Subgroups — #89342

Order 448.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×Q₈	—	2	1	no	—
5	C₇×C₄⋊C₄	—	2	1	no	—
6	C₇×C₄⋊C₄	C₄	1	2	yes	—
7	C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₇⋊C₈	—	2	1	no	—
9	C₂×C₇⋊C₈	—	2	1	no	—
10	#332	—	7	1	no	maximal, Sylow
11	C₇×Q₈	—	4	1	no	—
12	C₇×Q₈	—	4	1	no	—
13	C₂×C₄×C₇	—	2	2	no	—
14	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₇	D₄	1	1	yes	characteristic
17	C₇⋊C₈	—	4	1	no	—
18	C₇⋊C₈	—	4	1	no	—
19	C₄⋊Q₈	D₇	1	1	yes	characteristic
20	C₄⋊C₈	—	7	1	no	—
21	C₄⋊C₈	—	7	1	no	—
22	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₇	—	2	1	no	—
24	C₄×C₇	—	4	2	no	—
25	C₄×C₇	D₈	1	1	yes	characteristic
26	C₄×C₇	SD₁₆	1	1	yes	characteristic
27	C₄×C₇	SD₁₆	1	1	yes	characteristic
28	C₄×C₇	Q₁₆	1	1	yes	characteristic
29	C₂×Q₈	—	2	1	no	—
30	C₂×C₈	—	14	1	no	—
31	C₂×C₈	—	14	1	no	—
32	C₄⋊C₄	—	2	1	no	—
33	C₄⋊C₄	Dic₇	1	2	yes	—
34	C₄²	C₂×D₇	1	1	yes	characteristic
35	C₂×C₇	#101	1	1	yes	characteristic
36	C₂×C₇	#102	1	1	yes	characteristic
37	C₂×C₇	#103	1	1	yes	characteristic
38	Q₈	—	4	1	no	—
39	Q₈	—	4	1	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
42	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
44	C₈	—	28	1	no	—
45	C₈	—	28	1	no	—
46	C₇	#332	1	1	yes	characteristic, complemented, Sylow
47	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	4	2	no	—
50	C₄	#1141	1	1	yes	characteristic
51	C₄	#1142	1	1	yes	characteristic
52	C₄	#1143	1	1	yes	characteristic
53	C₄	#1144	1	1	yes	characteristic
54	C₂	#6406	1	1	yes	characteristic
55	C₂	#6409	1	1	yes	characteristic
56	C₂	#6410	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂×C₄
Frattini	C₄²	C₂×D₇
Fitting	C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₇	C₂²⋊C₄
cosocle	C₄²×C₇	C₂²
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#332
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
46. C₇	#332	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊Q₈	2
C₄⋊(C₇⋊C₈)	2
#332	7