Subgroups — #89361

Order 448.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#135	C₂	1	1	yes	maximal, characteristic
2	#6391	C₂	1	1	yes	maximal, characteristic, complemented
3	#6386	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#41	—	2	1	no	—
5	C₇×Q₁₆	C₄	1	1	yes	characteristic, complemented
6	C₇×SD₁₆	—	2	1	no	—
7	C₇×D₈	C₄	1	1	yes	characteristic, complemented
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₇	—	2	1	no	—
10	C₇⋊C₁₆	—	2	1	no	—
11	#360	—	7	1	no	maximal, Sylow
12	C₇×Q₈	—	2	1	no	—
13	C₇×D₄	—	2	1	no	—
14	C₇×D₄	—	4	1	no	—
15	C₂×C₄×C₇	—	4	1	no	—
16	C₂×C₄×C₇	D₄	1	1	yes	characteristic
17	C₂×Dic₇	—	4	1	no	—
18	C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₇×C₈	D₄	1	1	yes	characteristic
20	#135	D₇	1	1	yes	characteristic
21	M₃₂	—	7	1	no	—
22	#106	—	7	1	no	—
23	C₂²×C₇	—	4	1	no	—
24	C₂²×C₇	D₈	1	1	yes	characteristic
25	C₄×C₇	—	4	1	no	—
26	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×C₇	SD₁₆	1	1	yes	characteristic
28	Dic₇	—	8	1	no	—
29	#41	—	2	1	no	—
30	Q₁₆	Dic₇	1	1	yes	characteristic, complemented
31	SD₁₆	—	2	1	no	—
32	D₈	Dic₇	1	1	yes	characteristic, complemented
33	C₂×C₈	C₂×D₇	1	1	yes	characteristic
34	C₄⋊C₄	—	14	1	no	—
35	C₁₆	—	14	1	no	—
36	C₂×C₇	—	2	1	no	—
37	C₂×C₇	—	8	1	no	—
38	C₂×C₇	#102	1	1	yes	characteristic
39	Q₈	—	2	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	—	28	1	no	—
44	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
45	C₈	C₂×Dic₇	1	1	yes	characteristic
46	C₈	C₂²⋊D₇	1	1	yes	characteristic
47	C₇	#360	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	4	1	no	—
49	C₂²	#1141	1	1	yes	characteristic
50	C₄	—	4	1	no	—
51	C₄	—	56	1	no	—
52	C₄	#1142	1	1	yes	characteristic
53	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	1	no	—
56	C₂	#6406	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6406
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting, prime radical	C₇×#135	C₂
socle	C₂×C₇	#102
cosocle	C₂×C₇×C₈	C₂²
hypercenter	#135	D₇
hypocenter	C₇	#360
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #6391	C₂	semidirect
3. #6386	C₂	semidirect
5. C₇×Q₁₆	C₄	semidirect
7. C₇×D₈	C₄	semidirect
30. Q₁₆	Dic₇	semidirect
32. D₈	Dic₇	semidirect
47. C₇	#360	semidirect

Maximal subgroups

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

subgroup	index
C₇×#135	2
#6391	2
#6386	2
#360	7