Subgroups — C₇×#22654 #13199870

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 482 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×#106	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×#1852	C₂	1	2	yes	maximal, complemented
8	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1807	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1807	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#106	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	C₂²	1	2	yes	—
20	C₂×C₇×#107	C₂²	1	2	yes	—
21	C₂×C₇×#106	—	2	4	no	—
22	C₂×C₇×#106	C₂²	1	1	yes	characteristic
23	C₂×C₇×#106	C₂²	1	2	yes	—
24	C₂×C₇×#106	C₂²	1	1	yes	characteristic
25	C₂×C₇×#106	C₂²	1	1	yes	characteristic
26	C₂×C₇×#106	C₂²	1	1	yes	characteristic
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	#22654	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	4	no	—
78	C₇×#107	—	4	2	no	—
79	C₇×#106	—	2	2	no	—
80	C₇×#106	—	2	2	no	—
81	C₇×#106	—	2	2	no	—
82	C₇×#106	—	2	4	no	—
83	C₇×#106	D₄	1	4	yes	complemented
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×#106	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	#1852	C₂×C₇	1	2	yes	complemented
100	#1846	C₂×C₇	1	1	yes	characteristic, complemented
101	#1807	C₂×C₇	1	1	yes	characteristic, complemented
102	#1807	C₂×C₇	1	1	yes	characteristic, complemented
103	C₄×#106	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	4	no	—
107	C₂×C₇×D₄	C₄⋊C₄	1	4	yes	—
108	C₂²×C₄×C₇	—	2	1	no	—
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	2	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	—	4	4	no	—
121	C₂²×C₄×C₇	—	4	4	no	—
122	C₂²×C₄×C₇	—	8	2	no	—
123	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
130	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
131	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
132	C₂×C₇×C₈	—	2	1	no	—
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	8	no	—
139	C₂×C₇×C₈	—	2	8	no	—
140	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
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	C₂×D₄	1	2	yes	—
145	C₂×C₇×C₈	#41	1	2	yes	—
146	C₇×C₄⋊C₄	—	2	2	no	—
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	—	4	4	no	—
154	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
155	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
156	C₇×C₂²⋊C₄	—	4	4	no	—
157	C₇×C₂²⋊C₄	—	4	4	no	—
158	C₇×C₂²⋊C₄	—	8	2	no	—
159	C₇×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
160	C₇×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
161	C₄²×C₇	—	4	4	no	—
162	C₄²×C₇	—	8	2	no	—
163	C₄²×C₇	C₂²×C₄	1	2	yes	—
164	C₄²×C₇	C₂×Q₈	1	2	yes	—
165	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
166	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
167	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
168	C₂×#117	C₂²×C₇	1	2	yes	—
169	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
170	C₈×D₄	C₄×C₇	1	8	yes	complemented
171	#427	—	2	4	no	—
172	C₂×#107	C₂²×C₇	1	2	yes	—
173	C₂×#107	C₂²×C₇	1	2	yes	—
174	C₂×#106	—	2	4	no	—
175	C₂×#106	C₂²×C₇	1	1	yes	characteristic
176	C₂×#106	C₂²×C₇	1	1	yes	characteristic
177	C₂×#106	C₂²×C₇	1	1	yes	characteristic
178	C₂×#106	C₂²×C₇	1	2	yes	—
179	C₂×#106	C₂²×C₇	1	1	yes	characteristic
180	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
181	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
182	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
184	#388	—	2	2	no	—
185	#387	—	2	2	no	—
186	#386	—	2	2	no	—
187	#382	C₂²×C₇	1	2	yes	—
188	#380	C₂²×C₇	1	2	yes	—
189	#380	C₂²×C₇	1	2	yes	—
190	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
191	C₄×C₂²⋊C₄	—	2	2	no	—
192	#340	C₂²×C₇	1	2	yes	—
193	#340	C₂²×C₇	1	2	yes	—
194	#340	C₂²×C₇	1	2	yes	—
195	C₂³×C₇	—	2	1	no	—
196	C₂³×C₇	—	2	1	no	—
197	C₂³×C₇	—	2	2	no	—
198	C₂³×C₇	—	2	1	no	—
199	C₂³×C₇	—	2	1	no	—
200	C₂³×C₇	—	2	2	no	—
201	C₂³×C₇	—	2	2	no	—
202	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
203	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
204	C₂³×C₇	#135	1	4	yes	—
205	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
206	C₇×D₄	—	2	8	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	2	no	—
212	C₂×C₄×C₇	—	2	2	no	—
213	C₂×C₄×C₇	—	2	2	no	—
214	C₂×C₄×C₇	—	2	2	no	—
215	C₂×C₄×C₇	—	2	2	no	—
216	C₂×C₄×C₇	—	2	2	no	—
217	C₂×C₄×C₇	—	2	2	no	—
218	C₂×C₄×C₇	—	2	2	no	—
219	C₂×C₄×C₇	—	2	2	no	—
220	C₂×C₄×C₇	—	2	2	no	—
221	C₂×C₄×C₇	—	2	4	no	—
222	C₂×C₄×C₇	—	4	2	no	—
223	C₂×C₄×C₇	—	4	2	no	—
224	C₂×C₄×C₇	—	4	2	no	—
225	C₂×C₄×C₇	—	4	2	no	—
226	C₂×C₄×C₇	—	4	2	no	—
227	C₂×C₄×C₇	—	4	2	no	—
228	C₂×C₄×C₇	—	4	2	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	—	8	8	no	—
234	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
235	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
236	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
237	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
238	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
239	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
241	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
242	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
244	C₇×C₈	—	2	2	no	—
245	C₇×C₈	—	2	2	no	—
246	C₇×C₈	—	2	2	no	—
247	C₇×C₈	C₄×D₄	1	4	yes	—
248	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
249	C₂³×C₄	—	2	2	no	—
250	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
251	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
252	C₂²×C₈	—	2	2	no	—
253	C₂²×C₈	—	2	2	no	—
254	C₂²×C₈	—	2	2	no	—
255	C₂²×C₈	—	2	2	no	—
256	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
257	C₂²×C₈	C₂×C₄×C₇	1	4	yes	complemented
258	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
262	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
263	C₄×D₄	C₇×Q₈	1	4	yes	—
264	#117	—	2	4	no	—
265	#117	—	4	2	no	—
266	C₂×C₄⋊C₄	—	2	4	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	4	2	no	—
269	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
270	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
271	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
278	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
279	C₂×C₄²	—	2	2	no	—
280	C₂×C₄²	—	2	2	no	—
281	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
282	#107	—	2	4	no	—
283	#107	—	4	2	no	—
284	#106	—	2	2	no	—
285	#106	—	2	2	no	—
286	#106	—	2	2	no	—
287	#106	—	2	4	no	—
288	#106	C₇×D₄	1	4	yes	complemented
289	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
290	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	—
291	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
292	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	—	2	2	no	—
297	#95	—	2	2	no	—
298	C₂²×C₇	—	2	4	no	—
299	C₂²×C₇	—	2	2	no	—
300	C₂²×C₇	—	2	2	no	—
301	C₂²×C₇	—	2	4	no	—
302	C₂²×C₇	—	2	2	no	—
303	C₂²×C₇	—	2	2	no	—
304	C₂²×C₇	—	2	2	no	—
305	C₂²×C₇	—	4	2	no	—
306	C₂²×C₇	—	4	2	no	—
307	C₂²×C₇	#427	1	4	yes	complemented
308	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
309	C₂²×C₇	D₄²	1	1	yes	characteristic
310	C₂²×C₇	#547	1	1	yes	characteristic
311	C₂²×C₇	C₂×#135	1	2	yes	—
312	C₂²×C₇	#577	1	2	yes	—
313	C₄×C₇	—	2	2	no	—
314	C₄×C₇	—	2	2	no	—
315	C₄×C₇	—	2	2	no	—
316	C₄×C₇	—	2	2	no	—
317	C₄×C₇	—	4	4	no	—
318	C₄×C₇	—	8	2	no	—
319	C₄×C₇	—	8	2	no	—
320	C₄×C₇	—	8	2	no	—
321	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
322	C₄×C₇	#520	1	1	yes	characteristic
323	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
324	C₄×C₇	#554	1	1	yes	characteristic
325	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
326	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	C₇×C₄⋊C₄	1	4	yes	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	4	4	no	—
342	C₂²×C₄	—	4	4	no	—
343	C₂²×C₄	—	8	2	no	—
344	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
345	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
346	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
347	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
348	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
349	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
350	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
351	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	8	no	—
360	C₂×C₈	—	2	8	no	—
361	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
362	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
363	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
364	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
365	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
366	C₂×C₈	C₇×#41	1	2	yes	—
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	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	4	4	no	—
375	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
376	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₂²⋊C₄	—	4	4	no	—
379	C₂²⋊C₄	—	8	2	no	—
380	C₂²⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
381	C₂²⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
382	C₄²	—	4	4	no	—
383	C₄²	—	8	2	no	—
384	C₄²	C₂²×C₄×C₇	1	2	yes	—
385	C₄²	C₂×C₇×Q₈	1	2	yes	—
386	C₂×C₇	—	2	4	no	—
387	C₂×C₇	—	4	2	no	—
388	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
389	C₂×C₇	C₂×#427	1	1	yes	characteristic
390	C₂×C₇	#2912	1	1	yes	characteristic
391	C₂×C₇	#3282	1	2	yes	—
392	C₂×C₇	#3337	1	2	yes	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
401	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
402	C₂³	C₇×#135	1	4	yes	—
403	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
404	D₄	—	2	8	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	8	8	no	—
432	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
433	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
434	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
435	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
436	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
437	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
438	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
439	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
440	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
441	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
442	C₈	—	2	2	no	—
443	C₈	—	2	2	no	—
444	C₈	—	2	2	no	—
445	C₈	C₄×C₇×D₄	1	4	yes	—
446	C₇	#22654	1	1	yes	characteristic, direct product factor, Sylow
447	C₂²	—	2	2	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	C₇×#427	1	4	yes	complemented
457	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
458	C₂²	C₇×D₄²	1	1	yes	characteristic
459	C₂²	C₇×#547	1	1	yes	characteristic
460	C₂²	C₂×C₇×#135	1	2	yes	—
461	C₂²	C₇×#577	1	2	yes	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	4	no	—
467	C₄	—	8	2	no	—
468	C₄	—	8	2	no	—
469	C₄	—	8	2	no	—
470	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
471	C₄	C₇×#520	1	1	yes	characteristic
472	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
473	C₄	C₇×#554	1	1	yes	characteristic
474	C₂	—	2	4	no	—
475	C₂	—	4	2	no	—
476	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
477	C₂	C₂×C₇×#427	1	1	yes	characteristic
478	C₂	C₇×#2912	1	1	yes	characteristic
479	C₂	C₇×#3282	1	2	yes	—
480	C₂	C₇×#3337	1	2	yes	—

	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×#106	C₂	semidirect
5. C₇×#1949	C₂	semidirect
6. C₇×#1854	C₂	semidirect
7. C₇×#1852	C₂	semidirect
8. C₇×#1846	C₂	semidirect
9. C₇×#1807	C₂	semidirect
10. C₇×#1807	C₂	semidirect
11. C₄×C₇×#106	C₂	semidirect
17. C₇×C₈×D₄	C₄	semidirect
42. #22654	C₇	direct (N × H)
52. C₂²×C₇×C₈	C₂×C₄	semidirect
83. C₇×#106	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×#106	C₂×C₇	semidirect
97. #1949	C₂×C₇	semidirect
98. #1854	C₂×C₇	semidirect
99. #1852	C₂×C₇	semidirect
100. #1846	C₂×C₇	semidirect
101. #1807	C₂×C₇	semidirect
102. #1807	C₂×C₇	semidirect
103. C₄×#106	C₂×C₇	semidirect
170. C₈×D₄	C₄×C₇	semidirect
257. C₂²×C₈	C₂×C₄×C₇	semidirect
288. #106	C₇×D₄	semidirect
289. C₄⋊C₈	C₂×C₄×C₇	semidirect
291. C₂²⋊C₈	C₂×C₄×C₇	semidirect
292. C₄×C₈	C₂×C₄×C₇	semidirect
307. C₂²×C₇	#427	semidirect
446. C₇	#22654	direct (N × H)
456. 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×#106	2
C₇×#2352	2
C₇×#1949	2
C₇×#1854	2
C₇×#1852	2
C₇×#1846	2
C₇×#1807	2
C₄×C₇×#106	2
#22654	7

Subgroups — C7×#22654 #13199870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22654 #13199870