Group #11766550 — subgroups

Subgroups — C₉×#1829 #11766550

Order 1152.

408 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 150 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	C₉×#518	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#100	C₂	1	2	yes	maximal
3	C₉×#344	C₂	1	4	yes	maximal, complemented
4	C₃×#1829	C₃	1	1	yes	maximal, characteristic
5	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
6	C₂×C₉×M₁₆	C₂²	1	4	yes	—
7	C₄×D₄×C₉	—	2	4	no	—
8	C₉×#117	C₂²	1	2	yes	—
9	C₂×C₉×C₂²⋊C₄	C₄	1	4	yes	—
10	C₉×#100	—	2	4	no	—
11	C₃×#518	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃×#100	C₂×C₃	1	2	yes	—
13	C₃×#344	C₂×C₃	1	4	yes	—
14	C₂⁴×C₉	C₂×C₄	1	2	yes	—
15	C₂×D₄×C₉	—	2	4	no	—
16	C₂×D₄×C₉	D₄	1	4	yes	—
17	C₂²×C₄×C₉	—	2	4	no	—
18	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
19	C₉×M₁₆	—	2	4	no	—
20	C₉×M₁₆	—	2	4	no	—
21	C₂×C₈×C₉	—	2	4	no	—
22	C₉×C₄⋊C₄	—	2	2	no	—
23	C₉×C₂²⋊C₄	—	2	4	no	—
24	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
25	C₄²×C₉	—	2	2	no	—
26	#1829	C₉	1	1	yes	characteristic, direct product factor, Sylow
27	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
28	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
29	C₃×C₄×D₄	—	2	4	no	—
30	C₃×#117	C₂²×C₃	1	2	yes	—
31	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	4	yes	—
32	C₃×#100	—	2	4	no	—
33	C₂³×C₉	—	2	4	no	—
34	C₂³×C₉	—	8	2	no	—
35	C₂³×C₉	C₂²⋊C₄	1	4	yes	—
36	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
37	D₄×C₉	—	2	8	no	—
38	C₂×C₄×C₉	—	2	4	no	—
39	C₂×C₄×C₉	—	8	4	no	—
40	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₉	#41	1	4	yes	—
42	C₈×C₉	—	4	4	no	—
43	#518	C₂×C₉	1	1	yes	characteristic
44	C₂×#100	C₂×C₉	1	2	yes	—
45	#344	C₂×C₉	1	4	yes	complemented
46	C₂⁴×C₃	C₂×C₃×C₄	1	2	yes	—
47	C₂×C₃×D₄	—	2	4	no	—
48	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
49	C₂²×C₃×C₄	—	2	4	no	—
50	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
51	C₃×M₁₆	—	2	4	no	—
52	C₃×M₁₆	—	2	4	no	—
53	C₂×C₃×C₈	—	2	4	no	—
54	C₃×C₄⋊C₄	—	2	2	no	—
55	C₃×C₂²⋊C₄	—	2	4	no	—
56	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
57	C₃×C₄²	—	2	2	no	—
58	C₂²×C₉	—	2	4	no	—
59	C₂²×C₉	—	4	1	no	—
60	C₂²×C₉	—	4	4	no	—
61	C₂²×C₉	—	8	4	no	—
62	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₉	#117	1	2	yes	—
64	C₄×C₉	—	4	4	no	—
65	C₄×C₉	#123	1	4	yes	—
66	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
67	C₂×M₁₆	C₂²×C₉	1	4	yes	—
68	C₄×D₄	—	2	4	no	—
69	#117	C₂²×C₉	1	2	yes	—
70	C₂×C₂²⋊C₄	C₄×C₉	1	4	yes	—
71	#100	—	2	4	no	—
72	C₂³×C₃	—	2	4	no	—
73	C₂³×C₃	—	8	2	no	—
74	C₂³×C₃	C₃×C₂²⋊C₄	1	4	yes	—
75	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
76	C₃×D₄	—	2	8	no	—
77	C₂×C₃×C₄	—	2	4	no	—
78	C₂×C₃×C₄	—	8	4	no	—
79	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
80	C₂×C₃×C₄	C₃×#41	1	4	yes	—
81	C₃×C₈	—	4	4	no	—
82	C₂×C₉	—	2	1	no	—
83	C₂×C₉	—	2	2	no	—
84	C₂×C₉	—	4	4	no	—
85	C₂×C₉	#381	1	1	yes	characteristic
86	C₂⁴	C₂×C₄×C₉	1	2	yes	—
87	C₂×D₄	—	2	4	no	—
88	C₂×D₄	D₄×C₉	1	4	yes	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
91	M₁₆	—	2	4	no	—
92	M₁₆	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
97	C₄²	—	2	2	no	—
98	C₂²×C₃	—	2	4	no	—
99	C₂²×C₃	—	4	1	no	—
100	C₂²×C₃	—	4	4	no	—
101	C₂²×C₃	—	8	4	no	—
102	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₃	C₃×#117	1	2	yes	—
104	C₃×C₄	—	4	4	no	—
105	C₃×C₄	C₃×#123	1	4	yes	—
106	C₉	#1829	1	1	yes	characteristic, direct product factor, Sylow
107	C₂³	—	2	4	no	—
108	C₂³	—	8	2	no	—
109	C₂³	C₉×C₂²⋊C₄	1	4	yes	—
110	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
111	D₄	—	2	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	8	4	no	—
114	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
115	C₂×C₄	C₉×#41	1	4	yes	—
116	C₈	—	4	4	no	—
117	C₂×C₃	—	2	1	no	—
118	C₂×C₃	—	2	2	no	—
119	C₂×C₃	—	4	4	no	—
120	C₂×C₃	C₃×#381	1	1	yes	characteristic
121	C₂²	—	2	4	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	8	4	no	—
125	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²	C₉×#117	1	2	yes	—
127	C₄	—	4	4	no	—
128	C₄	C₉×#123	1	4	yes	—
129	C₃	C₃×#1829	1	1	yes	characteristic
130	C₂	—	2	1	no	—
131	C₂	—	2	2	no	—
132	C₂	—	4	4	no	—
133	C₂	C₉×#381	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#518

C₂

yes

maximal, characteristic

C₂×C₉×#100

C₂

yes

maximal

C₉×#344

C₂

yes

maximal, complemented

C₃×#1829

C₃

yes

maximal, characteristic

C₂²×D₄×C₉

C₂²

yes

characteristic

C₂×C₉×M₁₆

C₂²

yes

—

C₄×D₄×C₉

—

C₉×#117

C₂²

yes

—

C₂×C₉×C₂²⋊C₄

C₄

yes

—

C₉×#100

—

C₃×#518

C₂×C₃

yes

characteristic

C₂×C₃×#100

C₂×C₃

yes

—

C₃×#344

C₂×C₃

yes

—

C₂⁴×C₉

C₂×C₄

yes

—

C₂×D₄×C₉

—

C₂×D₄×C₉

D₄

yes

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₉×C₄⋊C₄

—

C₉×C₂²⋊C₄

—

C₉×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₉

—

#1829

C₉

yes

characteristic, direct product factor, Sylow

C₂²×C₃×D₄

C₂²×C₃

yes

characteristic

C₂×C₃×M₁₆

C₂²×C₃

yes

—

C₃×C₄×D₄

—

C₃×#117

C₂²×C₃

yes

—

C₂×C₃×C₂²⋊C₄

C₃×C₄

yes

—

C₃×#100

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₂²⋊C₄

yes

—

C₂³×C₉

C₂²×C₄

yes

characteristic

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

#41

yes

—

C₈×C₉

—

#518

C₂×C₉

yes

characteristic

C₂×#100

C₂×C₉

yes

—

#344

C₂×C₉

yes

complemented

C₂⁴×C₃

C₂×C₃×C₄

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃×D₄

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

C₂×C₃×C₄

yes

—

C₃×C₄²

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉

#117

yes

—

C₄×C₉

—

C₄×C₉

#123

yes

—

C₂²×D₄

C₂²×C₉

yes

characteristic

C₂×M₁₆

C₂²×C₉

yes

—

C₄×D₄

—

#117

C₂²×C₉

yes

—

C₂×C₂²⋊C₄

C₄×C₉

yes

—

#100

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₃×C₂²⋊C₄

yes

—

C₂³×C₃

C₂²×C₃×C₄

yes

characteristic

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

C₂×C₃×C₄

C₃×#41

yes

—

C₃×C₈

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

#381

yes

characteristic

C₂⁴

C₂×C₄×C₉

yes

—

C₂×D₄

—

C₂×D₄

D₄×C₉

yes

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×C₄×C₉

yes

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

100

C₂²×C₃

—

101

C₂²×C₃

—

102

C₂²×C₃

C₂×C₃×C₂²⋊C₄

yes

characteristic

103

C₂²×C₃

C₃×#117

yes

—

104

C₃×C₄

—

105

C₃×C₄

C₃×#123

yes

—

106

C₉

#1829

yes

characteristic, direct product factor, Sylow

107

C₂³

—

108

C₂³

—

109

C₂³

C₉×C₂²⋊C₄

yes

—

110

C₂³

C₂²×C₄×C₉

yes

characteristic

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×D₄×C₉

yes

—

115

C₂×C₄

C₉×#41

yes

—

116

C₈

—

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

—

120

C₂×C₃

C₃×#381

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×C₉×C₂²⋊C₄

yes

characteristic

126

C₂²

C₉×#117

yes

—

127

C₄

—

128

C₄

C₉×#123

yes

—

129

C₃

C₃×#1829

yes

characteristic

130

C₂

—

131

C₂

—

132

C₂

—

133

C₂

C₉×#381

yes

characteristic

	subgroup	G/H
center	C₂×C₉	#381
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#381
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

subgroup

G/H

center

C₂×C₉

#381

commutator

C₂³

C₂²×C₄×C₉

Frattini, cosocle

C₂²×C₃×C₄

C₂³×C₃

socle

C₂×C₃

C₃×#381

prime radical

C₂²×C₃×D₄

C₂²×C₃

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

N	H ≅ G/N	type
3. C₉×#344	C₂	semidirect
26. #1829	C₉	direct (N × H)
45. #344	C₂×C₉	semidirect
106. C₉	#1829	direct (N × H)

H ≅ G/N

type

3. C₉×#344

C₂

semidirect

26. #1829

C₉

direct (N × H)

45. #344

C₂×C₉

semidirect

106. C₉

#1829

direct (N × H)

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

subgroup	index
C₉×#518	2
C₂×C₉×#100	2
C₉×#344	2
C₃×#1829	3

subgroup

index

C₉×#518

C₂×C₉×#100

C₉×#344

C₃×#1829

Subgroups — C₉×#1829 #11766550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#1829 #11766550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1829 #11766550