁₆	C₂³	1	4	yes	—
84	C₂×C₇×M₁₆	C₂³	1	2	yes	—
85	C₂×C₇×M₁₆	C₂³	1	2	yes	—
86	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
87	C₂×C₇×M₁₆	C₂³	1	2	yes	—
88	C₂×C₇×M₁₆	C₂³	1	4	yes	—
89	C₂×C₇×M₁₆	C₂³	1	2	yes	—
90	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
91	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
92	C₂²×C₇×C₈	C₂³	1	2	yes	—
93	C₂²×C₇×C₈	C₂³	1	2	yes	—
94	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
95	C₂²×C₇×C₈	C₂³	1	2	yes	—
96	C₂²×C₇×C₈	C₂³	1	4	yes	—
97	C₂²×C₇×C₈	C₂³	1	4	yes	—
98	C₂²×C₇×C₈	C₂³	1	2	yes	—
99	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
100	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
101	C₇×#127	C₂×C₄	1	1	yes	characteristic
102	C₇×#127	C₂×C₄	1	2	yes	—
103	C₇×#126	C₂×C₄	1	4	yes	—
104	C₇×#126	C₂×C₄	1	4	yes	—
105	C₇×#126	C₂×C₄	1	4	yes	—
106	C₇×#125	C₂×C₄	1	2	yes	—
107	C₇×#125	C₂×C₄	1	4	yes	—
108	C₇×#124	C₂×C₄	1	2	yes	—
109	C₇×#124	C₂×C₄	1	4	yes	—
110	C₇×#124	C₂×C₄	1	1	yes	characteristic
111	C₇×#124	C₂×C₄	1	2	yes	—
112	C₇×#124	C₂×C₄	1	1	yes	characteristic
113	C₇×#124	C₂×C₄	1	2	yes	—
114	C₇×#123	C₂×C₄	1	2	yes	—
115	C₇×#123	C₂×C₄	1	4	yes	—
116	C₇×#123	C₂×C₄	1	2	yes	—
117	C₇×#123	C₂×C₄	1	4	yes	—
118	C₇×#123	C₂×C₄	1	4	yes	—
119	C₇×#123	C₂×C₄	1	4	yes	—
120	C₇×#123	C₂×C₄	1	4	yes	—
121	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
122	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
123	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
124	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
125	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
126	C₇×#121	C₂×C₄	1	4	yes	—
127	C₇×#121	C₂×C₄	1	4	yes	—
128	C₇×#121	C₂×C₄	1	4	yes	—
129	C₇×#121	C₂×C₄	1	2	yes	—
130	C₇×#121	C₂×C₄	1	4	yes	—
131	C₇×#121	C₂×C₄	1	4	yes	—
132	C₇×#121	C₂×C₄	1	4	yes	—
133	C₇×#121	C₂×C₄	1	4	yes	—
134	C₇×#120	C₂×C₄	1	2	yes	—
135	C₇×#120	C₂×C₄	1	4	yes	—
136	C₇×#120	C₂×C₄	1	2	yes	—
137	C₇×#120	C₂×C₄	1	4	yes	—
138	C₄×C₇×Q₈	C₂³	1	2	yes	—
139	C₄×C₇×D₄	C₂³	1	4	yes	—
140	C₄×C₇×D₄	C₂³	1	2	yes	—
141	C₄×C₇×D₄	C₂³	1	4	yes	—
142	C₄×C₇×D₄	C₂³	1	4	yes	—
143	C₄×C₇×D₄	C₂³	1	4	yes	—
144	C₇×#117	C₂³	1	2	yes	—
145	C₇×#117	C₂³	1	4	yes	—
146	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
147	C₂×C₄²×C₇	C₂³	1	2	yes	—
148	C₇×C₄⋊C₈	C₂³	1	4	yes	—
149	C₇×C₄⋊C₈	C₂³	1	4	yes	—
150	C₇×C₄⋊C₈	C₂³	1	4	yes	—
151	C₇×C₄⋊C₈	C₂³	1	4	yes	—
152	C₇×C₄⋊C₈	C₂³	1	2	yes	—
153	C₇×C₄⋊C₈	C₂³	1	4	yes	—
154	C₇×C₄⋊C₈	C₂³	1	2	yes	—
155	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
156	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
157	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
158	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
159	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
160	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
161	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
162	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
163	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
164	C₇×#97	C₂³	1	4	yes	—
165	C₇×#97	C₂³	1	4	yes	—
166	C₇×#97	C₂³	1	4	yes	—
167	C₄×C₇×C₈	C₂³	1	2	yes	—
168	C₄×C₇×C₈	C₂³	1	2	yes	—
169	C₄×C₇×C₈	C₂³	1	4	yes	—
170	C₄×C₇×C₈	C₂³	1	4	yes	—
171	#3529	C₂×C₇	1	1	yes	characteristic
172	#2995	C₂×C₇	1	2	yes	complemented
173	#2993	C₂×C₇	1	4	yes	complemented
174	#2980	C₂×C₇	1	1	yes	characteristic, complemented
175	#2979	C₂×C₇	1	2	yes	complemented
176	#2977	C₂×C₇	1	2	yes	complemented
177	#2976	C₂×C₇	1	4	yes	complemented
178	#2968	C₂×C₇	1	4	yes	complemented
179	#2966	C₂×C₇	1	2	yes	complemented
180	#2933	C₂×C₇	1	4	yes	complemented
181	#2932	C₂×C₇	1	2	yes	complemented
182	#2931	C₂×C₇	1	1	yes	characteristic, complemented
183	#2930	C₂×C₇	1	2	yes	complemented
184	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
185	C₇×#41	—	4	1	no	—
186	C₇×#41	—	4	1	no	—
187	C₇×#41	—	4	2	no	—
188	C₇×#41	—	4	2	no	—
189	C₇×#41	—	4	1	no	—
190	C₇×#41	—	4	1	no	—
191	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
192	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
193	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
194	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
195	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
196	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
197	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
198	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
199	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
200	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
201	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
202	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
203	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
204	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
205	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
206	C₂²×C₄×C₇	—	4	2	no	—
207	C₂²×C₄×C₇	—	4	2	no	—
208	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
209	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
210	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
211	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
212	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
214	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
216	C₇×M₁₆	—	2	8	no	—
217	C₇×M₁₆	—	4	4	no	—
218	C₇×M₁₆	—	4	2	no	—
219	C₇×M₁₆	—	4	1	no	—
220	C₇×M₁₆	—	4	2	no	—
221	C₇×M₁₆	—	4	2	no	—
222	C₇×M₁₆	—	4	1	no	—
223	C₇×M₁₆	—	4	2	no	—
224	C₂×C₇×C₈	—	2	4	no	—
225	C₂×C₇×C₈	—	2	4	no	—
226	C₂×C₇×C₈	—	2	2	no	—
227	C₂×C₇×C₈	—	4	1	no	—
228	C₂×C₇×C₈	—	4	4	no	—
229	C₂×C₇×C₈	—	4	2	no	—
230	C₂×C₇×C₈	—	4	2	no	—
231	C₂×C₇×C₈	—	4	4	no	—
232	C₂×C₇×C₈	C₂⁴	1	2	yes	—
233	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
234	C₂×C₇×C₈	C₂⁴	1	2	yes	—
235	C₂×C₇×C₈	C₂⁴	1	4	yes	—
236	C₂×C₇×C₈	C₂⁴	1	4	yes	—
237	C₂×C₇×C₈	C₂⁴	1	2	yes	—
238	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
239	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
240	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
241	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
242	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
243	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
244	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
245	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
246	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
247	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
248	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
249	C₇×C₂²⋊C₄	C₂²×C₄	1	2	yes	—
250	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
251	C₇×C₂²⋊C₄	C₂²×C₄	1	2	yes	—
252	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
253	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
254	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
255	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
256	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
257	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
258	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
259	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
260	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
261	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
262	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
263	C₄²×C₇	C₂²×C₄	1	2	yes	—
264	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
265	C₄²×C₇	C₂²×C₄	1	2	yes	—
266	C₄²×C₇	C₂⁴	1	2	yes	—
267	C₄²×C₇	C₂⁴	1	4	yes	—
268	C₂×#131	C₂²×C₇	1	1	yes	characteristic
269	C₂×#131	C₂²×C₇	1	2	yes	—
270	C₂×#131	C₂²×C₇	1	1	yes	characteristic, complemented
271	C₂²×M₁₆	C₂²×C₇	1	2	yes	complemented
272	#563	C₄×C₇	1	2	yes	—
273	#563	C₄×C₇	1	1	yes	characteristic
274	#562	C₄×C₇	1	4	yes	—
275	#562	C₄×C₇	1	2	yes	—
276	H(3,𝔽₄)	C₄×C₇	1	1	yes	characteristic
277	#560	C₄×C₇	1	4	yes	—
278	#560	C₄×C₇	1	2	yes	—
279	#532	C₂²×C₇	1	1	yes	characteristic
280	#532	C₂²×C₇	1	2	yes	—
281	#529	C₂²×C₇	1	2	yes	—
282	#529	C₂²×C₇	1	4	yes	—
283	#525	C₂²×C₇	1	2	yes	—
284	#525	C₂²×C₇	1	4	yes	—
285	#446	C₂²×C₇	1	2	yes	complemented
286	#446	C₂²×C₇	1	4	yes	complemented
287	C₈×Q₈	C₂²×C₇	1	2	yes	complemented
288	#436	C₂²×C₇	1	4	yes	—
289	#436	C₂²×C₇	1	2	yes	—
290	#436	C₂²×C₇	1	4	yes	complemented
291	#436	C₂²×C₇	1	4	yes	complemented
292	#436	C₂²×C₇	1	4	yes	complemented
293	#435	C₂²×C₇	1	4	yes	—
294	#435	C₂²×C₇	1	4	yes	—
295	#435	C₂²×C₇	1	4	yes	—
296	#435	C₂²×C₇	1	4	yes	complemented
297	#435	C₂²×C₇	1	4	yes	complemented
298	#435	C₂²×C₇	1	4	yes	complemented
299	#435	C₂²×C₇	1	4	yes	complemented
300	#435	C₂²×C₇	1	4	yes	complemented
301	#435	C₂²×C₇	1	4	yes	complemented
302	C₈×D₄	C₂²×C₇	1	4	yes	—
303	C₈×D₄	C₂²×C₇	1	2	yes	—
304	C₈×D₄	C₂²×C₇	1	4	yes	complemented
305	C₈×D₄	C₂²×C₇	1	4	yes	complemented
306	C₈×D₄	C₂²×C₇	1	4	yes	complemented
307	#433	C₂²×C₇	1	4	yes	complemented
308	#433	C₂²×C₇	1	4	yes	complemented
309	#433	C₂²×C₇	1	4	yes	complemented
310	#432	C₂²×C₇	1	2	yes	complemented
311	#431	C₂²×C₇	1	4	yes	complemented
312	#424	C₂²×C₇	1	2	yes	complemented
313	#424	C₂²×C₇	1	4	yes	complemented
314	#423	C₂²×C₇	1	1	yes	characteristic, complemented
315	#423	C₂²×C₇	1	2	yes	complemented
316	#408	C₂²×C₇	1	2	yes	—
317	#408	C₂²×C₇	1	1	yes	characteristic
318	#408	C₂²×C₇	1	2	yes	—
319	#408	C₂²×C₇	1	4	yes	—
320	#408	C₂²×C₇	1	1	yes	characteristic, complemented
321	#408	C₂²×C₇	1	2	yes	complemented
322	#407	C₂²×C₇	1	2	yes	complemented
323	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	complemented
324	#405	C₂²×C₇	1	2	yes	complemented
325	#405	C₂²×C₇	1	4	yes	complemented
326	C₂×#97	C₂²×C₇	1	2	yes	complemented
327	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
328	C₂³×C₇	—	4	2	no	—
329	C₂³×C₇	—	4	4	no	—
330	C₂³×C₇	C₂³×C₄	1	2	yes	—
331	C₂³×C₇	C₂³×C₄	1	4	yes	—
332	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
333	C₂³×C₇	C₂³×C₄	1	2	yes	—
334	C₇×Q₈	—	4	2	no	—
335	C₇×Q₈	—	4	1	no	—
336	C₇×Q₈	—	4	2	no	—
337	C₇×Q₈	—	4	1	no	—
338	C₇×D₄	—	4	4	no	—
339	C₇×D₄	—	4	4	no	—
340	C₇×D₄	—	4	4	no	—
341	C₇×D₄	—	4	4	no	—
342	C₇×D₄	—	4	2	no	—
343	C₇×D₄	—	4	2	no	—
344	C₇×D₄	—	4	4	no	—
345	C₇×D₄	—	4	1	no	—
346	C₇×D₄	—	4	2	no	—
347	C₇×D₄	—	4	2	no	—
348	C₇×D₄	—	4	1	no	—
349	C₂×C₄×C₇	—	2	2	no	—
350	C₂×C₄×C₇	—	2	2	no	—
351	C₂×C₄×C₇	—	2	1	no	—
352	C₂×C₄×C₇	—	2	4	no	—
353	C₂×C₄×C₇	—	2	4	no	—
354	C₂×C₄×C₇	—	2	1	no	—
355	C₂×C₄×C₇	—	2	2	no	—
356	C₂×C₄×C₇	—	2	2	no	—
357	C₂×C₄×C₇	—	4	4	no	—
358	C₂×C₄×C₇	—	4	2	no	—
359	C₂×C₄×C₇	—	4	2	no	—
360	C₂×C₄×C₇	—	4	4	no	—
361	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
362	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
363	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
364	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
365	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
366	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
367	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
368	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
369	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
370	C₇×C₈	—	2	2	no	—
371	C₇×C₈	—	2	2	no	—
372	C₇×C₈	—	2	1	no	—
373	C₇×C₈	—	2	4	no	—
374	C₇×C₈	—	2	2	no	—
375	C₇×C₈	—	2	4	no	—
376	C₇×C₈	H(5,𝔽₂)	1	2	yes	—
377	C₂×#41	C₂³×C₇	1	1	yes	characteristic
378	C₂×#41	C₂³×C₇	1	2	yes	—
379	C₂×#41	C₂³×C₇	1	1	yes	characteristic
380	C₂³×C₄	C₂³×C₇	1	2	yes	—
381	#131	—	4	2	no	—
382	#131	—	4	2	no	—
383	#131	—	4	4	no	—
384	C₂×M₁₆	—	4	4	no	—
385	C₂×M₁₆	C₂³×C₇	1	4	yes	—
386	C₂×M₁₆	C₂³×C₇	1	4	yes	—
387	C₂×M₁₆	C₂³×C₇	1	2	yes	—
388	C₂×M₁₆	C₂³×C₇	1	2	yes	—
389	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
390	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
391	C₂×M₁₆	C₂³×C₇	1	2	yes	—
392	C₂×M₁₆	C₂³×C₇	1	2	yes	—
393	C₂²×C₈	C₂³×C₇	1	2	yes	—
394	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
395	C₂²×C₈	C₂³×C₇	1	2	yes	—
396	C₂²×C₈	C₂³×C₇	1	2	yes	—
397	C₂²×C₈	C₂³×C₇	1	4	yes	—
398	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₇	1	2	yes	—
400	C₂²×C₈	C₂³×C₇	1	4	yes	—
401	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
402	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
403	#127	C₂×C₄×C₇	1	1	yes	characteristic
404	#127	C₂×C₄×C₇	1	2	yes	—
405	#126	C₂×C₄×C₇	1	4	yes	—
406	#126	C₂×C₄×C₇	1	4	yes	—
407	#126	C₂×C₄×C₇	1	4	yes	—
408	#125	C₂×C₄×C₇	1	2	yes	—
409	#125	C₂×C₄×C₇	1	4	yes	—
410	#124	C₂×C₄×C₇	1	2	yes	—
411	#124	C₂×C₄×C₇	1	4	yes	—
412	#124	C₂×C₄×C₇	1	1	yes	characteristic
413	#124	C₂×C₄×C₇	1	2	yes	—
414	#124	C₂×C₄×C₇	1	1	yes	characteristic
415	#124	C₂×C₄×C₇	1	2	yes	—
416	#123	C₂×C₄×C₇	1	2	yes	—
417	#123	C₂×C₄×C₇	1	4	yes	—
418	#123	C₂×C₄×C₇	1	2	yes	—
419	#123	C₂×C₄×C₇	1	4	yes	—
420	#123	C₂×C₄×C₇	1	4	yes	—
421	#123	C₂×C₄×C₇	1	4	yes	—
422	#123	C₂×C₄×C₇	1	4	yes	—
423	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
424	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
425	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
426	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
427	C₂²⋊Q₈	C₂×C₄×C₇	1	4	yes	—
428	#121	C₂×C₄×C₇	1	4	yes	—
429	#121	C₂×C₄×C₇	1	4	yes	—
430	#121	C₂×C₄×C₇	1	4	yes	—
431	#121	C₂×C₄×C₇	1	2	yes	—
432	#121	C₂×C₄×C₇	1	4	yes	—
433	#121	C₂×C₄×C₇	1	4	yes	—
434	#121	C₂×C₄×C₇	1	4	yes	—
435	#121	C₂×C₄×C₇	1	4	yes	—
436	#120	C₂×C₄×C₇	1	2	yes	—
437	#120	C₂×C₄×C₇	1	4	yes	—
438	#120	C₂×C₄×C₇	1	2	yes	—
439	#120	C₂×C₄×C₇	1	4	yes	—
440	C₄×Q₈	C₂³×C₇	1	2	yes	—
441	C₄×D₄	C₂³×C₇	1	4	yes	—
442	C₄×D₄	C₂³×C₇	1	2	yes	—
443	C₄×D₄	C₂³×C₇	1	4	yes	—
444	C₄×D₄	C₂³×C₇	1	4	yes	—
445	C₄×D₄	C₂³×C₇	1	4	yes	—
446	#117	C₂³×C₇	1	2	yes	—
447	#117	C₂³×C₇	1	4	yes	—
448	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
449	C₂×C₄²	C₂³×C₇	1	2	yes	—
450	C₄⋊C₈	C₂³×C₇	1	4	yes	—
451	C₄⋊C₈	C₂³×C₇	1	4	yes	—
452	C₄⋊C₈	C₂³×C₇	1	4	yes	—
453	C₄⋊C₈	C₂³×C₇	1	4	yes	—
454	C₄⋊C₈	C₂³×C₇	1	2	yes	—
455	C₄⋊C₈	C₂³×C₇	1	4	yes	—
456	C₄⋊C₈	C₂³×C₇	1	2	yes	—
457	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
458	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
459	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
460	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
461	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
462	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
463	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
464	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
465	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
466	#97	C₂³×C₇	1	4	yes	—
467	#97	C₂³×C₇	1	4	yes	—
468	#97	C₂³×C₇	1	4	yes	—
469	C₄×C₈	C₂³×C₇	1	2	yes	—
470	C₄×C₈	C₂³×C₇	1	2	yes	—
471	C₄×C₈	C₂³×C₇	1	4	yes	—
472	C₄×C₈	C₂³×C₇	1	4	yes	—
473	C₂²×C₇	—	2	2	no	—
474	C₂²×C₇	—	2	4	no	—
475	C₂²×C₇	—	2	2	no	—
476	C₂²×C₇	—	2	1	no	—
477	C₂²×C₇	—	2	2	no	—
478	C₂²×C₇	—	2	4	no	—
479	C₂²×C₇	—	2	4	no	—
480	C₂²×C₇	—	2	4	no	—
481	C₂²×C₇	—	2	4	no	—
482	C₂²×C₇	—	8	4	no	—
483	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
484	C₄×C₇	—	2	2	no	—
485	C₄×C₇	—	2	4	no	—
486	C₄×C₇	—	2	2	no	—
487	C₄×C₇	—	2	1	no	—
488	C₄×C₇	—	2	2	no	—
489	C₄×C₇	—	2	4	no	—
490	C₄×C₇	—	2	2	no	—
491	C₄×C₇	—	2	4	no	—
492	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
493	C₄×C₇	#585	1	1	yes	characteristic
494	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
495	#41	—	4	1	no	—
496	#41	—	4	1	no	—
497	#41	—	4	2	no	—
498	#41	—	4	2	no	—
499	#41	—	4	1	no	—
500	#41	—	4	1	no	—
501	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
502	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
503	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
504	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
505	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
506	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
507	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
508	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
509	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
510	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
511	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
512	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
513	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
514	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
515	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
516	C₂²×C₄	—	4	2	no	—
517	C₂²×C₄	—	4	2	no	—
518	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
519	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
520	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
521	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
522	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
524	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
525	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
526	M₁₆	—	2	8	no	—
527	M₁₆	—	4	4	no	—
528	M₁₆	—	4	2	no	—
529	M₁₆	—	4	1	no	—
530	M₁₆	—	4	2	no	—
531	M₁₆	—	4	2	no	—
532	M₁₆	—	4	1	no	—
533	M₁₆	—	4	2	no	—
534	C₂×C₈	—	2	4	no	—
535	C₂×C₈	—	2	4	no	—
536	C₂×C₈	—	2	2	no	—
537	C₂×C₈	—	4	1	no	—
538	C₂×C₈	—	4	4	no	—
539	C₂×C₈	—	4	2	no	—
540	C₂×C₈	—	4	2	no	—
541	C₂×C₈	—	4	4	no	—
542	C₂×C₈	C₂⁴×C₇	1	2	yes	—
543	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
544	C₂×C₈	C₂⁴×C₇	1	2	yes	—
545	C₂×C₈	C₂⁴×C₇	1	4	yes	—
546	C₂×C₈	C₂⁴×C₇	1	4	yes	—
547	C₂×C₈	C₂⁴×C₇	1	2	yes	—
548	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
549	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
550	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
551	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
552	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
553	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
554	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
555	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
556	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
557	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
558	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
559	C₂²⋊C₄	C₂²×C₄×C₇	1	2	yes	—
560	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
561	C₂²⋊C₄	C₂²×C₄×C₇	1	2	yes	—
562	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
563	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
564	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
565	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
566	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
567	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
568	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
569	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
570	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
571	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
572	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
573	C₄²	C₂²×C₄×C₇	1	2	yes	—
574	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
575	C₄²	C₂²×C₄×C₇	1	2	yes	—
576	C₄²	C₂⁴×C₇	1	2	yes	—
577	C₄²	C₂⁴×C₇	1	4	yes	—
578	C₂×C₇	—	4	2	no	—
579	C₂×C₇	—	4	1	no	—
580	C₂×C₇	—	4	2	no	—
581	C₂×C₇	—	4	4	no	—
582	C₂×C₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
583	C₂×C₇	#3573	1	2	yes	—
584	C₂³	—	4	2	no	—
585	C₂³	—	4	4	no	—
586	C₂³	C₂³×C₄×C₇	1	2	yes	—
587	C₂³	C₂³×C₄×C₇	1	4	yes	—
588	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
589	C₂³	C₂³×C₄×C₇	1	2	yes	—
590	Q₈	—	4	2	no	—
591	Q₈	—	4	1	no	—
592	Q₈	—	4	2	no	—
593	Q₈	—	4	1	no	—
594	D₄	—	4	4	no	—
595	D₄	—	4	4	no	—
596	D₄	—	4	4	no	—
597	D₄	—	4	4	no	—
598	D₄	—	4	2	no	—
599	D₄	—	4	2	no	—
600	D₄	—	4	4	no	—
601	D₄	—	4	1	no	—
602	D₄	—	4	2	no	—
603	D₄	—	4	2	no	—
604	D₄	—	4	1	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	4	4	no	—
614	C₂×C₄	—	4	2	no	—
615	C₂×C₄	—	4	2	no	—
616	C₂×C₄	—	4	4	no	—
617	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
618	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
619	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
620	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
621	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
622	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
623	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
624	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
625	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
626	C₈	—	2	2	no	—
627	C₈	—	2	2	no	—
628	C₈	—	2	1	no	—
629	C₈	—	2	4	no	—
630	C₈	—	2	2	no	—
631	C₈	—	2	4	no	—
632	C₈	C₇×H(5,𝔽₂)	1	2	yes	—
633	C₇	#60318	1	1	yes	characteristic, direct product factor, Sylow
634	C₂²	—	2	2	no	—
635	C₂²	—	2	4	no	—
636	C₂²	—	2	4	no	—
637	C₂²	—	2	4	no	—
638	C₂²	—	2	4	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	4	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	1	no	—
643	C₂²	—	8	4	no	—
644	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
645	C₄	—	2	2	no	—
646	C₄	—	2	4	no	—
647	C₄	—	2	2	no	—
648	C₄	—	2	4	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	4	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	1	no	—
653	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
654	C₄	C₇×#585	1	1	yes	characteristic
655	C₂	—	4	2	no	—
656	C₂	—	4	4	no	—
657	C₂	—	4	1	no	—
658	C₂	—	4	2	no	—
659	C₂	C₄×C₇×H(5,𝔽₂)	1	1	yes	characteristic
660	C₂	C₇×#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂⁴×C₄
prime radical	C₇×#3529	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2995	C₂	semidirect
3. C₇×#2993	C₂	semidirect
4. C₇×#2980	C₂	semidirect
5. C₇×#2979	C₂	semidirect
6. C₇×#2977	C₂	semidirect
7. C₇×#2976	C₂	semidirect
8. C₇×#2968	C₂	semidirect
9. C₇×#2966	C₂	semidirect
10. C₇×#2933	C₂	semidirect
11. C₇×#2932	C₂	semidirect
12. C₇×#2931	C₂	semidirect
13. C₇×#2930	C₂	semidirect
16. C₂×C₇×#131	C₂²	semidirect
17. C₂²×C₇×M₁₆	C₂²	semidirect
31. C₇×#446	C₂²	semidirect
32. C₇×#446	C₂²	semidirect
33. C₇×C₈×Q₈	C₂²	semidirect
36. C₇×#436	C₂²	semidirect
37. C₇×#436	C₂²	semidirect
38. C₇×#436	C₂²	semidirect
42. C₇×#435	C₂²	semidirect
43. C₇×#435	C₂²	semidirect
44. C₇×#435	C₂²	semidirect
45. C₇×#435	C₂²	semidirect
46. C₇×#435	C₂²	semidirect
47. C₇×#435	C₂²	semidirect
50. C₇×C₈×D₄	C₂²	semidirect
51. C₇×C₈×D₄	C₂²	semidirect
52. C₇×C₈×D₄	C₂²	semidirect
53. C₇×#433	C₂²	semidirect
54. C₇×#433	C₂²	semidirect
55. C₇×#433	C₂²	semidirect
56. C₇×#432	C₂²	semidirect
57. C₇×#431	C₂²	semidirect
58. C₇×#424	C₂²	semidirect
59. C₇×#424	C₂²	semidirect
60. C₇×#423	C₂²	semidirect
61. C₇×#423	C₂²	semidirect
66. C₇×#408	C₂²	semidirect
67. C₇×#408	C₂²	semidirect
68. C₇×#407	C₂²	semidirect
69. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
70. C₇×#405	C₂²	semidirect
71. C₇×#405	C₂²	semidirect
72. C₂×C₇×#97	C₂²	semidirect
73. C₂×C₄×C₇×C₈	C₂²	semidirect
74. #60318	C₇	direct (N × H)
172. #2995	C₂×C₇	semidirect
173. #2993	C₂×C₇	semidirect
174. #2980	C₂×C₇	semidirect
175. #2979	C₂×C₇	semidirect
176. #2977	C₂×C₇	semidirect
177. #2976	C₂×C₇	semidirect
178. #2968	C₂×C₇	semidirect
179. #2966	C₂×C₇	semidirect
180. #2933	C₂×C₇	semidirect
181. #2932	C₂×C₇	semidirect
182. #2931	C₂×C₇	semidirect
183. #2930	C₂×C₇	semidirect
270. C₂×#131	C₂²×C₇	semidirect
271. C₂²×M₁₆	C₂²×C₇	semidirect
285. #446	C₂²×C₇	semidirect
286. #446	C₂²×C₇	semidirect
287. C₈×Q₈	C₂²×C₇	semidirect
290. #436	C₂²×C₇	semidirect
291. #436	C₂²×C₇	semidirect
292. #436	C₂²×C₇	semidirect
296. #435	C₂²×C₇	semidirect
297. #435	C₂²×C₇	semidirect
298. #435	C₂²×C₇	semidirect
299. #435	C₂²×C₇	semidirect
300. #435	C₂²×C₇	semidirect
301. #435	C₂²×C₇	semidirect
304. C₈×D₄	C₂²×C₇	semidirect
305. C₈×D₄	C₂²×C₇	semidirect
306. C₈×D₄	C₂²×C₇	semidirect
307. #433	C₂²×C₇	semidirect
308. #433	C₂²×C₇	semidirect
309. #433	C₂²×C₇	semidirect
310. #432	C₂²×C₇	semidirect
311. #431	C₂²×C₇	semidirect
312. #424	C₂²×C₇	semidirect
313. #424	C₂²×C₇	semidirect
314. #423	C₂²×C₇	semidirect
315. #423	C₂²×C₇	semidirect
320. #408	C₂²×C₇	semidirect
321. #408	C₂²×C₇	semidirect
322. #407	C₂²×C₇	semidirect
323. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
324. #405	C₂²×C₇	semidirect
325. #405	C₂²×C₇	semidirect
326. C₂×#97	C₂²×C₇	semidirect
327. C₂×C₄×C₈	C₂²×C₇	semidirect
633. C₇	#60318	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×#3529	2
C₇×#2995	2
C₇×#2993	2
C₇×#2980	2
C₇×#2979	2
C₇×#2977	2
C₇×#2976	2
C₇×#2968	2
C₇×#2966	2
C₇×#2933	2
C₇×#2932	2
C₇×#2931	2
C₇×#2930	2
#60318	7

Subgroups — C7×#60318 #13237534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60318 #13237534