Subgroups — C₇×#22823 #13200039

Order 1792.

736 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#1985	C₂	1	2	yes	maximal
3	C₇×#1917	C₂	1	2	yes	maximal
4	C₇×#1916	C₂	1	2	yes	maximal
5	C₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₇×#1831	C₂	1	1	yes	maximal, characteristic
7	C₇×#1831	C₂	1	1	yes	maximal, characteristic
8	C₇×#1771	C₂	1	2	yes	maximal
9	C₇×#1767	C₂	1	1	yes	maximal, characteristic
10	C₇×#1767	C₂	1	1	yes	maximal, characteristic
11	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×#97	C₂²	1	2	yes	—
22	C₂×C₇×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
24	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
25	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
26	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
27	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
28	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
29	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
30	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₄³×C₇	C₂²	1	1	yes	characteristic
34	C₄³×C₇	C₂²	1	1	yes	characteristic
35	C₄³×C₇	C₂²	1	1	yes	characteristic
36	C₇×#336	C₂²	1	2	yes	—
37	C₇×#336	C₂²	1	2	yes	—
38	C₇×#336	C₂²	1	2	yes	—
39	C₇×#336	C₂²	1	2	yes	—
40	C₇×#336	C₂²	1	2	yes	—
41	C₇×#336	C₂²	1	2	yes	—
42	C₇×#336	C₂²	1	2	yes	—
43	#22823	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₇×C₈	C₂³	1	2	yes	—
46	C₂²×C₇×C₈	C₂³	1	2	yes	—
47	C₂²×C₇×C₈	C₂³	1	2	yes	—
48	C₂²×C₇×C₈	C₂³	1	2	yes	—
49	C₄×C₇×Q₈	—	2	4	no	—
50	C₄×C₇×Q₈	—	2	4	no	—
51	C₄×C₇×Q₈	—	2	2	no	—
52	C₄×C₇×Q₈	—	2	2	no	—
53	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
81	C₇×C₄⋊C₈	—	2	4	no	—
82	C₇×C₄⋊C₈	—	2	2	no	—
83	C₇×C₄⋊C₈	—	2	2	no	—
84	C₇×C₄⋊C₈	—	2	2	no	—
85	C₇×C₄⋊C₈	—	4	2	no	—
86	C₇×C₄⋊C₈	Q₈	1	4	yes	complemented
87	C₇×#97	—	2	4	no	—
88	C₇×#97	—	4	2	no	—
89	C₄×C₇×C₈	—	4	2	no	—
90	C₇×#95	C₂×C₄	1	1	yes	characteristic
91	C₇×#95	C₂×C₄	1	1	yes	characteristic
92	C₇×#95	C₂×C₄	1	1	yes	characteristic
93	C₇×#95	C₂×C₄	1	1	yes	characteristic
94	C₇×#95	C₂×C₄	1	1	yes	characteristic
95	C₇×#95	C₂×C₄	1	1	yes	characteristic
96	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic
97	#1985	C₂×C₇	1	2	yes	—
98	#1917	C₂×C₇	1	2	yes	—
99	#1916	C₂×C₇	1	2	yes	—
100	#1842	C₂×C₇	1	1	yes	characteristic
101	#1831	C₂×C₇	1	1	yes	characteristic
102	#1831	C₂×C₇	1	1	yes	characteristic
103	#1771	C₂×C₇	1	2	yes	—
104	#1767	C₂×C₇	1	1	yes	characteristic
105	#1767	C₂×C₇	1	1	yes	characteristic
106	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
107	C₂×C₇×Q₈	—	2	2	no	—
108	C₂×C₇×Q₈	—	2	2	no	—
109	C₂×C₇×Q₈	—	2	2	no	—
110	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
125	C₂×C₇×C₈	—	2	2	no	—
126	C₂×C₇×C₈	—	2	2	no	—
127	C₂×C₇×C₈	—	2	2	no	—
128	C₂×C₇×C₈	—	2	2	no	—
129	C₂×C₇×C₈	—	2	2	no	—
130	C₂×C₇×C₈	—	2	2	no	—
131	C₂×C₇×C₈	—	2	2	no	—
132	C₂×C₇×C₈	—	2	2	no	—
133	C₂×C₇×C₈	—	2	2	no	—
134	C₂×C₇×C₈	—	2	2	no	—
135	C₂×C₇×C₈	—	2	2	no	—
136	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
137	C₇×C₄⋊C₄	—	2	2	no	—
138	C₇×C₄⋊C₄	—	2	2	no	—
139	C₇×C₄⋊C₄	—	2	2	no	—
140	C₇×C₄⋊C₄	—	2	2	no	—
141	C₇×C₄⋊C₄	—	2	2	no	—
142	C₇×C₄⋊C₄	—	2	2	no	—
143	C₇×C₄⋊C₄	—	2	2	no	—
144	C₇×C₄⋊C₄	—	2	2	no	—
145	C₇×C₄⋊C₄	M₁₆	1	4	yes	—
146	C₄²×C₇	—	2	2	no	—
147	C₄²×C₇	—	2	2	no	—
148	C₄²×C₇	—	2	2	no	—
149	C₄²×C₇	—	2	2	no	—
150	C₄²×C₇	—	2	2	no	—
151	C₄²×C₇	—	2	2	no	—
152	C₄²×C₇	—	2	2	no	—
153	C₄²×C₇	—	2	2	no	—
154	C₄²×C₇	—	2	1	no	—
155	C₄²×C₇	—	2	1	no	—
156	C₄²×C₇	—	2	1	no	—
157	C₄²×C₇	—	2	1	no	—
158	C₄²×C₇	—	2	1	no	—
159	C₄²×C₇	—	2	1	no	—
160	C₄²×C₇	—	2	1	no	—
161	C₄²×C₇	—	2	1	no	—
162	C₄²×C₇	—	2	1	no	—
163	C₄²×C₇	—	2	1	no	—
164	C₄²×C₇	—	4	1	no	—
165	C₄²×C₇	—	4	1	no	—
166	C₄²×C₇	—	4	1	no	—
167	C₄²×C₇	M₁₆	1	4	yes	—
168	C₄²×C₇	C₂×Q₈	1	2	yes	—
169	C₄²×C₇	#41	1	2	yes	—
170	C₄²×C₇	#41	1	2	yes	—
171	C₄²×C₇	#41	1	2	yes	—
172	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
173	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
174	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
179	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
180	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
181	C₂×#97	C₂²×C₇	1	2	yes	—
182	C₂×#97	C₂²×C₇	1	2	yes	—
183	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
184	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
185	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
186	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
187	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
188	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
189	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
191	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
192	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
193	C₄³	C₂²×C₇	1	1	yes	characteristic
194	C₄³	C₂²×C₇	1	1	yes	characteristic
195	C₄³	C₂²×C₇	1	1	yes	characteristic
196	#336	C₂²×C₇	1	2	yes	—
197	#336	C₂²×C₇	1	2	yes	—
198	#336	C₂²×C₇	1	2	yes	—
199	#336	C₂²×C₇	1	2	yes	—
200	#336	C₂²×C₇	1	2	yes	—
201	#336	C₂²×C₇	1	2	yes	—
202	#336	C₂²×C₇	1	2	yes	—
203	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
204	C₇×Q₈	—	2	8	no	—
205	C₂×C₄×C₇	—	2	1	no	—
206	C₂×C₄×C₇	—	2	1	no	—
207	C₂×C₄×C₇	—	2	1	no	—
208	C₂×C₄×C₇	—	2	1	no	—
209	C₂×C₄×C₇	—	2	1	no	—
210	C₂×C₄×C₇	—	2	1	no	—
211	C₂×C₄×C₇	—	2	1	no	—
212	C₂×C₄×C₇	—	2	1	no	—
213	C₂×C₄×C₇	—	2	1	no	—
214	C₂×C₄×C₇	—	2	1	no	—
215	C₂×C₄×C₇	—	2	1	no	—
216	C₂×C₄×C₇	—	2	1	no	—
217	C₂×C₄×C₇	—	2	1	no	—
218	C₂×C₄×C₇	—	2	1	no	—
219	C₂×C₄×C₇	—	2	1	no	—
220	C₂×C₄×C₇	—	2	1	no	—
221	C₂×C₄×C₇	—	2	1	no	—
222	C₂×C₄×C₇	—	2	1	no	—
223	C₂×C₄×C₇	—	2	1	no	—
224	C₂×C₄×C₇	—	2	1	no	—
225	C₂×C₄×C₇	—	2	1	no	—
226	C₂×C₄×C₇	—	2	1	no	—
227	C₂×C₄×C₇	—	2	1	no	—
228	C₂×C₄×C₇	—	2	1	no	—
229	C₂×C₄×C₇	—	2	1	no	—
230	C₂×C₄×C₇	—	2	1	no	—
231	C₂×C₄×C₇	—	2	1	no	—
232	C₂×C₄×C₇	—	2	1	no	—
233	C₂×C₄×C₇	—	2	1	no	—
234	C₂×C₄×C₇	—	2	1	no	—
235	C₂×C₄×C₇	—	2	1	no	—
236	C₂×C₄×C₇	#117	1	2	yes	—
237	C₂×C₄×C₇	#117	1	2	yes	—
238	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
239	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
240	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
241	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
242	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
243	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
244	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
245	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
246	C₂×C₄×C₇	#131	1	2	yes	—
247	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
248	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
250	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
251	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
252	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
253	C₇×C₈	—	2	4	no	—
254	C₇×C₈	—	4	2	no	—
255	C₇×C₈	—	4	2	no	—
256	C₇×C₈	—	4	2	no	—
257	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₇	1	2	yes	—
260	C₂²×C₈	C₂³×C₇	1	2	yes	—
261	C₂²×C₈	C₂³×C₇	1	2	yes	—
262	C₄×Q₈	—	2	4	no	—
263	C₄×Q₈	—	2	4	no	—
264	C₄×Q₈	—	2	2	no	—
265	C₄×Q₈	—	2	2	no	—
266	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
268	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
269	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
270	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
275	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
276	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
277	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
278	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
279	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
280	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
281	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
282	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
283	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
284	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
285	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
286	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
290	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
294	C₄⋊C₈	—	2	4	no	—
295	C₄⋊C₈	—	2	2	no	—
296	C₄⋊C₈	—	2	2	no	—
297	C₄⋊C₈	—	2	2	no	—
298	C₄⋊C₈	—	4	2	no	—
299	C₄⋊C₈	C₇×Q₈	1	4	yes	complemented
300	#97	—	2	4	no	—
301	#97	—	4	2	no	—
302	C₄×C₈	—	4	2	no	—
303	#95	C₂×C₄×C₇	1	1	yes	characteristic
304	#95	C₂×C₄×C₇	1	1	yes	characteristic
305	#95	C₂×C₄×C₇	1	1	yes	characteristic
306	#95	C₂×C₄×C₇	1	1	yes	characteristic
307	#95	C₂×C₄×C₇	1	1	yes	characteristic
308	#95	C₂×C₄×C₇	1	1	yes	characteristic
309	C₂²×C₇	C₂×#117	1	1	yes	characteristic
310	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
311	C₂²×C₇	#520	1	1	yes	characteristic
312	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
313	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
314	C₂²×C₇	C₂×#131	1	1	yes	characteristic
315	C₂²×C₇	#568	1	1	yes	characteristic
316	C₄×C₇	—	2	2	no	—
317	C₄×C₇	—	2	2	no	—
318	C₄×C₇	—	2	2	no	—
319	C₄×C₇	—	2	2	no	—
320	C₄×C₇	—	2	2	no	—
321	C₄×C₇	—	2	2	no	—
322	C₄×C₇	—	2	2	no	—
323	C₄×C₇	—	2	2	no	—
324	C₄×C₇	—	4	1	no	—
325	C₄×C₇	—	4	1	no	—
326	C₄×C₇	—	4	1	no	—
327	C₄×C₇	—	4	1	no	—
328	C₄×C₇	—	4	1	no	—
329	C₄×C₇	#446	1	4	yes	—
330	C₄×C₇	#554	1	1	yes	characteristic
331	C₄×C₇	#556	1	1	yes	characteristic
332	C₄×C₇	#557	1	1	yes	characteristic
333	C₄×C₇	#559	1	1	yes	characteristic
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
338	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
339	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
340	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
341	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
342	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
343	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
344	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
345	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
346	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
348	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
349	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
350	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	C₇×M₁₆	1	4	yes	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	2	1	no	—
383	C₄²	—	2	1	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	4	1	no	—
392	C₄²	—	4	1	no	—
393	C₄²	—	4	1	no	—
394	C₄²	C₇×M₁₆	1	4	yes	—
395	C₄²	C₂×C₇×Q₈	1	2	yes	—
396	C₄²	C₇×#41	1	2	yes	—
397	C₄²	C₇×#41	1	2	yes	—
398	C₄²	C₇×#41	1	2	yes	—
399	C₂×C₇	#2355	1	1	yes	characteristic
400	C₂×C₇	#2920	1	1	yes	characteristic
401	C₂×C₇	#2925	1	1	yes	characteristic
402	C₂×C₇	C₂×#446	1	1	yes	characteristic
403	C₂×C₇	Q₈×M₁₆	1	1	yes	characteristic
404	C₂×C₇	#2989	1	1	yes	characteristic
405	C₂×C₇	#2990	1	1	yes	characteristic
406	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
407	Q₈	—	2	8	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	C₇×#117	1	2	yes	—
440	C₂×C₄	C₇×#117	1	2	yes	—
441	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
442	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
443	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
444	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
445	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
446	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
447	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
448	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
449	C₂×C₄	C₇×#131	1	2	yes	—
450	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
451	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
452	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
453	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
454	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
455	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
456	C₈	—	2	4	no	—
457	C₈	—	4	2	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	2	no	—
460	C₇	#22823	1	1	yes	characteristic, direct product factor, Sylow
461	C₂²	C₂×C₇×#117	1	1	yes	characteristic
462	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
463	C₂²	C₇×#520	1	1	yes	characteristic
464	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
465	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
466	C₂²	C₂×C₇×#131	1	1	yes	characteristic
467	C₂²	C₇×#568	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	4	1	no	—
478	C₄	—	4	1	no	—
479	C₄	—	4	1	no	—
480	C₄	—	4	1	no	—
481	C₄	C₇×#446	1	4	yes	—
482	C₄	C₇×#554	1	1	yes	characteristic
483	C₄	C₇×#556	1	1	yes	characteristic
484	C₄	C₇×#557	1	1	yes	characteristic
485	C₄	C₇×#559	1	1	yes	characteristic
486	C₂	C₇×#2355	1	1	yes	characteristic
487	C₂	C₇×#2920	1	1	yes	characteristic
488	C₂	C₇×#2925	1	1	yes	characteristic
489	C₂	C₂×C₇×#446	1	1	yes	characteristic
490	C₂	C₇×Q₈×M₁₆	1	1	yes	characteristic
491	C₂	C₇×#2989	1	1	yes	characteristic
492	C₂	C₇×#2990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
43. #22823	C₇	direct (N × H)
86. C₇×C₄⋊C₈	Q₈	semidirect
299. C₄⋊C₈	C₇×Q₈	semidirect
460. C₇	#22823	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₇×Q₈	2
C₇×#1985	2
C₇×#1917	2
C₇×#1916	2
C₇×#1842	2
C₇×#1831	2
C₇×#1771	2
C₇×#1767	2
C₄×C₇×C₄⋊C₈	2
#22823	7

Subgroups — C7×#22823 #13200039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22823 #13200039