Subgroups — C₇×#22766 #13199982

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2354	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1934	C₂	1	2	yes	maximal, complemented
5	C₇×#1918	C₂	1	2	yes	maximal, complemented
6	C₇×#1879	C₂	1	2	yes	maximal
7	C₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#102	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
14	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×#117	—	2	1	no	—
17	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
18	C₄×C₇×D₈	C₄	1	8	yes	complemented
19	C₂×C₇×#107	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×#102	C₂²	1	2	yes	—
22	C₂×C₇×#102	C₂²	1	2	yes	—
23	C₂×C₇×#102	C₂²	1	1	yes	characteristic
24	C₂×C₇×#102	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
27	C₇×#390	C₂²	1	2	yes	—
28	C₇×#389	C₂²	1	2	yes	—
29	C₇×#388	—	2	2	no	—
30	C₇×#385	—	2	2	no	—
31	C₇×#384	C₂²	1	1	yes	characteristic
32	C₇×#384	C₂²	1	1	yes	characteristic
33	C₇×#380	—	2	1	no	—
34	C₇×#380	—	2	1	no	—
35	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
36	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₇×#340	C₂²	1	2	yes	—
38	C₇×#340	C₂²	1	2	yes	—
39	C₇×#340	C₂²	1	2	yes	—
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#336	C₂²	1	2	yes	—
42	#22766	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
44	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₇	—	2	1	no	—
46	C₂³×C₄×C₇	—	2	1	no	—
47	C₂³×C₄×C₇	—	2	2	no	—
48	C₂×C₇×D₈	—	2	2	no	—
49	C₂×C₇×D₈	—	2	2	no	—
50	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
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	C₄×C₇×D₄	—	2	4	no	—
55	C₄×C₇×D₄	—	2	2	no	—
56	C₄×C₇×D₄	—	2	2	no	—
57	C₄×C₇×D₄	—	4	2	no	—
58	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
59	C₄×C₇×D₄	C₂×C₄	1	4	yes	complemented
60	C₇×#117	—	4	4	no	—
61	C₂×C₇×C₄⋊C₄	—	2	1	no	—
62	C₂×C₇×C₄⋊C₄	—	2	2	no	—
63	C₂×C₇×C₄⋊C₄	—	2	1	no	—
64	C₂×C₇×C₄⋊C₄	—	2	1	no	—
65	C₂×C₇×C₄⋊C₄	—	2	1	no	—
66	C₂×C₇×C₄⋊C₄	—	4	4	no	—
67	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
69	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
70	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
71	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
73	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
74	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
75	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
76	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
77	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
78	C₂×C₄²×C₇	—	2	2	no	—
79	C₂×C₄²×C₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
81	C₇×#107	C₂×C₄	1	4	yes	complemented
82	C₇×C₄⋊C₈	—	2	2	no	—
83	C₇×#102	—	2	4	no	—
84	C₇×#102	—	4	2	no	—
85	C₇×#102	C₂×C₄	1	4	yes	—
86	C₇×#102	C₂×C₄	1	4	yes	complemented
87	C₄×C₇×C₈	—	4	2	no	—
88	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
89	C₇×#95	—	2	2	no	—
90	C₇×#95	—	2	2	no	—
91	C₇×#95	D₄	1	2	yes	—
92	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
93	#2354	C₂×C₇	1	1	yes	characteristic, complemented
94	D₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
95	#1934	C₂×C₇	1	2	yes	complemented
96	#1918	C₂×C₇	1	2	yes	complemented
97	#1879	C₂×C₇	1	2	yes	—
98	#1834	C₂×C₇	1	1	yes	characteristic, complemented
99	#1802	C₂×C₇	1	1	yes	characteristic, complemented
100	#1802	C₂×C₇	1	1	yes	characteristic, complemented
101	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
102	C₄×#102	C₂×C₇	1	2	yes	complemented
103	C₂⁴×C₇	—	2	1	no	—
104	C₂⁴×C₇	—	2	1	no	—
105	C₂×C₇×D₄	—	2	1	no	—
106	C₂×C₇×D₄	—	2	1	no	—
107	C₂×C₇×D₄	—	2	1	no	—
108	C₂×C₇×D₄	—	2	1	no	—
109	C₂×C₇×D₄	—	2	1	no	—
110	C₂×C₇×D₄	—	2	1	no	—
111	C₂×C₇×D₄	—	2	1	no	—
112	C₂×C₇×D₄	—	2	1	no	—
113	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
114	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
115	C₂×C₇×D₄	#41	1	2	yes	—
116	C₂×C₇×D₄	#41	1	2	yes	—
117	C₂²×C₄×C₇	—	2	1	no	—
118	C₂²×C₄×C₇	—	2	1	no	—
119	C₂²×C₄×C₇	—	2	4	no	—
120	C₂²×C₄×C₇	—	2	4	no	—
121	C₂²×C₄×C₇	—	2	2	no	—
122	C₂²×C₄×C₇	—	2	2	no	—
123	C₂²×C₄×C₇	—	4	4	no	—
124	C₂²×C₄×C₇	—	4	2	no	—
125	C₂²×C₄×C₇	—	4	2	no	—
126	C₂²×C₄×C₇	—	4	2	no	—
127	C₂²×C₄×C₇	—	4	2	no	—
128	C₂²×C₄×C₇	—	4	4	no	—
129	C₂²×C₄×C₇	—	8	2	no	—
130	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
135	C₇×D₈	—	2	8	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	1	no	—
140	C₂×C₇×C₈	—	2	2	no	—
141	C₂×C₇×C₈	—	2	2	no	—
142	C₂×C₇×C₈	—	2	2	no	—
143	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
144	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
145	C₇×C₄⋊C₄	—	2	2	no	—
146	C₇×C₄⋊C₄	—	2	1	no	—
147	C₇×C₄⋊C₄	—	2	1	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	—	4	4	no	—
153	C₇×C₄⋊C₄	—	4	2	no	—
154	C₇×C₄⋊C₄	—	4	2	no	—
155	C₇×C₄⋊C₄	—	8	1	no	—
156	C₇×C₄⋊C₄	—	8	1	no	—
157	C₇×C₄⋊C₄	D₈	1	4	yes	complemented
158	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
159	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
160	C₇×C₂²⋊C₄	—	2	4	no	—
161	C₇×C₂²⋊C₄	—	2	4	no	—
162	C₇×C₂²⋊C₄	—	4	4	no	—
163	C₇×C₂²⋊C₄	—	4	4	no	—
164	C₇×C₂²⋊C₄	—	8	2	no	—
165	C₄²×C₇	—	2	2	no	—
166	C₄²×C₇	—	2	2	no	—
167	C₄²×C₇	—	4	4	no	—
168	C₄²×C₇	C₂²×C₄	1	2	yes	—
169	C₄²×C₇	#41	1	2	yes	—
170	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
171	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
172	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
173	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
174	C₂×#117	—	2	1	no	—
175	C₂²×C₄⋊C₄	—	2	1	no	—
176	C₄×D₈	C₄×C₇	1	8	yes	complemented
177	C₂×#107	C₂²×C₇	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
179	C₂×#102	C₂²×C₇	1	2	yes	—
180	C₂×#102	C₂²×C₇	1	2	yes	—
181	C₂×#102	C₂²×C₇	1	1	yes	characteristic
182	C₂×#102	C₂²×C₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
185	#390	C₂²×C₇	1	2	yes	—
186	#389	C₂²×C₇	1	2	yes	—
187	#388	—	2	2	no	—
188	#385	—	2	2	no	—
189	#384	C₂²×C₇	1	1	yes	characteristic
190	#384	C₂²×C₇	1	1	yes	characteristic
191	#380	—	2	1	no	—
192	#380	—	2	1	no	—
193	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
194	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
195	#340	C₂²×C₇	1	2	yes	—
196	#340	C₂²×C₇	1	2	yes	—
197	#340	C₂²×C₇	1	2	yes	—
198	#340	C₂²×C₇	1	2	yes	—
199	#336	C₂²×C₇	1	2	yes	—
200	C₂³×C₇	—	2	2	no	—
201	C₂³×C₇	—	2	2	no	—
202	C₂³×C₇	—	2	2	no	—
203	C₂³×C₇	—	2	2	no	—
204	C₂³×C₇	—	4	1	no	—
205	C₂³×C₇	—	4	1	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	—	4	1	no	—
213	C₂³×C₇	—	4	1	no	—
214	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
215	C₇×D₄	—	2	2	no	—
216	C₇×D₄	—	2	2	no	—
217	C₇×D₄	—	2	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	C₇×D₄	#117	1	4	yes	complemented
223	C₂×C₄×C₇	—	2	2	no	—
224	C₂×C₄×C₇	—	2	1	no	—
225	C₂×C₄×C₇	—	2	1	no	—
226	C₂×C₄×C₇	—	2	2	no	—
227	C₂×C₄×C₇	—	2	2	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	2	no	—
232	C₂×C₄×C₇	—	4	1	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	8	no	—
237	C₂×C₄×C₇	—	4	8	no	—
238	C₂×C₄×C₇	—	4	2	no	—
239	C₂×C₄×C₇	—	4	2	no	—
240	C₂×C₄×C₇	—	4	2	no	—
241	C₂×C₄×C₇	—	4	2	no	—
242	C₂×C₄×C₇	—	4	2	no	—
243	C₂×C₄×C₇	—	4	2	no	—
244	C₂×C₄×C₇	—	8	8	no	—
245	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
246	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
247	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
248	C₂×C₄×C₇	C₂×D₈	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	C₂×#41	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	—	2	2	no	—
258	C₇×C₈	—	4	2	no	—
259	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
260	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
261	C₂³×C₄	—	2	1	no	—
262	C₂³×C₄	—	2	1	no	—
263	C₂³×C₄	—	2	2	no	—
264	C₂×D₈	—	2	2	no	—
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
267	C₂²×C₈	C₂³×C₇	1	2	yes	—
268	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
270	C₄×D₄	—	2	4	no	—
271	C₄×D₄	—	2	2	no	—
272	C₄×D₄	—	2	2	no	—
273	C₄×D₄	—	4	2	no	—
274	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
275	C₄×D₄	C₂×C₄×C₇	1	4	yes	complemented
276	#117	—	4	4	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	4	4	no	—
283	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	1	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₄²	—	2	2	no	—
295	C₂×C₄²	C₂³×C₇	1	2	yes	—
296	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
297	#107	C₂×C₄×C₇	1	4	yes	complemented
298	C₄⋊C₈	—	2	2	no	—
299	#102	—	2	4	no	—
300	#102	—	4	2	no	—
301	#102	C₂×C₄×C₇	1	4	yes	—
302	#102	C₂×C₄×C₇	1	4	yes	complemented
303	C₄×C₈	—	4	2	no	—
304	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	#95	C₇×D₄	1	2	yes	—
308	C₂²×C₇	—	2	4	no	—
309	C₂²×C₇	—	4	2	no	—
310	C₂²×C₇	—	4	2	no	—
311	C₂²×C₇	—	4	2	no	—
312	C₂²×C₇	—	4	2	no	—
313	C₂²×C₇	—	4	2	no	—
314	C₂²×C₇	—	4	2	no	—
315	C₂²×C₇	—	4	2	no	—
316	C₂²×C₇	—	4	2	no	—
317	C₂²×C₇	—	4	2	no	—
318	C₂²×C₇	—	8	1	no	—
319	C₂²×C₇	—	8	1	no	—
320	C₂²×C₇	—	8	1	no	—
321	C₂²×C₇	—	8	1	no	—
322	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
323	C₂²×C₇	#546	1	1	yes	characteristic
324	C₂²×C₇	#548	1	1	yes	characteristic
325	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
326	C₂²×C₇	C₂×#135	1	1	yes	characteristic
327	C₂²×C₇	#577	1	1	yes	characteristic
328	C₂²×C₇	#577	1	1	yes	characteristic
329	C₄×C₇	—	2	2	no	—
330	C₄×C₇	—	2	4	no	—
331	C₄×C₇	—	4	2	no	—
332	C₄×C₇	—	4	2	no	—
333	C₄×C₇	—	4	2	no	—
334	C₄×C₇	—	8	2	no	—
335	C₄×C₇	—	8	2	no	—
336	C₄×C₇	C₄×D₈	1	4	yes	—
337	C₄×C₇	C₂×#117	1	1	yes	characteristic
338	C₄×C₇	#520	1	1	yes	characteristic
339	C₄×C₇	#552	1	1	yes	characteristic
340	C₄×C₇	#553	1	1	yes	characteristic
341	C₂⁴	—	2	1	no	—
342	C₂⁴	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
352	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
353	C₂×D₄	C₇×#41	1	2	yes	—
354	C₂×D₄	C₇×#41	1	2	yes	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	—	4	2	no	—
363	C₂²×C₄	—	4	2	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	8	2	no	—
368	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
369	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
370	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
371	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
372	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
373	D₈	—	2	8	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
382	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	4	4	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	8	1	no	—
394	C₄⋊C₄	—	8	1	no	—
395	C₄⋊C₄	C₇×D₈	1	4	yes	complemented
396	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
397	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
398	C₂²⋊C₄	—	2	4	no	—
399	C₂²⋊C₄	—	2	4	no	—
400	C₂²⋊C₄	—	4	4	no	—
401	C₂²⋊C₄	—	4	4	no	—
402	C₂²⋊C₄	—	8	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	4	4	no	—
406	C₄²	C₂²×C₄×C₇	1	2	yes	—
407	C₄²	C₇×#41	1	2	yes	—
408	C₂×C₇	—	4	4	no	—
409	C₂×C₇	—	8	2	no	—
410	C₂×C₇	#2352	1	1	yes	characteristic
411	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
412	C₂×C₇	#2952	1	1	yes	characteristic
413	C₂×C₇	#3294	1	1	yes	characteristic
414	C₂×C₇	#3303	1	1	yes	characteristic
415	C₂×C₇	#3358	1	1	yes	characteristic
416	C₂×C₇	#3360	1	1	yes	characteristic
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	1	no	—
426	C₂³	—	4	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	—	4	1	no	—
431	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
432	D₄	—	2	2	no	—
433	D₄	—	2	2	no	—
434	D₄	—	2	2	no	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	4	1	no	—
438	D₄	—	4	1	no	—
439	D₄	C₇×#117	1	4	yes	complemented
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	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	8	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	8	8	no	—
462	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
463	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
464	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
465	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
466	C₂×C₄	C₇×#135	1	2	yes	—
467	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
468	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
469	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
470	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
471	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
472	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
473	C₈	—	2	2	no	—
474	C₈	—	2	2	no	—
475	C₈	—	4	2	no	—
476	C₇	#22766	1	1	yes	characteristic, direct product factor, Sylow
477	C₂²	—	2	4	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	—	8	1	no	—
490	C₂²	—	8	1	no	—
491	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
492	C₂²	C₇×#546	1	1	yes	characteristic
493	C₂²	C₇×#548	1	1	yes	characteristic
494	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
495	C₂²	C₂×C₇×#135	1	1	yes	characteristic
496	C₂²	C₇×#577	1	1	yes	characteristic
497	C₂²	C₇×#577	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	4	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	8	2	no	—
504	C₄	—	8	2	no	—
505	C₄	C₄×C₇×D₈	1	4	yes	—
506	C₄	C₂×C₇×#117	1	1	yes	characteristic
507	C₄	C₇×#520	1	1	yes	characteristic
508	C₄	C₇×#552	1	1	yes	characteristic
509	C₄	C₇×#553	1	1	yes	characteristic
510	C₂	—	4	4	no	—
511	C₂	—	8	2	no	—
512	C₂	C₇×#2352	1	1	yes	characteristic
513	C₂	C₂×C₄×C₇×D₈	1	1	yes	characteristic
514	C₂	C₇×#2952	1	1	yes	characteristic
515	C₂	C₇×#3294	1	1	yes	characteristic
516	C₂	C₇×#3303	1	1	yes	characteristic
517	C₂	C₇×#3358	1	1	yes	characteristic
518	C₂	C₇×#3360	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×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2354	C₂	semidirect
3. C₇×D₄×C₄⋊C₄	C₂	semidirect
4. C₇×#1934	C₂	semidirect
5. C₇×#1918	C₂	semidirect
7. C₇×#1834	C₂	semidirect
8. C₇×#1802	C₂	semidirect
9. C₇×#1802	C₂	semidirect
10. C₇×C₈×C₄⋊C₄	C₂	semidirect
11. C₄×C₇×#102	C₂	semidirect
18. C₄×C₇×D₈	C₄	semidirect
42. #22766	C₇	direct (N × H)
59. C₄×C₇×D₄	C₂×C₄	semidirect
81. C₇×#107	C₂×C₄	semidirect
86. C₇×#102	C₂×C₄	semidirect
88. C₄×C₇×C₈	C₂×C₄	semidirect
93. #2354	C₂×C₇	semidirect
94. D₄×C₄⋊C₄	C₂×C₇	semidirect
95. #1934	C₂×C₇	semidirect
96. #1918	C₂×C₇	semidirect
98. #1834	C₂×C₇	semidirect
99. #1802	C₂×C₇	semidirect
100. #1802	C₂×C₇	semidirect
101. C₈×C₄⋊C₄	C₂×C₇	semidirect
102. C₄×#102	C₂×C₇	semidirect
157. C₇×C₄⋊C₄	D₈	semidirect
176. C₄×D₈	C₄×C₇	semidirect
222. C₇×D₄	#117	semidirect
275. C₄×D₄	C₂×C₄×C₇	semidirect
297. #107	C₂×C₄×C₇	semidirect
302. #102	C₂×C₄×C₇	semidirect
304. C₄×C₈	C₂×C₄×C₇	semidirect
395. C₄⋊C₄	C₇×D₈	semidirect
439. D₄	C₇×#117	semidirect
476. C₇	#22766	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×C₄×C₇×D₈	2
C₇×#2354	2
C₇×D₄×C₄⋊C₄	2
C₇×#1934	2
C₇×#1918	2
C₇×#1879	2
C₇×#1834	2
C₇×#1802	2
C₇×C₈×C₄⋊C₄	2
C₄×C₇×#102	2
#22766	7

Subgroups — C7×#22766 #13199982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22766 #13199982