Subgroups — C₇×#22543 #13199759

Order 1792.

1138 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#559	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, complemented
5	C₇×#1982	C₂	1	1	yes	maximal, characteristic
6	C₇×#1958	C₂	1	2	yes	maximal, complemented
7	C₇×#1957	C₂	1	1	yes	maximal, characteristic
8	C₇×#1804	C₂	1	2	yes	maximal, complemented
9	C₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#102	C₂	1	2	yes	maximal, complemented
12	C₇×#559	—	2	4	no	—
13	C₇×#559	C₄	1	8	yes	complemented
14	C₂×C₇×#127	C₂²	1	1	yes	characteristic
15	C₂×C₇×#124	—	2	1	no	—
16	C₂×C₇×#124	C₂²	1	2	yes	—
17	C₂×C₇×#123	—	2	2	no	—
18	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×D₄	—	2	1	no	—
21	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
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	1	yes	characteristic
26	C₂×C₇×#103	C₂²	1	1	yes	characteristic
27	C₂×C₇×#103	C₂²	1	2	yes	—
28	C₂×C₇×#102	C₂²	1	2	yes	—
29	C₂×C₇×#102	C₂²	1	2	yes	—
30	C₂×C₇×#102	C₂²	1	2	yes	—
31	C₂×C₇×#102	C₂²	1	2	yes	—
32	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
33	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
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	2	yes	—
43	C₇×#340	C₂²	1	1	yes	characteristic
44	#22543	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	1	yes	characteristic
47	C₂²×C₇×D₄	—	2	1	no	—
48	C₂²×C₇×D₄	—	2	1	no	—
49	C₂²×C₇×D₄	C₂³	1	2	yes	—
50	C₂³×C₄×C₇	—	2	2	no	—
51	C₂²×C₇×C₈	C₂³	1	2	yes	—
52	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₇×#127	—	2	2	no	—
55	C₇×#127	C₂×C₄	1	4	yes	—
56	C₇×#124	—	2	8	no	—
57	C₇×#124	—	2	4	no	—
58	C₇×#124	C₂×C₄	1	8	yes	—
59	C₇×#123	—	2	16	no	—
60	C₄×C₇×Q₈	—	2	2	no	—
61	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
62	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
63	C₄×C₇×Q₈	D₄	1	4	yes	—
64	C₄×C₇×D₄	—	2	8	no	—
65	C₄×C₇×D₄	—	2	4	no	—
66	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
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	2	no	—
78	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
79	C₂×C₄²×C₇	—	2	1	no	—
80	C₂×C₄²×C₇	—	2	1	no	—
81	C₂×C₄²×C₇	—	2	2	no	—
82	C₂×C₄²×C₇	C₂³	1	2	yes	—
83	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
84	C₇×C₄⋊C₈	—	2	4	no	—
85	C₇×C₄⋊C₈	—	2	2	no	—
86	C₇×#103	—	2	2	no	—
87	C₇×#103	—	2	2	no	—
88	C₇×#103	—	4	2	no	—
89	C₇×#102	—	2	4	no	—
90	C₇×#102	—	2	4	no	—
91	C₇×#102	—	4	2	no	—
92	C₇×#102	—	4	2	no	—
93	C₄×C₇×C₈	—	2	4	no	—
94	C₄×C₇×C₈	—	2	2	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×#559	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	complemented
103	#1982	C₂×C₇	1	1	yes	characteristic
104	#1958	C₂×C₇	1	2	yes	complemented
105	#1957	C₂×C₇	1	1	yes	characteristic
106	#1804	C₂×C₇	1	2	yes	complemented
107	#1803	C₂×C₇	1	1	yes	characteristic, complemented
108	C₄×#103	C₂×C₇	1	1	yes	characteristic, complemented
109	C₄×#102	C₂×C₇	1	2	yes	complemented
110	C₂⁴×C₇	—	2	2	no	—
111	C₂×C₇×Q₈	—	2	2	no	—
112	C₂×C₇×Q₈	—	2	2	no	—
113	C₂×C₇×Q₈	—	2	1	no	—
114	C₂×C₇×Q₈	—	2	1	no	—
115	C₂×C₇×Q₈	—	2	1	no	—
116	C₂×C₇×Q₈	—	2	1	no	—
117	C₂×C₇×Q₈	—	2	1	no	—
118	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
119	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
120	C₂×C₇×D₄	—	2	4	no	—
121	C₂×C₇×D₄	—	2	4	no	—
122	C₂×C₇×D₄	—	2	2	no	—
123	C₂×C₇×D₄	—	2	2	no	—
124	C₂×C₇×D₄	—	2	2	no	—
125	C₂×C₇×D₄	—	2	2	no	—
126	C₂×C₇×D₄	—	2	2	no	—
127	C₂×C₇×D₄	—	4	2	no	—
128	C₂×C₇×D₄	—	4	2	no	—
129	C₂×C₇×D₄	—	8	1	no	—
130	C₂×C₇×D₄	—	8	1	no	—
131	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
132	C₂²×C₄×C₇	—	2	1	no	—
133	C₂²×C₄×C₇	—	2	1	no	—
134	C₂²×C₄×C₇	—	2	1	no	—
135	C₂²×C₄×C₇	—	2	8	no	—
136	C₂²×C₄×C₇	—	2	1	no	—
137	C₂²×C₄×C₇	—	2	2	no	—
138	C₂²×C₄×C₇	—	8	2	no	—
139	C₂²×C₄×C₇	—	8	2	no	—
140	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
145	C₂×C₇×C₈	—	2	1	no	—
146	C₂×C₇×C₈	—	2	1	no	—
147	C₂×C₇×C₈	—	2	2	no	—
148	C₂×C₇×C₈	—	2	1	no	—
149	C₂×C₇×C₈	—	2	2	no	—
150	C₂×C₇×C₈	—	2	1	no	—
151	C₂×C₇×C₈	—	2	1	no	—
152	C₂×C₇×C₈	—	2	2	no	—
153	C₂×C₇×C₈	#41	1	2	yes	—
154	C₇×C₄⋊C₄	—	2	4	no	—
155	C₇×C₄⋊C₄	—	2	1	no	—
156	C₇×C₄⋊C₄	—	2	1	no	—
157	C₇×C₄⋊C₄	—	2	2	no	—
158	C₇×C₄⋊C₄	—	2	1	no	—
159	C₇×C₄⋊C₄	—	2	1	no	—
160	C₇×C₄⋊C₄	—	2	2	no	—
161	C₇×C₄⋊C₄	—	4	2	no	—
162	C₇×C₄⋊C₄	—	8	1	no	—
163	C₇×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
164	C₇×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
165	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
166	C₇×C₄⋊C₄	#41	1	2	yes	—
167	C₇×C₂²⋊C₄	—	2	8	no	—
168	C₇×C₂²⋊C₄	—	2	8	no	—
169	C₇×C₂²⋊C₄	—	2	8	no	—
170	C₄²×C₇	—	2	4	no	—
171	C₄²×C₇	—	4	2	no	—
172	C₄²×C₇	—	8	2	no	—
173	C₄²×C₇	C₂²×C₄	1	4	yes	—
174	C₄²×C₇	C₂²×C₄	1	2	yes	—
175	C₄²×C₇	C₂×D₄	1	4	yes	—
176	C₄²×C₇	C₂×D₄	1	2	yes	—
177	#559	—	2	4	no	—
178	#559	C₄×C₇	1	8	yes	complemented
179	C₂×#127	C₂²×C₇	1	1	yes	characteristic
180	C₂×#124	—	2	1	no	—
181	C₂×#124	C₂²×C₇	1	2	yes	—
182	C₂×#123	—	2	2	no	—
183	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
184	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
185	C₂×C₄×D₄	—	2	1	no	—
186	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
187	C₈×Q₈	—	2	4	no	—
188	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
190	C₂×#103	C₂²×C₇	1	1	yes	characteristic
191	C₂×#103	C₂²×C₇	1	1	yes	characteristic
192	C₂×#103	C₂²×C₇	1	2	yes	—
193	C₂×#102	C₂²×C₇	1	2	yes	—
194	C₂×#102	C₂²×C₇	1	2	yes	—
195	C₂×#102	C₂²×C₇	1	2	yes	—
196	C₂×#102	C₂²×C₇	1	2	yes	—
197	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
199	#391	—	2	1	no	—
200	#387	—	2	2	no	—
201	#384	C₂²×C₇	1	1	yes	characteristic
202	#382	C₂²×C₇	1	2	yes	—
203	C₄×C₄⋊C₄	—	2	1	no	—
204	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
205	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
206	#340	C₂²×C₇	1	2	yes	—
207	#340	C₂²×C₇	1	1	yes	characteristic
208	#340	C₂²×C₇	1	1	yes	characteristic
209	C₂³×C₇	—	2	4	no	—
210	C₂³×C₇	—	4	2	no	—
211	C₂³×C₇	—	4	2	no	—
212	C₂³×C₇	—	4	2	no	—
213	C₂³×C₇	—	4	2	no	—
214	C₂³×C₇	—	4	2	no	—
215	C₂³×C₇	—	4	2	no	—
216	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
217	C₇×Q₈	—	2	8	no	—
218	C₇×Q₈	—	2	2	no	—
219	C₇×Q₈	—	2	2	no	—
220	C₇×Q₈	—	4	1	no	—
221	C₇×Q₈	—	4	1	no	—
222	C₇×D₄	—	2	4	no	—
223	C₇×D₄	—	2	4	no	—
224	C₇×D₄	—	4	2	no	—
225	C₇×D₄	—	4	2	no	—
226	C₇×D₄	—	4	8	no	—
227	C₇×D₄	—	8	4	no	—
228	C₂×C₄×C₇	—	2	2	no	—
229	C₂×C₄×C₇	—	2	2	no	—
230	C₂×C₄×C₇	—	2	2	no	—
231	C₂×C₄×C₇	—	2	1	no	—
232	C₂×C₄×C₇	—	2	2	no	—
233	C₂×C₄×C₇	—	2	1	no	—
234	C₂×C₄×C₇	—	2	2	no	—
235	C₂×C₄×C₇	—	2	2	no	—
236	C₂×C₄×C₇	—	2	2	no	—
237	C₂×C₄×C₇	—	4	1	no	—
238	C₂×C₄×C₇	—	4	1	no	—
239	C₂×C₄×C₇	—	4	1	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	1	no	—
242	C₂×C₄×C₇	—	4	1	no	—
243	C₂×C₄×C₇	—	4	1	no	—
244	C₂×C₄×C₇	—	4	2	no	—
245	C₂×C₄×C₇	—	4	1	no	—
246	C₂×C₄×C₇	—	4	2	no	—
247	C₂×C₄×C₇	—	4	2	no	—
248	C₂×C₄×C₇	—	4	1	no	—
249	C₂×C₄×C₇	—	8	8	no	—
250	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
251	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
252	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
253	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
254	C₂×C₄×C₇	#135	1	2	yes	—
255	C₂×C₄×C₇	#135	1	2	yes	—
256	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
261	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
262	C₇×C₈	—	2	2	no	—
263	C₇×C₈	—	4	2	no	—
264	C₇×C₈	—	4	1	no	—
265	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
266	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
267	C₂²×D₄	—	2	1	no	—
268	C₂²×D₄	—	2	1	no	—
269	C₂²×D₄	C₂³×C₇	1	2	yes	—
270	C₂³×C₄	—	2	2	no	—
271	C₂²×C₈	C₂³×C₇	1	2	yes	—
272	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
274	#127	—	2	2	no	—
275	#127	C₂×C₄×C₇	1	4	yes	—
276	#124	—	2	8	no	—
277	#124	—	2	4	no	—
278	#124	C₂×C₄×C₇	1	8	yes	—
279	#123	—	2	16	no	—
280	C₄×Q₈	—	2	2	no	—
281	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
282	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
283	C₄×Q₈	C₇×D₄	1	4	yes	—
284	C₄×D₄	—	2	8	no	—
285	C₄×D₄	—	2	4	no	—
286	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	—	2	2	no	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₄²	—	2	1	no	—
300	C₂×C₄²	—	2	1	no	—
301	C₂×C₄²	—	2	2	no	—
302	C₂×C₄²	C₂³×C₇	1	2	yes	—
303	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
304	C₄⋊C₈	—	2	4	no	—
305	C₄⋊C₈	—	2	2	no	—
306	#103	—	2	2	no	—
307	#103	—	2	2	no	—
308	#103	—	4	2	no	—
309	#102	—	2	4	no	—
310	#102	—	2	4	no	—
311	#102	—	4	2	no	—
312	#102	—	4	2	no	—
313	C₄×C₈	—	2	4	no	—
314	C₄×C₈	—	2	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	1	no	—
317	#95	—	2	2	no	—
318	#95	—	2	2	no	—
319	C₂²×C₇	—	4	4	no	—
320	C₂²×C₇	—	4	4	no	—
321	C₂²×C₇	—	4	4	no	—
322	C₂²×C₇	—	8	2	no	—
323	C₂²×C₇	—	8	2	no	—
324	C₂²×C₇	—	8	2	no	—
325	C₂²×C₇	—	8	2	no	—
326	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
327	C₂²×C₇	#548	1	1	yes	characteristic
328	C₂²×C₇	#550	1	1	yes	characteristic
329	C₂²×C₇	C₂×#135	1	1	yes	characteristic
330	C₂²×C₇	C₂×#135	1	1	yes	characteristic
331	C₂²×C₇	#576	1	1	yes	characteristic
332	C₂²×C₇	#577	1	1	yes	characteristic
333	C₄×C₇	—	2	4	no	—
334	C₄×C₇	—	2	4	no	—
335	C₄×C₇	—	2	2	no	—
336	C₄×C₇	—	4	2	no	—
337	C₄×C₇	—	4	2	no	—
338	C₄×C₇	—	4	2	no	—
339	C₄×C₇	—	8	1	no	—
340	C₄×C₇	—	8	2	no	—
341	C₄×C₇	#416	1	4	yes	—
342	C₄×C₇	C₄×#41	1	1	yes	characteristic
343	C₄×C₇	#520	1	1	yes	characteristic
344	C₄×C₇	#548	1	1	yes	characteristic
345	C₄×C₇	#550	1	1	yes	characteristic
346	C₂⁴	—	2	2	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×Q₈	—	2	1	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
355	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
356	C₂×D₄	—	2	4	no	—
357	C₂×D₄	—	2	4	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	4	2	no	—
364	C₂×D₄	—	4	2	no	—
365	C₂×D₄	—	8	1	no	—
366	C₂×D₄	—	8	1	no	—
367	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	8	2	no	—
375	C₂²×C₄	—	8	2	no	—
376	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
377	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
378	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
379	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	C₇×#41	1	2	yes	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	4	2	no	—
398	C₄⋊C₄	—	8	1	no	—
399	C₄⋊C₄	C₇×C₂²⋊C₄	1	4	yes	—
400	C₄⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
401	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
402	C₄⋊C₄	C₇×#41	1	2	yes	—
403	C₂²⋊C₄	—	2	8	no	—
404	C₂²⋊C₄	—	2	8	no	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₄²	—	2	4	no	—
407	C₄²	—	4	2	no	—
408	C₄²	—	8	2	no	—
409	C₄²	C₂²×C₄×C₇	1	4	yes	—
410	C₄²	C₂²×C₄×C₇	1	2	yes	—
411	C₄²	C₂×C₇×D₄	1	4	yes	—
412	C₄²	C₂×C₇×D₄	1	2	yes	—
413	C₂×C₇	—	8	4	no	—
414	C₂×C₇	#2341	1	1	yes	characteristic
415	C₂×C₇	C₂×#416	1	1	yes	characteristic
416	C₂×C₇	#2897	1	1	yes	characteristic
417	C₂×C₇	#3361	1	1	yes	characteristic
418	C₂×C₇	#3365	1	1	yes	characteristic
419	C₂×C₇	#3394	1	1	yes	characteristic
420	C₂×C₇	#3396	1	1	yes	characteristic
421	C₂³	—	2	4	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
429	Q₈	—	2	8	no	—
430	Q₈	—	2	2	no	—
431	Q₈	—	2	2	no	—
432	Q₈	—	4	1	no	—
433	Q₈	—	4	1	no	—
434	D₄	—	2	4	no	—
435	D₄	—	2	4	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	8	no	—
439	D₄	—	8	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	8	8	no	—
462	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
463	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
464	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
465	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
466	C₂×C₄	C₇×#135	1	2	yes	—
467	C₂×C₄	C₇×#135	1	2	yes	—
468	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
469	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
470	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
471	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
472	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
473	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	4	2	no	—
476	C₈	—	4	1	no	—
477	C₇	#22543	1	1	yes	characteristic, direct product factor, Sylow
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	8	2	no	—
482	C₂²	—	8	2	no	—
483	C₂²	—	8	2	no	—
484	C₂²	—	8	2	no	—
485	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
486	C₂²	C₇×#548	1	1	yes	characteristic
487	C₂²	C₇×#550	1	1	yes	characteristic
488	C₂²	C₂×C₇×#135	1	1	yes	characteristic
489	C₂²	C₂×C₇×#135	1	1	yes	characteristic
490	C₂²	C₇×#576	1	1	yes	characteristic
491	C₂²	C₇×#577	1	1	yes	characteristic
492	C₄	—	2	4	no	—
493	C₄	—	2	4	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	1	no	—
499	C₄	—	8	2	no	—
500	C₄	C₇×#416	1	4	yes	—
501	C₄	C₄×C₇×#41	1	1	yes	characteristic
502	C₄	C₇×#520	1	1	yes	characteristic
503	C₄	C₇×#548	1	1	yes	characteristic
504	C₄	C₇×#550	1	1	yes	characteristic
505	C₂	—	8	4	no	—
506	C₂	C₇×#2341	1	1	yes	characteristic
507	C₂	C₂×C₇×#416	1	1	yes	characteristic
508	C₂	C₇×#2897	1	1	yes	characteristic
509	C₂	C₇×#3361	1	1	yes	characteristic
510	C₂	C₇×#3365	1	1	yes	characteristic
511	C₂	C₇×#3394	1	1	yes	characteristic
512	C₂	C₇×#3396	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	C₂×C₇×#127	C₂²
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
4. C₇×#1983	C₂	semidirect
6. C₇×#1958	C₂	semidirect
8. C₇×#1804	C₂	semidirect
9. C₇×#1803	C₂	semidirect
10. C₄×C₇×#103	C₂	semidirect
11. C₄×C₇×#102	C₂	semidirect
13. C₇×#559	C₄	semidirect
44. #22543	C₇	direct (N × H)
100. C₂×C₈×Q₈	C₂×C₇	semidirect
101. #2355	C₂×C₇	semidirect
102. #1983	C₂×C₇	semidirect
104. #1958	C₂×C₇	semidirect
106. #1804	C₂×C₇	semidirect
107. #1803	C₂×C₇	semidirect
108. C₄×#103	C₂×C₇	semidirect
109. C₄×#102	C₂×C₇	semidirect
178. #559	C₄×C₇	semidirect
477. C₇	#22543	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×#559	2
C₂×C₇×C₈×Q₈	2
C₇×#2355	2
C₇×#1983	2
C₇×#1982	2
C₇×#1958	2
C₇×#1957	2
C₇×#1804	2
C₇×#1803	2
C₄×C₇×#103	2
C₄×C₇×#102	2
#22543	7

Subgroups — C7×#22543 #13199759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22543 #13199759