Subgroups — C₄×C₇²×C₆₄ #72190885

Order 12544.

430 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇²×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇²×C₆₄	C₂	1	2	yes	maximal
3	C₄×C₇²×C₁₆	C₄	1	1	yes	characteristic
4	C₂×C₇²×C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₇²×C₃₂	C₂²	1	1	yes	characteristic
6	C₇²×C₆₄	C₄	1	4	yes	direct product factor
7	C₄×C₇×C₆₄	C₇	1	8	yes	maximal, direct product factor
8	C₂×C₇²×C₁₆	C₈	1	1	yes	characteristic
9	C₂×C₇²×C₁₆	C₂×C₄	1	1	yes	characteristic
10	C₄×C₇²×C₈	C₈	1	1	yes	characteristic
11	C₇²×C₃₂	C₈	1	2	yes	—
12	C₇²×C₃₂	C₂×C₄	1	2	yes	—
13	C₂×C₇×C₆₄	C₂×C₇	1	16	yes	—
14	C₄×C₇×C₃₂	C₂×C₇	1	8	yes	—
15	C₂×C₇²×C₈	C₁₆	1	1	yes	characteristic
16	C₂×C₇²×C₈	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₇²	C₁₆	1	1	yes	characteristic
18	C₇²×C₁₆	C₁₆	1	2	yes	—
19	C₇²×C₁₆	C₄²	1	1	yes	characteristic
20	C₇²×C₁₆	C₂×C₈	1	1	yes	characteristic
21	C₂×C₇×C₃₂	C₄×C₇	1	8	yes	—
22	C₂×C₇×C₃₂	C₂²×C₇	1	8	yes	—
23	C₄×C₇×C₁₆	C₄×C₇	1	8	yes	—
24	C₇×C₆₄	C₄×C₇	1	32	yes	direct product factor
25	C₂×C₄×C₇²	C₃₂	1	2	yes	—
26	C₂×C₄×C₇²	C₂×C₁₆	1	1	yes	characteristic
27	C₇²×C₈	C₃₂	1	2	yes	—
28	C₇²×C₈	C₄×C₈	1	1	yes	characteristic
29	C₇²×C₈	C₂×C₁₆	1	1	yes	characteristic
30	C₄×C₆₄	C₇²	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₇×C₁₆	C₇×C₈	1	8	yes	—
32	C₂×C₇×C₁₆	C₂×C₄×C₇	1	8	yes	—
33	C₄×C₇×C₈	C₇×C₈	1	8	yes	—
34	C₇×C₃₂	C₇×C₈	1	16	yes	—
35	C₇×C₃₂	C₂×C₄×C₇	1	16	yes	—
36	C₂²×C₇²	C₂×C₃₂	1	1	yes	characteristic
37	C₄×C₇²	C₆₄	1	4	yes	direct product factor
38	C₄×C₇²	C₄×C₁₆	1	1	yes	characteristic
39	C₄×C₇²	C₂×C₃₂	1	1	yes	characteristic
40	C₂×C₆₄	C₂×C₇²	1	2	yes	—
41	C₄×C₃₂	C₂×C₇²	1	1	yes	characteristic
42	C₂×C₇×C₈	C₇×C₁₆	1	8	yes	—
43	C₂×C₇×C₈	C₂×C₇×C₈	1	8	yes	—
44	C₄²×C₇	C₇×C₁₆	1	8	yes	—
45	C₇×C₁₆	C₇×C₁₆	1	16	yes	—
46	C₇×C₁₆	C₄²×C₇	1	8	yes	—
47	C₇×C₁₆	C₂×C₇×C₈	1	8	yes	—
48	C₂×C₇²	C₄×C₃₂	1	1	yes	characteristic
49	C₂×C₇²	C₂×C₆₄	1	2	yes	—
50	C₂×C₃₂	C₄×C₇²	1	1	yes	characteristic
51	C₂×C₃₂	C₂²×C₇²	1	1	yes	characteristic
52	C₄×C₁₆	C₄×C₇²	1	1	yes	characteristic
53	C₆₄	C₄×C₇²	1	4	yes	direct product factor
54	C₂×C₄×C₇	C₇×C₃₂	1	16	yes	—
55	C₂×C₄×C₇	C₂×C₇×C₁₆	1	8	yes	—
56	C₇×C₈	C₇×C₃₂	1	16	yes	—
57	C₇×C₈	C₄×C₇×C₈	1	8	yes	—
58	C₇×C₈	C₂×C₇×C₁₆	1	8	yes	—
59	C₇²	C₄×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₁₆	C₇²×C₈	1	1	yes	characteristic
61	C₂×C₁₆	C₂×C₄×C₇²	1	1	yes	characteristic
62	C₄×C₈	C₇²×C₈	1	1	yes	characteristic
63	C₃₂	C₇²×C₈	1	2	yes	—
64	C₃₂	C₂×C₄×C₇²	1	2	yes	—
65	C₂²×C₇	C₂×C₇×C₃₂	1	8	yes	—
66	C₄×C₇	C₇×C₆₄	1	32	yes	direct product factor
67	C₄×C₇	C₄×C₇×C₁₆	1	8	yes	—
68	C₄×C₇	C₂×C₇×C₃₂	1	8	yes	—
69	C₂×C₈	C₇²×C₁₆	1	1	yes	characteristic
70	C₂×C₈	C₂×C₇²×C₈	1	1	yes	characteristic
71	C₄²	C₇²×C₁₆	1	1	yes	characteristic
72	C₁₆	C₇²×C₁₆	1	2	yes	—
73	C₁₆	C₄²×C₇²	1	1	yes	characteristic
74	C₁₆	C₂×C₇²×C₈	1	1	yes	characteristic
75	C₂×C₇	C₄×C₇×C₃₂	1	8	yes	—
76	C₂×C₇	C₂×C₇×C₆₄	1	16	yes	—
77	C₂×C₄	C₇²×C₃₂	1	2	yes	—
78	C₂×C₄	C₂×C₇²×C₁₆	1	1	yes	characteristic
79	C₈	C₇²×C₃₂	1	2	yes	—
80	C₈	C₄×C₇²×C₈	1	1	yes	characteristic
81	C₈	C₂×C₇²×C₁₆	1	1	yes	characteristic
82	C₇	C₄×C₇×C₆₄	1	8	yes	direct product factor
83	C₂²	C₂×C₇²×C₃₂	1	1	yes	characteristic
84	C₄	C₇²×C₆₄	1	4	yes	direct product factor
85	C₄	C₂×C₇²×C₃₂	1	1	yes	characteristic
86	C₄	C₄×C₇²×C₁₆	1	1	yes	characteristic
87	C₂	C₂×C₇²×C₆₄	1	2	yes	—
88	C₂	C₄×C₇²×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃₂	C₂²×C₇²
socle, prime radical	C₂²×C₇²	C₂×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₇²×C₆₄	C₄	direct (N × H)
7. C₄×C₇×C₆₄	C₇	direct (N × H)
24. C₇×C₆₄	C₄×C₇	direct (N × H)
30. C₄×C₆₄	C₇²	direct (N × H)
37. C₄×C₇²	C₆₄	direct (N × H)
53. C₆₄	C₄×C₇²	direct (N × H)
59. C₇²	C₄×C₆₄	direct (N × H)
66. C₄×C₇	C₇×C₆₄	direct (N × H)
82. C₇	C₄×C₇×C₆₄	direct (N × H)
84. C₄	C₇²×C₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇²×C₃₂	2
C₂×C₇²×C₆₄	2
C₄×C₇×C₆₄	7

Subgroups — C4×C72×C64 #72190885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇²×C₆₄ #72190885