Subgroups — C₇×#60309 #13237525

Order 1792.

2142 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₇×#2990	C₂	1	4	yes	maximal, complemented
3	C₇×#2989	C₂	1	2	yes	maximal, complemented
4	C₇×#2974	C₂	1	4	yes	maximal, complemented
5	C₇×#2973	C₂	1	4	yes	maximal, complemented
6	C₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2970	C₂	1	4	yes	maximal, complemented
8	C₇×#2965	C₂	1	2	yes	maximal, complemented
9	C₇×D₄×M₁₆	C₂	1	4	yes	maximal, complemented
10	C₂×C₇×#435	C₂	1	4	yes	maximal, complemented
11	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×H(5,𝔽₂)	C₄	1	1	yes	characteristic
13	C₂²×C₇×M₁₆	C₂²	1	4	yes	complemented
14	C₂³×C₇×C₈	C₂²	1	2	yes	complemented
15	C₇×#520	C₄	1	4	yes	—
16	C₇×#520	C₄	1	2	yes	—
17	C₇×#518	C₄	1	4	yes	—
18	C₇×#518	C₄	1	4	yes	—
19	C₇×#518	C₄	1	1	yes	characteristic
20	C₄×C₇×#41	C₂²	1	2	yes	—
21	C₄×C₇×#41	C₂²	1	4	yes	—
22	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
24	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
25	C₇×#446	C₂²	1	2	yes	complemented
26	C₇×#446	C₂²	1	2	yes	complemented
27	C₇×#446	C₂²	1	4	yes	complemented
28	C₇×#436	C₂²	1	2	yes	complemented
29	C₇×#436	C₂²	1	4	yes	complemented
30	C₇×#436	C₂²	1	2	yes	complemented
31	C₇×#435	C₂²	1	4	yes	complemented
32	C₇×#435	C₂²	1	8	yes	complemented
33	C₇×#435	C₂²	1	8	yes	complemented
34	C₇×#435	C₂²	1	8	yes	complemented
35	C₇×#435	C₂²	1	8	yes	complemented
36	C₇×#435	C₂²	1	4	yes	complemented
37	C₇×#435	C₂²	1	8	yes	complemented
38	C₇×C₈×D₄	C₂²	1	4	yes	complemented
39	C₇×C₈×D₄	C₂²	1	4	yes	complemented
40	C₇×C₈×D₄	C₂²	1	4	yes	complemented
41	C₇×C₈×D₄	C₂²	1	4	yes	complemented
42	C₇×#432	C₂²	1	8	yes	complemented
43	C₇×#432	C₂²	1	4	yes	complemented
44	C₇×#431	C₂²	1	2	yes	complemented
45	C₇×#431	C₂²	1	4	yes	complemented
46	C₇×#423	C₂²	1	4	yes	complemented
47	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	complemented
49	C₇×#407	C₂²	1	8	yes	complemented
50	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
51	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	complemented
52	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	complemented
53	C₄×C₇×M₁₆	C₂²	1	4	yes	complemented
54	C₂×C₇×#97	C₂²	1	4	yes	complemented
55	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
56	#60309	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₇×H(5,𝔽₂)	—	2	8	no	—
58	C₂×C₇×#41	C₂×C₄	1	4	yes	—
59	C₂×C₇×#41	C₂×C₄	1	2	yes	—
60	C₂²×C₇×D₄	C₂×C₄	1	4	yes	—
61	C₂²×C₇×D₄	C₂×C₄	1	4	yes	—
62	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₇	C₂³	1	4	yes	—
65	C₂×C₇×M₁₆	—	4	8	no	—
66	C₂×C₇×M₁₆	C₂³	1	4	yes	—
67	C₂×C₇×M₁₆	C₂³	1	4	yes	—
68	C₂×C₇×M₁₆	C₂³	1	8	yes	—
69	C₂²×C₇×C₈	—	4	4	no	—
70	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₇×C₈	C₂³	1	4	yes	—
72	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₇×C₈	C₂³	1	4	yes	—
74	C₂²×C₇×C₈	C₂³	1	4	yes	—
75	C₂²×C₇×C₈	C₂³	1	2	yes	—
76	C₂²×C₇×C₈	C₂³	1	4	yes	—
77	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
78	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
79	C₄×C₇×Q₈	C₂³	1	2	yes	—
80	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
81	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
82	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
83	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
84	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
85	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
86	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
87	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
88	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
89	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
90	C₄×C₇×D₄	C₂³	1	4	yes	—
91	C₄×C₇×D₄	C₂³	1	4	yes	—
92	C₄×C₇×D₄	C₂³	1	8	yes	—
93	C₄×C₇×D₄	C₂³	1	2	yes	—
94	C₇×#117	C₂×C₄	1	4	yes	—
95	C₇×#117	C₂×C₄	1	4	yes	—
96	C₇×#117	C₂×C₄	1	8	yes	—
97	C₇×#117	C₂×C₄	1	2	yes	—
98	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
99	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
100	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
101	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
102	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	8	yes	—
103	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
104	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
105	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₇	C₂³	1	4	yes	—
107	C₂×C₄²×C₇	C₂³	1	4	yes	—
108	C₇×C₄⋊C₈	C₂³	1	4	yes	—
109	C₇×C₄⋊C₈	C₂³	1	4	yes	—
110	C₇×C₄⋊C₈	C₂³	1	8	yes	—
111	C₇×C₄⋊C₈	C₂³	1	2	yes	—
112	C₇×C₄⋊C₈	C₂³	1	2	yes	—
113	C₇×C₄⋊C₈	C₂³	1	4	yes	—
114	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
115	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
116	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
117	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
118	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
119	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
120	C₇×#97	C₂³	1	4	yes	—
121	C₇×#97	C₂³	1	8	yes	—
122	C₇×#97	C₂³	1	4	yes	—
123	C₄×C₇×C₈	C₂³	1	2	yes	—
124	C₄×C₇×C₈	C₂³	1	2	yes	—
125	C₄×C₇×C₈	C₂³	1	4	yes	—
126	C₄×H(5,𝔽₂)	C₂×C₇	1	1	yes	characteristic
127	#2990	C₂×C₇	1	4	yes	complemented
128	#2989	C₂×C₇	1	2	yes	complemented
129	#2974	C₂×C₇	1	4	yes	complemented
130	#2973	C₂×C₇	1	4	yes	complemented
131	#2972	C₂×C₇	1	1	yes	characteristic, complemented
132	#2970	C₂×C₇	1	4	yes	complemented
133	#2965	C₂×C₇	1	2	yes	complemented
134	D₄×M₁₆	C₂×C₇	1	4	yes	complemented
135	C₂×#435	C₂×C₇	1	4	yes	complemented
136	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic, complemented
137	C₂⁴×C₇	C₂²×C₄	1	4	yes	—
138	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
139	C₇×#41	—	2	16	no	—
140	C₇×#41	—	2	8	no	—
141	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
142	C₂×C₇×D₄	—	2	16	no	—
143	C₂×C₇×D₄	—	2	16	no	—
144	C₂×C₇×D₄	M₁₆	1	8	yes	complemented
145	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
146	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
147	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
148	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₇	—	4	2	no	—
150	C₂²×C₄×C₇	—	4	4	no	—
151	C₂²×C₄×C₇	—	4	2	no	—
152	C₂²×C₄×C₇	—	4	4	no	—
153	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
155	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
159	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
161	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
162	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
166	C₇×M₁₆	—	2	8	no	—
167	C₇×M₁₆	—	4	8	no	—
168	C₇×M₁₆	—	4	4	no	—
169	C₂×C₇×C₈	—	2	2	no	—
170	C₂×C₇×C₈	—	2	8	no	—
171	C₂×C₇×C₈	—	4	4	no	—
172	C₂×C₇×C₈	—	4	2	no	—
173	C₂×C₇×C₈	—	4	4	no	—
174	C₂×C₇×C₈	—	4	1	no	—
175	C₂×C₇×C₈	—	4	4	no	—
176	C₂×C₇×C₈	C₂⁴	1	2	yes	—
177	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
178	C₂×C₇×C₈	C₂⁴	1	4	yes	—
179	C₂×C₇×C₈	C₂⁴	1	4	yes	—
180	C₂×C₇×C₈	C₂⁴	1	4	yes	—
181	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
182	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
183	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
184	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
185	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
186	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
187	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
188	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
189	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
190	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
191	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
192	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
193	C₇×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
194	C₄²×C₇	C₂²×C₄	1	8	yes	—
195	C₄²×C₇	C₂²×C₄	1	4	yes	—
196	C₄²×C₇	C₂²×C₄	1	2	yes	—
197	C₄²×C₇	C₂²×C₄	1	4	yes	—
198	C₄²×C₇	C₂⁴	1	2	yes	—
199	C₄²×C₇	C₂⁴	1	4	yes	—
200	C₂×H(5,𝔽₂)	C₄×C₇	1	1	yes	characteristic
201	C₂²×M₁₆	C₂²×C₇	1	4	yes	complemented
202	C₂³×C₈	C₂²×C₇	1	2	yes	complemented
203	#520	C₄×C₇	1	4	yes	—
204	#520	C₄×C₇	1	2	yes	—
205	#518	C₄×C₇	1	4	yes	—
206	#518	C₄×C₇	1	4	yes	—
207	#518	C₄×C₇	1	1	yes	characteristic
208	C₄×#41	C₂²×C₇	1	2	yes	—
209	C₄×#41	C₂²×C₇	1	4	yes	—
210	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
211	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
212	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
213	#446	C₂²×C₇	1	2	yes	complemented
214	#446	C₂²×C₇	1	2	yes	complemented
215	#446	C₂²×C₇	1	4	yes	complemented
216	#436	C₂²×C₇	1	2	yes	complemented
217	#436	C₂²×C₇	1	4	yes	complemented
218	#436	C₂²×C₇	1	2	yes	complemented
219	#435	C₂²×C₇	1	4	yes	complemented
220	#435	C₂²×C₇	1	8	yes	complemented
221	#435	C₂²×C₇	1	8	yes	complemented
222	#435	C₂²×C₇	1	8	yes	complemented
223	#435	C₂²×C₇	1	8	yes	complemented
224	#435	C₂²×C₇	1	4	yes	complemented
225	#435	C₂²×C₇	1	8	yes	complemented
226	C₈×D₄	C₂²×C₇	1	4	yes	complemented
227	C₈×D₄	C₂²×C₇	1	4	yes	complemented
228	C₈×D₄	C₂²×C₇	1	4	yes	complemented
229	C₈×D₄	C₂²×C₇	1	4	yes	complemented
230	#432	C₂²×C₇	1	8	yes	complemented
231	#432	C₂²×C₇	1	4	yes	complemented
232	#431	C₂²×C₇	1	2	yes	complemented
233	#431	C₂²×C₇	1	4	yes	complemented
234	#423	C₂²×C₇	1	4	yes	complemented
235	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
236	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	complemented
237	#407	C₂²×C₇	1	8	yes	complemented
238	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	complemented
239	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	complemented
240	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	complemented
241	C₄×M₁₆	C₂²×C₇	1	4	yes	complemented
242	C₂×#97	C₂²×C₇	1	4	yes	complemented
243	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
244	C₂³×C₇	—	2	8	no	—
245	C₂³×C₇	—	4	2	no	—
246	C₂³×C₇	—	4	2	no	—
247	C₂³×C₇	—	4	4	no	—
248	C₂³×C₇	—	4	4	no	—
249	C₂³×C₇	C₂×M₁₆	1	8	yes	complemented
250	C₂³×C₇	C₂³×C₄	1	4	yes	—
251	C₂³×C₇	C₂³×C₄	1	4	yes	—
252	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
253	C₇×Q₈	—	2	4	no	—
254	C₇×D₄	—	2	16	no	—
255	C₇×D₄	—	2	8	no	—
256	C₇×D₄	—	2	4	no	—
257	C₇×D₄	C₂×M₁₆	1	16	yes	complemented
258	C₂×C₄×C₇	—	2	8	no	—
259	C₂×C₄×C₇	—	2	8	no	—
260	C₂×C₄×C₇	—	2	1	no	—
261	C₂×C₄×C₇	—	2	4	no	—
262	C₂×C₄×C₇	—	2	4	no	—
263	C₂×C₄×C₇	—	2	1	no	—
264	C₂×C₄×C₇	—	2	4	no	—
265	C₂×C₄×C₇	—	2	4	no	—
266	C₂×C₄×C₇	—	4	2	no	—
267	C₂×C₄×C₇	—	4	4	no	—
268	C₂×C₄×C₇	—	4	4	no	—
269	C₂×C₄×C₇	—	4	8	no	—
270	C₂×C₄×C₇	—	4	1	no	—
271	C₂×C₄×C₇	—	4	4	no	—
272	C₂×C₄×C₇	—	4	4	no	—
273	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	complemented
274	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
275	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
276	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
277	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
278	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
279	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
280	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
281	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
282	C₇×C₈	—	2	1	no	—
283	C₇×C₈	—	2	4	no	—
284	C₇×C₈	—	2	4	no	—
285	C₇×C₈	—	2	4	no	—
286	C₇×C₈	—	2	2	no	—
287	C₇×C₈	H(5,𝔽₂)	1	2	yes	complemented
288	H(5,𝔽₂)	—	2	8	no	—
289	C₂×#41	C₂×C₄×C₇	1	4	yes	—
290	C₂×#41	C₂×C₄×C₇	1	2	yes	—
291	C₂²×D₄	C₂×C₄×C₇	1	4	yes	—
292	C₂²×D₄	C₂×C₄×C₇	1	4	yes	—
293	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
294	C₂³×C₄	C₂³×C₇	1	2	yes	—
295	C₂³×C₄	C₂³×C₇	1	4	yes	—
296	C₂×M₁₆	—	4	8	no	—
297	C₂×M₁₆	C₂³×C₇	1	4	yes	—
298	C₂×M₁₆	C₂³×C₇	1	8	yes	—
299	C₂×M₁₆	C₂³×C₇	1	4	yes	—
300	C₂²×C₈	—	4	4	no	—
301	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₇	1	4	yes	—
303	C₂²×C₈	C₂³×C₇	1	4	yes	—
304	C₂²×C₈	C₂³×C₇	1	2	yes	—
305	C₂²×C₈	C₂³×C₇	1	4	yes	—
306	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₇	1	4	yes	—
308	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
309	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
310	C₄×Q₈	C₂³×C₇	1	2	yes	—
311	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
312	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
313	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
314	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
315	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
316	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
317	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
318	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
319	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
320	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
321	C₄×D₄	C₂³×C₇	1	4	yes	—
322	C₄×D₄	C₂³×C₇	1	4	yes	—
323	C₄×D₄	C₂³×C₇	1	8	yes	—
324	C₄×D₄	C₂³×C₇	1	2	yes	—
325	#117	C₂×C₄×C₇	1	4	yes	—
326	#117	C₂×C₄×C₇	1	4	yes	—
327	#117	C₂×C₄×C₇	1	8	yes	—
328	#117	C₂×C₄×C₇	1	2	yes	—
329	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
330	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
331	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
333	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	8	yes	—
334	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
335	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
336	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
337	C₂×C₄²	C₂³×C₇	1	4	yes	—
338	C₂×C₄²	C₂³×C₇	1	4	yes	—
339	C₄⋊C₈	C₂³×C₇	1	4	yes	—
340	C₄⋊C₈	C₂³×C₇	1	4	yes	—
341	C₄⋊C₈	C₂³×C₇	1	8	yes	—
342	C₄⋊C₈	C₂³×C₇	1	2	yes	—
343	C₄⋊C₈	C₂³×C₇	1	2	yes	—
344	C₄⋊C₈	C₂³×C₇	1	4	yes	—
345	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
346	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
347	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
348	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
349	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
350	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
351	#97	C₂³×C₇	1	4	yes	—
352	#97	C₂³×C₇	1	8	yes	—
353	#97	C₂³×C₇	1	4	yes	—
354	C₄×C₈	C₂³×C₇	1	2	yes	—
355	C₄×C₈	C₂³×C₇	1	2	yes	—
356	C₄×C₈	C₂³×C₇	1	4	yes	—
357	C₂²×C₇	—	2	1	no	—
358	C₂²×C₇	—	2	4	no	—
359	C₂²×C₇	—	2	1	no	—
360	C₂²×C₇	—	2	4	no	—
361	C₂²×C₇	—	2	4	no	—
362	C₂²×C₇	—	2	4	no	—
363	C₂²×C₇	—	2	4	no	—
364	C₂²×C₇	—	4	8	no	—
365	C₂²×C₇	—	8	8	no	—
366	C₂²×C₇	C₂²×M₁₆	1	8	yes	—
367	C₂²×C₇	C₂²×M₁₆	1	2	yes	—
368	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
369	C₄×C₇	—	2	1	no	—
370	C₄×C₇	—	2	4	no	—
371	C₄×C₇	—	2	4	no	—
372	C₄×C₇	—	2	4	no	—
373	C₄×C₇	—	2	4	no	—
374	C₄×C₇	—	2	2	no	—
375	C₄×C₇	C₂²×M₁₆	1	4	yes	—
376	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
377	C₄×C₇	#585	1	1	yes	characteristic
378	C₂⁴	C₂²×C₄×C₇	1	4	yes	—
379	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
380	#41	—	2	16	no	—
381	#41	—	2	8	no	—
382	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
383	C₂×D₄	—	2	16	no	—
384	C₂×D₄	—	2	16	no	—
385	C₂×D₄	C₇×M₁₆	1	8	yes	complemented
386	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
387	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
388	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
389	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
395	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
396	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
397	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
398	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
399	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
400	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
401	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
402	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
403	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
404	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
407	M₁₆	—	2	8	no	—
408	M₁₆	—	4	8	no	—
409	M₁₆	—	4	4	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	8	no	—
412	C₂×C₈	—	4	4	no	—
413	C₂×C₈	—	4	2	no	—
414	C₂×C₈	—	4	4	no	—
415	C₂×C₈	—	4	1	no	—
416	C₂×C₈	—	4	4	no	—
417	C₂×C₈	C₂⁴×C₇	1	2	yes	—
418	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
419	C₂×C₈	C₂⁴×C₇	1	4	yes	—
420	C₂×C₈	C₂⁴×C₇	1	4	yes	—
421	C₂×C₈	C₂⁴×C₇	1	4	yes	—
422	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
423	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
424	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
425	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
426	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
427	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
428	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
429	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
430	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
431	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
432	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
433	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
434	C₂²⋊C₄	C₂²×C₄×C₇	1	4	yes	—
435	C₄²	C₂²×C₄×C₇	1	8	yes	—
436	C₄²	C₂²×C₄×C₇	1	4	yes	—
437	C₄²	C₂²×C₄×C₇	1	2	yes	—
438	C₄²	C₂²×C₄×C₇	1	4	yes	—
439	C₄²	C₂⁴×C₇	1	2	yes	—
440	C₄²	C₂⁴×C₇	1	4	yes	—
441	C₂×C₇	—	2	2	no	—
442	C₂×C₇	—	2	8	no	—
443	C₂×C₇	—	4	4	no	—
444	C₂×C₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
445	C₂×C₇	C₂³×M₁₆	1	1	yes	characteristic
446	C₂×C₇	#3573	1	1	yes	characteristic
447	C₂³	—	2	8	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	4	no	—
451	C₂³	—	4	4	no	—
452	C₂³	C₂×C₇×M₁₆	1	8	yes	complemented
453	C₂³	C₂³×C₄×C₇	1	4	yes	—
454	C₂³	C₂³×C₄×C₇	1	4	yes	—
455	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
456	Q₈	—	2	4	no	—
457	D₄	—	2	16	no	—
458	D₄	—	2	8	no	—
459	D₄	—	2	4	no	—
460	D₄	C₂×C₇×M₁₆	1	16	yes	complemented
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	8	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	4	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	8	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	complemented
477	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
478	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
479	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
480	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
481	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
482	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
483	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
484	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
485	C₈	—	2	1	no	—
486	C₈	—	2	4	no	—
487	C₈	—	2	4	no	—
488	C₈	—	2	4	no	—
489	C₈	—	2	2	no	—
490	C₈	C₇×H(5,𝔽₂)	1	2	yes	complemented
491	C₇	#60309	1	1	yes	characteristic, direct product factor, Sylow
492	C₂²	—	2	1	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	4	no	—
496	C₂²	—	2	4	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	2	4	no	—
499	C₂²	—	4	8	no	—
500	C₂²	—	8	8	no	—
501	C₂²	C₂²×C₇×M₁₆	1	8	yes	—
502	C₂²	C₂²×C₇×M₁₆	1	2	yes	—
503	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
504	C₄	—	2	4	no	—
505	C₄	—	2	4	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	1	no	—
508	C₄	—	2	4	no	—
509	C₄	—	2	4	no	—
510	C₄	C₂²×C₇×M₁₆	1	4	yes	—
511	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
512	C₄	C₇×#585	1	1	yes	characteristic
513	C₂	—	2	2	no	—
514	C₂	—	2	8	no	—
515	C₂	—	4	4	no	—
516	C₂	C₄×C₇×H(5,𝔽₂)	1	1	yes	characteristic
517	C₂	C₂³×C₇×M₁₆	1	1	yes	characteristic
518	C₂	C₇×#3573	1	1	yes	characteristic

	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	C₂×C₇×H(5,𝔽₂)	C₄
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×#2990	C₂	semidirect
3. C₇×#2989	C₂	semidirect
4. C₇×#2974	C₂	semidirect
5. C₇×#2973	C₂	semidirect
6. C₇×#2972	C₂	semidirect
7. C₇×#2970	C₂	semidirect
8. C₇×#2965	C₂	semidirect
9. C₇×D₄×M₁₆	C₂	semidirect
10. C₂×C₇×#435	C₂	semidirect
11. C₂×C₇×C₈×D₄	C₂	semidirect
13. C₂²×C₇×M₁₆	C₂²	semidirect
14. C₂³×C₇×C₈	C₂²	semidirect
25. C₇×#446	C₂²	semidirect
26. C₇×#446	C₂²	semidirect
27. C₇×#446	C₂²	semidirect
28. C₇×#436	C₂²	semidirect
29. C₇×#436	C₂²	semidirect
30. C₇×#436	C₂²	semidirect
31. C₇×#435	C₂²	semidirect
32. C₇×#435	C₂²	semidirect
33. C₇×#435	C₂²	semidirect
34. C₇×#435	C₂²	semidirect
35. C₇×#435	C₂²	semidirect
36. C₇×#435	C₂²	semidirect
37. C₇×#435	C₂²	semidirect
38. C₇×C₈×D₄	C₂²	semidirect
39. C₇×C₈×D₄	C₂²	semidirect
40. C₇×C₈×D₄	C₂²	semidirect
41. C₇×C₈×D₄	C₂²	semidirect
42. C₇×#432	C₂²	semidirect
43. C₇×#432	C₂²	semidirect
44. C₇×#431	C₂²	semidirect
45. C₇×#431	C₂²	semidirect
46. C₇×#423	C₂²	semidirect
47. C₂×C₇×C₄⋊C₈	C₂²	semidirect
48. C₂×C₇×C₄⋊C₈	C₂²	semidirect
49. C₇×#407	C₂²	semidirect
50. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
51. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
52. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
53. C₄×C₇×M₁₆	C₂²	semidirect
54. C₂×C₇×#97	C₂²	semidirect
55. C₂×C₄×C₇×C₈	C₂²	semidirect
56. #60309	C₇	direct (N × H)
127. #2990	C₂×C₇	semidirect
128. #2989	C₂×C₇	semidirect
129. #2974	C₂×C₇	semidirect
130. #2973	C₂×C₇	semidirect
131. #2972	C₂×C₇	semidirect
132. #2970	C₂×C₇	semidirect
133. #2965	C₂×C₇	semidirect
134. D₄×M₁₆	C₂×C₇	semidirect
135. C₂×#435	C₂×C₇	semidirect
136. C₂×C₈×D₄	C₂×C₇	semidirect
144. C₂×C₇×D₄	M₁₆	semidirect
201. C₂²×M₁₆	C₂²×C₇	semidirect
202. C₂³×C₈	C₂²×C₇	semidirect
213. #446	C₂²×C₇	semidirect
214. #446	C₂²×C₇	semidirect
215. #446	C₂²×C₇	semidirect
216. #436	C₂²×C₇	semidirect
217. #436	C₂²×C₇	semidirect
218. #436	C₂²×C₇	semidirect
219. #435	C₂²×C₇	semidirect
220. #435	C₂²×C₇	semidirect
221. #435	C₂²×C₇	semidirect
222. #435	C₂²×C₇	semidirect
223. #435	C₂²×C₇	semidirect
224. #435	C₂²×C₇	semidirect
225. #435	C₂²×C₇	semidirect
226. C₈×D₄	C₂²×C₇	semidirect
227. C₈×D₄	C₂²×C₇	semidirect
228. C₈×D₄	C₂²×C₇	semidirect
229. C₈×D₄	C₂²×C₇	semidirect
230. #432	C₂²×C₇	semidirect
231. #432	C₂²×C₇	semidirect
232. #431	C₂²×C₇	semidirect
233. #431	C₂²×C₇	semidirect
234. #423	C₂²×C₇	semidirect
235. C₂×C₄⋊C₈	C₂²×C₇	semidirect
236. C₂×C₄⋊C₈	C₂²×C₇	semidirect
237. #407	C₂²×C₇	semidirect
238. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
239. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
240. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
241. C₄×M₁₆	C₂²×C₇	semidirect
242. C₂×#97	C₂²×C₇	semidirect
243. C₂×C₄×C₈	C₂²×C₇	semidirect
249. C₂³×C₇	C₂×M₁₆	semidirect
257. C₇×D₄	C₂×M₁₆	semidirect
273. C₂×C₄×C₇	C₂×M₁₆	semidirect
287. C₇×C₈	H(5,𝔽₂)	semidirect
385. C₂×D₄	C₇×M₁₆	semidirect
452. C₂³	C₂×C₇×M₁₆	semidirect
460. D₄	C₂×C₇×M₁₆	semidirect
476. C₂×C₄	C₂×C₇×M₁₆	semidirect
490. C₈	C₇×H(5,𝔽₂)	semidirect
491. C₇	#60309	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₇×H(5,𝔽₂)	2
C₇×#2990	2
C₇×#2989	2
C₇×#2974	2
C₇×#2973	2
C₇×#2972	2
C₇×#2970	2
C₇×#2965	2
C₇×D₄×M₁₆	2
C₂×C₇×#435	2
C₂×C₇×C₈×D₄	2
#60309	7

Subgroups — C7×#60309 #13237525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60309 #13237525