Subgroups — C₂²×C₇²×C₆₄ #72190887

Order 12544.

710 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇²×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇²×C₆₄	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₇²×C₁₆	C₄	1	1	yes	characteristic
4	C₂×C₇²×C₃₂	C₄	1	3	yes	—
5	C₂×C₇²×C₃₂	C₂²	1	3	yes	—
6	C₇²×C₆₄	C₂²	1	4	yes	direct product factor
7	C₂²×C₇×C₆₄	C₇	1	8	yes	maximal, direct product factor
8	C₂²×C₇²×C₈	C₈	1	1	yes	characteristic
9	C₂×C₇²×C₁₆	C₈	1	3	yes	—
10	C₂×C₇²×C₁₆	C₂×C₄	1	3	yes	—
11	C₇²×C₃₂	C₂×C₄	1	3	yes	—
12	C₇²×C₃₂	C₂³	1	1	yes	characteristic
13	C₂²×C₇×C₃₂	C₂×C₇	1	8	yes	—
14	C₂×C₇×C₆₄	C₂×C₇	1	48	yes	direct product factor
15	C₂²×C₄×C₇²	C₁₆	1	1	yes	characteristic
16	C₂×C₇²×C₈	C₁₆	1	3	yes	—
17	C₂×C₇²×C₈	C₂×C₈	1	3	yes	—
18	C₇²×C₁₆	C₂×C₈	1	3	yes	—
19	C₇²×C₁₆	C₂²×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₁₆	C₄×C₇	1	8	yes	—
21	C₂×C₇×C₃₂	C₄×C₇	1	24	yes	—
22	C₂×C₇×C₃₂	C₂²×C₇	1	24	yes	—
23	C₇×C₆₄	C₂²×C₇	1	32	yes	direct product factor
24	C₂³×C₇²	C₃₂	1	1	yes	characteristic
25	C₂×C₄×C₇²	C₃₂	1	3	yes	—
26	C₂×C₄×C₇²	C₂×C₁₆	1	3	yes	—
27	C₇²×C₈	C₂×C₁₆	1	3	yes	—
28	C₇²×C₈	C₂²×C₈	1	1	yes	characteristic
29	C₂²×C₆₄	C₇²	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²×C₇×C₈	C₇×C₈	1	8	yes	—
31	C₂×C₇×C₁₆	C₇×C₈	1	24	yes	—
32	C₂×C₇×C₁₆	C₂×C₄×C₇	1	24	yes	—
33	C₇×C₃₂	C₂×C₄×C₇	1	24	yes	—
34	C₇×C₃₂	C₂³×C₇	1	8	yes	—
35	C₂²×C₇²	C₆₄	1	4	yes	direct product factor
36	C₂²×C₇²	C₂×C₃₂	1	3	yes	—
37	C₄×C₇²	C₂×C₃₂	1	3	yes	—
38	C₄×C₇²	C₂²×C₁₆	1	1	yes	characteristic
39	C₂²×C₃₂	C₂×C₇²	1	1	yes	characteristic
40	C₂×C₆₄	C₂×C₇²	1	6	yes	direct product factor
41	C₂²×C₄×C₇	C₇×C₁₆	1	8	yes	—
42	C₂×C₇×C₈	C₇×C₁₆	1	24	yes	—
43	C₂×C₇×C₈	C₂×C₇×C₈	1	24	yes	—
44	C₇×C₁₆	C₂×C₇×C₈	1	24	yes	—
45	C₇×C₁₆	C₂²×C₄×C₇	1	8	yes	—
46	C₂×C₇²	C₂×C₆₄	1	6	yes	direct product factor
47	C₂×C₇²	C₂²×C₃₂	1	1	yes	characteristic
48	C₂²×C₁₆	C₄×C₇²	1	1	yes	characteristic
49	C₂×C₃₂	C₄×C₇²	1	3	yes	—
50	C₂×C₃₂	C₂²×C₇²	1	3	yes	—
51	C₆₄	C₂²×C₇²	1	4	yes	direct product factor
52	C₂³×C₇	C₇×C₃₂	1	8	yes	—
53	C₂×C₄×C₇	C₇×C₃₂	1	24	yes	—
54	C₂×C₄×C₇	C₂×C₇×C₁₆	1	24	yes	—
55	C₇×C₈	C₂×C₇×C₁₆	1	24	yes	—
56	C₇×C₈	C₂²×C₇×C₈	1	8	yes	—
57	C₇²	C₂²×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₈	C₇²×C₈	1	1	yes	characteristic
59	C₂×C₁₆	C₇²×C₈	1	3	yes	—
60	C₂×C₁₆	C₂×C₄×C₇²	1	3	yes	—
61	C₃₂	C₂×C₄×C₇²	1	3	yes	—
62	C₃₂	C₂³×C₇²	1	1	yes	characteristic
63	C₂²×C₇	C₇×C₆₄	1	32	yes	direct product factor
64	C₂²×C₇	C₂×C₇×C₃₂	1	24	yes	—
65	C₄×C₇	C₂×C₇×C₃₂	1	24	yes	—
66	C₄×C₇	C₂²×C₇×C₁₆	1	8	yes	—
67	C₂²×C₄	C₇²×C₁₆	1	1	yes	characteristic
68	C₂×C₈	C₇²×C₁₆	1	3	yes	—
69	C₂×C₈	C₂×C₇²×C₈	1	3	yes	—
70	C₁₆	C₂×C₇²×C₈	1	3	yes	—
71	C₁₆	C₂²×C₄×C₇²	1	1	yes	characteristic
72	C₂×C₇	C₂×C₇×C₆₄	1	48	yes	direct product factor
73	C₂×C₇	C₂²×C₇×C₃₂	1	8	yes	—
74	C₂³	C₇²×C₃₂	1	1	yes	characteristic
75	C₂×C₄	C₇²×C₃₂	1	3	yes	—
76	C₂×C₄	C₂×C₇²×C₁₆	1	3	yes	—
77	C₈	C₂×C₇²×C₁₆	1	3	yes	—
78	C₈	C₂²×C₇²×C₈	1	1	yes	characteristic
79	C₇	C₂²×C₇×C₆₄	1	8	yes	direct product factor
80	C₂²	C₇²×C₆₄	1	4	yes	direct product factor
81	C₂²	C₂×C₇²×C₃₂	1	3	yes	—
82	C₄	C₂×C₇²×C₃₂	1	3	yes	—
83	C₄	C₂²×C₇²×C₁₆	1	1	yes	characteristic
84	C₂	C₂×C₇²×C₆₄	1	6	yes	direct product factor
85	C₂	C₂²×C₇²×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂³×C₇²
socle, prime radical	C₂³×C₇²	C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇²×C₆₄	C₂	direct (N × H)
6. C₇²×C₆₄	C₂²	direct (N × H)
7. C₂²×C₇×C₆₄	C₇	direct (N × H)
14. C₂×C₇×C₆₄	C₂×C₇	direct (N × H)
23. C₇×C₆₄	C₂²×C₇	direct (N × H)
29. C₂²×C₆₄	C₇²	direct (N × H)
35. C₂²×C₇²	C₆₄	direct (N × H)
40. C₂×C₆₄	C₂×C₇²	direct (N × H)
46. C₂×C₇²	C₂×C₆₄	direct (N × H)
51. C₆₄	C₂²×C₇²	direct (N × H)
57. C₇²	C₂²×C₆₄	direct (N × H)
63. C₂²×C₇	C₇×C₆₄	direct (N × H)
72. C₂×C₇	C₂×C₇×C₆₄	direct (N × H)
79. C₇	C₂²×C₇×C₆₄	direct (N × H)
80. C₂²	C₇²×C₆₄	direct (N × H)
84. C₂	C₂×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
C₂²×C₇²×C₃₂	2
C₂×C₇²×C₆₄	2
C₂²×C₇×C₆₄	7

Subgroups — C22×C72×C64 #72190887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇²×C₆₄ #72190887