Group #11923437 — subgroups

Subgroups — C₂⁴×(C₂²⋊C₉)⋊C₂ #11923437

Order 1152.

9506 conjugacy classes of subgroups; only the 642 normal subgroups are shown.

normal subgroups only

This group has 9506 conjugacy classes of subgroups — too many to draw. Only its 642 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×C₂²⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×(C₂²⋊C₉)⋊C₂	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×#67	—	3	1	no	maximal
4	C₂⁴×D₉	—	4	1	no	maximal
5	C₂³×C₂²⋊C₉	C₂²	1	15	yes	complemented
6	C₂²×(C₂²⋊C₉)⋊C₂	C₂²	1	140	yes	direct product factor
7	C₂⁶×C₃	D₃	1	1	yes	characteristic
8	C₂⁵×D₃	—	3	1	no	—
9	C₂³×#67	—	3	30	no	—
10	C₂³×#67	—	3	30	no	—
11	C₂⁴×Dic₃	—	3	1	no	—
12	C₂⁴×C₉	—	4	1	no	—
13	C₂³×D₉	—	4	30	no	—
14	C₂²×C₂²⋊C₉	C₂³	1	35	yes	complemented
15	C₂×(C₂²⋊C₉)⋊C₂	C₂³	1	120	yes	direct product factor
16	C₂⁴×D₄	—	9	1	no	Sylow
17	C₂⁵×C₃	—	3	1	no	—
18	C₂⁵×C₃	—	3	15	no	—
19	C₂⁵×C₃	C₂×D₃	1	15	yes	—
20	C₂⁴×D₃	—	3	30	no	—
21	C₂⁴×D₃	—	6	15	no	—
22	C₂⁴×D₃	—	6	1	no	—
23	C₂²×#67	—	3	420	no	—
24	C₂²×#67	—	3	140	no	—
25	C₂³×Dic₃	—	3	30	no	—
26	C₂³×C₉	—	4	15	no	—
27	C₂²×D₉	—	4	140	no	—
28	C₂×C₂²⋊C₉	C₂⁴	1	15	yes	complemented
29	(C₂²⋊C₉)⋊C₂	C₂⁴	1	16	yes	direct product factor
30	C₂⁶	—	9	1	no	—
31	C₂⁶	D₉	1	1	yes	characteristic, complemented
32	C₂³×D₄	—	9	30	no	—
33	C₂³×D₄	—	9	30	no	—
34	C₂⁴×C₄	—	9	1	no	—
35	C₂⁴×C₃	—	3	15	no	—
36	C₂⁴×C₃	—	3	15	no	—
37	C₂⁴×C₃	—	3	105	no	—
38	C₂⁴×C₃	—	6	35	no	—
39	C₂⁴×C₃	S₄	1	1	yes	characteristic
40	C₂⁴×C₃	C₂²×D₃	1	35	yes	—
41	C₂³×D₃	—	3	140	no	—
42	C₂³×D₃	—	6	210	no	—
43	C₂³×D₃	—	6	30	no	—
44	C₂×#67	—	3	840	no	—
45	C₂×#67	—	3	120	no	—
46	C₂²×Dic₃	—	3	140	no	—
47	C₂²×C₉	—	4	35	no	—
48	C₂×D₉	—	4	120	no	—
49	C₂²⋊C₉	C₂⁵	1	1	yes	characteristic, complemented
50	C₂⁵	—	3	1	no	—
51	C₂⁵	—	3	15	no	—
52	C₂⁵	—	9	30	no	—
53	C₂⁵	—	18	15	no	—
54	C₂⁵	—	18	1	no	—
55	C₂⁵	C₂×D₉	1	15	yes	complemented
56	C₂²×D₄	—	9	420	no	—
57	C₂²×D₄	—	9	140	no	—
58	C₂³×C₄	—	9	30	no	—
59	C₂³×C₃	—	3	35	no	—
60	C₂³×C₃	—	3	105	no	—
61	C₂³×C₃	—	3	105	no	—
62	C₂³×C₃	—	6	105	no	—
63	C₂³×C₃	C₂×S₄	1	15	yes	—
64	C₂³×C₃	C₂³×D₃	1	15	yes	—
65	C₂²×D₃	—	3	120	no	—
66	C₂²×D₃	—	6	420	no	—
67	C₂²×D₃	—	6	140	no	—
68	#67	—	3	240	no	—
69	#67	—	3	16	no	—
70	C₂×Dic₃	—	3	120	no	—
71	C₂×C₉	—	4	15	no	—
72	D₉	—	4	16	no	—
73	C₂⁴	—	3	15	no	—
74	C₂⁴	—	3	15	no	—
75	C₂⁴	—	3	105	no	—
76	C₂⁴	—	6	35	no	—
77	C₂⁴	—	9	140	no	—
78	C₂⁴	—	18	210	no	—
79	C₂⁴	—	18	30	no	—
80	C₂⁴	(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic, direct product factor
81	C₂⁴	C₂²×D₉	1	35	yes	complemented
82	C₂×D₄	—	9	840	no	—
83	C₂×D₄	—	9	120	no	—
84	C₂²×C₄	—	9	140	no	—
85	C₂²×C₃	—	3	15	no	—
86	C₂²×C₃	—	3	15	no	—
87	C₂²×C₃	—	3	105	no	—
88	C₂²×C₃	—	6	35	no	—
89	C₂²×C₃	C₂²×S₄	1	35	yes	—
90	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
91	C₂×D₃	—	3	16	no	—
92	C₂×D₃	—	6	120	no	—
93	C₂×D₃	—	6	120	no	—
94	Dic₃	—	3	16	no	—
95	C₉	—	4	1	no	Sylow
96	C₂³	—	3	35	no	—
97	C₂³	—	3	105	no	—
98	C₂³	—	3	105	no	—
99	C₂³	—	6	105	no	—
100	C₂³	—	9	120	no	—
101	C₂³	—	18	420	no	—
102	C₂³	—	18	140	no	—
103	C₂³	C₂×(C₂²⋊C₉)⋊C₂	1	15	yes	direct product factor
104	C₂³	C₂³×D₉	1	15	yes	complemented
105	D₄	—	9	240	no	—
106	D₄	—	9	16	no	—
107	C₂×C₄	—	9	120	no	—
108	C₂×C₃	—	3	1	no	—
109	C₂×C₃	—	3	15	no	—
110	C₂×C₃	C₂³×S₄	1	15	yes	—
111	D₃	—	6	16	no	—
112	C₂²	—	3	15	no	—
113	C₂²	—	3	15	no	—
114	C₂²	—	3	105	no	—
115	C₂²	—	6	35	no	—
116	C₂²	—	9	16	no	—
117	C₂²	—	18	120	no	—
118	C₂²	—	18	120	no	—
119	C₂²	C₂²×(C₂²⋊C₉)⋊C₂	1	35	yes	direct product factor
120	C₂²	C₂⁴×D₉	1	1	yes	characteristic, complemented
121	C₄	—	9	16	no	—
122	C₃	C₂⁴×S₄	1	1	yes	characteristic
123	C₂	—	3	1	no	—
124	C₂	—	3	15	no	—
125	C₂	—	18	16	no	—
126	C₂	C₂³×(C₂²⋊C₉)⋊C₂	1	15	yes	direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×C₂²⋊C₉

C₂

yes

maximal, characteristic, complemented

C₂³×(C₂²⋊C₉)⋊C₂

C₂

yes

maximal, direct product factor

C₂⁴×#67

—

maximal

C₂⁴×D₉

—

maximal

C₂³×C₂²⋊C₉

C₂²

yes

complemented

C₂²×(C₂²⋊C₉)⋊C₂

C₂²

140

yes

direct product factor

C₂⁶×C₃

D₃

yes

characteristic

C₂⁵×D₃

—

C₂³×#67

—

C₂³×#67

—

C₂⁴×Dic₃

—

C₂⁴×C₉

—

C₂³×D₉

—

C₂²×C₂²⋊C₉

C₂³

yes

complemented

C₂×(C₂²⋊C₉)⋊C₂

C₂³

120

yes

direct product factor

C₂⁴×D₄

—

Sylow

C₂⁵×C₃

—

C₂⁵×C₃

—

C₂⁵×C₃

C₂×D₃

yes

—

C₂⁴×D₃

—

C₂⁴×D₃

—

C₂⁴×D₃

—

C₂²×#67

—

420

—

C₂²×#67

—

140

—

C₂³×Dic₃

—

C₂³×C₉

—

C₂²×D₉

—

140

—

C₂×C₂²⋊C₉

C₂⁴

yes

complemented

(C₂²⋊C₉)⋊C₂

C₂⁴

yes

direct product factor

C₂⁶

—

C₂⁶

D₉

yes

characteristic, complemented

C₂³×D₄

—

C₂³×D₄

—

C₂⁴×C₄

—

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

—

105

—

C₂⁴×C₃

—

C₂⁴×C₃

S₄

yes

characteristic

C₂⁴×C₃

C₂²×D₃

yes

—

C₂³×D₃

—

140

—

C₂³×D₃

—

210

—

C₂³×D₃

—

C₂×#67

—

840

—

C₂×#67

—

120

—

C₂²×Dic₃

—

140

—

C₂²×C₉

—

C₂×D₉

—

120

—

C₂²⋊C₉

C₂⁵

yes

characteristic, complemented

C₂⁵

—

C₂⁵

—

C₂⁵

—

C₂⁵

—

C₂⁵

—

C₂⁵

C₂×D₉

yes

complemented

C₂²×D₄

—

420

—

C₂²×D₄

—

140

—

C₂³×C₄

—

C₂³×C₃

—

C₂³×C₃

—

105

—

C₂³×C₃

—

105

—

C₂³×C₃

—

105

—

C₂³×C₃

C₂×S₄

yes

—

C₂³×C₃

C₂³×D₃

yes

—

C₂²×D₃

—

120

—

C₂²×D₃

—

420

—

C₂²×D₃

—

140

—

#67

—

240

—

#67

—

C₂×Dic₃

—

120

—

C₂×C₉

—

D₉

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

105

—

C₂⁴

—

C₂⁴

—

140

—

C₂⁴

—

210

—

C₂⁴

—

C₂⁴

(C₂²⋊C₉)⋊C₂

yes

characteristic, direct product factor

C₂⁴

C₂²×D₉

yes

complemented

C₂×D₄

—

840

—

C₂×D₄

—

120

—

C₂²×C₄

—

140

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

105

—

C₂²×C₃

—

C₂²×C₃

C₂²×S₄

yes

—

C₂²×C₃

C₂⁴×D₃

yes

characteristic

C₂×D₃

—

C₂×D₃

—

120

—

C₂×D₃

—

120

—

Dic₃

—

C₉

—

Sylow

C₂³

—

C₂³

—

105

—

C₂³

—

105

—

C₂³

—

105

—

100

C₂³

—

120

—

101

C₂³

—

420

—

102

C₂³

—

140

—

103

C₂³

C₂×(C₂²⋊C₉)⋊C₂

yes

direct product factor

104

C₂³

C₂³×D₉

yes

complemented

105

D₄

—

240

—

106

D₄

—

107

C₂×C₄

—

120

—

108

C₂×C₃

—

109

C₂×C₃

—

110

C₂×C₃

C₂³×S₄

yes

—

111

D₃

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

105

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

120

—

118

C₂²

—

120

—

119

C₂²

C₂²×(C₂²⋊C₉)⋊C₂

yes

direct product factor

120

C₂²

C₂⁴×D₉

yes

characteristic, complemented

121

C₄

—

122

C₃

C₂⁴×S₄

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂³×(C₂²⋊C₉)⋊C₂

yes

direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂⁵
Frattini	C₃	C₂⁴×S₄
Fitting, socle	C₂⁶×C₃	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₂⁴

subgroup

G/H

center, hypercenter

C₂⁴

(C₂²⋊C₉)⋊C₂

commutator, cosocle, hypocenter

C₂²⋊C₉

C₂⁵

Frattini

C₃

C₂⁴×S₄

Fitting, socle

C₂⁶×C₃

D₃

LCS

G ▷ C₂²⋊C₉

UCS

C₁ ◁ C₂⁴

N	H ≅ G/N	type
1. C₂⁴×C₂²⋊C₉	C₂	semidirect
2. C₂³×(C₂²⋊C₉)⋊C₂	C₂	direct (N × H)
5. C₂³×C₂²⋊C₉	C₂²	semidirect
6. C₂²×(C₂²⋊C₉)⋊C₂	C₂²	direct (N × H)
14. C₂²×C₂²⋊C₉	C₂³	semidirect
15. C₂×(C₂²⋊C₉)⋊C₂	C₂³	direct (N × H)
28. C₂×C₂²⋊C₉	C₂⁴	semidirect
29. (C₂²⋊C₉)⋊C₂	C₂⁴	direct (N × H)
31. C₂⁶	D₉	semidirect
49. C₂²⋊C₉	C₂⁵	semidirect
55. C₂⁵	C₂×D₉	semidirect
80. C₂⁴	(C₂²⋊C₉)⋊C₂	direct (N × H)
81. C₂⁴	C₂²×D₉	semidirect
103. C₂³	C₂×(C₂²⋊C₉)⋊C₂	direct (N × H)
104. C₂³	C₂³×D₉	semidirect
119. C₂²	C₂²×(C₂²⋊C₉)⋊C₂	direct (N × H)
120. C₂²	C₂⁴×D₉	semidirect
126. C₂	C₂³×(C₂²⋊C₉)⋊C₂	direct (N × H)

H ≅ G/N

type

1. C₂⁴×C₂²⋊C₉

C₂

semidirect

2. C₂³×(C₂²⋊C₉)⋊C₂

C₂

direct (N × H)

5. C₂³×C₂²⋊C₉

C₂²

semidirect

6. C₂²×(C₂²⋊C₉)⋊C₂

C₂²

direct (N × H)

14. C₂²×C₂²⋊C₉

C₂³

semidirect

15. C₂×(C₂²⋊C₉)⋊C₂

C₂³

direct (N × H)

28. C₂×C₂²⋊C₉

C₂⁴

semidirect

29. (C₂²⋊C₉)⋊C₂

C₂⁴

direct (N × H)

31. C₂⁶

D₉

semidirect

49. C₂²⋊C₉

C₂⁵

semidirect

55. C₂⁵

C₂×D₉

semidirect

80. C₂⁴

(C₂²⋊C₉)⋊C₂

direct (N × H)

81. C₂⁴

C₂²×D₉

semidirect

103. C₂³

C₂×(C₂²⋊C₉)⋊C₂

direct (N × H)

104. C₂³

C₂³×D₉

semidirect

119. C₂²

C₂²×(C₂²⋊C₉)⋊C₂

direct (N × H)

120. C₂²

C₂⁴×D₉

semidirect

126. C₂

C₂³×(C₂²⋊C₉)⋊C₂

direct (N × H)

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

subgroup	index
C₂⁴×C₂²⋊C₉	2
C₂³×(C₂²⋊C₉)⋊C₂	2
C₂⁴×#67	3
C₂⁴×D₉	4

subgroup

index

C₂⁴×C₂²⋊C₉

C₂³×(C₂²⋊C₉)⋊C₂

C₂⁴×#67

C₂⁴×D₉

Subgroups — C₂⁴×(C₂²⋊C₉)⋊C₂ #11923437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C24×(C22⋊C9)⋊C2 #11923437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×(C₂²⋊C₉)⋊C₂ #11923437