Group #512549 — subgroups

Subgroups — C₂²×C₂²⋊C₃₂ #512549

Order 512.

934 conjugacy classes of subgroups; only the 442 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 442 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	C₂⁴×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₃₂	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₁₆	—	2	8	no	—
6	C₂³×C₁₆	C₄	1	6	yes	—
7	C₂³×C₁₆	C₄	1	1	yes	characteristic
8	C₂³×C₁₆	C₂²	1	6	yes	—
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃₂	—	2	8	no	—
11	C₂²×C₃₂	C₂²	1	12	yes	complemented
12	C₂²⋊C₃₂	C₂²	1	16	yes	direct product factor
13	C₂⁴×C₄	C₈	1	1	yes	characteristic
14	C₂³×C₈	—	2	4	no	—
15	C₂³×C₈	—	2	4	no	—
16	C₂³×C₈	C₈	1	1	yes	characteristic
17	C₂³×C₈	C₈	1	6	yes	—
18	C₂³×C₈	C₂×C₄	1	6	yes	—
19	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	—	2	4	no	—
21	C₂²×C₁₆	—	2	24	no	—
22	C₂²×C₁₆	—	2	24	no	—
23	C₂²×C₁₆	C₂×C₄	1	6	yes	—
24	C₂²×C₁₆	C₂×C₄	1	12	yes	—
25	C₂²×C₁₆	C₂×C₄	1	3	yes	—
26	C₂²×C₁₆	D₄	1	8	yes	—
27	C₂²×C₁₆	C₂³	1	4	yes	—
28	C₂²×C₁₆	C₂³	1	3	yes	—
29	C₂×C₃₂	—	2	24	no	—
30	C₂×C₃₂	C₂³	1	8	yes	complemented
31	C₂⁵	C₁₆	1	1	yes	characteristic
32	C₂³×C₄	—	2	4	no	—
33	C₂³×C₄	—	2	4	no	—
34	C₂³×C₄	C₁₆	1	1	yes	characteristic
35	C₂³×C₄	C₁₆	1	6	yes	—
36	C₂³×C₄	C₂×C₈	1	6	yes	—
37	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₈	—	2	4	no	—
39	C₂²×C₈	—	2	24	no	—
40	C₂²×C₈	—	2	12	no	—
41	C₂²×C₈	—	2	12	no	—
42	C₂²×C₈	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₈	C₂×C₈	1	6	yes	—
44	C₂²×C₈	C₂×C₈	1	12	yes	—
45	C₂²×C₈	C₂×C₈	1	3	yes	—
46	C₂²×C₈	M₁₆	1	4	yes	—
47	C₂²×C₈	C₂²×C₄	1	4	yes	—
48	C₂²×C₈	C₂²×C₄	1	3	yes	—
49	C₂×C₁₆	—	2	12	no	—
50	C₂×C₁₆	—	2	24	no	—
51	C₂×C₁₆	—	2	8	no	—
52	C₂×C₁₆	C₂²⋊C₄	1	12	yes	—
53	C₂×C₁₆	C₂²×C₄	1	4	yes	—
54	C₂×C₁₆	C₂²×C₄	1	3	yes	—
55	C₂×C₁₆	C₂×D₄	1	12	yes	—
56	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
57	C₃₂	—	2	8	no	—
58	C₂⁴	—	2	4	no	—
59	C₂⁴	—	2	4	no	—
60	C₂⁴	C₃₂	1	8	yes	complemented
61	C₂⁴	C₂×C₁₆	1	6	yes	—
62	C₂⁴	C₂×C₁₆	1	1	yes	characteristic
63	C₂²×C₄	—	2	4	no	—
64	C₂²×C₄	—	2	24	no	—
65	C₂²×C₄	—	2	12	no	—
66	C₂²×C₄	—	2	12	no	—
67	C₂²×C₄	C₂²⋊C₈	1	4	yes	—
68	C₂²×C₄	C₂×C₁₆	1	6	yes	—
69	C₂²×C₄	C₂×C₁₆	1	12	yes	—
70	C₂²×C₄	C₂×C₁₆	1	3	yes	—
71	C₂²×C₄	M₃₂	1	4	yes	—
72	C₂²×C₄	C₂²×C₈	1	4	yes	—
73	C₂²×C₄	C₂²×C₈	1	3	yes	—
74	C₂×C₈	—	2	12	no	—
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	24	no	—
78	C₂×C₈	C₂²⋊C₈	1	12	yes	—
79	C₂×C₈	C₂×C₂²⋊C₄	1	6	yes	—
80	C₂×C₈	C₂²×C₈	1	4	yes	—
81	C₂×C₈	C₂²×C₈	1	3	yes	—
82	C₂×C₈	C₂×M₁₆	1	6	yes	—
83	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
84	C₁₆	—	2	4	no	—
85	C₁₆	C₂×C₂²⋊C₄	1	6	yes	—
86	C₁₆	C₂²×D₄	1	2	yes	—
87	C₂³	—	2	12	no	—
88	C₂³	—	2	32	no	—
89	C₂³	—	2	12	no	—
90	C₂³	C₂²⋊C₁₆	1	4	yes	—
91	C₂³	C₂×C₃₂	1	4	yes	—
92	C₂³	C₂×C₃₂	1	24	yes	complemented
93	C₂³	M₆₄	1	4	yes	—
94	C₂³	C₂²×C₁₆	1	4	yes	—
95	C₂³	C₂²×C₁₆	1	3	yes	—
96	C₂×C₄	—	2	12	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	24	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	C₂²⋊C₁₆	1	12	yes	—
101	C₂×C₄	C₂×C₂²⋊C₈	1	6	yes	—
102	C₂×C₄	C₂²×C₁₆	1	4	yes	—
103	C₂×C₄	C₂²×C₁₆	1	3	yes	—
104	C₂×C₄	C₂×M₃₂	1	6	yes	—
105	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
106	C₈	—	2	4	no	—
107	C₈	C₂×C₂²⋊C₈	1	6	yes	—
108	C₈	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₈	C₂²×M₁₆	1	1	yes	characteristic
110	C₂²	—	2	4	no	—
111	C₂²	—	2	4	no	—
112	C₂²	—	2	48	no	—
113	C₂²	C₂²⋊C₃₂	1	16	yes	direct product factor
114	C₂²	C₂×C₂²⋊C₁₆	1	6	yes	—
115	C₂²	C₂²×C₃₂	1	6	yes	—
116	C₂²	C₂²×C₃₂	1	8	yes	complemented
117	C₂²	C₂×M₆₄	1	6	yes	—
118	C₂²	C₂³×C₁₆	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	C₂×C₂²⋊C₁₆	1	6	yes	—
121	C₄	C₂²×C₂²⋊C₈	1	1	yes	characteristic
122	C₄	C₂²×M₃₂	1	1	yes	characteristic
123	C₂	—	2	8	no	—
124	C₂	C₂×C₂²⋊C₃₂	1	12	yes	direct product factor
125	C₂	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic
126	C₂	C₂³×C₃₂	1	1	yes	characteristic
127	C₂	C₂²×M₆₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×C₁₆

C₂

yes

maximal, characteristic

C₂³×C₃₂

C₂

yes

maximal, complemented

C₂×C₂²⋊C₃₂

C₂

yes

maximal, direct product factor

C₂⁴×C₈

C₄

yes

characteristic

C₂³×C₁₆

—

C₂³×C₁₆

C₄

yes

—

C₂³×C₁₆

C₄

yes

characteristic

C₂³×C₁₆

C₂²

yes

—

C₂³×C₁₆

C₂²

yes

characteristic

C₂²×C₃₂

—

C₂²×C₃₂

C₂²

yes

complemented

C₂²⋊C₃₂

C₂²

yes

direct product factor

C₂⁴×C₄

C₈

yes

characteristic

C₂³×C₈

—

C₂³×C₈

—

C₂³×C₈

C₈

yes

characteristic

C₂³×C₈

C₈

yes

—

C₂³×C₈

C₂×C₄

yes

—

C₂³×C₈

C₂×C₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

D₄

yes

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

C₂×C₃₂

—

C₂×C₃₂

C₂³

yes

complemented

C₂⁵

C₁₆

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₁₆

yes

characteristic

C₂³×C₄

C₁₆

yes

—

C₂³×C₄

C₂×C₈

yes

—

C₂³×C₄

C₂×C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

C₂²×C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

characteristic

C₃₂

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₃₂

yes

complemented

C₂⁴

C₂×C₁₆

yes

—

C₂⁴

C₂×C₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₈

yes

—

C₂²×C₄

C₂×C₁₆

yes

—

C₂²×C₄

C₂×C₁₆

yes

—

C₂²×C₄

C₂×C₁₆

yes

—

C₂²×C₄

M₃₂

yes

—

C₂²×C₄

C₂²×C₈

yes

—

C₂²×C₄

C₂²×C₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂×C₂²⋊C₄

yes

—

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂³×C₄

yes

characteristic

C₁₆

—

C₁₆

C₂×C₂²⋊C₄

yes

—

C₁₆

C₂²×D₄

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₁₆

yes

—

C₂³

C₂×C₃₂

yes

—

C₂³

C₂×C₃₂

yes

complemented

C₂³

M₆₄

yes

—

C₂³

C₂²×C₁₆

yes

—

C₂³

C₂²×C₁₆

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊C₁₆

yes

—

101

C₂×C₄

C₂×C₂²⋊C₈

yes

—

102

C₂×C₄

C₂²×C₁₆

yes

—

103

C₂×C₄

C₂²×C₁₆

yes

—

104

C₂×C₄

C₂×M₃₂

yes

—

105

C₂×C₄

C₂³×C₈

yes

characteristic

106

C₈

—

107

C₈

C₂×C₂²⋊C₈

yes

—

108

C₈

C₂²×C₂²⋊C₄

yes

characteristic

109

C₈

C₂²×M₁₆

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂²⋊C₃₂

yes

direct product factor

114

C₂²

C₂×C₂²⋊C₁₆

yes

—

115

C₂²

C₂²×C₃₂

yes

—

116

C₂²

C₂²×C₃₂

yes

complemented

117

C₂²

C₂×M₆₄

yes

—

118

C₂²

C₂³×C₁₆

yes

characteristic

119

C₄

—

120

C₄

C₂×C₂²⋊C₁₆

yes

—

121

C₄

C₂²×C₂²⋊C₈

yes

characteristic

122

C₄

C₂²×M₃₂

yes

characteristic

123

C₂

—

124

C₂

C₂×C₂²⋊C₃₂

yes

direct product factor

125

C₂

C₂²×C₂²⋊C₁₆

yes

characteristic

126

C₂

C₂³×C₃₂

yes

characteristic

127

C₂

C₂²×M₆₄

yes

characteristic

	subgroup	G/H
center	C₂³×C₁₆	C₂²
commutator	C₂	C₂³×C₃₂
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂⁴	C₂×C₁₆
prime radical	C₂⁵	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂³×C₁₆

C₂²

commutator

C₂

C₂³×C₃₂

Frattini, cosocle

C₂×C₁₆

C₂⁴

socle

C₂⁴

C₂×C₁₆

prime radical

C₂⁵

C₁₆

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₆ ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂³×C₃₂	C₂	semidirect
3. C₂×C₂²⋊C₃₂	C₂	direct (N × H)
11. C₂²×C₃₂	C₂²	semidirect
12. C₂²⋊C₃₂	C₂²	direct (N × H)
30. C₂×C₃₂	C₂³	semidirect
60. C₂⁴	C₃₂	semidirect
92. C₂³	C₂×C₃₂	semidirect
113. C₂²	C₂²⋊C₃₂	direct (N × H)
116. C₂²	C₂²×C₃₂	semidirect
124. C₂	C₂×C₂²⋊C₃₂	direct (N × H)

H ≅ G/N

type

2. C₂³×C₃₂

C₂

semidirect

3. C₂×C₂²⋊C₃₂

C₂

direct (N × H)

11. C₂²×C₃₂

C₂²

semidirect

12. C₂²⋊C₃₂

C₂²

direct (N × H)

30. C₂×C₃₂

C₂³

semidirect

60. C₂⁴

C₃₂

semidirect

92. C₂³

C₂×C₃₂

semidirect

113. C₂²

C₂²⋊C₃₂

direct (N × H)

116. C₂²

C₂²×C₃₂

semidirect

124. C₂

C₂×C₂²⋊C₃₂

direct (N × H)

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

subgroup	index
C₂⁴×C₁₆	2
C₂³×C₃₂	2
C₂×C₂²⋊C₃₂	2

subgroup

index

C₂⁴×C₁₆

C₂³×C₃₂

C₂×C₂²⋊C₃₂

Subgroups — C₂²×C₂²⋊C₃₂ #512549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C22⋊C32 #512549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊C₃₂ #512549