Subgroups — #89770

Order 448.

114 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	#6489	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#6382	C₂	1	2	yes	maximal, complemented
5	#6381	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₇×C₄⋊C₄	C₂²	1	2	yes	—
8	C₇×C₂²⋊C₄	—	2	1	no	—
9	C₄²×C₇	—	2	1	no	—
10	C₂²⋊Dic₇	—	2	1	no	—
11	C₄⋊Dic₇	C₂²	1	2	yes	—
12	C₄×Dic₇	—	2	1	no	—
13	C₂×C₇⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	#427	—	7	1	no	maximal, Sylow
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	Q₈	1	2	yes	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇	—	2	2	no	—
24	C₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈	—	7	1	no	—
27	#117	—	7	1	no	—
28	#117	D₇	1	1	yes	characteristic
29	#107	—	7	2	no	—
30	#106	—	7	2	no	—
31	C₂²×C₇	—	2	2	no	—
32	C₂²×C₇	C₄⋊C₄	1	2	yes	—
33	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₇	—	4	2	no	—
35	C₄×C₇	C₄⋊C₄	1	2	yes	—
36	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
38	Dic₇	—	4	2	no	—
39	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
40	C₂×C₈	—	7	4	no	—
41	C₂×C₈	—	7	1	no	—
42	C₂×C₈	—	7	1	no	—
43	C₄⋊C₄	—	7	2	no	—
44	C₄⋊C₄	C₂×D₇	1	2	yes	—
45	C₂²⋊C₄	—	2	1	no	—
46	C₂²⋊C₄	—	14	1	no	—
47	C₄²	—	2	1	no	—
48	C₄²	—	14	1	no	—
49	C₂×C₇	—	2	2	no	—
50	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇	#135	1	2	yes	—
52	C₂³	C₂²⋊D₇	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	14	2	no	—
55	C₂×C₄	Dic₁₄	1	2	yes	—
56	C₂×C₄	C₄×D₇	1	2	yes	—
57	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
58	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
59	C₈	—	7	2	no	—
60	C₈	—	7	2	no	—
61	C₇	#427	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	2	2	no	—
63	C₂²	Dic₇⋊C₄	1	2	yes	—
64	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
65	C₄	—	4	2	no	—
66	C₄	—	28	2	no	—
67	C₄	Dic₇⋊C₄	1	2	yes	—
68	C₄	C₂×Dic₁₄	1	1	yes	characteristic
69	C₄	C₂×C₄×D₇	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	C₂×Dic₇⋊C₄	1	1	yes	characteristic
72	C₂	#6513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#427
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #6382	C₂	semidirect
5. #6381	C₂	semidirect
13. C₂×C₇⋊C₈	C₄	semidirect
61. C₇	#427	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
#6489	2
C₂²×C₇⋊C₈	2
#6382	2
#6381	2
#427	7