Subgroups — #11739655

Order 896.

402 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#117	C₂	1	1	yes	maximal, characteristic
2	#89979	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#89772	C₂	1	4	yes	maximal, complemented
5	#89771	C₂	1	2	yes	maximal, complemented
6	#89769	C₂	1	2	yes	maximal, complemented
7	#89698	C₂	1	2	yes	maximal, complemented
8	#89697	C₂	1	2	yes	maximal, complemented
9	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₇×#117	C₄	1	1	yes	characteristic
11	C₇×#117	C₄	1	1	yes	characteristic
12	C₇×#117	C₄	1	2	yes	—
13	C₇×#117	C₂²	1	1	yes	characteristic
14	C₇×#117	C₂²	1	1	yes	characteristic
15	C₇×#117	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₄	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	2	yes	—
20	C₂×#1136	C₂²	1	2	yes	—
21	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
22	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
24	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
26	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
27	C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
28	C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
29	#6377	C₂²	1	4	yes	complemented
30	C₄×C₇⋊C₈	C₂²	1	4	yes	complemented
31	#2924	—	7	1	no	maximal, Sylow
32	C₂⁴×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇	—	2	4	no	—
34	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
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	C₄²×C₇	C₂×C₄	1	2	yes	—
53	C₄²×C₇	C₂×C₄	1	2	yes	—
54	C₄²×C₇	C₂³	1	2	yes	—
55	C₄²×C₇	C₂³	1	2	yes	—
56	#1136	—	2	4	no	—
57	C₂×C₇⋊C₈	—	2	4	no	—
58	C₂×C₇⋊C₈	C₂³	1	2	yes	—
59	C₂×C₇⋊C₈	C₂³	1	2	yes	—
60	C₂×C₇⋊C₈	C₂³	1	4	yes	—
61	C₂×#117	D₇	1	1	yes	characteristic
62	#433	—	7	4	no	—
63	#432	—	7	2	no	—
64	#431	—	7	2	no	—
65	#424	—	7	2	no	—
66	#407	—	7	1	no	—
67	C₂×C₂²⋊C₈	—	7	1	no	—
68	#405	—	7	2	no	—
69	C₂³×C₇	—	2	2	no	—
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	—	4	1	no	—
72	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	4	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	4	2	no	—
82	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	#41	1	4	yes	—
90	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
91	C₇⋊C₈	—	2	2	no	—
92	C₇⋊C₈	—	2	2	no	—
93	C₇⋊C₈	—	2	4	no	—
94	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
95	C₂×M₁₆	—	7	2	no	—
96	C₂²×C₈	—	7	2	no	—
97	#117	Dic₇	1	1	yes	characteristic
98	#117	Dic₇	1	1	yes	characteristic
99	#117	Dic₇	1	2	yes	—
100	#117	C₂×D₇	1	1	yes	characteristic
101	#117	C₂×D₇	1	1	yes	characteristic
102	#117	C₂×D₇	1	2	yes	—
103	C₂×C₄⋊C₄	Dic₇	1	1	yes	characteristic
104	C₂×C₄⋊C₄	Dic₇	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	Dic₇	1	2	yes	—
106	C₂×C₄²	C₂×D₇	1	2	yes	—
107	C₄⋊C₈	—	7	4	no	—
108	C₄⋊C₈	—	7	4	no	—
109	C₂²⋊C₈	—	7	1	no	—
110	C₂²⋊C₈	—	7	1	no	—
111	C₂²⋊C₈	—	7	2	no	—
112	C₂²⋊C₈	—	7	2	no	—
113	C₂²⋊C₈	—	7	2	no	—
114	#97	—	7	4	no	—
115	C₄×C₈	—	7	4	no	—
116	C₂²×C₇	—	2	1	no	—
117	C₂²×C₇	—	2	1	no	—
118	C₂²×C₇	—	2	1	no	—
119	C₂²×C₇	—	2	1	no	—
120	C₂²×C₇	—	2	1	no	—
121	C₂²×C₇	—	2	1	no	—
122	C₂²×C₇	—	2	1	no	—
123	C₂²×C₇	—	4	4	no	—
124	C₂²×C₇	#117	1	2	yes	—
125	C₂²×C₇	#131	1	2	yes	—
126	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
127	C₄×C₇	—	2	1	no	—
128	C₄×C₇	—	2	1	no	—
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	2	2	no	—
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	—	2	2	no	—
133	C₄×C₇	#117	1	2	yes	—
134	C₄×C₇	C₂×#41	1	2	yes	—
135	C₂⁴	C₂×Dic₇	1	1	yes	characteristic
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×Dic₇	1	2	yes	—
138	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
141	C₂²×C₄	C₂×Dic₇	1	2	yes	—
142	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
145	M₁₆	—	14	4	no	—
146	C₂×C₈	—	7	2	no	—
147	C₂×C₈	—	7	4	no	—
148	C₂×C₈	—	7	2	no	—
149	C₂×C₈	—	14	4	no	—
150	C₄⋊C₄	C₂×Dic₇	1	2	yes	—
151	C₄⋊C₄	C₂×Dic₇	1	2	yes	—
152	C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
153	C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
154	C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
155	C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
156	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
158	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
159	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
160	C₄²	C₂×Dic₇	1	2	yes	—
161	C₄²	C₂×Dic₇	1	2	yes	—
162	C₄²	C₂²×D₇	1	2	yes	—
163	C₄²	C₂²×D₇	1	2	yes	—
164	C₂×C₇	—	2	2	no	—
165	C₂×C₇	—	2	2	no	—
166	C₂×C₇	—	4	1	no	—
167	C₂×C₇	C₂×#117	1	1	yes	characteristic
168	C₂×C₇	C₂×#131	1	1	yes	characteristic
169	C₂×C₇	#568	1	1	yes	characteristic
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	1	no	—
173	C₂³	C₂²×Dic₇	1	1	yes	characteristic
174	C₂³	C₂²×Dic₇	1	1	yes	characteristic
175	C₂³	C₂²×Dic₇	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	#1157	1	4	yes	—
184	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₇	1	2	yes	—
186	C₂×C₄	C₂²×Dic₇	1	2	yes	—
187	C₂×C₄	C₂²×Dic₇	1	2	yes	—
188	C₂×C₄	C₂²×Dic₇	1	2	yes	—
189	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
191	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
192	C₈	—	14	2	no	—
193	C₈	—	14	2	no	—
194	C₈	—	14	4	no	—
195	C₇	#2924	1	1	yes	characteristic, complemented, Sylow
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	4	no	—
204	C₂²	#6489	1	2	yes	—
205	C₂²	#6511	1	2	yes	—
206	C₂²	C₂³×Dic₇	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	#6489	1	2	yes	—
214	C₄	C₂×#1157	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	2	no	—
217	C₂	—	4	1	no	—
218	C₂	C₂×#6489	1	1	yes	characteristic
219	C₂	C₂×#6511	1	1	yes	characteristic
220	C₂	#90510	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₂×C₇×#117	C₂
prime radical	C₂⁴×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₂×#117	D₇
hypocenter	C₇	#2924
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
4. #89772	C₂	semidirect
5. #89771	C₂	semidirect
6. #89769	C₂	semidirect
7. #89698	C₂	semidirect
8. #89697	C₂	semidirect
27. C₄⋊(C₇⋊C₈)	C₂²	semidirect
28. C₄⋊(C₇⋊C₈)	C₂²	semidirect
29. #6377	C₂²	semidirect
30. C₄×C₇⋊C₈	C₂²	semidirect
195. C₇	#2924	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#117	2
#89979	2
C₂×C₂²⋊(C₇⋊C₈)	2
#89772	2
#89771	2
#89769	2
#89698	2
#89697	2
#2924	7