Group #11799708 — subgroups

Subgroups — #11799708

Order 1152.

189 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 64 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	#10588380	C₂	1	1	yes	maximal, characteristic
2	#10588360	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588194	C₂	1	2	yes	maximal
5	#10588190	C₂	1	2	yes	maximal, complemented
6	#69307	—	3	1	no	maximal
7	#63467	C₂²	1	1	yes	characteristic
8	#63464	—	2	1	no	—
9	#63431	—	2	1	no	—
10	C₉×M₃₂	C₂²	1	2	yes	—
11	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
12	#63377	—	2	1	no	—
13	#63377	C₂²	1	2	yes	—
14	#63372	C₂²	1	1	yes	characteristic
15	#63364	—	2	1	no	—
16	#4700	—	3	1	no	—
17	#4680	—	3	1	no	—
18	C₃×#364	D₃	1	1	yes	characteristic
19	#4518	—	3	2	no	—
20	#4513	—	3	2	no	—
21	#3697	—	2	1	no	—
22	C₂×Dic₁₈	—	4	1	no	—
23	C₂×C₈×C₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
25	C₄²×C₉	D₄	1	1	yes	characteristic
26	C₄⋊Dic₉	—	2	2	no	—
27	C₄⋊Dic₉	—	4	1	no	—
28	#3667	—	2	1	no	—
29	D₇₂	Q₈	1	1	yes	characteristic, complemented
30	#3664	—	2	1	no	—
31	#3663	—	4	1	no	—
32	C₈×D₉	—	4	1	no	—
33	Dic₃₆	Q₈	1	1	yes	characteristic, complemented
34	C₉×C₁₆	—	2	2	no	—
35	#2230	—	9	1	no	Sylow
36	#904	—	3	1	no	—
37	#901	—	6	1	no	—
38	#869	—	6	1	no	—
39	C₃×M₃₂	C₂×D₃	1	2	yes	—
40	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	#819	—	3	1	no	—
42	#817	—	3	2	no	—
43	#817	—	6	1	no	—
44	#805	—	6	1	no	—
45	C₂×C₄×C₉	—	2	1	no	—
46	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
47	#611	—	4	1	no	—
48	C₂×Dic₉	—	4	2	no	—
49	D₃₆	—	2	1	no	—
50	C₄×D₉	—	4	1	no	—
51	Dic₁₈	—	2	1	no	—
52	Dic₁₈	—	8	2	no	—
53	C₈×C₉	SD₁₆	1	1	yes	characteristic
54	C₈×C₉	SD₁₆	1	1	yes	characteristic
55	C₈×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₈×C₉	#41	1	1	yes	characteristic
57	C₉⋊C₈	—	4	1	no	—
58	#498	—	9	1	no	—
59	#443	—	9	1	no	—
60	#365	—	9	2	no	—
61	#364	D₉	1	1	yes	characteristic, complemented
62	#360	—	9	2	no	—
63	#235	—	6	1	no	—
64	C₂×Dic₆	—	12	1	no	—
65	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
67	C₃×C₄²	D₁₂	1	1	yes	characteristic
68	C₄⋊Dic₃	—	6	2	no	—
69	C₄⋊Dic₃	—	12	1	no	—
70	#208	—	6	1	no	—
71	Dic₁₂	—	3	1	no	—
72	D₂₄	—	3	1	no	—
73	#204	—	6	1	no	—
74	ΓL(1,𝔽₂₅)	—	12	1	no	—
75	D₃×C₈	—	12	1	no	—
76	C₃×C₁₆	—	2	2	no	—
77	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×D₉	—	4	1	no	—
79	C₄×C₉	—	2	2	no	—
80	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
82	Dic₉	—	4	1	no	—
83	Dic₉	—	8	2	no	—
84	#135	—	9	1	no	—
85	#131	—	18	1	no	—
86	C₄⋊Q₈	—	18	1	no	—
87	M₃₂	C₂×D₉	1	2	yes	—
88	#108	—	9	1	no	—
89	#106	—	9	2	no	—
90	#106	—	18	1	no	—
91	#104	—	18	1	no	—
92	C₄×C₈	C₂×D₉	1	1	yes	characteristic
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
95	#67	—	12	1	no	—
96	C₂×Dic₃	—	12	2	no	—
97	D₁₂	—	6	1	no	—
98	C₄×D₃	—	12	1	no	—
99	Dic₆	—	6	1	no	—
100	Dic₆	—	24	2	no	—
101	C₃×C₈	#204	1	1	yes	characteristic
102	C₃×C₈	#204	1	1	yes	characteristic
103	C₃×C₈	#237	1	1	yes	characteristic
104	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
105	C₃⋊C₈	—	12	1	no	—
106	C₂×C₉	—	2	1	no	—
107	C₂×C₉	#476	1	1	yes	characteristic
108	D₉	—	8	1	no	—
109	#41	—	18	1	no	—
110	C₂×Q₈	—	36	1	no	—
111	Q₁₆	—	9	1	no	—
112	SD₁₆	—	18	1	no	—
113	D₈	—	9	1	no	—
114	M₁₆	—	18	1	no	—
115	M₁₆	—	36	1	no	—
116	C₂×C₈	—	36	1	no	—
117	C₂×C₈	D₃₆	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
119	C₄⋊C₄	—	18	2	no	—
120	C₄⋊C₄	—	36	1	no	—
121	C₄²	D₃₆	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₂²×C₃	#908	1	1	yes	characteristic
124	C₂×D₃	—	12	1	no	—
125	C₃×C₄	—	2	2	no	—
126	C₃×C₄	#897	1	1	yes	characteristic
127	C₃×C₄	C₂×#204	1	1	yes	characteristic
128	Dic₃	—	12	1	no	—
129	Dic₃	—	24	2	no	—
130	C₉	#2230	1	1	yes	characteristic, complemented, Sylow
131	Q₈	—	18	1	no	—
132	Q₈	—	72	2	no	—
133	D₄	—	18	1	no	—
134	D₄	—	36	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	36	1	no	—
137	C₂×C₄	—	36	2	no	—
138	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
139	C₈	—	36	1	no	—
140	C₈	#3664	1	1	yes	characteristic
141	C₈	#3664	1	1	yes	characteristic
142	C₈	#3699	1	1	yes	characteristic
143	C₈	Q₈×D₉	1	1	yes	characteristic
144	C₂×C₃	—	2	1	no	—
145	C₂×C₃	#4842	1	1	yes	characteristic
146	D₃	—	24	1	no	—
147	C₂²	—	36	1	no	—
148	C₂²	#63470	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	36	1	no	—
151	C₄	—	72	2	no	—
152	C₄	#63459	1	1	yes	characteristic
153	C₄	C₂×#3664	1	1	yes	characteristic
154	C₃	#69307	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	72	1	no	—
157	C₂	#10588526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588380

C₂

yes

maximal, characteristic

#10588360

C₂

yes

maximal, characteristic, complemented

C₉×#364

C₂

yes

maximal, characteristic, complemented

#10588194

C₂

yes

maximal

#10588190

C₂

yes

maximal, complemented

#69307

—

maximal

#63467

C₂²

yes

characteristic

#63464

—

#63431

—

C₉×M₃₂

C₂²

yes

—

C₄×C₈×C₉

C₂²

yes

characteristic

#63377

—

#63377

C₂²

yes

—

#63372

C₂²

yes

—

—

—

D₃

yes

—

—

—

C₂×Dic₁₈

—

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

C₂³

yes

characteristic

C₄²×C₉

D₄

yes

characteristic

C₄⋊Dic₉

—

C₄⋊Dic₉

—

#3667

—

D₇₂

Q₈

yes

characteristic, complemented

#3664

—

#3663

—

C₈×D₉

—

Dic₃₆

Q₈

yes

characteristic, complemented

C₉×C₁₆

—

—

—

—

—

C₃×M₃₂

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

#611

—

C₂×Dic₉

—

D₃₆

—

C₄×D₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

SD₁₆

yes

characteristic

C₈×C₉

SD₁₆

yes

characteristic

C₈×C₉

C₂×Q₈

yes

characteristic

C₈×C₉

#41

yes

characteristic

C₉⋊C₈

—

—

—

—

D₉

yes

characteristic, complemented

#360

—

#235

—

C₂×Dic₆

—

C₂×C₃×C₈

D₁₂

yes

characteristic

C₂×C₃×C₈

C₂²×D₃

yes

characteristic

C₃×C₄²

D₁₂

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

#208

—

Dic₁₂

—

D₂₄

—

#204

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₃×C₁₆

—

C₂²×C₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₉

—

C₄×C₉

—

C₄×C₉

C₂²⋊Q₈

yes

characteristic

C₄×C₉

C₂×SD₁₆

yes

characteristic

Dic₉

—

Dic₉

—

#135

—

#131

—

C₄⋊Q₈

—

M₃₂

C₂×D₉

yes

—

—

—

—

—

C₄×C₈

C₂×D₉

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₁₂

yes

characteristic

#67

—

C₂×Dic₃

—

D₁₂

—

C₄×D₃

—

Dic₆

—

100

Dic₆

—

101

C₃×C₈

#204

yes

characteristic

102

C₃×C₈

#204

yes

characteristic

103

C₃×C₈

#237

yes

characteristic

104

C₃×C₈

D₃×Q₈

yes

characteristic

105

C₃⋊C₈

—

106

C₂×C₉

—

107

C₂×C₉

#476

yes

characteristic

108

D₉

—

109

#41

—

110

C₂×Q₈

—

111

Q₁₆

—

112

SD₁₆

—

113

D₈

—

114

M₁₆

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

D₃₆

yes

characteristic

118

C₂×C₈

C₂²×D₉

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

D₃₆

yes

characteristic

122

C₁₆

—

123

C₂²×C₃

#908

yes

characteristic

124

C₂×D₃

—

125

C₃×C₄

—

126

C₃×C₄

#897

yes

characteristic

127

C₃×C₄

C₂×#204

yes

characteristic

128

Dic₃

—

129

Dic₃

—

130

C₉

#2230

yes

characteristic, complemented, Sylow

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×D₃₆

yes

characteristic

139

C₈

—

140

C₈

#3664

yes

characteristic

141

C₈

#3664

yes

characteristic

142

C₈

#3699

yes

characteristic

143

C₈

Q₈×D₉

yes

characteristic

144

C₂×C₃

—

145

C₂×C₃

#4842

yes

characteristic

146

D₃

—

147

C₂²

—

148

C₂²

#63470

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#63459

yes

characteristic

153

C₄

C₂×#3664

yes

characteristic

154

C₃

#69307

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#10588526

yes

characteristic

	subgroup	G/H
center	C₂	#10588526
commutator, cosocle	C₂×C₈×C₉	C₂³
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₉×#364	C₂
socle	C₂×C₃	#4842
prime radical	#63467	C₂²
hypercenter	#364	D₉
hypocenter	C₉	#2230
LCS	G ▷ C₂×C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#10588526

commutator, cosocle

C₂×C₈×C₉

C₂³

Frattini

C₂×C₃×C₈

C₂²×D₃

C₂

C₂×C₃

C₂²

D₉

C₉

G ▷ C₂×C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
2. #10588360	C₂	semidirect
3. C₉×#364	C₂	semidirect
5. #10588190	C₂	semidirect
29. D₇₂	Q₈	semidirect
33. Dic₃₆	Q₈	semidirect
61. #364	D₉	semidirect
130. C₉	#2230	semidirect

H ≅ G/N

type

2. #10588360

C₂

semidirect

3. C₉×#364

C₂

semidirect

5. #10588190

C₂

semidirect

29. D₇₂

Q₈

semidirect

33. Dic₃₆

Q₈

semidirect

61. #364

D₉

semidirect

130. C₉

#2230

semidirect

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

subgroup	index
#10588380	2
#10588360	2
C₉×#364	2
#10588194	2
#10588190	2
#69307	3

subgroup

index

Subgroups — #11799708

Conjugacy classes of subgroups

Complements

Maximal subgroups