Group #11781961 — subgroups

Subgroups — #11781961

Order 1152.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#10588513	C₂	1	2	yes	maximal
2	#10588362	C₂	1	1	yes	maximal, characteristic
3	#10588359	C₂	1	1	yes	maximal, characteristic
4	C₉×#334	C₂	1	1	yes	maximal, characteristic
5	#10588159	C₂	1	2	yes	maximal
6	#68552	—	3	1	no	maximal
7	C₂×Dic₃₆	—	2	2	no	—
8	#63431	C₂²	1	2	yes	—
9	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
10	C₉×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
12	#63378	—	2	1	no	—
13	#63378	—	2	1	no	—
14	#63376	—	2	2	no	—
15	#63375	—	2	1	no	—
16	Dic₉⋊C₈	—	2	1	no	—
17	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
18	#4849	—	3	2	no	—
19	#4682	—	3	1	no	—
20	#4679	—	3	1	no	—
21	C₃×#334	D₃	1	1	yes	characteristic
22	#4482	—	3	2	no	—
23	C₂×Dic₁₈	—	2	2	no	—
24	C₂×Dic₁₈	D₄	1	1	yes	characteristic
25	C₂×C₈×C₉	—	2	2	no	—
26	C₂×C₈×C₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₉	D₄	1	1	yes	characteristic
28	C₄²×C₉	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₉	—	2	2	no	—
30	C₄⋊Dic₉	—	2	2	no	—
31	C₄⋊Dic₉	D₄	1	1	yes	characteristic
32	Dic₉⋊C₄	—	2	1	no	—
33	C₄×Dic₉	—	2	1	no	—
34	C₂×C₉⋊C₈	—	2	1	no	—
35	Dic₃₆	—	4	4	no	—
36	#1694	—	9	1	no	Sylow
37	C₂×Dic₁₂	—	6	2	no	—
38	#869	—	3	2	no	—
39	C₄×Dic₆	—	3	1	no	—
40	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
41	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
42	#818	—	6	1	no	—
43	#818	—	6	1	no	—
44	#816	—	6	2	no	—
45	#815	—	6	1	no	—
46	Dic₃⋊C₈	—	6	1	no	—
47	C₄⋊(C₃⋊C₈)	—	3	1	no	—
48	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₉	#41	1	1	yes	characteristic
51	C₂×Dic₉	—	2	1	no	—
52	C₂×Dic₉	—	2	1	no	—
53	C₂×Dic₉	—	4	2	no	—
54	Dic₁₈	—	2	2	no	—
55	Dic₁₈	—	4	2	no	—
56	Dic₁₈	—	4	2	no	—
57	C₈×C₉	—	2	1	no	—
58	C₈×C₉	—	4	2	no	—
59	C₈×C₉	Q₁₆	1	2	yes	complemented
60	C₉⋊C₈	—	4	1	no	—
61	#500	—	9	1	no	—
62	#462	—	9	2	no	—
63	#446	—	9	1	no	—
64	#334	D₉	1	1	yes	characteristic
65	#332	—	9	2	no	—
66	C₂×Dic₆	—	3	1	no	—
67	C₂×Dic₆	—	6	2	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
70	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
71	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
72	C₄⋊Dic₃	—	3	1	no	—
73	C₄⋊Dic₃	—	6	2	no	—
74	C₄⋊Dic₃	—	6	2	no	—
75	Dic₃⋊C₄	—	6	1	no	—
76	C₄×Dic₃	—	6	1	no	—
77	C₂×C₃⋊C₈	—	6	1	no	—
78	Dic₁₂	—	12	4	no	—
79	C₂²×C₉	#121	1	1	yes	characteristic
80	C₄×C₉	—	2	1	no	—
81	C₄×C₉	C₂×Q₁₆	1	1	yes	characteristic
82	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₄×C₉	#137	1	1	yes	characteristic
84	Dic₉	—	4	1	no	—
85	Dic₉	—	4	1	no	—
86	Dic₉	—	8	2	no	—
87	C₂×Q₁₆	—	18	2	no	—
88	C₄⋊Q₈	—	9	2	no	—
89	C₄×Q₈	—	9	1	no	—
90	#107	—	18	1	no	—
91	#107	—	18	1	no	—
92	C₄⋊C₈	—	9	1	no	—
93	C₄⋊C₈	—	18	1	no	—
94	C₄⋊C₈	C₂×D₉	1	2	yes	—
95	#103	—	18	2	no	—
96	#97	—	18	1	no	—
97	C₄×C₈	C₂×D₉	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
99	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
100	C₂×C₃×C₄	#239	1	1	yes	characteristic
101	C₂×Dic₃	—	6	1	no	—
102	C₂×Dic₃	—	6	1	no	—
103	C₂×Dic₃	—	12	2	no	—
104	Dic₆	—	6	2	no	—
105	Dic₆	—	12	2	no	—
106	Dic₆	—	12	2	no	—
107	C₃×C₈	—	2	1	no	—
108	C₃×C₈	—	4	2	no	—
109	C₃×C₈	Dic₁₂	1	2	yes	—
110	C₃⋊C₈	—	12	1	no	—
111	C₂×C₉	#462	1	1	yes	characteristic
112	C₂×C₉	#469	1	1	yes	characteristic
113	C₂×C₉	#473	1	1	yes	characteristic
114	C₂×Q₈	—	9	1	no	—
115	C₂×Q₈	—	18	2	no	—
116	Q₁₆	—	36	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	18	1	no	—
119	C₂×C₈	D₃₆	1	1	yes	characteristic
120	C₂×C₈	D₃₆	1	1	yes	characteristic
121	C₄⋊C₄	—	9	1	no	—
122	C₄⋊C₄	—	18	1	no	—
123	C₄⋊C₄	—	18	2	no	—
124	C₄⋊C₄	—	18	2	no	—
125	C₄²	—	18	1	no	—
126	C₄²	C₂²×D₉	1	1	yes	characteristic
127	C₂²×C₃	#895	1	1	yes	characteristic
128	C₃×C₄	—	2	1	no	—
129	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
130	C₃×C₄	#909	1	1	yes	characteristic
131	C₃×C₄	#914	1	2	yes	—
132	Dic₃	—	12	1	no	—
133	Dic₃	—	12	1	no	—
134	Dic₃	—	24	2	no	—
135	C₉	#1694	1	1	yes	characteristic, complemented, Sylow
136	Q₈	—	18	2	no	—
137	Q₈	—	36	2	no	—
138	Q₈	—	36	2	no	—
139	C₂×C₄	—	18	1	no	—
140	C₂×C₄	—	18	1	no	—
141	C₂×C₄	—	36	2	no	—
142	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
143	C₂×C₄	D₄×D₉	1	1	yes	characteristic
144	C₂×C₄	#3701	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	4	2	no	—
147	C₈	—	36	1	no	—
148	C₈	Dic₃₆	1	2	yes	complemented
149	C₂×C₃	#4849	1	1	yes	characteristic
150	C₂×C₃	#4865	1	1	yes	characteristic
151	C₂×C₃	#4896	1	1	yes	characteristic
152	C₂²	#63457	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	36	1	no	—
155	C₄	—	36	1	no	—
156	C₄	—	72	2	no	—
157	C₄	C₂×Dic₃₆	1	1	yes	characteristic
158	C₄	#63471	1	1	yes	characteristic
159	C₄	#63476	1	2	yes	—
160	C₃	#68552	1	1	yes	characteristic
161	C₂	#10588513	1	1	yes	characteristic
162	C₂	#10588545	1	1	yes	characteristic
163	C₂	#10588576	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588513

C₂

yes

maximal

#10588362

C₂

yes

maximal, characteristic

#10588359

C₂

yes

maximal, characteristic

C₉×#334

C₂

yes

maximal, characteristic

#10588159

C₂

yes

maximal

#68552

—

maximal

C₂×Dic₃₆

—

#63431

C₂²

yes

—

C₄×Dic₁₈

C₂²

yes

characteristic

C₉×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₉

C₂²

yes

—

—

—

—

Dic₉⋊C₈

—

C₄⋊(C₉⋊C₈)

C₂²

yes

—

—

—

D₃

yes

characteristic

#4482

—

C₂×Dic₁₈

—

C₂×Dic₁₈

D₄

yes

characteristic

C₂×C₈×C₉

—

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

characteristic

C₄²×C₉

C₂³

yes

characteristic

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₄⋊Dic₉

D₄

yes

characteristic

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂×C₉⋊C₈

—

Dic₃₆

—

#1694

—

Sylow

C₂×Dic₁₂

—

#869

—

C₄×Dic₆

—

C₃×C₄⋊C₈

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

—

—

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

Q₁₆

yes

complemented

C₉⋊C₈

—

—

—

—

D₉

yes

characteristic

#332

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₁₂

yes

characteristic

C₂×C₃×C₈

D₁₂

yes

characteristic

C₃×C₄²

C₂²×D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₉

#121

yes

characteristic

C₄×C₉

—

C₄×C₉

C₂×Q₁₆

yes

characteristic

C₄×C₉

AGL(1,ℤ/8)

yes

—

C₄×C₉

#137

yes

characteristic

Dic₉

—

Dic₉

—

Dic₉

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₉

yes

—

#103

—

#97

—

C₄×C₈

C₂×D₉

yes

characteristic

C₂×C₃×C₄

C₂×D₁₂

yes

characteristic

C₂×C₃×C₄

D₃×D₄

yes

characteristic

100

C₂×C₃×C₄

#239

yes

characteristic

101

C₂×Dic₃

—

102

C₂×Dic₃

—

103

C₂×Dic₃

—

104

Dic₆

—

105

Dic₆

—

106

Dic₆

—

107

C₃×C₈

—

108

C₃×C₈

—

109

C₃×C₈

Dic₁₂

yes

—

110

C₃⋊C₈

—

111

C₂×C₉

#462

yes

characteristic

112

C₂×C₉

#469

yes

characteristic

113

C₂×C₉

#473

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

Q₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

D₃₆

yes

characteristic

120

C₂×C₈

D₃₆

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

—

126

C₄²

C₂²×D₉

yes

characteristic

127

C₂²×C₃

#895

yes

characteristic

128

C₃×C₄

—

129

C₃×C₄

C₂×Dic₁₂

yes

characteristic

130

C₃×C₄

#909

yes

characteristic

131

C₃×C₄

#914

yes

—

132

Dic₃

—

133

Dic₃

—

134

Dic₃

—

135

C₉

#1694

yes

characteristic, complemented, Sylow

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×D₃₆

yes

characteristic

143

C₂×C₄

D₄×D₉

yes

characteristic

144

C₂×C₄

#3701

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

Dic₃₆

yes

complemented

149

C₂×C₃

#4849

yes

characteristic

150

C₂×C₃

#4865

yes

characteristic

151

C₂×C₃

#4896

yes

characteristic

152

C₂²

#63457

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₂×Dic₃₆

yes

characteristic

158

C₄

#63471

yes

characteristic

159

C₄

#63476

yes

—

160

C₃

#68552

yes

characteristic

161

C₂

#10588513

yes

characteristic

162

C₂

#10588545

yes

characteristic

163

C₂

#10588576

yes

characteristic

	subgroup	G/H
center	C₂²	#63457
commutator, cosocle	C₄²×C₉	C₂³
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#334	C₂
socle, prime radical	C₂²×C₃	#895
hypercenter	#334	D₉
hypocenter	C₉	#1694
LCS	G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#63457

commutator, cosocle

C₄²×C₉

C₂³

Frattini

C₃×C₄²

C₂²×D₃

C₂

C₂²×C₃

D₉

C₉

G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
59. C₈×C₉	Q₁₆	semidirect
135. C₉	#1694	semidirect
148. C₈	Dic₃₆	semidirect

H ≅ G/N

type

59. C₈×C₉

Q₁₆

semidirect

135. C₉

#1694

semidirect

148. C₈

Dic₃₆

semidirect

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

subgroup	index
#10588513	2
#10588362	2
#10588359	2
C₉×#334	2
#10588159	2
#68552	3

subgroup

index

Subgroups — #11781961

Conjugacy classes of subgroups

Complements

Maximal subgroups