Subgroups — C₇²×C₈×C₃₂ #72190889

Order 12544.

670 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇²×C₃₂	C₂	1	2	yes	maximal
3	C₇²×C₈²	C₄	1	1	yes	characteristic
4	C₄×C₇²×C₁₆	C₄	1	1	yes	characteristic
5	C₄×C₇²×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₇²×C₃₂	C₄	1	4	yes	—
7	C₇×C₈×C₃₂	C₇	1	8	yes	maximal, direct product factor
8	C₂×C₇²×C₁₆	C₈	1	2	yes	—
9	C₂×C₇²×C₁₆	C₂×C₄	1	2	yes	—
10	C₄×C₇²×C₈	C₈	1	2	yes	—
11	C₄×C₇²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₇²×C₃₂	C₈	1	8	yes	direct product factor
13	C₄×C₇×C₃₂	C₂×C₇	1	16	yes	—
14	C₇×C₈×C₁₆	C₂×C₇	1	8	yes	—
15	C₂×C₇²×C₈	C₁₆	1	4	yes	—
16	C₂×C₇²×C₈	C₄²	1	1	yes	characteristic
17	C₂×C₇²×C₈	C₂×C₈	1	1	yes	characteristic
18	C₄²×C₇²	C₂×C₈	1	1	yes	characteristic
19	C₇²×C₁₆	C₁₆	1	4	yes	—
20	C₇²×C₁₆	C₂×C₈	1	4	yes	—
21	C₂×C₇×C₃₂	C₄×C₇	1	32	yes	—
22	C₄×C₇×C₁₆	C₄×C₇	1	8	yes	—
23	C₄×C₇×C₁₆	C₂²×C₇	1	8	yes	—
24	C₇×C₈²	C₄×C₇	1	8	yes	—
25	C₂×C₄×C₇²	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₇²	C₂×C₁₆	1	2	yes	—
27	C₇²×C₈	C₃₂	1	8	yes	direct product factor
28	C₇²×C₈	C₄×C₈	1	2	yes	—
29	C₇²×C₈	C₂×C₁₆	1	2	yes	—
30	C₈×C₃₂	C₇²	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₇×C₁₆	C₇×C₈	1	16	yes	—
32	C₂×C₇×C₁₆	C₂×C₄×C₇	1	16	yes	—
33	C₄×C₇×C₈	C₇×C₈	1	16	yes	—
34	C₄×C₇×C₈	C₂×C₄×C₇	1	8	yes	—
35	C₇×C₃₂	C₇×C₈	1	64	yes	direct product factor
36	C₂²×C₇²	C₄×C₁₆	1	1	yes	characteristic
37	C₄×C₇²	C₈²	1	1	yes	characteristic
38	C₄×C₇²	C₄×C₁₆	1	1	yes	characteristic
39	C₄×C₇²	C₂×C₃₂	1	4	yes	—
40	C₄×C₃₂	C₂×C₇²	1	2	yes	—
41	C₈×C₁₆	C₂×C₇²	1	1	yes	characteristic
42	C₂×C₇×C₈	C₇×C₁₆	1	32	yes	—
43	C₂×C₇×C₈	C₄²×C₇	1	8	yes	—
44	C₂×C₇×C₈	C₂×C₇×C₈	1	8	yes	—
45	C₄²×C₇	C₂×C₇×C₈	1	8	yes	—
46	C₇×C₁₆	C₇×C₁₆	1	32	yes	—
47	C₇×C₁₆	C₂×C₇×C₈	1	32	yes	—
48	C₂×C₇²	C₈×C₁₆	1	1	yes	characteristic
49	C₂×C₇²	C₄×C₃₂	1	2	yes	—
50	C₂×C₃₂	C₄×C₇²	1	4	yes	—
51	C₄×C₁₆	C₄×C₇²	1	1	yes	characteristic
52	C₄×C₁₆	C₂²×C₇²	1	1	yes	characteristic
53	C₈²	C₄×C₇²	1	1	yes	characteristic
54	C₂×C₄×C₇	C₄×C₇×C₈	1	8	yes	—
55	C₂×C₄×C₇	C₂×C₇×C₁₆	1	16	yes	—
56	C₇×C₈	C₇×C₃₂	1	64	yes	direct product factor
57	C₇×C₈	C₄×C₇×C₈	1	16	yes	—
58	C₇×C₈	C₂×C₇×C₁₆	1	16	yes	—
59	C₇²	C₈×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₁₆	C₇²×C₈	1	2	yes	—
61	C₂×C₁₆	C₂×C₄×C₇²	1	2	yes	—
62	C₄×C₈	C₇²×C₈	1	2	yes	—
63	C₄×C₈	C₂×C₄×C₇²	1	1	yes	characteristic
64	C₃₂	C₇²×C₈	1	8	yes	direct product factor
65	C₂²×C₇	C₄×C₇×C₁₆	1	8	yes	—
66	C₄×C₇	C₇×C₈²	1	8	yes	—
67	C₄×C₇	C₄×C₇×C₁₆	1	8	yes	—
68	C₄×C₇	C₂×C₇×C₃₂	1	32	yes	—
69	C₂×C₈	C₇²×C₁₆	1	4	yes	—
70	C₂×C₈	C₄²×C₇²	1	1	yes	characteristic
71	C₂×C₈	C₂×C₇²×C₈	1	1	yes	characteristic
72	C₄²	C₂×C₇²×C₈	1	1	yes	characteristic
73	C₁₆	C₇²×C₁₆	1	4	yes	—
74	C₁₆	C₂×C₇²×C₈	1	4	yes	—
75	C₂×C₇	C₇×C₈×C₁₆	1	8	yes	—
76	C₂×C₇	C₄×C₇×C₃₂	1	16	yes	—
77	C₂×C₄	C₄×C₇²×C₈	1	1	yes	characteristic
78	C₂×C₄	C₂×C₇²×C₁₆	1	2	yes	—
79	C₈	C₇²×C₃₂	1	8	yes	direct product factor
80	C₈	C₄×C₇²×C₈	1	2	yes	—
81	C₈	C₂×C₇²×C₁₆	1	2	yes	—
82	C₇	C₇×C₈×C₃₂	1	8	yes	direct product factor
83	C₂²	C₄×C₇²×C₁₆	1	1	yes	characteristic
84	C₄	C₂×C₇²×C₃₂	1	4	yes	—
85	C₄	C₄×C₇²×C₁₆	1	1	yes	characteristic
86	C₄	C₇²×C₈²	1	1	yes	characteristic
87	C₂	C₄×C₇²×C₃₂	1	2	yes	—
88	C₂	C₇²×C₈×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₁₆	C₂²×C₇²
socle, prime radical	C₂²×C₇²	C₄×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₇×C₈×C₃₂	C₇	direct (N × H)
12. C₇²×C₃₂	C₈	direct (N × H)
27. C₇²×C₈	C₃₂	direct (N × H)
30. C₈×C₃₂	C₇²	direct (N × H)
35. C₇×C₃₂	C₇×C₈	direct (N × H)
56. C₇×C₈	C₇×C₃₂	direct (N × H)
59. C₇²	C₈×C₃₂	direct (N × H)
64. C₃₂	C₇²×C₈	direct (N × H)
79. C₈	C₇²×C₃₂	direct (N × H)
82. C₇	C₇×C₈×C₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×C₈×C₁₆	2
C₄×C₇²×C₃₂	2
C₇×C₈×C₃₂	7

Subgroups — C72×C8×C32 #72190889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×C₈×C₃₂ #72190889