Subgroups — #89769

Order 448.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#1136	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#6377	C₂	1	2	yes	maximal, complemented
5	C₄×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₇×C₄⋊C₄	C₄	1	2	yes	—
8	C₇×C₂²⋊C₄	C₄	1	2	yes	—
9	C₄²×C₇	C₂²	1	2	yes	—
10	#1136	C₄	1	4	yes	complemented
11	C₂×C₇⋊C₈	C₄	1	4	yes	complemented
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#405	—	7	1	no	maximal, Sylow
16	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂×M₁₆	—	7	1	no	—
26	C₂²×C₈	—	7	1	no	—
27	#117	D₇	1	1	yes	characteristic
28	#97	—	7	2	no	—
29	C₄×C₈	—	7	2	no	—
30	C₂²×C₇	—	2	2	no	—
31	C₂²×C₇	C₄²	1	2	yes	—
32	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₇	—	2	4	no	—
34	C₄×C₇	C₄²	1	2	yes	—
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
38	M₁₆	—	7	4	no	—
39	C₂×C₈	—	7	4	no	—
40	C₂×C₈	—	7	1	no	—
41	C₂×C₈	—	7	2	no	—
42	C₂×C₈	—	7	1	no	—
43	C₄⋊C₄	Dic₇	1	2	yes	—
44	C₂²⋊C₄	Dic₇	1	2	yes	—
45	C₄²	C₂×D₇	1	2	yes	—
46	C₂×C₇	—	2	2	no	—
47	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₇	#131	1	2	yes	—
49	C₂³	C₂×Dic₇	1	1	yes	characteristic
50	C₂×C₄	C₄×D₇	1	2	yes	—
51	C₂×C₄	C₄×D₇	1	2	yes	—
52	C₂×C₄	C₂×Dic₇	1	4	yes	—
53	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
55	C₈	—	7	2	no	—
56	C₈	—	7	2	no	—
57	C₈	—	7	4	no	—
58	C₇	#405	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	2	no	—
60	C₂²	C₄×Dic₇	1	2	yes	—
61	C₂²	C₂²×Dic₇	1	1	yes	characteristic
62	C₄	—	2	4	no	—
63	C₄	C₄×Dic₇	1	2	yes	—
64	C₄	C₂×C₄×D₇	1	1	yes	characteristic
65	C₄	C₂×C₄×D₇	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	C₂×C₄×Dic₇	1	1	yes	characteristic
68	C₂	#6511	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂×C₄²
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #6377	C₂	semidirect
5. C₄×C₇⋊C₈	C₂	semidirect
10. #1136	C₄	semidirect
11. C₂×C₇⋊C₈	C₄	semidirect
58. C₇	#405	semidirect

Maximal subgroups

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

subgroup	index
C₇×#117	2
C₂×#1136	2
C₂²×C₇⋊C₈	2
#6377	2
C₄×C₇⋊C₈	2
#405	7