Group #11771768 — subgroups

Subgroups — #11771768

Order 1152.

150 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 44 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	#10588807	C₂	1	1	yes	maximal, characteristic
2	C₉×#349	C₂	1	1	yes	maximal, characteristic
3	C₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	#67537	—	3	1	no	maximal
5	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
6	#63486	—	2	1	no	—
7	C₂×#3667	—	2	1	no	—
8	C₂×Dic₃₆	C₄	1	1	yes	characteristic
9	C₉×M₃₂	—	2	1	no	—
10	#63377	C₄	1	1	yes	characteristic
11	#63376	—	2	1	no	—
12	#5137	—	3	1	no	—
13	C₃×#349	D₃	1	1	yes	characteristic
14	C₃⋊#338	—	3	1	no	—
15	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
16	C₂×Dic₁₈	—	2	1	no	—
17	C₉×M₁₆	—	4	1	no	—
18	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₉	D₄	1	1	yes	characteristic
20	#3676	—	4	1	no	—
21	C₄⋊Dic₉	—	2	1	no	—
22	Dic₉⋊C₄	—	4	1	no	—
23	#3667	—	4	1	no	—
24	#3667	—	4	1	no	—
25	C₂×C₉⋊C₈	—	2	2	no	—
26	Dic₃₆	—	4	1	no	—
27	C₉×C₁₆	—	4	1	no	—
28	#1342	—	9	1	no	Sylow
29	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
30	#924	—	6	1	no	—
31	C₂×#208	—	6	1	no	—
32	C₂×Dic₁₂	—	3	1	no	—
33	C₃×M₃₂	—	2	1	no	—
34	#817	—	3	1	no	—
35	#816	—	6	1	no	—
36	C₂³×C₉	SD₁₆	1	1	yes	characteristic
37	C₂×C₄×C₉	—	2	1	no	—
38	C₂×C₄×C₉	—	2	1	no	—
39	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₉	D₈	1	1	yes	characteristic
41	C₂×Dic₉	—	4	1	no	—
42	C₂×Dic₉	—	4	1	no	—
43	Dic₁₈	—	4	1	no	—
44	Dic₁₈	—	4	1	no	—
45	C₈×C₉	—	2	1	no	—
46	C₈×C₉	—	2	1	no	—
47	C₉⋊C₈	—	4	2	no	—
48	#470	—	9	1	no	—
49	#349	D₉	1	1	yes	characteristic
50	#338	—	9	1	no	—
51	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
52	C₂×Dic₆	—	6	1	no	—
53	C₃×M₁₆	—	4	1	no	—
54	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
55	C₂×C₃×C₈	#67	1	1	yes	characteristic
56	#217	—	12	1	no	—
57	C₄⋊Dic₃	—	6	1	no	—
58	Dic₃⋊C₄	—	12	1	no	—
59	#208	—	12	1	no	—
60	#208	—	12	1	no	—
61	C₂×C₃⋊C₈	—	6	2	no	—
62	Dic₁₂	—	12	1	no	—
63	C₃×C₁₆	—	4	1	no	—
64	C₂²×C₉	—	2	1	no	—
65	C₂²×C₉	—	4	1	no	—
66	C₂²×C₉	#102	1	1	yes	characteristic
67	C₄×C₉	—	2	1	no	—
68	C₄×C₉	#99	1	1	yes	characteristic
69	C₄×C₉	#104	1	1	yes	characteristic
70	Dic₉	—	8	1	no	—
71	Dic₉	—	8	1	no	—
72	C₂×Q₁₆	—	9	1	no	—
73	C₂×M₁₆	—	18	1	no	—
74	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
75	C₂²⋊Q₈	—	18	1	no	—
76	M₃₂	—	2	1	no	—
77	#106	—	9	1	no	—
78	#103	—	18	1	no	—
79	C₂³×C₃	#204	1	1	yes	characteristic
80	C₂×C₃×C₄	—	2	1	no	—
81	C₂×C₃×C₄	—	2	1	no	—
82	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
83	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
84	C₂×Dic₃	—	12	1	no	—
85	C₂×Dic₃	—	12	1	no	—
86	Dic₆	—	12	1	no	—
87	Dic₆	—	12	1	no	—
88	C₃×C₈	—	2	1	no	—
89	C₃×C₈	—	2	1	no	—
90	C₃⋊C₈	—	12	2	no	—
91	C₂×C₉	—	2	1	no	—
92	C₂×C₉	—	4	1	no	—
93	C₂×C₉	#327	1	1	yes	characteristic
94	C₂×Q₈	—	18	1	no	—
95	C₂²×C₄	D₃₆	1	1	yes	characteristic
96	Q₁₆	—	36	1	no	—
97	M₁₆	—	4	1	no	—
98	M₁₆	—	36	1	no	—
99	M₁₆	—	36	1	no	—
100	C₂×C₈	—	18	2	no	—
101	C₂×C₈	C₄×D₉	1	1	yes	characteristic
102	C₂×C₈	#611	1	1	yes	characteristic
103	C₄⋊C₄	—	18	1	no	—
104	C₄⋊C₄	—	36	1	no	—
105	C₂²⋊C₄	—	36	1	no	—
106	C₁₆	—	4	1	no	—
107	C₂²×C₃	—	2	1	no	—
108	C₂²×C₃	—	4	1	no	—
109	C₂²×C₃	#821	1	1	yes	characteristic
110	C₃×C₄	—	2	1	no	—
111	C₃×C₄	#806	1	1	yes	characteristic
112	C₃×C₄	#825	1	1	yes	characteristic
113	Dic₃	—	24	1	no	—
114	Dic₃	—	24	1	no	—
115	C₉	#1342	1	1	yes	characteristic, complemented, Sylow
116	C₂³	#3664	1	1	yes	characteristic
117	Q₈	—	36	1	no	—
118	Q₈	—	36	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	36	1	no	—
122	C₂×C₄	—	36	1	no	—
123	C₂×C₄	D₇₂	1	1	yes	characteristic
124	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	36	2	no	—
128	C₂×C₃	—	2	1	no	—
129	C₂×C₃	—	4	1	no	—
130	C₂×C₃	#4470	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	4	1	no	—
133	C₂²	#63380	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	72	1	no	—
136	C₄	—	72	1	no	—
137	C₄	#63365	1	1	yes	characteristic
138	C₄	#63384	1	1	yes	characteristic
139	C₃	#67537	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	4	1	no	—
142	C₂	#10588147	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588807

C₂

yes

maximal, characteristic

C₉×#349

C₂

yes

maximal, characteristic

C₉⋊#338

C₂

yes

maximal, characteristic

#67537

—

maximal

C₂×C₉×M₁₆

C₂²

yes

characteristic

#63486

—

C₂×#3667

—

C₂×Dic₃₆

C₄

yes

characteristic

C₉×M₃₂

—

#63377

C₄

yes

—

—

D₃

yes

characteristic

C₃⋊#338

—

C₂²×C₄×C₉

D₄

yes

characteristic

C₂×Dic₁₈

—

C₉×M₁₆

—

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

characteristic

#3676

—

C₄⋊Dic₉

—

Dic₉⋊C₄

—

#3667

—

#3667

—

C₂×C₉⋊C₈

—

Dic₃₆

—

C₉×C₁₆

—

#1342

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

characteristic

#924

—

C₂×#208

—

C₂×Dic₁₂

—

C₃×M₃₂

—

#817

—

#816

—

C₂³×C₉

SD₁₆

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉

D₈

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

—

C₉⋊C₈

—

#470

—

#349

D₉

yes

characteristic

#338

—

C₂²×C₃×C₄

D₁₂

yes

characteristic

C₂×Dic₆

—

C₃×M₁₆

—

C₂×C₃×C₈

C₄×D₃

yes

C₂×C₃×C₈

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

#208

—

#208

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₃×C₁₆

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

#102

yes

characteristic

C₄×C₉

—

C₄×C₉

yes

C₄×C₉

yes

Dic₉

—

Dic₉

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉

yes

characteristic

C₂²⋊Q₈

—

M₃₂

—

#106

—

#103

—

C₂³×C₃

#204

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₂₄

yes

characteristic

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

#327

yes

characteristic

C₂×Q₈

—

C₂²×C₄

D₃₆

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

C₄×D₉

yes

characteristic

102

C₂×C₈

#611

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₁₆

—

107

C₂²×C₃

—

108

C₂²×C₃

—

109

C₂²×C₃

#821

yes

characteristic

110

C₃×C₄

—

111

C₃×C₄

#806

yes

characteristic

112

C₃×C₄

#825

yes

characteristic

113

Dic₃

—

114

Dic₃

—

115

C₉

#1342

yes

characteristic, complemented, Sylow

116

C₂³

#3664

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₇₂

yes

characteristic

124

C₂×C₄

(C₂×D₉)⋊C₄

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

#4470

yes

characteristic

131

C₂²

—

132

C₂²

—

133

C₂²

#63380

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#63365

yes

characteristic

138

C₄

#63384

yes

characteristic

139

C₃

#67537

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#10588147

yes

characteristic

	subgroup	G/H
center	C₂	#10588147
commutator	C₂×C₈×C₉	C₂×C₄
Frattini	C₂×C₃×M₁₆	C₂×D₃
Fitting	C₉×#349	C₂
socle	C₂×C₃	#4470
prime radical	C₂³×C₃	#204
cosocle	C₂×C₉×M₁₆	C₂²
hypercenter	#349	D₉
hypocenter	C₉	#1342
LCS	G ▷ C₂×C₈×C₉ ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₂

#10588147

commutator

C₂×C₈×C₉

C₂×C₄

Frattini

C₂×C₃×M₁₆

C₂×D₃

C₂

C₂×C₃

C₂³×C₃

C₂×C₉×M₁₆

C₂²

D₉

C₉

G ▷ C₂×C₈×C₉ ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
115. C₉	#1342	semidirect

H ≅ G/N

type

115. C₉

#1342

semidirect

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

subgroup	index
#10588807	2
C₉×#349	2
C₉⋊#338	2
#67537	3

subgroup

index

Subgroups — #11771768

Conjugacy classes of subgroups

Complements

Maximal subgroups