Subgroups — #11736985

Order 896.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇×M₁₆	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	#89899	C₂	1	2	yes	maximal, complemented
5	#89784	C₂	1	1	yes	maximal, characteristic, complemented
6	#89609	C₂	1	1	yes	maximal, characteristic, complemented
7	#89606	C₂	1	1	yes	maximal, characteristic, complemented
8	#89500	C₂	1	2	yes	maximal, complemented
9	#89498	C₂	1	2	yes	maximal, complemented
10	C₄×#1131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₇	C₂²	1	2	yes	—
14	C₂×C₇×M₁₆	C₂²	1	2	yes	complemented
15	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
16	D₄×Dic₇	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₇	C₂²	1	2	yes	—
18	C₂×#1136	C₂²	1	2	yes	complemented
19	D₇×M₁₆	—	2	8	no	—
20	C₂×#1131	C₂²	1	2	yes	—
21	C₂×#1131	C₂²	1	1	yes	characteristic
22	C₂×#1131	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₇	C₂²	1	2	yes	complemented
24	C₂×C₈×D₇	C₂²	1	2	yes	complemented
25	#6456	C₄	1	1	yes	characteristic
26	D₇×C₄⋊C₄	C₄	1	1	yes	characteristic
27	(C₂²⋊D₇)⋊C₄	C₄	1	2	yes	—
28	D₇×C₂²⋊C₄	C₄	1	2	yes	—
29	C₄×D₂₈	C₂²	1	1	yes	characteristic
30	C₄²×D₇	C₂²	1	1	yes	characteristic
31	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
34	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
35	(C₂×D₇)⋊C₈	C₂²	1	2	yes	complemented
36	(C₂×D₇)⋊C₈	C₂²	1	2	yes	complemented
37	#6389	C₂²	1	1	yes	characteristic, complemented
38	#6389	C₂²	1	1	yes	characteristic, complemented
39	Dic₇⋊C₈	C₂²	1	2	yes	complemented
40	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
41	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	D₄×M₁₆	—	7	1	no	maximal, Sylow
43	C₂³×D₇	C₂×C₄	1	2	yes	—
44	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₂²⋊D₇	C₂×C₄	1	2	yes	—
47	C₂²×Dic₇	C₂×C₄	1	2	yes	—
48	D₄×D₇	—	2	4	no	—
49	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₇	—	2	4	no	—
51	C₂×C₄×D₇	—	2	4	no	—
52	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
55	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₇	C₂³	1	2	yes	—
57	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
58	C₇×M₁₆	—	2	4	no	—
59	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂³	1	2	yes	—
61	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
63	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
64	C₄²×C₇	C₂³	1	1	yes	characteristic
65	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
66	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
67	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
68	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
69	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
70	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₇	C₂³	1	1	yes	characteristic
73	#1136	—	2	4	no	—
74	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₇⋊C₈	C₂³	1	2	yes	—
76	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
77	#1131	—	2	4	no	—
78	#1131	—	2	2	no	—
79	#1131	D₄	1	4	yes	complemented
80	C₈×D₇	—	2	4	no	—
81	C₈×D₇	—	2	4	no	—
82	C₂²×M₁₆	—	7	2	no	—
83	C₂×C₄×D₄	—	7	1	no	—
84	#436	—	7	1	no	—
85	#436	D₇	1	1	yes	characteristic, complemented
86	#435	—	7	2	no	—
87	#435	—	7	2	no	—
88	C₈×D₄	—	7	1	no	—
89	C₈×D₄	—	7	1	no	—
90	#423	—	7	1	no	—
91	#407	—	7	2	no	—
92	C₄×M₁₆	—	7	1	no	—
93	C₂³×C₇	C₂²×C₄	1	2	yes	—
94	C₂²×D₇	—	2	4	no	—
95	C₂²×D₇	—	2	4	no	—
96	C₂²×D₇	—	4	2	no	—
97	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×D₇	C₂²×C₄	1	2	yes	—
99	C₇×D₄	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₇	—	2	4	no	—
107	C₂×Dic₇	—	2	4	no	—
108	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×Dic₇	C₂²×C₄	1	2	yes	—
111	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
112	D₂₈	M₁₆	1	4	yes	complemented
113	C₄×D₇	—	2	4	no	—
114	C₄×D₇	—	2	2	no	—
115	C₄×D₇	—	2	2	no	—
116	C₄×D₇	—	2	2	no	—
117	C₄×D₇	—	2	4	no	—
118	C₄×D₇	—	4	1	no	—
119	C₄×D₇	C₂×D₄	1	2	yes	—
120	C₄×D₇	#41	1	2	yes	—
121	C₇×C₈	—	2	2	no	—
122	C₇×C₈	—	2	1	no	—
123	C₇×C₈	C₂×D₄	1	2	yes	complemented
124	C₇⋊C₈	—	2	2	no	—
125	C₇⋊C₈	—	2	1	no	—
126	C₇⋊C₈	C₂×D₄	1	2	yes	complemented
127	C₂²×D₄	—	7	1	no	—
128	C₂³×C₄	—	7	2	no	—
129	C₂×M₁₆	—	7	2	no	—
130	C₂×M₁₆	—	7	1	no	—
131	C₂×M₁₆	—	7	2	no	—
132	C₂×M₁₆	—	7	1	no	—
133	C₂×M₁₆	—	14	8	no	—
134	C₂×M₁₆	C₂×D₇	1	2	yes	complemented
135	C₂²×C₈	—	7	2	no	—
136	C₂²×C₈	—	7	2	no	—
137	C₄×D₄	—	7	1	no	—
138	C₄×D₄	—	7	2	no	—
139	C₄×D₄	—	7	1	no	—
140	C₄×D₄	—	7	2	no	—
141	C₄×D₄	—	7	1	no	—
142	C₄×D₄	C₂×D₇	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	7	1	no	—
144	C₂×C₂²⋊C₄	—	7	2	no	—
145	C₂×C₄²	—	7	1	no	—
146	C₄⋊C₈	—	7	2	no	—
147	C₄⋊C₈	—	7	1	no	—
148	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic, complemented
149	C₂²⋊C₈	—	7	2	no	—
150	C₂²⋊C₈	—	7	2	no	—
151	C₂²⋊C₈	—	7	2	no	—
152	C₂²⋊C₈	C₂×D₇	1	2	yes	complemented
153	#97	—	7	1	no	—
154	#97	—	7	1	no	—
155	C₄×C₈	—	7	1	no	—
156	C₄×C₈	C₂×D₇	1	1	yes	characteristic, complemented
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	—	2	2	no	—
160	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
161	C₂×D₇	—	2	2	no	—
162	C₂×D₇	—	2	1	no	—
163	C₂×D₇	—	2	1	no	—
164	C₂×D₇	—	2	2	no	—
165	C₂×D₇	—	4	8	no	—
166	C₂×D₇	C₄×D₄	1	2	yes	—
167	C₂×D₇	C₂×M₁₆	1	4	yes	complemented
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	C₂×M₁₆	1	2	yes	complemented
171	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
172	C₄×C₇	C₂×#41	1	1	yes	characteristic
173	Dic₇	—	2	2	no	—
174	Dic₇	—	2	1	no	—
175	Dic₇	—	2	1	no	—
176	Dic₇	C₄×D₄	1	2	yes	—
177	C₂⁴	—	7	2	no	—
178	C₂×D₄	—	7	2	no	—
179	C₂×D₄	—	7	1	no	—
180	C₂×D₄	—	14	4	no	—
181	C₂×D₄	C₄×D₇	1	1	yes	characteristic
182	C₂²×C₄	—	7	1	no	—
183	C₂²×C₄	—	7	2	no	—
184	C₂²×C₄	—	7	1	no	—
185	C₂²×C₄	—	7	2	no	—
186	C₂²×C₄	—	7	2	no	—
187	C₂²×C₄	—	7	2	no	—
188	C₂²×C₄	—	7	1	no	—
189	C₂²×C₄	—	14	4	no	—
190	C₂²×C₄	—	14	4	no	—
191	C₂²×C₄	C₂²×D₇	1	2	yes	—
192	M₁₆	—	2	4	no	—
193	M₁₆	—	7	4	no	—
194	M₁₆	—	14	2	no	—
195	M₁₆	—	14	4	no	—
196	M₁₆	—	14	4	no	—
197	C₂×C₈	—	7	1	no	—
198	C₂×C₈	—	7	1	no	—
199	C₂×C₈	—	7	2	no	—
200	C₂×C₈	—	14	4	no	—
201	C₂×C₈	—	14	4	no	—
202	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₇	1	2	yes	—
204	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
205	C₄⋊C₄	—	7	2	no	—
206	C₄⋊C₄	—	7	1	no	—
207	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
208	C₂²⋊C₄	—	7	2	no	—
209	C₂²⋊C₄	—	7	2	no	—
210	C₂²⋊C₄	—	7	2	no	—
211	C₂²⋊C₄	C₄×D₇	1	2	yes	—
212	C₄²	—	7	1	no	—
213	C₄²	—	7	1	no	—
214	C₄²	—	7	1	no	—
215	C₄²	C₂²×D₇	1	1	yes	characteristic
216	C₂×C₇	—	4	2	no	—
217	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
219	C₂×C₇	#568	1	1	yes	characteristic
220	D₇	—	2	2	no	—
221	D₇	—	2	4	no	—
222	C₂³	—	7	2	no	—
223	C₂³	—	7	1	no	—
224	C₂³	—	14	4	no	—
225	C₂³	—	14	4	no	—
226	C₂³	—	28	2	no	—
227	C₂³	C₂×C₄×D₇	1	2	yes	—
228	D₄	—	2	2	no	—
229	D₄	—	7	4	no	—
230	D₄	—	14	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	7	1	no	—
234	C₂×C₄	—	7	2	no	—
235	C₂×C₄	—	7	2	no	—
236	C₂×C₄	—	7	1	no	—
237	C₂×C₄	—	7	1	no	—
238	C₂×C₄	—	7	2	no	—
239	C₂×C₄	—	14	4	no	—
240	C₂×C₄	—	14	4	no	—
241	C₂×C₄	—	14	4	no	—
242	C₂×C₄	—	14	2	no	—
243	C₂×C₄	—	14	2	no	—
244	C₂×C₄	—	14	2	no	—
245	C₂×C₄	—	28	1	no	—
246	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
248	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
249	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	1	no	—
252	C₈	—	7	2	no	—
253	C₈	—	14	2	no	—
254	C₈	—	14	1	no	—
255	C₈	D₄×D₇	1	2	yes	complemented
256	C₇	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	7	4	no	—
261	C₂²	—	7	2	no	—
262	C₂²	—	14	1	no	—
263	C₂²	—	14	1	no	—
264	C₂²	—	14	2	no	—
265	C₂²	—	14	2	no	—
266	C₂²	—	28	8	no	—
267	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	7	2	no	—
271	C₄	—	14	1	no	—
272	C₄	—	14	1	no	—
273	C₄	—	14	2	no	—
274	C₄	D₇×M₁₆	1	2	yes	complemented
275	C₄	C₂×D₄×D₇	1	1	yes	characteristic
276	C₄	D₇×#41	1	1	yes	characteristic
277	C₂	—	4	2	no	—
278	C₂	—	14	2	no	—
279	C₂	—	14	4	no	—
280	C₂	C₄×D₄×D₇	1	1	yes	characteristic
281	C₂	C₂×D₇×M₁₆	1	1	yes	characteristic
282	C₂	#90441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#436	C₂
prime radical	C₂×D₄×D₇	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#436	D₇
hypocenter	C₇	D₄×M₁₆
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×D₇×M₁₆	C₂	semidirect
3. C₇×#436	C₂	semidirect
4. #89899	C₂	semidirect
5. #89784	C₂	semidirect
6. #89609	C₂	semidirect
7. #89606	C₂	semidirect
8. #89500	C₂	semidirect
9. #89498	C₂	semidirect
10. C₄×#1131	C₂	semidirect
11. C₈×D₂₈	C₂	semidirect
14. C₂×C₇×M₁₆	C₂²	semidirect
18. C₂×#1136	C₂²	semidirect
23. C₂×C₈×D₇	C₂²	semidirect
24. C₂×C₈×D₇	C₂²	semidirect
31. C₇×C₄⋊C₈	C₂²	semidirect
32. C₇×C₂²⋊C₈	C₂²	semidirect
33. C₄×C₇×C₈	C₂²	semidirect
34. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
35. (C₂×D₇)⋊C₈	C₂²	semidirect
36. (C₂×D₇)⋊C₈	C₂²	semidirect
37. #6389	C₂²	semidirect
38. #6389	C₂²	semidirect
39. Dic₇⋊C₈	C₂²	semidirect
40. C₄⋊(C₇⋊C₈)	C₂²	semidirect
41. C₄×C₇⋊C₈	C₂²	semidirect
79. #1131	D₄	semidirect
85. #436	D₇	semidirect
112. D₂₈	M₁₆	semidirect
123. C₇×C₈	C₂×D₄	semidirect
126. C₇⋊C₈	C₂×D₄	semidirect
134. C₂×M₁₆	C₂×D₇	semidirect
148. C₄⋊C₈	C₂×D₇	semidirect
152. C₂²⋊C₈	C₂×D₇	semidirect
156. C₄×C₈	C₂×D₇	semidirect
167. C₂×D₇	C₂×M₁₆	semidirect
170. C₄×C₇	C₂×M₁₆	semidirect
255. C₈	D₄×D₇	semidirect
256. C₇	D₄×M₁₆	semidirect
274. C₄	D₇×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₇×M₁₆	2
C₄×D₄×D₇	2
C₇×#436	2
#89899	2
#89784	2
#89609	2
#89606	2
#89500	2
#89498	2
C₄×#1131	2
C₈×D₂₈	2
D₄×M₁₆	7