Subgroups — #89292

Order 448.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6467	C₂	1	1	yes	maximal, characteristic
2	C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#1157	—	2	1	no	—
5	C₇×M₁₆	—	2	1	no	—
6	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
7	Dic₂₈	C₄	1	1	yes	characteristic
8	D₅₆	C₄	1	1	yes	characteristic
9	#1132	—	2	1	no	—
10	C₇⋊C₁₆	—	2	1	no	—
11	#359	—	7	1	no	maximal, Sylow
12	C₂×C₄×C₇	D₄	1	1	yes	characteristic
13	C₂²⋊D₇	—	4	1	no	—
14	D₂₈	—	2	1	no	—
15	C₄×D₇	—	4	1	no	—
16	Dic₁₄	—	2	1	no	—
17	C₇×C₈	—	4	1	no	—
18	C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₇×C₈	D₄	1	1	yes	characteristic
20	#135	—	7	1	no	—
21	C₂×C₁₆	—	7	1	no	—
22	#108	D₇	1	1	yes	characteristic, complemented
23	C₂²×C₇	SD₁₆	1	1	yes	characteristic
24	C₂×D₇	—	4	1	no	—
25	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₇	D₈	1	1	yes	characteristic
27	Dic₇	—	4	1	no	—
28	#41	—	14	1	no	—
29	Q₁₆	—	7	1	no	—
30	SD₁₆	—	14	1	no	—
31	D₈	—	7	1	no	—
32	M₁₆	—	2	1	no	—
33	C₂×C₈	C₂×D₇	1	1	yes	characteristic
34	C₁₆	—	14	1	no	—
35	C₂×C₇	—	2	1	no	—
36	C₂×C₇	#102	1	1	yes	characteristic
37	D₇	—	8	1	no	—
38	Q₈	—	14	1	no	—
39	D₄	—	14	1	no	—
40	D₄	—	28	1	no	—
41	C₂×C₄	—	28	1	no	—
42	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
43	C₈	—	4	1	no	—
44	C₈	C₄×D₇	1	1	yes	characteristic
45	C₈	D₂₈	1	1	yes	characteristic
46	C₇	#359	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	28	1	no	—
48	C₂²	#1143	1	1	yes	characteristic
49	C₄	—	28	1	no	—
50	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
51	C₄	#1141	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	56	1	no	—
54	C₂	#6383	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1141
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	#102
prime radical	#6467	C₂
cosocle	C₂×C₇×C₈	C₂²
hypercenter	#108	D₇
hypocenter	C₇	#359
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₇×#108	C₂	semidirect
3. C₂×C₇⋊C₁₆	C₂	semidirect
22. #108	D₇	semidirect
46. C₇	#359	semidirect

Maximal subgroups

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

subgroup	index
#6467	2
C₇×#108	2
C₂×C₇⋊C₁₆	2
#359	7