Subgroups — D₇×M₃₂ #89678

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#6386	C₂	1	1	yes	maximal, characteristic, complemented
4	#6372	C₂	1	2	yes	maximal, complemented
5	D₇×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₇	C₄	1	1	yes	characteristic
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇⋊C₈	C₄	1	1	yes	characteristic
9	C₈×D₇	C₄	1	1	yes	characteristic
10	C₈×D₇	C₄	1	1	yes	characteristic
11	C₈×D₇	C₂²	1	1	yes	characteristic
12	C₈×D₇	C₂²	1	1	yes	characteristic
13	C₇×C₁₆	C₂²	1	2	yes	complemented
14	C₇⋊C₁₆	C₂²	1	2	yes	complemented
15	C₂×M₃₂	—	7	1	no	maximal, Sylow
16	C₂²×D₇	C₈	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₇	C₈	1	1	yes	characteristic
19	C₄×D₇	C₈	1	1	yes	characteristic
20	C₄×D₇	C₈	1	1	yes	characteristic
21	C₄×D₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₇	C₂×C₄	1	1	yes	characteristic
23	C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₇×C₈	C₂³	1	1	yes	characteristic
25	C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	—	7	1	no	—
28	M₃₂	—	7	1	no	—
29	M₃₂	—	7	2	no	—
30	M₃₂	D₇	1	1	yes	characteristic, direct product factor
31	C₂×C₁₆	—	7	2	no	—
32	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
33	C₂×D₇	—	2	2	no	—
34	C₂×D₇	C₂×C₈	1	1	yes	characteristic
35	C₂×D₇	C₂×C₈	1	1	yes	characteristic
36	C₄×C₇	C₂×C₈	1	1	yes	characteristic
37	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
38	Dic₇	C₂×C₈	1	1	yes	characteristic
39	Dic₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄	—	7	1	no	—
41	C₂×C₈	—	7	1	no	—
42	C₂×C₈	—	7	1	no	—
43	C₂×C₈	—	7	1	no	—
44	C₂×C₈	—	7	1	no	—
45	C₂×C₈	—	7	1	no	—
46	C₂×C₈	C₂×D₇	1	1	yes	characteristic
47	C₁₆	—	7	2	no	—
48	C₁₆	C₂×D₇	1	2	yes	complemented
49	C₂×C₇	—	2	1	no	—
50	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
51	D₇	—	2	1	no	—
52	D₇	M₃₂	1	2	yes	direct product factor
53	C₂³	—	7	1	no	—
54	C₂×C₄	—	7	1	no	—
55	C₂×C₄	—	7	1	no	—
56	C₂×C₄	—	7	1	no	—
57	C₂×C₄	—	7	1	no	—
58	C₂×C₄	—	7	1	no	—
59	C₂×C₄	C₄×D₇	1	1	yes	characteristic
60	C₈	—	7	1	no	—
61	C₈	—	7	1	no	—
62	C₈	C₄×D₇	1	1	yes	characteristic
63	C₈	C₂²×D₇	1	1	yes	characteristic
64	C₇	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	7	1	no	—
66	C₂²	—	7	1	no	—
67	C₂²	—	14	2	no	—
68	C₂²	C₈×D₇	1	1	yes	characteristic
69	C₄	—	7	1	no	—
70	C₄	—	7	1	no	—
71	C₄	C₈×D₇	1	1	yes	characteristic
72	C₄	C₂×C₄×D₇	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	7	2	no	—
75	C₂	—	14	1	no	—
76	C₂	C₂×C₈×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×D₇
commutator, socle	C₂×C₇	C₂²×C₈
Fitting	C₇×M₃₂	C₂
prime radical	C₂²×D₇	C₈
cosocle	C₇×C₈	C₂³
hypercenter	M₃₂	D₇
hypocenter	C₇	C₂×M₃₂
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₇×M₃₂	C₂	semidirect
3. #6386	C₂	semidirect
4. #6372	C₂	semidirect
5. D₇×C₁₆	C₂	semidirect
13. C₇×C₁₆	C₂²	semidirect
14. C₇⋊C₁₆	C₂²	semidirect
30. M₃₂	D₇	direct (N × H)
48. C₁₆	C₂×D₇	semidirect
52. D₇	M₃₂	direct (N × H)
64. C₇	C₂×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₇	2
C₇×M₃₂	2
#6386	2
#6372	2
D₇×C₁₆	2
C₂×M₃₂	7

Subgroups — D7×M32 #89678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×M₃₂ #89678