Subgroups — C₂³×C₇²×C₃₂ #72190893

Order 12544.

2710 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇²×C₃₂	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₇²×C₈	C₄	1	1	yes	characteristic
4	C₂²×C₇²×C₁₆	C₄	1	7	yes	—
5	C₂²×C₇²×C₁₆	C₂²	1	7	yes	—
6	C₂×C₇²×C₃₂	C₂²	1	28	yes	direct product factor
7	C₂³×C₇×C₃₂	C₇	1	8	yes	maximal, direct product factor
8	C₂³×C₄×C₇²	C₈	1	1	yes	characteristic
9	C₂²×C₇²×C₈	C₈	1	7	yes	—
10	C₂²×C₇²×C₈	C₂×C₄	1	7	yes	—
11	C₂×C₇²×C₁₆	C₂×C₄	1	21	yes	—
12	C₂×C₇²×C₁₆	C₂³	1	7	yes	—
13	C₇²×C₃₂	C₂³	1	8	yes	direct product factor
14	C₂³×C₇×C₁₆	C₂×C₇	1	8	yes	—
15	C₂²×C₇×C₃₂	C₂×C₇	1	112	yes	direct product factor
16	C₂⁴×C₇²	C₁₆	1	1	yes	characteristic
17	C₂²×C₄×C₇²	C₁₆	1	7	yes	—
18	C₂²×C₄×C₇²	C₂×C₈	1	7	yes	—
19	C₂×C₇²×C₈	C₂×C₈	1	21	yes	—
20	C₂×C₇²×C₈	C₂²×C₄	1	7	yes	—
21	C₇²×C₁₆	C₂²×C₄	1	7	yes	—
22	C₇²×C₁₆	C₂⁴	1	1	yes	characteristic
23	C₂³×C₇×C₈	C₄×C₇	1	8	yes	—
24	C₂²×C₇×C₁₆	C₄×C₇	1	56	yes	—
25	C₂²×C₇×C₁₆	C₂²×C₇	1	56	yes	—
26	C₂×C₇×C₃₂	C₂²×C₇	1	224	yes	direct product factor
27	C₂³×C₇²	C₃₂	1	8	yes	direct product factor
28	C₂³×C₇²	C₂×C₁₆	1	7	yes	—
29	C₂×C₄×C₇²	C₂×C₁₆	1	21	yes	—
30	C₂×C₄×C₇²	C₂²×C₈	1	7	yes	—
31	C₇²×C₈	C₂²×C₈	1	7	yes	—
32	C₇²×C₈	C₂³×C₄	1	1	yes	characteristic
33	C₂³×C₃₂	C₇²	1	1	yes	characteristic, direct product factor, Sylow
34	C₂³×C₄×C₇	C₇×C₈	1	8	yes	—
35	C₂²×C₇×C₈	C₇×C₈	1	56	yes	—
36	C₂²×C₇×C₈	C₂×C₄×C₇	1	56	yes	—
37	C₂×C₇×C₁₆	C₂×C₄×C₇	1	168	yes	—
38	C₂×C₇×C₁₆	C₂³×C₇	1	56	yes	—
39	C₇×C₃₂	C₂³×C₇	1	64	yes	direct product factor
40	C₂²×C₇²	C₂×C₃₂	1	28	yes	direct product factor
41	C₂²×C₇²	C₂²×C₁₆	1	7	yes	—
42	C₄×C₇²	C₂²×C₁₆	1	7	yes	—
43	C₄×C₇²	C₂³×C₈	1	1	yes	characteristic
44	C₂³×C₁₆	C₂×C₇²	1	1	yes	characteristic
45	C₂²×C₃₂	C₂×C₇²	1	14	yes	direct product factor
46	C₂⁴×C₇	C₇×C₁₆	1	8	yes	—
47	C₂²×C₄×C₇	C₇×C₁₆	1	56	yes	—
48	C₂²×C₄×C₇	C₂×C₇×C₈	1	56	yes	—
49	C₂×C₇×C₈	C₂×C₇×C₈	1	168	yes	—
50	C₂×C₇×C₈	C₂²×C₄×C₇	1	56	yes	—
51	C₇×C₁₆	C₂²×C₄×C₇	1	56	yes	—
52	C₇×C₁₆	C₂⁴×C₇	1	8	yes	—
53	C₂×C₇²	C₂²×C₃₂	1	14	yes	direct product factor
54	C₂×C₇²	C₂³×C₁₆	1	1	yes	characteristic
55	C₂³×C₈	C₄×C₇²	1	1	yes	characteristic
56	C₂²×C₁₆	C₄×C₇²	1	7	yes	—
57	C₂²×C₁₆	C₂²×C₇²	1	7	yes	—
58	C₂×C₃₂	C₂²×C₇²	1	28	yes	direct product factor
59	C₂³×C₇	C₇×C₃₂	1	64	yes	direct product factor
60	C₂³×C₇	C₂×C₇×C₁₆	1	56	yes	—
61	C₂×C₄×C₇	C₂×C₇×C₁₆	1	168	yes	—
62	C₂×C₄×C₇	C₂²×C₇×C₈	1	56	yes	—
63	C₇×C₈	C₂²×C₇×C₈	1	56	yes	—
64	C₇×C₈	C₂³×C₄×C₇	1	8	yes	—
65	C₇²	C₂³×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
66	C₂³×C₄	C₇²×C₈	1	1	yes	characteristic
67	C₂²×C₈	C₇²×C₈	1	7	yes	—
68	C₂²×C₈	C₂×C₄×C₇²	1	7	yes	—
69	C₂×C₁₆	C₂×C₄×C₇²	1	21	yes	—
70	C₂×C₁₆	C₂³×C₇²	1	7	yes	—
71	C₃₂	C₂³×C₇²	1	8	yes	direct product factor
72	C₂²×C₇	C₂×C₇×C₃₂	1	224	yes	direct product factor
73	C₂²×C₇	C₂²×C₇×C₁₆	1	56	yes	—
74	C₄×C₇	C₂²×C₇×C₁₆	1	56	yes	—
75	C₄×C₇	C₂³×C₇×C₈	1	8	yes	—
76	C₂⁴	C₇²×C₁₆	1	1	yes	characteristic
77	C₂²×C₄	C₇²×C₁₆	1	7	yes	—
78	C₂²×C₄	C₂×C₇²×C₈	1	7	yes	—
79	C₂×C₈	C₂×C₇²×C₈	1	21	yes	—
80	C₂×C₈	C₂²×C₄×C₇²	1	7	yes	—
81	C₁₆	C₂²×C₄×C₇²	1	7	yes	—
82	C₁₆	C₂⁴×C₇²	1	1	yes	characteristic
83	C₂×C₇	C₂²×C₇×C₃₂	1	112	yes	direct product factor
84	C₂×C₇	C₂³×C₇×C₁₆	1	8	yes	—
85	C₂³	C₇²×C₃₂	1	8	yes	direct product factor
86	C₂³	C₂×C₇²×C₁₆	1	7	yes	—
87	C₂×C₄	C₂×C₇²×C₁₆	1	21	yes	—
88	C₂×C₄	C₂²×C₇²×C₈	1	7	yes	—
89	C₈	C₂²×C₇²×C₈	1	7	yes	—
90	C₈	C₂³×C₄×C₇²	1	1	yes	characteristic
91	C₇	C₂³×C₇×C₃₂	1	8	yes	direct product factor
92	C₂²	C₂×C₇²×C₃₂	1	28	yes	direct product factor
93	C₂²	C₂²×C₇²×C₁₆	1	7	yes	—
94	C₄	C₂²×C₇²×C₁₆	1	7	yes	—
95	C₄	C₂³×C₇²×C₈	1	1	yes	characteristic
96	C₂	C₂²×C₇²×C₃₂	1	14	yes	direct product factor
97	C₂	C₂³×C₇²×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂⁴×C₇²
socle, prime radical	C₂⁴×C₇²	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇²×C₃₂	C₂	direct (N × H)
6. C₂×C₇²×C₃₂	C₂²	direct (N × H)
7. C₂³×C₇×C₃₂	C₇	direct (N × H)
13. C₇²×C₃₂	C₂³	direct (N × H)
15. C₂²×C₇×C₃₂	C₂×C₇	direct (N × H)
26. C₂×C₇×C₃₂	C₂²×C₇	direct (N × H)
27. C₂³×C₇²	C₃₂	direct (N × H)
33. C₂³×C₃₂	C₇²	direct (N × H)
39. C₇×C₃₂	C₂³×C₇	direct (N × H)
40. C₂²×C₇²	C₂×C₃₂	direct (N × H)
45. C₂²×C₃₂	C₂×C₇²	direct (N × H)
53. C₂×C₇²	C₂²×C₃₂	direct (N × H)
58. C₂×C₃₂	C₂²×C₇²	direct (N × H)
59. C₂³×C₇	C₇×C₃₂	direct (N × H)
65. C₇²	C₂³×C₃₂	direct (N × H)
71. C₃₂	C₂³×C₇²	direct (N × H)
72. C₂²×C₇	C₂×C₇×C₃₂	direct (N × H)
83. C₂×C₇	C₂²×C₇×C₃₂	direct (N × H)
85. C₂³	C₇²×C₃₂	direct (N × H)
91. C₇	C₂³×C₇×C₃₂	direct (N × H)
92. C₂²	C₂×C₇²×C₃₂	direct (N × H)
96. C₂	C₂²×C₇²×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
C₂³×C₇²×C₁₆	2
C₂²×C₇²×C₃₂	2
C₂³×C₇×C₃₂	7

Subgroups — C23×C72×C32 #72190893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇²×C₃₂ #72190893