Group #11766056 — subgroups

Subgroups — C₉×#1335 #11766056

Order 1152.

150 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 66 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×#443	C₂	1	1	yes	maximal, characteristic
2	C₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1335	C₃	1	1	yes	maximal, characteristic
5	C₉×#135	C₄	1	1	yes	characteristic
6	C₉×#131	—	2	1	no	—
7	C₉×M₃₂	—	2	1	no	—
8	C₉×#108	C₄	1	1	yes	characteristic
9	C₉×#104	—	2	1	no	—
10	C₄×C₈×C₉	—	2	1	no	—
11	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₃×#443	C₂×C₃	1	1	yes	characteristic
13	C₃×#364	C₂×C₃	1	1	yes	characteristic
14	C₃×C₈²	C₂×C₃	1	1	yes	characteristic
15	C₉×#41	—	2	1	no	—
16	C₉×Q₁₆	C₈	1	1	yes	characteristic, complemented
17	C₉×SD₁₆	—	2	1	no	—
18	C₉×D₈	C₈	1	1	yes	characteristic, complemented
19	C₉×M₁₆	—	2	1	no	—
20	C₉×M₁₆	—	4	1	no	—
21	C₂×C₈×C₉	—	2	2	no	—
22	C₂×C₈×C₉	—	4	1	no	—
23	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₉	D₄	1	1	yes	characteristic
25	C₄²×C₉	D₄	1	1	yes	characteristic
26	C₉×C₁₆	—	4	1	no	—
27	#1335	C₉	1	1	yes	characteristic, direct product factor, Sylow
28	C₃×#135	C₃×C₄	1	1	yes	characteristic
29	C₃×#131	—	2	1	no	—
30	C₃×M₃₂	—	2	1	no	—
31	C₃×#108	C₃×C₄	1	1	yes	characteristic
32	C₃×#104	—	2	1	no	—
33	C₃×C₄×C₈	—	2	1	no	—
34	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
35	Q₈×C₉	—	2	1	no	—
36	D₄×C₉	—	2	1	no	—
37	D₄×C₉	—	4	1	no	—
38	C₂×C₄×C₉	—	2	1	no	—
39	C₂×C₄×C₉	—	4	1	no	—
40	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×C₉	—	2	4	no	—
42	C₈×C₉	—	4	1	no	—
43	C₈×C₉	C₂×C₈	1	1	yes	characteristic
44	C₈×C₉	M₁₆	1	1	yes	characteristic
45	C₈×C₉	D₈	1	1	yes	characteristic
46	C₈×C₉	SD₁₆	1	1	yes	characteristic
47	#443	C₂×C₉	1	1	yes	characteristic
48	#364	C₂×C₉	1	1	yes	characteristic, complemented
49	C₈²	C₂×C₉	1	1	yes	characteristic, complemented
50	C₃×#41	—	2	1	no	—
51	C₃×Q₁₆	C₃×C₈	1	1	yes	characteristic
52	C₃×SD₁₆	—	2	1	no	—
53	C₃×D₈	C₃×C₈	1	1	yes	characteristic
54	C₃×M₁₆	—	2	1	no	—
55	C₃×M₁₆	—	4	1	no	—
56	C₂×C₃×C₈	—	2	2	no	—
57	C₂×C₃×C₈	—	4	1	no	—
58	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
59	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
60	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
61	C₃×C₁₆	—	4	1	no	—
62	C₂²×C₉	—	4	1	no	—
63	C₂²×C₉	#104	1	1	yes	characteristic
64	C₄×C₉	—	2	2	no	—
65	C₄×C₉	—	4	1	no	—
66	C₄×C₉	C₂²⋊C₈	1	1	yes	characteristic
67	C₄×C₉	#102	1	1	yes	characteristic
68	#135	C₄×C₉	1	1	yes	characteristic
69	#131	—	2	1	no	—
70	M₃₂	—	2	1	no	—
71	#108	C₄×C₉	1	1	yes	characteristic
72	#104	—	2	1	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
75	C₃×Q₈	—	2	1	no	—
76	C₃×D₄	—	2	1	no	—
77	C₃×D₄	—	4	1	no	—
78	C₂×C₃×C₄	—	2	1	no	—
79	C₂×C₃×C₄	—	4	1	no	—
80	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
81	C₃×C₈	—	2	4	no	—
82	C₃×C₈	—	4	1	no	—
83	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
84	C₃×C₈	C₃×M₁₆	1	1	yes	characteristic
85	C₃×C₈	C₃×D₈	1	1	yes	characteristic
86	C₃×C₈	C₃×SD₁₆	1	1	yes	characteristic
87	C₂×C₉	—	2	1	no	—
88	C₂×C₉	—	8	1	no	—
89	C₂×C₉	D₄⋊C₈	1	1	yes	characteristic
90	#41	—	2	1	no	—
91	Q₁₆	C₈×C₉	1	1	yes	characteristic, complemented
92	SD₁₆	—	2	1	no	—
93	D₈	C₈×C₉	1	1	yes	characteristic, complemented
94	M₁₆	—	2	1	no	—
95	M₁₆	—	4	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
99	C₂×C₈	D₄×C₉	1	1	yes	characteristic
100	C₄²	D₄×C₉	1	1	yes	characteristic
101	C₁₆	—	4	1	no	—
102	C₂²×C₃	—	4	1	no	—
103	C₂²×C₃	C₃×#104	1	1	yes	characteristic
104	C₃×C₄	—	2	2	no	—
105	C₃×C₄	—	4	1	no	—
106	C₃×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
107	C₃×C₄	C₃×#102	1	1	yes	characteristic
108	C₉	#1335	1	1	yes	characteristic, direct product factor, Sylow
109	Q₈	—	2	1	no	—
110	D₄	—	2	1	no	—
111	D₄	—	4	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	—	4	1	no	—
117	C₈	C₂×C₈×C₉	1	1	yes	characteristic
118	C₈	C₉×M₁₆	1	1	yes	characteristic
119	C₈	C₉×D₈	1	1	yes	characteristic
120	C₈	C₉×SD₁₆	1	1	yes	characteristic
121	C₂×C₃	—	2	1	no	—
122	C₂×C₃	—	8	1	no	—
123	C₂×C₃	C₃×D₄⋊C₈	1	1	yes	characteristic
124	C₂²	—	4	1	no	—
125	C₂²	C₉×#104	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	1	no	—
128	C₄	C₉×C₂²⋊C₈	1	1	yes	characteristic
129	C₄	C₉×#102	1	1	yes	characteristic
130	C₃	C₃×#1335	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	8	1	no	—
133	C₂	C₉×D₄⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#443

C₂

yes

maximal, characteristic

C₉×#364

C₂

yes

maximal, characteristic, complemented

C₈²×C₉

C₂

yes

maximal, characteristic, complemented

C₃×#1335

C₃

yes

maximal, characteristic

C₉×#135

C₄

yes

characteristic

C₉×#131

—

C₉×M₃₂

—

C₉×#108

C₄

yes

characteristic

C₉×#104

—

C₄×C₈×C₉

—

C₄×C₈×C₉

C₂²

yes

characteristic

C₃×#443

C₂×C₃

yes

characteristic

C₃×#364

C₂×C₃

yes

characteristic

C₃×C₈²

C₂×C₃

yes

characteristic

C₉×#41

—

C₉×Q₁₆

C₈

yes

characteristic, complemented

C₉×SD₁₆

—

C₉×D₈

C₈

yes

characteristic, complemented

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

characteristic

C₄²×C₉

D₄

yes

characteristic

C₉×C₁₆

—

#1335

C₉

yes

characteristic, direct product factor, Sylow

C₃×#135

C₃×C₄

yes

characteristic

C₃×#131

—

C₃×M₃₂

—

C₃×#108

C₃×C₄

yes

characteristic

C₃×#104

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂²×C₃

yes

characteristic

Q₈×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₈×C₉

—

C₈×C₉

—

C₈×C₉

C₂×C₈

yes

characteristic

C₈×C₉

M₁₆

yes

characteristic

C₈×C₉

D₈

yes

characteristic

C₈×C₉

SD₁₆

yes

characteristic

#443

C₂×C₉

yes

characteristic

#364

C₂×C₉

yes

characteristic, complemented

C₈²

C₂×C₉

yes

characteristic, complemented

C₃×#41

—

C₃×Q₁₆

C₃×C₈

yes

characteristic

C₃×SD₁₆

—

C₃×D₈

C₃×C₈

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×C₄

yes

characteristic

C₂×C₃×C₈

C₃×D₄

yes

characteristic

C₃×C₄²

C₃×D₄

yes

characteristic

C₃×C₁₆

—

C₂²×C₉

—

C₂²×C₉

#104

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂²⋊C₈

yes

C₄×C₉

yes

C₄×C₉

yes

characteristic

#131

—

M₃₂

—

#108

C₄×C₉

yes

characteristic

#104

—

C₄×C₈

—

C₄×C₈

C₂²×C₉

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×C₂²⋊C₄

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×C₃×C₈

yes

characteristic

C₃×C₈

C₃×M₁₆

yes

characteristic

C₃×C₈

C₃×D₈

yes

characteristic

C₃×C₈

C₃×SD₁₆

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

D₄⋊C₈

yes

characteristic

#41

—

Q₁₆

C₈×C₉

yes

characteristic, complemented

SD₁₆

—

D₈

C₈×C₉

yes

characteristic, complemented

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₉

yes

characteristic

C₂×C₈

D₄×C₉

yes

characteristic

100

C₄²

D₄×C₉

yes

characteristic

101

C₁₆

—

102

C₂²×C₃

—

103

C₂²×C₃

C₃×#104

yes

characteristic

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

C₃×C₂²⋊C₈

yes

characteristic

107

C₃×C₄

C₃×#102

yes

characteristic

108

C₉

#1335

yes

characteristic, direct product factor, Sylow

109

Q₈

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₉×C₂²⋊C₄

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

C₂×C₈×C₉

yes

characteristic

118

C₈

C₉×M₁₆

yes

characteristic

119

C₈

C₉×D₈

yes

characteristic

120

C₈

C₉×SD₁₆

yes

characteristic

121

C₂×C₃

—

122

C₂×C₃

—

123

C₂×C₃

C₃×D₄⋊C₈

yes

characteristic

124

C₂²

—

125

C₂²

C₉×#104

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

C₉×C₂²⋊C₈

yes

characteristic

129

C₄

C₉×#102

yes

characteristic

130

C₃

C₃×#1335

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

C₉×D₄⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈×C₉	D₈
commutator	C₈	C₂×C₈×C₉
Frattini, cosocle	C₃×C₄×C₈	C₂²×C₃
socle	C₂×C₃	C₃×D₄⋊C₈
prime radical	C₃×#135	C₃×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₉ ◁ C₂×C₈×C₉ ◁ C₄×C₈×C₉ ◁ G

subgroup

G/H

center

C₈×C₉

D₈

commutator

C₈

C₂×C₈×C₉

Frattini, cosocle

C₃×C₄×C₈

C₂²×C₃

socle

C₂×C₃

C₃×D₄⋊C₈

prime radical

C₃×#135

C₃×C₄

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈×C₉ ◁ C₂×C₈×C₉ ◁ C₄×C₈×C₉ ◁ G

N	H ≅ G/N	type
2. C₉×#364	C₂	semidirect
3. C₈²×C₉	C₂	semidirect
16. C₉×Q₁₆	C₈	semidirect
18. C₉×D₈	C₈	semidirect
27. #1335	C₉	direct (N × H)
48. #364	C₂×C₉	semidirect
49. C₈²	C₂×C₉	semidirect
91. Q₁₆	C₈×C₉	semidirect
93. D₈	C₈×C₉	semidirect
108. C₉	#1335	direct (N × H)

H ≅ G/N

type

2. C₉×#364

C₂

semidirect

3. C₈²×C₉

C₂

semidirect

16. C₉×Q₁₆

C₈

semidirect

18. C₉×D₈

C₈

semidirect

27. #1335

C₉

direct (N × H)

48. #364

C₂×C₉

semidirect

49. C₈²

C₂×C₉

semidirect

91. Q₁₆

C₈×C₉

semidirect

93. D₈

C₈×C₉

semidirect

108. C₉

#1335

direct (N × H)

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

subgroup	index
C₉×#443	2
C₉×#364	2
C₈²×C₉	2
C₃×#1335	3

subgroup

index

C₉×#443

C₉×#364

C₈²×C₉

C₃×#1335

Subgroups — C₉×#1335 #11766056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#1335 #11766056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1335 #11766056