Subgroups — #11739748

Order 896.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	#89979	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#89784	C₂	1	2	yes	maximal, complemented
5	#89783	C₂	1	4	yes	maximal, complemented
6	D₄×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
7	#89698	C₂	1	1	yes	maximal, characteristic, complemented
8	#89697	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₄	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₇	C₂²	1	2	yes	—
11	C₄×C₇×D₄	C₄	1	2	yes	—
12	C₄×C₇×D₄	C₄	1	1	yes	characteristic
13	C₄×C₇×D₄	C₄	1	1	yes	characteristic
14	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₄×C₇×D₄	C₂²	1	2	yes	—
16	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₇×C₂²⋊C₄	C₄	1	2	yes	—
19	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₂×#1136	C₂²	1	4	yes	complemented
21	C₂²×C₇⋊C₈	C₂²	1	4	yes	complemented
22	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
25	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
26	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
27	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
28	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
29	#6377	C₂²	1	2	yes	complemented
30	C₄×C₇⋊C₈	C₂²	1	2	yes	complemented
31	#2974	—	7	1	no	maximal, Sylow
32	C₂⁴×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₇×D₄	—	2	4	no	—
34	C₂×C₇×D₄	C₂×C₄	1	2	yes	—
35	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇	—	4	4	no	—
38	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
50	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₄²×C₇	C₂³	1	1	yes	characteristic
57	#1136	—	4	4	no	—
58	C₂×C₇⋊C₈	—	4	4	no	—
59	C₂×C₇⋊C₈	C₂³	1	2	yes	—
60	C₂×C₇⋊C₈	C₂³	1	4	yes	—
61	C₂×C₇⋊C₈	C₂³	1	2	yes	—
62	C₂×C₄×D₄	D₇	1	1	yes	characteristic
63	#436	—	7	2	no	—
64	#435	—	7	4	no	—
65	C₈×D₄	—	7	2	no	—
66	#432	—	7	1	no	—
67	#431	—	7	1	no	—
68	#407	—	7	2	no	—
69	C₂×C₂²⋊C₈	—	7	2	no	—
70	C₂³×C₇	—	2	4	no	—
71	C₂³×C₇	—	2	4	no	—
72	C₂³×C₇	—	4	2	no	—
73	C₂³×C₇	C₂²×C₄	1	2	yes	—
74	C₂³×C₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₇×D₄	—	2	4	no	—
77	C₇×D₄	—	2	2	no	—
78	C₇×D₄	M₁₆	1	4	yes	complemented
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	2	4	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	4	2	no	—
85	C₂×C₄×C₇	—	4	2	no	—
86	C₂×C₄×C₇	—	4	2	no	—
87	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
95	C₇⋊C₈	—	2	2	no	—
96	C₇⋊C₈	—	2	4	no	—
97	C₇⋊C₈	—	2	2	no	—
98	C₂²×D₄	Dic₇	1	1	yes	characteristic
99	C₂³×C₄	C₂×D₇	1	2	yes	—
100	C₂×M₁₆	—	7	4	no	—
101	C₂²×C₈	—	7	4	no	—
102	C₄×D₄	Dic₇	1	2	yes	—
103	C₄×D₄	Dic₇	1	1	yes	characteristic
104	C₄×D₄	Dic₇	1	1	yes	characteristic
105	C₄×D₄	C₂×D₇	1	1	yes	characteristic
106	C₄×D₄	C₂×D₇	1	2	yes	—
107	C₄×D₄	C₂×D₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	Dic₇	1	1	yes	characteristic
109	C₂×C₂²⋊C₄	Dic₇	1	2	yes	—
110	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
111	C₄⋊C₈	—	7	2	no	—
112	C₄⋊C₈	—	7	2	no	—
113	C₂²⋊C₈	—	7	1	no	—
114	C₂²⋊C₈	—	7	2	no	—
115	C₂²⋊C₈	—	7	1	no	—
116	C₂²⋊C₈	—	7	4	no	—
117	C₂²⋊C₈	—	7	4	no	—
118	#97	—	7	2	no	—
119	C₄×C₈	—	7	2	no	—
120	C₂²×C₇	—	2	1	no	—
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	—	2	1	no	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	—	2	2	no	—
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	—	4	8	no	—
128	C₂²×C₇	C₂×M₁₆	1	4	yes	—
129	C₂²×C₇	#131	1	2	yes	—
130	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₇	—	2	1	no	—
132	C₄×C₇	—	2	2	no	—
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	—	2	1	no	—
135	C₄×C₇	—	2	1	no	—
136	C₄×C₇	—	2	1	no	—
137	C₄×C₇	C₂×M₁₆	1	2	yes	—
138	C₄×C₇	H(5,𝔽₂)	1	2	yes	—
139	C₂⁴	C₂×Dic₇	1	2	yes	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	C₂×Dic₇	1	2	yes	—
142	C₂×D₄	C₂×Dic₇	1	1	yes	characteristic
143	C₂×D₄	C₂×Dic₇	1	1	yes	characteristic
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₇	1	2	yes	—
148	C₂²×C₄	C₂×Dic₇	1	2	yes	—
149	C₂²×C₄	C₂×Dic₇	1	2	yes	—
150	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₇	1	2	yes	—
152	C₂²×C₄	C₂²×D₇	1	2	yes	—
153	M₁₆	—	28	4	no	—
154	C₂×C₈	—	7	2	no	—
155	C₂×C₈	—	7	2	no	—
156	C₂×C₈	—	7	4	no	—
157	C₂×C₈	—	28	4	no	—
158	C₄⋊C₄	C₂×Dic₇	1	2	yes	—
159	C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
161	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
162	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
163	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
164	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
165	C₄²	C₂×Dic₇	1	1	yes	characteristic
166	C₄²	C₂×Dic₇	1	1	yes	characteristic
167	C₄²	C₂²×D₇	1	1	yes	characteristic
168	C₄²	C₂²×D₇	1	1	yes	characteristic
169	C₂×C₇	—	2	2	no	—
170	C₂×C₇	—	2	4	no	—
171	C₂×C₇	—	4	2	no	—
172	C₂×C₇	#518	1	1	yes	characteristic
173	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
174	C₂×C₇	C₂×#131	1	1	yes	characteristic
175	C₂³	—	2	4	no	—
176	C₂³	—	2	4	no	—
177	C₂³	—	4	2	no	—
178	C₂³	C₂²×Dic₇	1	2	yes	—
179	C₂³	C₂²×Dic₇	1	2	yes	—
180	C₂³	C₂²×Dic₇	1	1	yes	characteristic
181	D₄	—	2	4	no	—
182	D₄	—	2	2	no	—
183	D₄	#1136	1	4	yes	complemented
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₇	1	2	yes	—
197	C₂×C₄	C₂²×Dic₇	1	2	yes	—
198	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
199	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
200	C₈	—	14	2	no	—
201	C₈	—	14	4	no	—
202	C₈	—	14	2	no	—
203	C₇	#2974	1	1	yes	characteristic, complemented, Sylow
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	8	no	—
212	C₂²	C₂×#1136	1	4	yes	—
213	C₂²	#6511	1	2	yes	—
214	C₂²	C₂³×Dic₇	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	C₂×#1136	1	2	yes	—
222	C₄	#6548	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	4	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×#1136	1	1	yes	characteristic
227	C₂	#90489	1	1	yes	characteristic
228	C₂	C₂×#6511	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₂×C₄×C₇×D₄	C₂
prime radical	C₂²×C₇×D₄	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₇
hypocenter	C₇	#2974
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #89979	C₂	semidirect
3. C₂×C₂²⋊(C₇⋊C₈)	C₂	semidirect
4. #89784	C₂	semidirect
5. #89783	C₂	semidirect
6. D₄×C₇⋊C₈	C₂	semidirect
7. #89698	C₂	semidirect
8. #89697	C₂	semidirect
20. C₂×#1136	C₂²	semidirect
21. C₂²×C₇⋊C₈	C₂²	semidirect
25. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
26. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
27. C₄⋊(C₇⋊C₈)	C₂²	semidirect
28. C₄⋊(C₇⋊C₈)	C₂²	semidirect
29. #6377	C₂²	semidirect
30. C₄×C₇⋊C₈	C₂²	semidirect
78. C₇×D₄	M₁₆	semidirect
183. D₄	#1136	semidirect
203. C₇	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×D₄	2
#89979	2
C₂×C₂²⋊(C₇⋊C₈)	2
#89784	2
#89783	2
D₄×C₇⋊C₈	2
#89698	2
#89697	2
#2974	7