Subgroups — C₇×#22656 #13199872

Order 1792.

1010 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	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2352	C₂	1	2	yes	maximal
5	C₇×#1949	C₂	1	2	yes	maximal, complemented
6	C₇×#1854	C₂	1	2	yes	maximal, complemented
7	C₇×#1853	C₂	1	2	yes	maximal, complemented
8	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×#117	C₂²	1	2	yes	—
16	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₇×C₈×D₄	C₄	1	8	yes	complemented
18	C₇×#427	—	2	4	no	—
19	C₂×C₇×#107	—	2	4	no	—
20	C₂×C₇×#107	C₂²	1	1	yes	characteristic
21	C₂×C₇×#107	C₂²	1	2	yes	—
22	C₂×C₇×#107	C₂²	1	1	yes	characteristic
23	C₂×C₇×#107	C₂²	1	1	yes	characteristic
24	C₂×C₇×#107	C₂²	1	1	yes	characteristic
25	C₂×C₇×#106	C₂²	1	2	yes	—
26	C₂×C₇×#106	C₂²	1	2	yes	—
27	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#388	—	2	2	no	—
32	C₇×#387	—	2	2	no	—
33	C₇×#386	—	2	2	no	—
34	C₇×#382	C₂²	1	2	yes	—
35	C₇×#380	C₂²	1	2	yes	—
36	C₇×#380	C₂²	1	2	yes	—
37	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
38	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
39	C₇×#340	C₂²	1	2	yes	—
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	2	yes	—
42	#22656	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₇	—	2	2	no	—
45	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₇×C₈	—	2	2	no	—
48	C₂²×C₇×C₈	—	2	2	no	—
49	C₂²×C₇×C₈	—	2	2	no	—
50	C₂²×C₇×C₈	—	2	2	no	—
51	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
52	C₂²×C₇×C₈	C₂×C₄	1	4	yes	complemented
53	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
58	C₄×C₇×D₄	Q₈	1	4	yes	—
59	C₇×#117	—	2	4	no	—
60	C₇×#117	—	4	2	no	—
61	C₂×C₇×C₄⋊C₄	—	2	4	no	—
62	C₂×C₇×C₄⋊C₄	—	2	2	no	—
63	C₂×C₇×C₄⋊C₄	—	4	2	no	—
64	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
66	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
67	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
68	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
70	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
72	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
73	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₇	—	2	2	no	—
75	C₂×C₄²×C₇	—	2	2	no	—
76	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
77	C₇×#107	—	2	2	no	—
78	C₇×#107	—	2	2	no	—
79	C₇×#107	—	2	2	no	—
80	C₇×#107	—	2	4	no	—
81	C₇×#107	D₄	1	4	yes	complemented
82	C₇×#106	—	2	4	no	—
83	C₇×#106	—	4	2	no	—
84	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
85	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
86	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
87	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
88	C₇×#95	—	2	2	no	—
89	C₇×#95	—	2	2	no	—
90	C₇×#95	—	2	2	no	—
91	C₇×#95	—	2	2	no	—
92	C₇×#95	—	2	2	no	—
93	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
94	C₂×#427	C₂×C₇	1	1	yes	characteristic, complemented
95	C₂²×#107	C₂×C₇	1	1	yes	characteristic, complemented
96	#2352	C₂×C₇	1	2	yes	—
97	#1949	C₂×C₇	1	2	yes	complemented
98	#1854	C₂×C₇	1	2	yes	complemented
99	#1853	C₂×C₇	1	2	yes	complemented
100	#1846	C₂×C₇	1	1	yes	characteristic, complemented
101	#1808	C₂×C₇	1	1	yes	characteristic, complemented
102	#1808	C₂×C₇	1	1	yes	characteristic, complemented
103	C₄×#107	C₂×C₇	1	1	yes	characteristic, complemented
104	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
105	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₇×D₄	—	2	2	no	—
107	C₂×C₇×D₄	—	2	2	no	—
108	C₂×C₇×D₄	C₄⋊C₄	1	4	yes	—
109	C₂²×C₄×C₇	—	2	1	no	—
110	C₂²×C₄×C₇	—	2	1	no	—
111	C₂²×C₄×C₇	—	2	1	no	—
112	C₂²×C₄×C₇	—	2	1	no	—
113	C₂²×C₄×C₇	—	2	1	no	—
114	C₂²×C₄×C₇	—	2	1	no	—
115	C₂²×C₄×C₇	—	2	1	no	—
116	C₂²×C₄×C₇	—	2	1	no	—
117	C₂²×C₄×C₇	—	2	2	no	—
118	C₂²×C₄×C₇	—	2	2	no	—
119	C₂²×C₄×C₇	—	2	2	no	—
120	C₂²×C₄×C₇	—	2	2	no	—
121	C₂²×C₄×C₇	—	4	4	no	—
122	C₂²×C₄×C₇	—	4	4	no	—
123	C₂²×C₄×C₇	—	8	2	no	—
124	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
131	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
132	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
133	C₂×C₇×C₈	—	2	1	no	—
134	C₂×C₇×C₈	—	2	1	no	—
135	C₂×C₇×C₈	—	2	1	no	—
136	C₂×C₇×C₈	—	2	1	no	—
137	C₂×C₇×C₈	—	2	1	no	—
138	C₂×C₇×C₈	—	2	1	no	—
139	C₂×C₇×C₈	—	2	8	no	—
140	C₂×C₇×C₈	—	2	8	no	—
141	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
142	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
143	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
144	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
145	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
146	C₂×C₇×C₈	#41	1	2	yes	—
147	C₇×C₄⋊C₄	—	2	2	no	—
148	C₇×C₄⋊C₄	—	2	2	no	—
149	C₇×C₄⋊C₄	—	2	2	no	—
150	C₇×C₄⋊C₄	—	2	2	no	—
151	C₇×C₄⋊C₄	—	2	2	no	—
152	C₇×C₄⋊C₄	—	2	2	no	—
153	C₇×C₄⋊C₄	—	2	2	no	—
154	C₇×C₄⋊C₄	—	4	4	no	—
155	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
156	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
157	C₇×C₂²⋊C₄	—	4	4	no	—
158	C₇×C₂²⋊C₄	—	4	4	no	—
159	C₇×C₂²⋊C₄	—	8	2	no	—
160	C₇×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
161	C₇×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
162	C₄²×C₇	—	4	4	no	—
163	C₄²×C₇	—	8	2	no	—
164	C₄²×C₇	C₂²×C₄	1	2	yes	—
165	C₄²×C₇	C₂×Q₈	1	2	yes	—
166	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
167	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
168	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
169	C₂×#117	C₂²×C₇	1	2	yes	—
170	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
171	C₈×D₄	C₄×C₇	1	8	yes	complemented
172	#427	—	2	4	no	—
173	C₂×#107	—	2	4	no	—
174	C₂×#107	C₂²×C₇	1	1	yes	characteristic
175	C₂×#107	C₂²×C₇	1	1	yes	characteristic
176	C₂×#107	C₂²×C₇	1	1	yes	characteristic
177	C₂×#107	C₂²×C₇	1	2	yes	—
178	C₂×#107	C₂²×C₇	1	1	yes	characteristic
179	C₂×#106	C₂²×C₇	1	2	yes	—
180	C₂×#106	C₂²×C₇	1	2	yes	—
181	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
182	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
183	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
185	#388	—	2	2	no	—
186	#387	—	2	2	no	—
187	#386	—	2	2	no	—
188	#382	C₂²×C₇	1	2	yes	—
189	#380	C₂²×C₇	1	2	yes	—
190	#380	C₂²×C₇	1	2	yes	—
191	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
192	C₄×C₂²⋊C₄	—	2	2	no	—
193	#340	C₂²×C₇	1	2	yes	—
194	#340	C₂²×C₇	1	2	yes	—
195	#340	C₂²×C₇	1	2	yes	—
196	C₂³×C₇	—	2	1	no	—
197	C₂³×C₇	—	2	1	no	—
198	C₂³×C₇	—	2	1	no	—
199	C₂³×C₇	—	2	1	no	—
200	C₂³×C₇	—	2	1	no	—
201	C₂³×C₇	—	2	1	no	—
202	C₂³×C₇	—	2	1	no	—
203	C₂³×C₇	—	2	1	no	—
204	C₂³×C₇	—	2	1	no	—
205	C₂³×C₇	—	2	1	no	—
206	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
207	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
208	C₂³×C₇	#135	1	2	yes	—
209	C₂³×C₇	#135	1	2	yes	—
210	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
211	C₇×D₄	—	2	8	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	2	no	—
221	C₂×C₄×C₇	—	2	2	no	—
222	C₂×C₄×C₇	—	2	2	no	—
223	C₂×C₄×C₇	—	2	2	no	—
224	C₂×C₄×C₇	—	2	2	no	—
225	C₂×C₄×C₇	—	2	2	no	—
226	C₂×C₄×C₇	—	2	2	no	—
227	C₂×C₄×C₇	—	2	2	no	—
228	C₂×C₄×C₇	—	2	4	no	—
229	C₂×C₄×C₇	—	4	2	no	—
230	C₂×C₄×C₇	—	4	2	no	—
231	C₂×C₄×C₇	—	4	2	no	—
232	C₂×C₄×C₇	—	4	2	no	—
233	C₂×C₄×C₇	—	4	2	no	—
234	C₂×C₄×C₇	—	4	2	no	—
235	C₂×C₄×C₇	—	4	2	no	—
236	C₂×C₄×C₇	—	4	2	no	—
237	C₂×C₄×C₇	—	4	2	no	—
238	C₂×C₄×C₇	—	4	2	no	—
239	C₂×C₄×C₇	—	4	2	no	—
240	C₂×C₄×C₇	—	8	8	no	—
241	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
242	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
243	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
244	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
245	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
248	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
251	C₇×C₈	—	2	2	no	—
252	C₇×C₈	—	2	2	no	—
253	C₇×C₈	—	2	2	no	—
254	C₇×C₈	C₄×D₄	1	4	yes	—
255	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
256	C₂³×C₄	—	2	2	no	—
257	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
258	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
259	C₂²×C₈	—	2	2	no	—
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
264	C₂²×C₈	C₂×C₄×C₇	1	4	yes	complemented
265	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
269	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
270	C₄×D₄	C₇×Q₈	1	4	yes	—
271	#117	—	2	4	no	—
272	#117	—	4	2	no	—
273	C₂×C₄⋊C₄	—	2	4	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	4	2	no	—
276	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
277	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
285	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	—	2	2	no	—
288	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
289	#107	—	2	2	no	—
290	#107	—	2	2	no	—
291	#107	—	2	2	no	—
292	#107	—	2	4	no	—
293	#107	C₇×D₄	1	4	yes	complemented
294	#106	—	2	4	no	—
295	#106	—	4	2	no	—
296	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
297	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	—
298	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
299	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	C₂²×C₇	—	2	2	no	—
306	C₂²×C₇	—	2	2	no	—
307	C₂²×C₇	—	2	2	no	—
308	C₂²×C₇	—	2	2	no	—
309	C₂²×C₇	—	2	2	no	—
310	C₂²×C₇	—	2	2	no	—
311	C₂²×C₇	—	2	2	no	—
312	C₂²×C₇	—	2	2	no	—
313	C₂²×C₇	—	2	2	no	—
314	C₂²×C₇	—	4	1	no	—
315	C₂²×C₇	—	4	1	no	—
316	C₂²×C₇	—	4	1	no	—
317	C₂²×C₇	—	4	1	no	—
318	C₂²×C₇	#427	1	4	yes	complemented
319	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
320	C₂²×C₇	D₄²	1	1	yes	characteristic
321	C₂²×C₇	#547	1	1	yes	characteristic
322	C₂²×C₇	C₂×#135	1	1	yes	characteristic
323	C₂²×C₇	C₂×#135	1	1	yes	characteristic
324	C₂²×C₇	#576	1	1	yes	characteristic
325	C₂²×C₇	#578	1	1	yes	characteristic
326	C₄×C₇	—	2	2	no	—
327	C₄×C₇	—	2	2	no	—
328	C₄×C₇	—	2	2	no	—
329	C₄×C₇	—	2	2	no	—
330	C₄×C₇	—	4	4	no	—
331	C₄×C₇	—	8	2	no	—
332	C₄×C₇	—	8	2	no	—
333	C₄×C₇	—	8	2	no	—
334	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
335	C₄×C₇	#520	1	1	yes	characteristic
336	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
337	C₄×C₇	#554	1	1	yes	characteristic
338	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
339	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	C₇×C₄⋊C₄	1	4	yes	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	4	4	no	—
356	C₂²×C₄	—	4	4	no	—
357	C₂²×C₄	—	8	2	no	—
358	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
359	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
360	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
361	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
362	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
363	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
364	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
365	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
366	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	8	no	—
374	C₂×C₈	—	2	8	no	—
375	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
376	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
377	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
378	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
379	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
380	C₂×C₈	C₇×#41	1	2	yes	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	4	4	no	—
389	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
390	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₂²⋊C₄	—	4	4	no	—
393	C₂²⋊C₄	—	8	2	no	—
394	C₂²⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
395	C₂²⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
396	C₄²	—	4	4	no	—
397	C₄²	—	8	2	no	—
398	C₄²	C₂²×C₄×C₇	1	2	yes	—
399	C₄²	C₂×C₇×Q₈	1	2	yes	—
400	C₂×C₇	—	2	4	no	—
401	C₂×C₇	—	4	2	no	—
402	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
403	C₂×C₇	C₂×#427	1	1	yes	characteristic
404	C₂×C₇	#2913	1	1	yes	characteristic
405	C₂×C₇	#3280	1	1	yes	characteristic
406	C₂×C₇	#3285	1	1	yes	characteristic
407	C₂×C₇	#3336	1	1	yes	characteristic
408	C₂×C₇	#3339	1	1	yes	characteristic
409	C₂³	—	2	1	no	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
420	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
421	C₂³	C₇×#135	1	2	yes	—
422	C₂³	C₇×#135	1	2	yes	—
423	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
424	D₄	—	2	8	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	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	8	8	no	—
454	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
455	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
456	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
457	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
458	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
460	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
461	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
462	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
463	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
464	C₈	—	2	2	no	—
465	C₈	—	2	2	no	—
466	C₈	—	2	2	no	—
467	C₈	C₄×C₇×D₄	1	4	yes	—
468	C₇	#22656	1	1	yes	characteristic, direct product factor, Sylow
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	4	1	no	—
479	C₂²	—	4	1	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	C₇×#427	1	4	yes	complemented
483	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
484	C₂²	C₇×D₄²	1	1	yes	characteristic
485	C₂²	C₇×#547	1	1	yes	characteristic
486	C₂²	C₂×C₇×#135	1	1	yes	characteristic
487	C₂²	C₂×C₇×#135	1	1	yes	characteristic
488	C₂²	C₇×#576	1	1	yes	characteristic
489	C₂²	C₇×#578	1	1	yes	characteristic
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	4	no	—
495	C₄	—	8	2	no	—
496	C₄	—	8	2	no	—
497	C₄	—	8	2	no	—
498	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
499	C₄	C₇×#520	1	1	yes	characteristic
500	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
501	C₄	C₇×#554	1	1	yes	characteristic
502	C₂	—	2	4	no	—
503	C₂	—	4	2	no	—
504	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
505	C₂	C₂×C₇×#427	1	1	yes	characteristic
506	C₂	C₇×#2913	1	1	yes	characteristic
507	C₂	C₇×#3280	1	1	yes	characteristic
508	C₂	C₇×#3285	1	1	yes	characteristic
509	C₂	C₇×#3336	1	1	yes	characteristic
510	C₂	C₇×#3339	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#427	C₂	semidirect
3. C₂²×C₇×#107	C₂	semidirect
5. C₇×#1949	C₂	semidirect
6. C₇×#1854	C₂	semidirect
7. C₇×#1853	C₂	semidirect
8. C₇×#1846	C₂	semidirect
9. C₇×#1808	C₂	semidirect
10. C₇×#1808	C₂	semidirect
11. C₄×C₇×#107	C₂	semidirect
17. C₇×C₈×D₄	C₄	semidirect
42. #22656	C₇	direct (N × H)
52. C₂²×C₇×C₈	C₂×C₄	semidirect
81. C₇×#107	D₄	semidirect
84. C₇×C₄⋊C₈	C₂×C₄	semidirect
86. C₇×C₂²⋊C₈	C₂×C₄	semidirect
87. C₄×C₇×C₈	C₂×C₄	semidirect
94. C₂×#427	C₂×C₇	semidirect
95. C₂²×#107	C₂×C₇	semidirect
97. #1949	C₂×C₇	semidirect
98. #1854	C₂×C₇	semidirect
99. #1853	C₂×C₇	semidirect
100. #1846	C₂×C₇	semidirect
101. #1808	C₂×C₇	semidirect
102. #1808	C₂×C₇	semidirect
103. C₄×#107	C₂×C₇	semidirect
171. C₈×D₄	C₄×C₇	semidirect
264. C₂²×C₈	C₂×C₄×C₇	semidirect
293. #107	C₇×D₄	semidirect
296. C₄⋊C₈	C₂×C₄×C₇	semidirect
298. C₂²⋊C₈	C₂×C₄×C₇	semidirect
299. C₄×C₈	C₂×C₄×C₇	semidirect
318. C₂²×C₇	#427	semidirect
468. C₇	#22656	direct (N × H)
482. C₂²	C₇×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×D₄	2
C₂×C₇×#427	2
C₂²×C₇×#107	2
C₇×#2352	2
C₇×#1949	2
C₇×#1854	2
C₇×#1853	2
C₇×#1846	2
C₇×#1808	2
C₄×C₇×#107	2
#22656	7

Subgroups — C7×#22656 #13199872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22656 #13199872