Subgroups — #89264

Order 448.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₂²×Dic₇	C₄	1	1	yes	characteristic
6	C₂²×Dic₇	C₄	1	1	yes	characteristic
7	C₂×C₇×C₈	—	2	1	no	—
8	Dic₇⋊C₄	—	2	1	no	—
9	Dic₇⋊C₄	—	2	1	no	—
10	C₂×C₇⋊C₈	—	2	1	no	—
11	#324	—	7	1	no	maximal, Sylow
12	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₇	—	4	1	no	—
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×Dic₇	—	2	1	no	—
17	C₂×Dic₇	—	2	1	no	—
18	C₂×Dic₇	—	2	1	no	—
19	C₂×Dic₇	—	4	1	no	—
20	C₂×Dic₇	C₈	1	2	yes	complemented
21	C₇×C₈	—	4	1	no	—
22	C₇⋊C₈	—	4	1	no	—
23	C₂×C₄⋊C₄	—	7	1	no	—
24	C₂²⋊C₈	—	7	1	no	—
25	C₂²⋊C₈	D₇	1	1	yes	characteristic
26	C₂²×C₇	—	2	1	no	—
27	C₂²×C₇	—	2	1	no	—
28	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₇	M₁₆	1	1	yes	characteristic
31	C₄×C₇	—	2	1	no	—
32	C₄×C₇	—	2	1	no	—
33	Dic₇	—	2	2	no	—
34	Dic₇	—	4	1	no	—
35	C₂²×C₄	—	7	1	no	—
36	C₂²×C₄	—	7	1	no	—
37	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	14	1	no	—
40	C₄⋊C₄	—	14	1	no	—
41	C₄⋊C₄	—	14	1	no	—
42	C₂×C₇	—	2	1	no	—
43	C₂×C₇	—	2	1	no	—
44	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₇	#99	1	1	yes	characteristic
46	C₂×C₇	#101	1	1	yes	characteristic
47	C₂³	C₄×D₇	1	1	yes	characteristic
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	—	7	2	no	—
50	C₂×C₄	—	14	1	no	—
51	C₂×C₄	—	14	1	no	—
52	C₂×C₄	—	14	1	no	—
53	C₂×C₄	—	28	1	no	—
54	C₂×C₄	D₂₈	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₈	—	28	1	no	—
58	C₇	#324	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	C₈×D₇	1	1	yes	characteristic
62	C₂²	#1131	1	1	yes	characteristic
63	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	14	2	no	—
67	C₄	—	28	1	no	—
68	C₂	—	2	1	no	—
69	C₂	—	2	1	no	—
70	C₂	#6380	1	1	yes	characteristic
71	C₂	(C₂×D₇)⋊C₈	1	1	yes	characteristic
72	C₂	#6398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₇)⋊C₄
commutator	C₂²×C₇	C₂×C₈
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₇×C₂²⋊C₈	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂²⋊C₈	D₇
hypocenter	C₇	#324
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
20. C₂×Dic₇	C₈	semidirect
58. C₇	#324	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇⋊C₄	2
C₇×C₂²⋊C₈	2
C₂²⋊(C₇⋊C₈)	2
#324	7