Subgroups — #89325

Order 448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#1136	C₂	1	2	yes	maximal
3	C₂²×C₄×C₇	C₄	1	2	yes	—
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₇×C₄⋊C₄	—	2	2	no	—
6	#1136	—	2	2	no	—
7	#1136	C₄	1	4	yes	complemented
8	C₂×C₇⋊C₈	—	2	2	no	—
9	#343	—	7	1	no	maximal, Sylow
10	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₇	—	2	2	no	—
12	C₂×C₄×C₇	—	4	2	no	—
13	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₇	D₄	1	2	yes	—
16	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
17	C₇⋊C₈	—	2	4	no	—
18	C₂×M₁₆	—	7	2	no	—
19	C₂×C₄⋊C₄	D₇	1	1	yes	characteristic
20	C₂²×C₇	—	2	1	no	—
21	C₂²×C₇	—	2	1	no	—
22	C₂²×C₇	C₄²	1	1	yes	characteristic
23	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
25	C₄×C₇	—	4	2	no	—
26	C₄×C₇	C₂²⋊C₄	1	2	yes	—
27	C₄×C₇	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄	Dic₇	1	2	yes	—
29	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
30	M₁₆	—	7	4	no	—
31	M₁₆	—	14	2	no	—
32	C₂×C₈	—	14	2	no	—
33	C₄⋊C₄	—	2	2	no	—
34	C₂×C₇	—	2	1	no	—
35	C₂×C₇	—	2	1	no	—
36	C₂×C₇	#95	1	1	yes	characteristic
37	C₂×C₇	#100	1	1	yes	characteristic
38	C₂×C₇	#101	1	1	yes	characteristic
39	C₂³	C₂×Dic₇	1	1	yes	characteristic
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	4	2	no	—
42	C₂×C₄	Dic₁₄	1	1	yes	characteristic
43	C₂×C₄	C₄×D₇	1	2	yes	—
44	C₂×C₄	D₂₈	1	1	yes	characteristic
45	C₂×C₄	C₂²⋊D₇	1	2	yes	—
46	C₈	—	14	4	no	—
47	C₇	#343	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₄×Dic₇	1	1	yes	characteristic
51	C₂²	C₄⋊Dic₇	1	1	yes	characteristic
52	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	Dic₇⋊C₄	1	2	yes	—
55	C₄	(C₂×D₇)⋊C₄	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₇⋊#95	1	1	yes	characteristic
59	C₂	#6407	1	1	yes	characteristic
60	C₂	#6410	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₇
commutator	C₂²×C₇	C₄²
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₂×C₇×C₄⋊C₄	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	#343
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. #1136	C₄	semidirect
47. C₇	#343	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊C₄	2
C₂×#1136	2
#343	7