Subgroups — #89698

Order 448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
3	C₄⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#6377	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₄²×C₇	C₄	1	1	yes	characteristic
9	C₄²×C₇	C₄	1	1	yes	characteristic
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₈	C₂²	1	4	yes	—
13	#432	—	7	1	no	maximal, Sylow
14	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇	—	2	2	no	—
16	C₂×C₄×C₇	—	2	2	no	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₇⋊C₈	—	2	4	no	—
25	C₂×C₄²	D₇	1	1	yes	characteristic
26	C₄⋊C₈	—	7	2	no	—
27	C₂²⋊C₈	—	7	2	no	—
28	#97	—	7	2	no	—
29	C₂²×C₇	—	2	1	no	—
30	C₂²×C₇	—	2	1	no	—
31	C₂²×C₇	M₁₆	1	2	yes	—
32	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₇	—	2	1	no	—
34	C₄×C₇	—	2	1	no	—
35	C₄×C₇	—	2	1	no	—
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	M₁₆	1	2	yes	—
38	C₄×C₇	#41	1	2	yes	—
39	C₂²×C₄	Dic₇	1	1	yes	characteristic
40	C₂²×C₄	Dic₇	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
42	C₂×C₈	—	7	4	no	—
43	C₄²	Dic₇	1	1	yes	characteristic
44	C₄²	Dic₇	1	1	yes	characteristic
45	C₄²	C₂×D₇	1	1	yes	characteristic
46	C₄²	C₂×D₇	1	1	yes	characteristic
47	C₂×C₇	—	2	2	no	—
48	C₂×C₇	#117	1	1	yes	characteristic
49	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
51	C₂³	C₂×Dic₇	1	1	yes	characteristic
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
56	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
57	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
58	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
59	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
61	C₈	—	14	4	no	—
62	C₇	#432	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	#1136	1	2	yes	—
66	C₂²	C₂²×Dic₇	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	#1136	1	2	yes	—
72	C₄	#1157	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	C₂×#1136	1	1	yes	characteristic
75	C₂	C₂×#1136	1	1	yes	characteristic
76	C₂	#6489	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₂×C₄²×C₇	C₂
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	#432
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₇⋊C₈)	C₂	semidirect
4. #6377	C₂	semidirect
62. C₇	#432	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₇	2
C₂²⋊(C₇⋊C₈)	2
C₄⋊(C₇⋊C₈)	2
#6377	2
#432	7