Subgroups — #11729338

Order 896.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
3	#89319	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₇×#117	—	2	4	no	—
6	C₂×C₇×C₄⋊C₄	C₄	1	2	yes	—
7	C₂×C₇×C₂²⋊C₄	C₄	1	2	yes	—
8	C₂×C₄²×C₇	C₂²	1	2	yes	—
9	C₂²×C₇⋊C₈	C₂²	1	4	yes	—
10	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
11	#1818	—	7	1	no	maximal, Sylow
12	C₂⁴×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₇	—	2	4	no	—
14	C₂²×C₄×C₇	C₂×C₄	1	4	yes	—
15	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₇	D₄	1	4	yes	—
17	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₇×C₄⋊C₄	—	2	4	no	—
19	C₇×C₂²⋊C₄	—	2	4	no	—
20	C₄²×C₇	—	2	4	no	—
21	C₂×C₇⋊C₈	—	2	4	no	—
22	C₂×C₇⋊C₈	—	2	8	no	—
23	C₂×#117	D₇	1	1	yes	characteristic
24	C₂×C₂²⋊C₈	—	7	2	no	—
25	#336	—	7	4	no	—
26	C₂³×C₇	—	2	2	no	—
27	C₂³×C₇	—	2	4	no	—
28	C₂³×C₇	C₂²⋊C₄	1	2	yes	—
29	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₇	—	2	2	no	—
31	C₂×C₄×C₇	—	2	8	no	—
32	C₂×C₄×C₇	—	2	4	no	—
33	C₂×C₄×C₇	—	2	8	no	—
34	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₇	#41	1	4	yes	—
37	C₇⋊C₈	—	4	4	no	—
38	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
39	C₂²×C₈	—	7	4	no	—
40	#117	—	2	4	no	—
41	C₂×C₄⋊C₄	Dic₇	1	2	yes	—
42	C₂×C₂²⋊C₄	Dic₇	1	2	yes	—
43	C₂×C₄²	C₂×D₇	1	2	yes	—
44	C₂²⋊C₈	—	14	4	no	—
45	C₂²×C₇	—	2	2	no	—
46	C₂²×C₇	—	2	4	no	—
47	C₂²×C₇	—	4	4	no	—
48	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₇	#117	1	2	yes	—
50	C₂²×C₇	#131	1	4	yes	—
51	C₄×C₇	—	2	2	no	—
52	C₄×C₇	—	4	4	no	—
53	C₄×C₇	#123	1	4	yes	—
54	C₂⁴	C₂×Dic₇	1	1	yes	characteristic
55	C₂²×C₄	—	2	4	no	—
56	C₂²×C₄	C₂×Dic₇	1	4	yes	—
57	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
58	C₂²×C₄	C₂²⋊D₇	1	4	yes	—
59	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
60	C₂×C₈	—	14	8	no	—
61	C₂×C₈	—	14	4	no	—
62	C₄⋊C₄	—	2	4	no	—
63	C₂²⋊C₄	—	2	4	no	—
64	C₄²	—	2	4	no	—
65	C₂×C₇	—	4	2	no	—
66	C₂×C₇	#381	1	1	yes	characteristic
67	C₂×C₇	#408	1	2	yes	—
68	C₂×C₇	#433	1	4	yes	—
69	C₂³	—	2	2	no	—
70	C₂³	—	2	4	no	—
71	C₂³	C₂²⋊Dic₇	1	2	yes	—
72	C₂³	C₂²×Dic₇	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	8	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	C₂²⋊Dic₇	1	2	yes	—
78	C₂×C₄	#1157	1	4	yes	—
79	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
80	C₈	—	28	4	no	—
81	C₇	#1818	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	2	2	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	4	4	no	—
85	C₂²	#6489	1	2	yes	—
86	C₂²	#6511	1	4	yes	—
87	C₂²	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#6492	1	4	yes	—
91	C₂	—	4	2	no	—
92	C₂	#89772	1	4	yes	—
93	C₂	#89983	1	1	yes	characteristic
94	C₂	#90006	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇
commutator, socle	C₂³×C₇	C₂²×C₄
Fitting	C₂×C₇×#117	C₂
prime radical	C₂⁴×C₇	C₂×C₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×#117	D₇
hypocenter	C₇	#1818
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #89319	C₂	semidirect
81. C₇	#1818	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
C₂×C₂²⋊(C₇⋊C₈)	2
#89319	2
#1818	7