Group #30624524 — subgroups

Subgroups — C₂²×D₆₁⋊C₃₂ #30624524

Order 15616.

542 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 320 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₁⋊C₃₂	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#14519827	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
4	D₆₁⋊C₃₂	C₂²	1	16	yes	direct product factor
5	C₂×#14519827	C₂²	1	12	yes	complemented
6	C₂²×C₈×D₆₁	C₄	1	1	yes	characteristic
7	C₂²×#11758185	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₆₁	C₄	1	6	yes	—
9	C₂×C₁₆×D₆₁	C₂²	1	6	yes	—
10	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic
11	#14519827	C₂³	1	8	yes	complemented
12	C₂²×C₄×D₆₁	C₈	1	1	yes	characteristic
13	C₂²×#92418	C₈	1	1	yes	characteristic
14	C₂×C₈×D₆₁	C₈	1	6	yes	—
15	C₂×C₈×D₆₁	C₂×C₄	1	6	yes	—
16	C₂×#11758185	C₂×C₄	1	3	yes	—
17	C₂×#11758185	C₂³	1	3	yes	—
18	C₁₆×D₆₁	C₂×C₄	1	12	yes	—
19	C₁₆×D₆₁	C₂³	1	4	yes	—
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₁	C₂×C₄	1	6	yes	—
22	C₂³×D₆₁	C₁₆	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
24	C₂²×Dic₆₁	C₁₆	1	1	yes	characteristic
25	C₂×C₄×D₆₁	C₁₆	1	6	yes	—
26	C₂×C₄×D₆₁	C₂×C₈	1	6	yes	—
27	C₂×C₈×C₆₁	C₂×C₈	1	3	yes	—
28	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
29	C₂×#92418	C₂×C₈	1	6	yes	—
30	C₈×D₆₁	C₂×C₈	1	12	yes	—
31	C₈×D₆₁	C₂²×C₄	1	4	yes	—
32	C₁₆×C₆₁	C₂²×C₄	1	4	yes	—
33	#11758185	C₂²×C₄	1	3	yes	—
34	#11758185	C₂⁴	1	1	yes	characteristic
35	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
36	C₂²×D₆₁	C₃₂	1	8	yes	complemented
37	C₂²×D₆₁	C₂×C₁₆	1	6	yes	—
38	C₂×C₄×C₆₁	C₂×C₁₆	1	3	yes	—
39	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
40	C₂×Dic₆₁	C₂×C₁₆	1	6	yes	—
41	C₄×D₆₁	C₂×C₁₆	1	12	yes	—
42	C₄×D₆₁	C₂²×C₈	1	4	yes	—
43	C₈×C₆₁	C₂²×C₈	1	3	yes	—
44	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
45	#92418	C₂²×C₈	1	4	yes	—
46	C₂³×C₃₂	—	61	1	no	maximal, Sylow
47	C₂²×C₆₁	C₂×C₃₂	1	4	yes	complemented
48	C₂²×C₆₁	C₂²×C₁₆	1	3	yes	—
49	C₂×D₆₁	C₂×C₃₂	1	24	yes	complemented
50	C₂×D₆₁	C₂²×C₁₆	1	4	yes	—
51	C₄×C₆₁	C₂²×C₁₆	1	3	yes	—
52	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
53	Dic₆₁	C₂²×C₁₆	1	4	yes	—
54	C₂³×C₁₆	—	61	1	no	—
55	C₂²×C₃₂	—	61	2	no	—
56	C₂²×C₃₂	—	61	12	no	—
57	C₂×C₆₁	C₂²×C₃₂	1	6	yes	complemented
58	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
59	D₆₁	C₂²×C₃₂	1	8	yes	complemented
60	C₂³×C₈	—	61	1	no	—
61	C₂²×C₁₆	—	61	1	no	—
62	C₂²×C₁₆	—	61	6	no	—
63	C₂²×C₁₆	—	61	6	no	—
64	C₂²×C₁₆	#6923	1	1	yes	characteristic
65	C₂×C₃₂	—	61	12	no	—
66	C₂×C₃₂	—	61	16	no	—
67	C₆₁	C₂³×C₃₂	1	1	yes	characteristic, complemented, Sylow
68	C₂³×C₄	—	61	1	no	—
69	C₂²×C₈	—	61	1	no	—
70	C₂²×C₈	—	61	6	no	—
71	C₂²×C₈	—	61	6	no	—
72	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
73	C₂×C₁₆	—	61	3	no	—
74	C₂×C₁₆	—	61	3	no	—
75	C₂×C₁₆	—	61	4	no	—
76	C₂×C₁₆	—	61	12	no	—
77	C₂×C₁₆	C₂×#6923	1	6	yes	—
78	C₃₂	—	61	8	no	—
79	C₂⁴	—	61	1	no	—
80	C₂²×C₄	—	61	1	no	—
81	C₂²×C₄	—	61	6	no	—
82	C₂²×C₄	—	61	6	no	—
83	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
84	C₂×C₈	—	61	6	no	—
85	C₂×C₈	—	61	12	no	—
86	C₂×C₈	—	61	4	no	—
87	C₂×C₈	D₆₁⋊C₈	1	3	yes	—
88	C₂×C₈	C₂²×#6923	1	3	yes	—
89	C₁₆	—	61	1	no	—
90	C₁₆	—	61	3	no	—
91	C₁₆	C₂²×#6923	1	4	yes	—
92	C₂³	—	61	6	no	—
93	C₂³	—	61	8	no	—
94	C₂³	D₆₁⋊C₁₆	1	1	yes	characteristic
95	C₂×C₄	—	61	6	no	—
96	C₂×C₄	—	61	12	no	—
97	C₂×C₄	—	61	4	no	—
98	C₂×C₄	D₆₁⋊C₁₆	1	3	yes	—
99	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
100	C₈	—	61	4	no	—
101	C₈	C₂×D₆₁⋊C₈	1	3	yes	—
102	C₈	C₂³×#6923	1	1	yes	characteristic
103	C₂²	—	61	24	no	—
104	C₂²	—	61	4	no	—
105	C₂²	D₆₁⋊C₃₂	1	4	yes	direct product factor
106	C₂²	C₂×D₆₁⋊C₁₆	1	3	yes	—
107	C₄	—	61	4	no	—
108	C₄	C₂×D₆₁⋊C₁₆	1	3	yes	—
109	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
110	C₂	—	61	8	no	—
111	C₂	C₂×D₆₁⋊C₃₂	1	6	yes	direct product factor
112	C₂	C₂²×D₆₁⋊C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₆₁⋊C₃₂

C₂

yes

maximal, direct product factor

C₂²×#14519827

C₂

yes

maximal, complemented

C₂²×C₁₆×D₆₁

C₂

yes

maximal, characteristic

D₆₁⋊C₃₂

C₂²

yes

direct product factor

C₂×#14519827

C₂²

yes

complemented

C₂²×C₈×D₆₁

C₄

yes

characteristic

C₂²×#11758185

C₂²

yes

characteristic

C₂×C₁₆×D₆₁

C₄

yes

—

C₂×C₁₆×D₆₁

C₂²

yes

—

C₂²×C₁₆×C₆₁

C₄

yes

characteristic

#14519827

C₂³

yes

complemented

C₂²×C₄×D₆₁

C₈

yes

characteristic

C₂²×#92418

C₈

yes

characteristic

C₂×C₈×D₆₁

C₈

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₂×#11758185

C₂×C₄

yes

—

C₂×#11758185

C₂³

yes

—

C₁₆×D₆₁

C₂×C₄

yes

—

C₁₆×D₆₁

C₂³

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₂³×D₆₁

C₁₆

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×Dic₆₁

C₁₆

yes

characteristic

C₂×C₄×D₆₁

C₁₆

yes

—

C₂×C₄×D₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₂×#92418

C₂×C₈

yes

—

C₈×D₆₁

C₂×C₈

yes

—

C₈×D₆₁

C₂²×C₄

yes

—

C₁₆×C₆₁

C₂²×C₄

yes

—

#11758185

C₂²×C₄

yes

—

#11758185

C₂⁴

yes

characteristic

C₂³×C₆₁

C₂×C₁₆

yes

characteristic

C₂²×D₆₁

C₃₂

yes

complemented

C₂²×D₆₁

C₂×C₁₆

yes

—

C₂×C₄×C₆₁

C₂×C₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×Dic₆₁

C₂×C₁₆

yes

—

C₄×D₆₁

C₂×C₁₆

yes

—

C₄×D₆₁

C₂²×C₈

yes

—

C₈×C₆₁

C₂²×C₈

yes

—

C₈×C₆₁

C₂³×C₄

yes

characteristic

#92418

C₂²×C₈

yes

—

C₂³×C₃₂

—

maximal, Sylow

C₂²×C₆₁

C₂×C₃₂

yes

complemented

C₂²×C₆₁

C₂²×C₁₆

yes

—

C₂×D₆₁

C₂×C₃₂

yes

complemented

C₂×D₆₁

C₂²×C₁₆

yes

—

C₄×C₆₁

C₂²×C₁₆

yes

—

C₄×C₆₁

C₂³×C₈

yes

characteristic

Dic₆₁

C₂²×C₁₆

yes

—

C₂³×C₁₆

—

C₂²×C₃₂

—

C₂²×C₃₂

—

C₂×C₆₁

C₂²×C₃₂

yes

complemented

C₂×C₆₁

C₂³×C₁₆

yes

characteristic

D₆₁

C₂²×C₃₂

yes

complemented

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

#6923

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₆₁

C₂³×C₃₂

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6923

yes

—

C₃₂

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₆₁⋊C₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₆₁⋊C₈

yes

—

C₂×C₈

C₂²×#6923

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₂²×#6923

yes

—

C₂³

—

C₂³

—

C₂³

D₆₁⋊C₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₆₁⋊C₁₆

yes

—

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

100

C₈

—

101

C₈

C₂×D₆₁⋊C₈

yes

—

102

C₈

C₂³×#6923

yes

characteristic

103

C₂²

—

104

C₂²

—

105

C₂²

D₆₁⋊C₃₂

yes

direct product factor

106

C₂²

C₂×D₆₁⋊C₁₆

yes

—

107

C₄

—

108

C₄

C₂×D₆₁⋊C₁₆

yes

—

109

C₄

C₂²×D₆₁⋊C₈

yes

characteristic

110

C₂

—

111

C₂

C₂×D₆₁⋊C₃₂

yes

direct product factor

112

C₂

C₂²×D₆₁⋊C₁₆

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	#6923
commutator, hypocenter	C₆₁	C₂³×C₃₂
Frattini	C₈	C₂³×#6923
Fitting	C₂²×C₁₆×C₆₁	C₄
socle	C₂³×C₆₁	C₂×C₁₆
prime radical	C₂³×D₆₁	C₁₆
cosocle	#11758185	C₂⁴
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂²×C₁₆

subgroup

G/H

center, hypercenter

C₂²×C₁₆

#6923

commutator, hypocenter

C₆₁

C₂³×C₃₂

Frattini

C₈

C₂³×#6923

Fitting

C₂²×C₁₆×C₆₁

C₄

socle

C₂³×C₆₁

C₂×C₁₆

prime radical

C₂³×D₆₁

C₁₆

C₂⁴

G ▷ C₆₁

C₁ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂×D₆₁⋊C₃₂	C₂	direct (N × H)
2. C₂²×#14519827	C₂	semidirect
4. D₆₁⋊C₃₂	C₂²	direct (N × H)
5. C₂×#14519827	C₂²	semidirect
11. #14519827	C₂³	semidirect
36. C₂²×D₆₁	C₃₂	semidirect
47. C₂²×C₆₁	C₂×C₃₂	semidirect
49. C₂×D₆₁	C₂×C₃₂	semidirect
57. C₂×C₆₁	C₂²×C₃₂	semidirect
59. D₆₁	C₂²×C₃₂	semidirect
67. C₆₁	C₂³×C₃₂	semidirect
105. C₂²	D₆₁⋊C₃₂	direct (N × H)
111. C₂	C₂×D₆₁⋊C₃₂	direct (N × H)

H ≅ G/N

type

1. C₂×D₆₁⋊C₃₂

C₂

direct (N × H)

2. C₂²×#14519827

C₂

semidirect

4. D₆₁⋊C₃₂

C₂²

direct (N × H)

5. C₂×#14519827

C₂²

semidirect

11. #14519827

C₂³

semidirect

36. C₂²×D₆₁

C₃₂

semidirect

47. C₂²×C₆₁

C₂×C₃₂

semidirect

49. C₂×D₆₁

C₂×C₃₂

semidirect

57. C₂×C₆₁

C₂²×C₃₂

semidirect

59. D₆₁

C₂²×C₃₂

semidirect

67. C₆₁

C₂³×C₃₂

semidirect

105. C₂²

D₆₁⋊C₃₂

direct (N × H)

111. C₂

C₂×D₆₁⋊C₃₂

direct (N × H)

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

subgroup	index
C₂×D₆₁⋊C₃₂	2
C₂²×#14519827	2
C₂²×C₁₆×D₆₁	2
C₂³×C₃₂	61

subgroup

index

C₂×D₆₁⋊C₃₂

C₂²×#14519827

C₂²×C₁₆×D₆₁

C₂³×C₃₂

Subgroups — C₂²×D₆₁⋊C₃₂ #30624524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×D61⋊C32 #30624524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₆₁⋊C₃₂ #30624524