Group #11786189 — subgroups

Subgroups — #11786189

Order 1152.

768 conjugacy classes of subgroups; only the 281 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 281 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	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63641	C₂	1	4	yes	maximal
3	#10592631	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
5	#70446	—	3	4	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	2	yes	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₃²×C₄⋊C₈	—	2	2	no	—
11	#63658	C₄	1	2	yes	—
12	#63658	C₄	1	2	yes	—
13	#63641	—	2	8	no	—
14	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
15	C₂×C₃×C₄⋊C₈	D₃	1	4	yes	—
16	C₂×Dic₃⋊C₈	—	3	16	no	—
17	#4932	—	3	4	no	—
18	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
19	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
22	C₂²×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
23	C₂×C₃²×C₈	—	2	4	no	—
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₃²×C₄²	D₄	1	2	yes	—
26	C₃²×C₄²	Q₈	1	2	yes	—
27	C₄×C₃⋊Dic₃	D₄	1	4	yes	—
28	C₄×C₃⋊Dic₃	Q₈	1	4	yes	—
29	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
30	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
31	#1843	—	9	1	no	Sylow
32	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
33	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
34	C₂×C₄×Dic₃	—	3	8	no	—
35	C₂²×C₃⋊C₈	—	3	8	no	—
36	C₃×C₄⋊C₈	—	2	8	no	—
37	C₃⋊#95	—	3	8	no	—
38	C₃⋊#95	—	3	8	no	—
39	Dic₃⋊C₈	—	6	32	no	—
40	C₄⋊(C₃⋊C₈)	—	6	8	no	—
41	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃²×C₄	—	2	2	no	—
43	C₂×C₃²×C₄	—	2	2	no	—
44	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
45	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
46	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
47	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
50	C₂×C₃⋊Dic₃	—	2	4	no	—
51	C₂×C₃⋊Dic₃	—	2	4	no	—
52	C₂×C₃⋊Dic₃	C₄⋊C₄	1	8	yes	—
53	C₃²×C₈	—	4	2	no	—
54	C₃⋊(C₃⋊C₈)	—	4	2	no	—
55	C₂×C₄⋊C₈	—	9	4	no	—
56	C₂×C₄⋊C₈	—	9	1	no	—
57	C₂×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
58	#376	—	9	1	no	—
59	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
60	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
61	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
62	C₂²×Dic₃	—	3	16	no	—
63	C₂×C₃×C₈	—	2	16	no	—
64	C₂×C₃×C₈	—	2	8	no	—
65	C₃×C₄²	Dic₆	1	8	yes	—
66	C₃×C₄²	#67	1	8	yes	—
67	C₄×Dic₃	—	3	16	no	—
68	C₄×Dic₃	—	3	16	no	—
69	C₂×C₃⋊C₈	—	6	16	no	—
70	C₂×C₃⋊C₈	—	6	8	no	—
71	C₂²×C₃²	C₂×C₄⋊C₄	1	2	yes	—
72	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₃²	#131	1	2	yes	—
74	C₂²×C₃²	#131	1	2	yes	—
75	C₃²×C₄	—	2	2	no	—
76	C₃²×C₄	—	2	2	no	—
77	C₃²×C₄	#127	1	1	yes	characteristic
78	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
79	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
80	C₃⋊Dic₃	—	2	8	no	—
81	C₂²×C₈	—	9	2	no	—
82	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
83	C₂×C₄²	—	9	2	no	—
84	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
85	C₄⋊C₈	—	2	2	no	—
86	C₄⋊C₈	—	18	8	no	—
87	C₄⋊C₈	—	18	2	no	—
88	#95	—	9	2	no	—
89	#95	—	9	2	no	—
90	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
91	C₂×C₃×C₄	—	2	8	no	—
92	C₂×C₃×C₄	—	2	8	no	—
93	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
94	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
95	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
96	C₂×C₃×C₄	D₃×D₄	1	8	yes	—
97	C₂×C₃×C₄	D₃×Q₈	1	8	yes	—
98	C₂×C₃×C₄	C₂×#67	1	4	yes	—
99	C₂×Dic₃	—	3	32	no	—
100	C₂×Dic₃	—	6	16	no	—
101	C₂×Dic₃	—	6	16	no	—
102	C₃×C₈	—	4	8	no	—
103	C₃⋊C₈	—	12	8	no	—
104	C₂×C₃²	#384	1	1	yes	characteristic
105	C₂×C₃²	#424	1	4	yes	—
106	C₂×C₃²	#424	1	1	yes	characteristic
107	C₂×C₃²	#424	1	1	yes	characteristic
108	C₂²×C₄	—	9	4	no	—
109	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
110	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	18	4	no	—
115	C₂×C₈	—	18	2	no	—
116	C₄²	—	9	4	no	—
117	C₄²	—	9	4	no	—
118	C₄²	#634	1	2	yes	—
119	C₄²	#638	1	2	yes	—
120	C₂²×C₃	D₃×C₄⋊C₄	1	8	yes	—
121	C₂²×C₃	#901	1	8	yes	—
122	C₂²×C₃	#907	1	8	yes	—
123	C₂²×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
124	C₃×C₄	—	2	8	no	—
125	C₃×C₄	—	2	8	no	—
126	C₃×C₄	#888	1	8	yes	—
127	C₃×C₄	#940	1	4	yes	—
128	C₃×C₄	#944	1	4	yes	—
129	Dic₃	—	6	32	no	—
130	C₃²	#1843	1	1	yes	characteristic, complemented, Sylow
131	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	9	8	no	—
135	C₂×C₄	—	18	4	no	—
136	C₂×C₄	—	18	4	no	—
137	C₂×C₄	#3750	1	2	yes	—
138	C₂×C₄	#3750	1	2	yes	—
139	C₂×C₄	C₂×#634	1	1	yes	characteristic
140	C₂×C₄	D₄×C₃⋊D₃	1	2	yes	—
141	C₂×C₄	Q₈×C₃⋊D₃	1	2	yes	—
142	C₂×C₄	C₂×#638	1	1	yes	characteristic
143	C₈	—	4	2	no	—
144	C₈	—	36	2	no	—
145	C₂×C₃	#4839	1	16	yes	—
146	C₂×C₃	#4975	1	4	yes	—
147	C₂×C₃	#5102	1	4	yes	—
148	C₂×C₃	#5119	1	4	yes	—
149	C₂²	C₄⋊C₄×C₃⋊D₃	1	2	yes	—
150	C₂²	#64110	1	2	yes	—
151	C₂²	#64116	1	2	yes	—
152	C₂²	C₂×#3750	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	18	8	no	—
156	C₄	#64097	1	2	yes	—
157	C₄	#64149	1	1	yes	characteristic
158	C₄	#64153	1	1	yes	characteristic
159	C₃	#70446	1	4	yes	complemented
160	C₂	#10592538	1	4	yes	—
161	C₂	#10592674	1	1	yes	characteristic
162	C₂	#10592801	1	1	yes	characteristic
163	C₂	#10592818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃²×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×#63641

C₂

yes

maximal

#10592631

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊(C₃⋊C₈))

C₂

yes

maximal, characteristic

#70446

—

maximal

C₂²×C₃²×C₈

C₂²

yes

—

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₂×C₄×C₃⋊Dic₃

C₂²

yes

—

C₂²×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃²×C₄⋊C₈

—

#63658

C₄

yes

—

#63658

C₄

yes

—

#63641

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₂×C₃×C₄⋊C₈

D₃

yes

—

C₂×Dic₃⋊C₈

—

#4932

—

C₂×C₄⋊(C₃⋊C₈)

—

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₂²×C₃⋊Dic₃

C₂×C₄

yes

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

—

C₃²×C₄²

D₄

yes

—

C₃²×C₄²

Q₈

yes

—

C₄×C₃⋊Dic₃

D₄

yes

—

C₄×C₃⋊Dic₃

Q₈

yes

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

#1843

—

Sylow

C₂²×C₃×C₈

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

C₃×C₄⋊C₈

—

C₃⋊#95

—

C₃⋊#95

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₃²

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×Q₈

yes

characteristic

C₂×C₃²×C₄

C₂×Q₈

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₃⋊D₃

yes

characteristic

#376

—

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

Dic₆

yes

—

C₃×C₄²

#67

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃²

#131

yes

—

C₂²×C₃²

#131

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#127

yes

characteristic

C₃²×C₄

C₄⋊Q₈

yes

—

C₃²×C₄

C₄⋊Q₈

yes

characteristic

C₃⋊Dic₃

—

C₂²×C₈

—

C₂²×C₈

C₂×C₃⋊D₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

#95

—

C₂³×C₃

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

C₂×Dic₆

yes

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

D₃×Q₈

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

C₂×Dic₃

—

100

C₂×Dic₃

—

101

C₂×Dic₃

—

102

C₃×C₈

—

103

C₃⋊C₈

—

104

C₂×C₃²

#384

yes

characteristic

105

C₂×C₃²

#424

yes

—

106

C₂×C₃²

#424

yes

characteristic

107

C₂×C₃²

#424

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

C₄×C₃⋊D₃

yes

characteristic

110

C₂²×C₄

C₄×C₃⋊D₃

yes

characteristic

111

C₂²×C₄

C₂²×C₃⋊D₃

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄²

—

117

C₄²

—

118

C₄²

#634

yes

—

119

C₄²

#638

yes

—

120

C₂²×C₃

D₃×C₄⋊C₄

yes

—

121

C₂²×C₃

#901

yes

—

122

C₂²×C₃

#907

yes

—

123

C₂²×C₃

C₂×Dic₃⋊C₄

yes

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₃×C₄

#888

yes

—

127

C₃×C₄

#940

yes

—

128

C₃×C₄

#944

yes

—

129

Dic₃

—

130

C₃²

#1843

yes

characteristic, complemented, Sylow

131

C₂³

C₂×C₄×C₃⋊D₃

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#3750

yes

—

138

C₂×C₄

#3750

yes

—

139

C₂×C₄

C₂×#634

yes

characteristic

140

C₂×C₄

D₄×C₃⋊D₃

yes

—

141

C₂×C₄

Q₈×C₃⋊D₃

yes

—

142

C₂×C₄

C₂×#638

yes

characteristic

143

C₈

—

144

C₈

—

145

C₂×C₃

#4839

yes

—

146

C₂×C₃

#4975

yes

—

147

C₂×C₃

#5102

yes

—

148

C₂×C₃

#5119

yes

—

149

C₂²

C₄⋊C₄×C₃⋊D₃

yes

—

150

C₂²

#64110

yes

—

151

C₂²

#64116

yes

—

152

C₂²

C₂×#3750

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#64097

yes

—

157

C₄

#64149

yes

characteristic

158

C₄

#64153

yes

characteristic

159

C₃

#70446

yes

complemented

160

C₂

#10592538

yes

—

161

C₂

#10592674

yes

characteristic

162

C₂

#10592801

yes

characteristic

163

C₂

#10592818

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator, socle, prime radical	C₂³×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄⋊C₈	C₂
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	#1843
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×C₃⋊D₃

commutator, socle, prime radical

C₂³×C₃²

C₂²×C₄

Fitting

C₂×C₃²×C₄⋊C₈

C₂

cosocle

C₂²×C₃²×C₄

C₂³

hypercenter

C₂×C₄⋊C₈

C₃⋊D₃

C₃²

G ▷ C₂³×C₃² ▷ C₃²

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
130. C₃²	#1843	semidirect
159. C₃	#70446	semidirect

H ≅ G/N

type

130. C₃²

#1843

semidirect

159. C₃

#70446

semidirect

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

subgroup	index
C₂×C₃²×C₄⋊C₈	2
C₂×#63641	2
#10592631	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#70446	3

subgroup

index

C₂×C₃²×C₄⋊C₈

C₂×#63641

#10592631

C₂×C₄⋊(C₃⋊(C₃⋊C₈))

#70446

Subgroups — #11786189

Conjugacy classes of subgroups

Complements

Maximal subgroups