Group #11800013 — subgroups

Subgroups — #11800013

Order 1152.

186 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10588800	C₂	1	1	yes	maximal, characteristic
3	C₉×#349	C₂	1	1	yes	maximal, characteristic
4	#10588195	C₂	1	2	yes	maximal, complemented
5	#10588190	C₂	1	2	yes	maximal, complemented
6	#69117	—	3	1	no	maximal
7	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
8	#63486	—	2	1	no	—
9	C₂×#3667	—	2	1	no	—
10	C₂×Dic₃₆	C₂²	1	1	yes	characteristic
11	C₉×M₃₂	C₂²	1	2	yes	—
12	#63377	C₂²	1	1	yes	characteristic
13	#63376	—	2	1	no	—
14	#63372	—	2	1	no	—
15	#63372	C₂²	1	2	yes	—
16	#5137	—	3	1	no	—
17	#5125	—	3	1	no	—
18	C₃×#349	D₃	1	1	yes	characteristic
19	#4517	—	3	2	no	—
20	#4513	—	3	2	no	—
21	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
22	C₂×Dic₁₈	—	2	1	no	—
23	C₉×M₁₆	—	2	2	no	—
24	C₂×C₈×C₉	D₄	1	1	yes	characteristic
25	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
26	#3676	—	4	1	no	—
27	C₄⋊Dic₉	—	2	1	no	—
28	Dic₉⋊C₄	—	4	1	no	—
29	#3667	—	2	2	no	—
30	#3667	—	4	1	no	—
31	C₂×C₉⋊C₈	—	4	1	no	—
32	Dic₃₆	—	2	2	no	—
33	C₉×C₁₆	—	2	2	no	—
34	#2239	—	9	1	no	Sylow
35	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
36	#924	—	6	1	no	—
37	C₂×#208	—	6	1	no	—
38	C₂×Dic₁₂	—	3	1	no	—
39	C₃×M₃₂	C₂×D₃	1	2	yes	—
40	#819	—	3	2	no	—
41	#819	—	6	1	no	—
42	#817	—	3	1	no	—
43	#816	—	6	1	no	—
44	C₂³×C₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₄×C₉	—	2	1	no	—
46	C₂×C₄×C₉	—	2	1	no	—
47	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₉	—	4	1	no	—
50	C₂×Dic₉	—	4	1	no	—
51	Dic₁₈	—	4	1	no	—
52	Dic₁₈	—	4	1	no	—
53	C₈×C₉	—	2	1	no	—
54	C₈×C₉	#41	1	2	yes	—
55	C₉⋊C₈	—	4	2	no	—
56	#470	—	9	1	no	—
57	#430	—	9	1	no	—
58	#365	—	9	2	no	—
59	#362	—	9	2	no	—
60	#349	D₉	1	1	yes	characteristic
61	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
62	C₂×Dic₆	—	6	1	no	—
63	C₃×M₁₆	—	2	2	no	—
64	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
66	#217	—	12	1	no	—
67	C₄⋊Dic₃	—	6	1	no	—
68	Dic₃⋊C₄	—	12	1	no	—
69	#208	—	6	2	no	—
70	#208	—	12	1	no	—
71	C₂×C₃⋊C₈	—	12	1	no	—
72	Dic₁₂	—	6	2	no	—
73	C₃×C₁₆	—	2	2	no	—
74	C₂²×C₉	—	2	1	no	—
75	C₂²×C₉	—	4	1	no	—
76	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₉	—	2	1	no	—
78	C₄×C₉	#123	1	1	yes	characteristic
79	C₄×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
80	Dic₉	—	8	1	no	—
81	Dic₉	—	8	1	no	—
82	C₂×Q₁₆	—	9	1	no	—
83	C₂×M₁₆	—	18	1	no	—
84	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
85	C₂²⋊Q₈	—	18	1	no	—
86	M₃₂	C₂×D₉	1	2	yes	—
87	#108	—	9	2	no	—
88	#108	—	18	1	no	—
89	#106	—	9	1	no	—
90	#103	—	18	1	no	—
91	C₂³×C₃	#204	1	1	yes	characteristic
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	#204	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
96	C₂×Dic₃	—	12	1	no	—
97	C₂×Dic₃	—	12	1	no	—
98	Dic₆	—	12	1	no	—
99	Dic₆	—	12	1	no	—
100	C₃×C₈	—	2	1	no	—
101	C₃×C₈	#237	1	2	yes	—
102	C₃⋊C₈	—	12	2	no	—
103	C₂×C₉	—	2	1	no	—
104	C₂×C₉	—	4	1	no	—
105	C₂×C₉	#481	1	1	yes	characteristic
106	C₂×Q₈	—	18	1	no	—
107	C₂²×C₄	D₃₆	1	1	yes	characteristic
108	Q₁₆	—	18	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	18	2	no	—
111	M₁₆	—	36	1	no	—
112	C₂×C₈	—	36	1	no	—
113	C₂×C₈	D₃₆	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
115	C₄⋊C₄	—	18	1	no	—
116	C₄⋊C₄	—	36	1	no	—
117	C₂²⋊C₄	—	36	1	no	—
118	C₁₆	—	2	2	no	—
119	C₂²×C₃	—	2	1	no	—
120	C₂²×C₃	—	4	1	no	—
121	C₂²×C₃	C₂×#204	1	1	yes	characteristic
122	C₃×C₄	—	2	1	no	—
123	C₃×C₄	#886	1	1	yes	characteristic
124	C₃×C₄	#908	1	1	yes	characteristic
125	Dic₃	—	24	1	no	—
126	Dic₃	—	24	1	no	—
127	C₉	#2239	1	1	yes	characteristic, complemented, Sylow
128	C₂³	#3664	1	1	yes	characteristic
129	Q₈	—	36	1	no	—
130	Q₈	—	36	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	36	1	no	—
134	C₂×C₄	—	36	1	no	—
135	C₂×C₄	#3664	1	1	yes	characteristic
136	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	36	2	no	—
139	C₈	#3699	1	2	yes	—
140	C₂×C₃	—	2	1	no	—
141	C₂×C₃	—	4	1	no	—
142	C₂×C₃	#4735	1	1	yes	characteristic
143	C₂²	—	2	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	C₂×#3664	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	72	1	no	—
148	C₄	—	72	1	no	—
149	C₄	#63448	1	1	yes	characteristic
150	C₄	#63470	1	1	yes	characteristic
151	C₃	#69117	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	4	1	no	—
154	C₂	#10588417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588807

C₂

yes

maximal, characteristic

#10588800

C₂

yes

maximal, characteristic

C₉×#349

C₂

yes

maximal, characteristic

#10588195

C₂

yes

maximal, complemented

#10588190

C₂

yes

maximal, complemented

#69117

—

maximal

C₂×C₉×M₁₆

C₂²

yes

characteristic

#63486

—

C₂×#3667

—

C₂×Dic₃₆

C₂²

yes

characteristic

C₉×M₃₂

C₂²

yes

—

#63377

C₂²

yes

—

—

C₂²

yes

—

#5137

—

#5125

—

C₃×#349

D₃

yes

characteristic

#4517

—

#4513

—

C₂²×C₄×C₉

D₄

yes

characteristic

C₂×Dic₁₈

—

C₉×M₁₆

—

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

C₂³

yes

characteristic

#3676

—

C₄⋊Dic₉

—

Dic₉⋊C₄

—

#3667

—

#3667

—

C₂×C₉⋊C₈

—

Dic₃₆

—

C₉×C₁₆

—

#2239

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

characteristic

#924

—

C₂×#208

—

C₂×Dic₁₂

—

C₃×M₃₂

C₂×D₃

yes

—

—

—

—

—

C₂³×C₉

SD₁₆

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

SD₁₆

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

#41

yes

—

C₉⋊C₈

—

—

—

—

—

D₉

yes

characteristic

C₂²×C₃×C₄

D₁₂

yes

characteristic

C₂×Dic₆

—

C₃×M₁₆

—

C₂×C₃×C₈

D₁₂

yes

characteristic

C₂×C₃×C₈

C₂²×D₃

yes

characteristic

#217

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

#208

—

#208

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₃×C₁₆

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×SD₁₆

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

C₂×D₉

yes

—

—

—

—

—

C₂³×C₃

#204

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#204

yes

characteristic

C₂×C₃×C₄

C₂×D₁₂

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

100

C₃×C₈

—

101

C₃×C₈

#237

yes

—

102

C₃⋊C₈

—

103

C₂×C₉

—

104

C₂×C₉

—

105

C₂×C₉

#481

yes

characteristic

106

C₂×Q₈

—

107

C₂²×C₄

D₃₆

yes

characteristic

108

Q₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

D₃₆

yes

characteristic

114

C₂×C₈

C₂²×D₉

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₁₆

—

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

C₂×#204

yes

characteristic

122

C₃×C₄

—

123

C₃×C₄

#886

yes

characteristic

124

C₃×C₄

#908

yes

characteristic

125

Dic₃

—

126

Dic₃

—

127

C₉

#2239

yes

characteristic, complemented, Sylow

128

C₂³

#3664

yes

characteristic

129

Q₈

—

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

#3664

yes

characteristic

136

C₂×C₄

C₂×D₃₆

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#3699

yes

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

#4735

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×#3664

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#63448

yes

characteristic

150

C₄

#63470

yes

characteristic

151

C₃

#69117

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

#10588417

yes

characteristic

	subgroup	G/H
center	C₂	#10588417
commutator, cosocle	C₂×C₈×C₉	C₂³
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₉×#349	C₂
socle	C₂×C₃	#4735
prime radical	C₂³×C₃	#204
hypercenter	#349	D₉
hypocenter	C₉	#2239
LCS	G ▷ C₂×C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₂

#10588417

commutator, cosocle

C₂×C₈×C₉

C₂³

Frattini

C₂×C₃×C₈

C₂²×D₃

C₂

C₂×C₃

C₂³×C₃

D₉

C₉

G ▷ C₂×C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
4. #10588195	C₂	semidirect
5. #10588190	C₂	semidirect
127. C₉	#2239	semidirect

H ≅ G/N

type

4. #10588195

C₂

semidirect

5. #10588190

C₂

semidirect

127. C₉

#2239

semidirect

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

subgroup	index
#10588807	2
#10588800	2
C₉×#349	2
#10588195	2
#10588190	2
#69117	3

subgroup

index

Subgroups — #11800013

Conjugacy classes of subgroups

Complements

Maximal subgroups