Subgroups — C₇×#22541 #13199757

Order 1792.

1074 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1983	C₂	1	2	yes	maximal
5	C₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1958	C₂	1	2	yes	maximal
7	C₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1805	C₂	1	2	yes	maximal, complemented
9	C₇×#1802	C₂	1	1	yes	maximal, characteristic
10	C₄×C₇×#103	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×#102	C₂	1	1	yes	maximal, characteristic
12	C₇×#556	—	2	4	no	—
13	C₇×#556	C₄	1	8	yes	complemented
14	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×#126	—	2	2	no	—
16	C₂×C₇×#124	C₂²	1	2	yes	—
17	C₂×C₇×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₇×#117	—	2	1	no	—
22	C₇×C₈×Q₈	—	2	4	no	—
23	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₇×#103	C₂²	1	2	yes	—
26	C₂×C₇×#103	C₂²	1	2	yes	—
27	C₂×C₇×#103	C₂²	1	2	yes	—
28	C₂×C₇×#103	C₂²	1	2	yes	—
29	C₂×C₇×#102	C₂²	1	1	yes	characteristic
30	C₂×C₇×#102	C₂²	1	1	yes	characteristic
31	C₂×C₇×#102	C₂²	1	2	yes	—
32	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
34	C₇×#391	—	2	1	no	—
35	C₇×#387	—	2	2	no	—
36	C₇×#384	C₂²	1	1	yes	characteristic
37	C₇×#382	C₂²	1	2	yes	—
38	C₄×C₇×C₄⋊C₄	—	2	1	no	—
39	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	1	yes	characteristic
42	C₇×#340	C₂²	1	1	yes	characteristic
43	C₇×#340	C₂²	1	2	yes	—
44	#22541	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
46	C₂²×C₇×Q₈	C₂³	1	2	yes	—
47	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₇	—	2	1	no	—
49	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₇×C₈	C₂³	1	2	yes	—
51	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₇×C₄⋊Q₈	—	2	2	no	—
53	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
54	C₇×#126	—	2	16	no	—
55	C₇×#124	—	2	4	no	—
56	C₇×#124	C₂×C₄	1	8	yes	—
57	C₇×C₂²⋊Q₈	—	2	8	no	—
58	C₄×C₇×Q₈	—	2	4	no	—
59	C₄×C₇×Q₈	C₂×C₄	1	8	yes	—
60	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
61	C₄×C₇×Q₈	D₄	1	4	yes	—
62	C₄×C₇×D₄	—	2	2	no	—
63	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
64	C₇×#117	—	2	8	no	—
65	C₂×C₇×C₄⋊C₄	—	2	1	no	—
66	C₂×C₇×C₄⋊C₄	—	2	2	no	—
67	C₂×C₇×C₄⋊C₄	—	2	1	no	—
68	C₂×C₇×C₄⋊C₄	—	2	1	no	—
69	C₂×C₇×C₄⋊C₄	—	2	1	no	—
70	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
72	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
73	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
74	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
77	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
78	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
79	C₂×C₄²×C₇	—	2	2	no	—
80	C₂×C₄²×C₇	—	2	1	no	—
81	C₂×C₄²×C₇	—	2	2	no	—
82	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₇	C₂³	1	2	yes	—
84	C₇×C₄⋊C₈	—	2	2	no	—
85	C₇×C₄⋊C₈	—	2	4	no	—
86	C₇×#103	—	2	4	no	—
87	C₇×#103	—	2	4	no	—
88	C₇×#103	—	4	2	no	—
89	C₇×#103	—	4	2	no	—
90	C₇×#102	—	2	2	no	—
91	C₇×#102	—	2	2	no	—
92	C₇×#102	—	4	2	no	—
93	C₄×C₇×C₈	—	2	2	no	—
94	C₄×C₇×C₈	—	2	4	no	—
95	C₇×#95	—	2	2	no	—
96	C₇×#95	—	2	1	no	—
97	C₇×#95	—	2	2	no	—
98	C₇×#95	—	2	2	no	—
99	C₂×#556	C₂×C₇	1	1	yes	characteristic
100	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
101	#2355	C₂×C₇	1	1	yes	characteristic, complemented
102	#1983	C₂×C₇	1	2	yes	—
103	#1981	C₂×C₇	1	1	yes	characteristic, complemented
104	#1958	C₂×C₇	1	2	yes	—
105	#1956	C₂×C₇	1	1	yes	characteristic, complemented
106	#1805	C₂×C₇	1	2	yes	complemented
107	#1802	C₂×C₇	1	1	yes	characteristic
108	C₄×#103	C₂×C₇	1	2	yes	complemented
109	C₄×#102	C₂×C₇	1	1	yes	characteristic
110	C₂⁴×C₇	—	2	1	no	—
111	C₂×C₇×Q₈	—	2	2	no	—
112	C₂×C₇×Q₈	—	2	2	no	—
113	C₂×C₇×Q₈	—	2	2	no	—
114	C₂×C₇×Q₈	—	2	2	no	—
115	C₂×C₇×Q₈	—	2	2	no	—
116	C₂×C₇×Q₈	—	2	2	no	—
117	C₂×C₇×Q₈	—	2	2	no	—
118	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
119	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
120	C₂×C₇×D₄	—	2	1	no	—
121	C₂×C₇×D₄	—	2	1	no	—
122	C₂×C₇×D₄	—	2	1	no	—
123	C₂×C₇×D₄	—	2	1	no	—
124	C₂×C₇×D₄	—	2	1	no	—
125	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₇	—	2	1	no	—
127	C₂²×C₄×C₇	—	2	1	no	—
128	C₂²×C₄×C₇	—	2	2	no	—
129	C₂²×C₄×C₇	—	2	4	no	—
130	C₂²×C₄×C₇	—	2	1	no	—
131	C₂²×C₄×C₇	—	2	2	no	—
132	C₂²×C₄×C₇	—	4	2	no	—
133	C₂²×C₄×C₇	—	4	2	no	—
134	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
139	C₂×C₇×C₈	—	2	1	no	—
140	C₂×C₇×C₈	—	2	1	no	—
141	C₂×C₇×C₈	—	2	1	no	—
142	C₂×C₇×C₈	—	2	2	no	—
143	C₂×C₇×C₈	—	2	1	no	—
144	C₂×C₇×C₈	—	2	2	no	—
145	C₂×C₇×C₈	—	2	2	no	—
146	C₂×C₇×C₈	—	2	1	no	—
147	C₂×C₇×C₈	#41	1	2	yes	—
148	C₇×C₄⋊C₄	—	2	4	no	—
149	C₇×C₄⋊C₄	—	2	8	no	—
150	C₇×C₄⋊C₄	—	2	4	no	—
151	C₇×C₄⋊C₄	—	2	1	no	—
152	C₇×C₄⋊C₄	—	2	1	no	—
153	C₇×C₄⋊C₄	—	2	1	no	—
154	C₇×C₄⋊C₄	—	2	2	no	—
155	C₇×C₄⋊C₄	—	2	2	no	—
156	C₇×C₄⋊C₄	—	2	1	no	—
157	C₇×C₄⋊C₄	—	4	2	no	—
158	C₇×C₄⋊C₄	—	8	1	no	—
159	C₇×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
160	C₇×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
161	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
162	C₇×C₄⋊C₄	#41	1	2	yes	—
163	C₇×C₂²⋊C₄	—	2	8	no	—
164	C₇×C₂²⋊C₄	—	2	4	no	—
165	C₇×C₂²⋊C₄	—	2	8	no	—
166	C₄²×C₇	—	2	8	no	—
167	C₄²×C₇	—	4	2	no	—
168	C₄²×C₇	—	8	2	no	—
169	C₄²×C₇	C₂²×C₄	1	2	yes	—
170	C₄²×C₇	C₂²×C₄	1	4	yes	—
171	C₄²×C₇	C₂×D₄	1	2	yes	—
172	C₄²×C₇	C₂×D₄	1	4	yes	—
173	#556	—	2	4	no	—
174	#556	C₄×C₇	1	8	yes	complemented
175	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
176	C₂×#126	—	2	2	no	—
177	C₂×#124	C₂²×C₇	1	2	yes	—
178	C₂×C₂²⋊Q₈	—	2	1	no	—
179	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
180	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
181	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
182	C₂×#117	—	2	1	no	—
183	C₈×Q₈	—	2	4	no	—
184	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
185	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
186	C₂×#103	C₂²×C₇	1	2	yes	—
187	C₂×#103	C₂²×C₇	1	2	yes	—
188	C₂×#103	C₂²×C₇	1	2	yes	—
189	C₂×#103	C₂²×C₇	1	2	yes	—
190	C₂×#102	C₂²×C₇	1	1	yes	characteristic
191	C₂×#102	C₂²×C₇	1	1	yes	characteristic
192	C₂×#102	C₂²×C₇	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
195	#391	—	2	1	no	—
196	#387	—	2	2	no	—
197	#384	C₂²×C₇	1	1	yes	characteristic
198	#382	C₂²×C₇	1	2	yes	—
199	C₄×C₄⋊C₄	—	2	1	no	—
200	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
201	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
202	#340	C₂²×C₇	1	2	yes	—
203	#340	C₂²×C₇	1	1	yes	characteristic
204	#340	C₂²×C₇	1	1	yes	characteristic
205	C₂³×C₇	—	2	2	no	—
206	C₂³×C₇	—	4	1	no	—
207	C₂³×C₇	—	4	1	no	—
208	C₂³×C₇	—	4	1	no	—
209	C₂³×C₇	—	4	1	no	—
210	C₂³×C₇	—	4	1	no	—
211	C₂³×C₇	—	4	1	no	—
212	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
213	C₇×Q₈	—	2	8	no	—
214	C₇×Q₈	—	2	4	no	—
215	C₇×Q₈	—	2	4	no	—
216	C₇×Q₈	—	4	2	no	—
217	C₇×Q₈	—	4	2	no	—
218	C₇×D₄	—	2	2	no	—
219	C₇×D₄	—	2	2	no	—
220	C₇×D₄	—	4	1	no	—
221	C₇×D₄	—	4	1	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	1	no	—
226	C₂×C₄×C₇	—	2	2	no	—
227	C₂×C₄×C₇	—	2	1	no	—
228	C₂×C₄×C₇	—	2	2	no	—
229	C₂×C₄×C₇	—	2	4	no	—
230	C₂×C₄×C₇	—	2	2	no	—
231	C₂×C₄×C₇	—	4	2	no	—
232	C₂×C₄×C₇	—	4	2	no	—
233	C₂×C₄×C₇	—	4	1	no	—
234	C₂×C₄×C₇	—	4	1	no	—
235	C₂×C₄×C₇	—	4	1	no	—
236	C₂×C₄×C₇	—	4	1	no	—
237	C₂×C₄×C₇	—	4	8	no	—
238	C₂×C₄×C₇	—	4	1	no	—
239	C₂×C₄×C₇	—	4	2	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	2	no	—
242	C₂×C₄×C₇	—	4	2	no	—
243	C₂×C₄×C₇	—	4	1	no	—
244	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
245	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
246	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
247	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄×C₇	#135	1	2	yes	—
249	C₂×C₄×C₇	#135	1	2	yes	—
250	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
255	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
256	C₇×C₈	—	2	2	no	—
257	C₇×C₈	—	4	1	no	—
258	C₇×C₈	—	4	2	no	—
259	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
260	C₂²×Q₈	C₂³×C₇	1	2	yes	—
261	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
262	C₂³×C₄	—	2	1	no	—
263	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₇	1	2	yes	—
265	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
266	C₄⋊Q₈	—	2	2	no	—
267	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	—
268	#126	—	2	16	no	—
269	#124	—	2	4	no	—
270	#124	C₂×C₄×C₇	1	8	yes	—
271	C₂²⋊Q₈	—	2	8	no	—
272	C₄×Q₈	—	2	4	no	—
273	C₄×Q₈	C₂×C₄×C₇	1	8	yes	—
274	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
275	C₄×Q₈	C₇×D₄	1	4	yes	—
276	C₄×D₄	—	2	2	no	—
277	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
278	#117	—	2	8	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	—	2	2	no	—
296	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
297	C₂×C₄²	C₂³×C₇	1	2	yes	—
298	C₄⋊C₈	—	2	2	no	—
299	C₄⋊C₈	—	2	4	no	—
300	#103	—	2	4	no	—
301	#103	—	2	4	no	—
302	#103	—	4	2	no	—
303	#103	—	4	2	no	—
304	#102	—	2	2	no	—
305	#102	—	2	2	no	—
306	#102	—	4	2	no	—
307	C₄×C₈	—	2	2	no	—
308	C₄×C₈	—	2	4	no	—
309	#95	—	2	2	no	—
310	#95	—	2	1	no	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	C₂²×C₇	—	4	2	no	—
314	C₂²×C₇	—	4	2	no	—
315	C₂²×C₇	—	4	2	no	—
316	C₂²×C₇	—	8	1	no	—
317	C₂²×C₇	—	8	1	no	—
318	C₂²×C₇	—	8	1	no	—
319	C₂²×C₇	—	8	1	no	—
320	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
321	C₂²×C₇	#548	1	1	yes	characteristic
322	C₂²×C₇	#550	1	1	yes	characteristic
323	C₂²×C₇	C₂×#135	1	1	yes	characteristic
324	C₂²×C₇	C₂×#135	1	1	yes	characteristic
325	C₂²×C₇	#577	1	1	yes	characteristic
326	C₂²×C₇	#578	1	1	yes	characteristic
327	C₄×C₇	—	2	4	no	—
328	C₄×C₇	—	2	4	no	—
329	C₄×C₇	—	2	2	no	—
330	C₄×C₇	—	4	2	no	—
331	C₄×C₇	—	4	4	no	—
332	C₄×C₇	—	4	2	no	—
333	C₄×C₇	—	8	1	no	—
334	C₄×C₇	—	8	2	no	—
335	C₄×C₇	#416	1	4	yes	—
336	C₄×C₇	C₄×#41	1	1	yes	characteristic
337	C₄×C₇	#520	1	1	yes	characteristic
338	C₄×C₇	#548	1	1	yes	characteristic
339	C₄×C₇	#550	1	1	yes	characteristic
340	C₂⁴	—	2	1	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
349	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	4	2	no	—
363	C₂²×C₄	—	4	2	no	—
364	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
365	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
366	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
367	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	C₇×#41	1	2	yes	—
378	C₄⋊C₄	—	2	4	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	8	1	no	—
389	C₄⋊C₄	C₇×C₂²⋊C₄	1	4	yes	—
390	C₄⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
391	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
392	C₄⋊C₄	C₇×#41	1	2	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	2	8	no	—
396	C₄²	—	2	8	no	—
397	C₄²	—	4	2	no	—
398	C₄²	—	8	2	no	—
399	C₄²	C₂²×C₄×C₇	1	2	yes	—
400	C₄²	C₂²×C₄×C₇	1	4	yes	—
401	C₄²	C₂×C₇×D₄	1	2	yes	—
402	C₄²	C₂×C₇×D₄	1	4	yes	—
403	C₂×C₇	—	8	2	no	—
404	C₂×C₇	#2341	1	1	yes	characteristic
405	C₂×C₇	C₂×#416	1	1	yes	characteristic
406	C₂×C₇	#2898	1	1	yes	characteristic
407	C₂×C₇	#3360	1	1	yes	characteristic
408	C₂×C₇	#3364	1	1	yes	characteristic
409	C₂×C₇	#3393	1	1	yes	characteristic
410	C₂×C₇	#3394	1	1	yes	characteristic
411	C₂³	—	2	2	no	—
412	C₂³	—	4	1	no	—
413	C₂³	—	4	1	no	—
414	C₂³	—	4	1	no	—
415	C₂³	—	4	1	no	—
416	C₂³	—	4	1	no	—
417	C₂³	—	4	1	no	—
418	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
419	Q₈	—	2	8	no	—
420	Q₈	—	2	4	no	—
421	Q₈	—	2	4	no	—
422	Q₈	—	4	2	no	—
423	Q₈	—	4	2	no	—
424	D₄	—	2	2	no	—
425	D₄	—	2	2	no	—
426	D₄	—	4	1	no	—
427	D₄	—	4	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	8	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
451	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
452	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
453	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
454	C₂×C₄	C₇×#135	1	2	yes	—
455	C₂×C₄	C₇×#135	1	2	yes	—
456	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
457	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
458	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
459	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
460	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
461	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
462	C₈	—	2	2	no	—
463	C₈	—	4	1	no	—
464	C₈	—	4	2	no	—
465	C₇	#22541	1	1	yes	characteristic, direct product factor, Sylow
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	8	1	no	—
470	C₂²	—	8	1	no	—
471	C₂²	—	8	1	no	—
472	C₂²	—	8	1	no	—
473	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
474	C₂²	C₇×#548	1	1	yes	characteristic
475	C₂²	C₇×#550	1	1	yes	characteristic
476	C₂²	C₂×C₇×#135	1	1	yes	characteristic
477	C₂²	C₂×C₇×#135	1	1	yes	characteristic
478	C₂²	C₇×#577	1	1	yes	characteristic
479	C₂²	C₇×#578	1	1	yes	characteristic
480	C₄	—	2	4	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	1	no	—
487	C₄	—	8	2	no	—
488	C₄	C₇×#416	1	4	yes	—
489	C₄	C₄×C₇×#41	1	1	yes	characteristic
490	C₄	C₇×#520	1	1	yes	characteristic
491	C₄	C₇×#548	1	1	yes	characteristic
492	C₄	C₇×#550	1	1	yes	characteristic
493	C₂	—	8	2	no	—
494	C₂	C₇×#2341	1	1	yes	characteristic
495	C₂	C₂×C₇×#416	1	1	yes	characteristic
496	C₂	C₇×#2898	1	1	yes	characteristic
497	C₂	C₇×#3360	1	1	yes	characteristic
498	C₂	C₇×#3364	1	1	yes	characteristic
499	C₂	C₇×#3393	1	1	yes	characteristic
500	C₂	C₇×#3394	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	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₈×Q₈	C₂	semidirect
3. C₇×#2355	C₂	semidirect
5. C₇×#1981	C₂	semidirect
7. C₇×#1956	C₂	semidirect
8. C₇×#1805	C₂	semidirect
10. C₄×C₇×#103	C₂	semidirect
13. C₇×#556	C₄	semidirect
44. #22541	C₇	direct (N × H)
100. C₂×C₈×Q₈	C₂×C₇	semidirect
101. #2355	C₂×C₇	semidirect
103. #1981	C₂×C₇	semidirect
105. #1956	C₂×C₇	semidirect
106. #1805	C₂×C₇	semidirect
108. C₄×#103	C₂×C₇	semidirect
174. #556	C₄×C₇	semidirect
465. C₇	#22541	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#556	2
C₂×C₇×C₈×Q₈	2
C₇×#2355	2
C₇×#1983	2
C₇×#1981	2
C₇×#1958	2
C₇×#1956	2
C₇×#1805	2
C₇×#1802	2
C₄×C₇×#103	2
C₄×C₇×#102	2
#22541	7

Subgroups — C7×#22541 #13199757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22541 #13199757