Group #11797792 — subgroups

Subgroups — #11797792

Order 1152.

323 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₃×C₄×Dic₁₂	C₂	1	2	yes	maximal
2	C₂×#63550	C₂	1	1	yes	maximal, characteristic
3	#10589327	C₂	1	1	yes	maximal, characteristic
4	#10589142	C₂	1	1	yes	maximal, characteristic
5	#10589117	C₂	1	2	yes	maximal
6	#70909	—	3	2	no	maximal
7	C₂×C₃×Dic₁₂	C₂²	1	2	yes	—
8	C₃×C₄×Dic₆	—	2	2	no	—
9	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
12	C₃×#818	C₂²	1	2	yes	—
13	C₃×#816	—	2	2	no	—
14	#63550	C₄	1	4	yes	complemented
15	C₃×C₄×Q₁₆	—	3	2	no	—
16	C₂×#829	—	3	2	no	—
17	C₄×Dic₁₂	D₃	1	2	yes	—
18	#4564	—	3	2	no	—
19	#4492	—	3	2	no	—
20	#4490	—	3	2	no	—
21	#4461	—	3	2	no	—
22	#4461	—	6	1	no	—
23	C₂×C₃×Dic₆	—	2	2	no	—
24	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
26	C₃²×C₄²	D₄	1	1	yes	characteristic
27	C₃×C₄⋊Dic₃	—	2	2	no	—
28	C₃×Dic₃⋊C₄	—	4	2	no	—
29	C₃×C₄×Dic₃	—	4	2	no	—
30	C₃×Dic₁₂	—	2	2	no	—
31	C₃×Dic₁₂	C₂×C₄	1	4	yes	—
32	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
33	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
34	#2176	—	9	1	no	Sylow
35	C₂×C₃×Q₁₆	—	3	2	no	—
36	C₃×C₄×Q₈	—	6	2	no	—
37	C₂×Dic₁₂	C₂×D₃	1	2	yes	—
38	C₄×Dic₆	—	2	2	no	—
39	C₃×#107	—	3	2	no	—
40	C₃×#103	—	6	2	no	—
41	C₃×C₄×C₈	—	2	1	no	—
42	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
43	#829	—	3	8	no	—
44	#818	C₂×D₃	1	2	yes	—
45	#816	—	2	2	no	—
46	C₂×C₃⋊C₁₆	—	3	2	no	—
47	C₂×C₃⋊C₁₆	—	3	2	no	—
48	C₂×C₃⋊C₁₆	—	6	1	no	—
49	C₂×C₃⋊C₁₆	—	6	1	no	—
50	C₂×C₃²×C₄	D₈	1	2	yes	—
51	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×Dic₃	—	4	2	no	—
53	C₂×C₃×Dic₃	—	4	2	no	—
54	C₃×Dic₆	—	2	4	no	—
55	C₃×Dic₆	—	4	2	no	—
56	C₃²×C₈	—	2	1	no	—
57	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₃²×C₈	#41	1	1	yes	characteristic
59	C₂×Q₃₂	—	9	1	no	—
60	C₄×Q₁₆	—	3	2	no	—
61	#367	—	9	1	no	—
62	#358	—	9	2	no	—
63	#346	—	9	1	no	—
64	C₂×C₃×Q₈	—	6	2	no	—
65	C₂×Dic₆	—	2	2	no	—
66	C₃×Q₁₆	—	3	4	no	—
67	C₃×Q₁₆	—	6	2	no	—
68	C₂×C₃×C₈	—	2	1	no	—
69	C₂×C₃×C₈	—	2	1	no	—
70	C₂×C₃×C₈	#67	1	2	yes	—
71	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
72	C₃×C₄⋊C₄	—	6	2	no	—
73	C₃×C₄⋊C₄	—	12	2	no	—
74	C₃×C₄²	—	2	1	no	—
75	C₃×C₄²	—	12	2	no	—
76	C₃×C₄²	#67	1	2	yes	—
77	C₄⋊Dic₃	—	2	2	no	—
78	Dic₃⋊C₄	—	4	2	no	—
79	C₄×Dic₃	—	4	2	no	—
80	Dic₁₂	—	2	2	no	—
81	Dic₁₂	C₄×D₃	1	4	yes	—
82	C₃⋊C₁₆	—	3	4	no	—
83	C₃⋊C₁₆	—	6	2	no	—
84	C₃⋊C₁₆	—	6	2	no	—
85	C₃⋊C₁₆	—	6	2	no	—
86	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
87	C₃²×C₄	—	2	2	no	—
88	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
89	C₃²×C₄	#135	1	1	yes	characteristic
90	C₃×Dic₃	—	4	4	no	—
91	C₃×Dic₃	—	8	2	no	—
92	C₂×Q₁₆	—	3	2	no	—
93	C₄×Q₈	—	6	2	no	—
94	Q₃₂	—	9	4	no	—
95	C₂×C₁₆	—	9	1	no	—
96	C₂×C₁₆	—	9	1	no	—
97	#107	—	3	2	no	—
98	#103	—	6	2	no	—
99	C₄×C₈	D₃²	1	1	yes	characteristic
100	C₃×Q₈	—	6	4	no	—
101	C₃×Q₈	—	12	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	1	no	—
104	C₂×C₃×C₄	—	12	2	no	—
105	C₂×C₃×C₄	—	12	2	no	—
106	C₂×C₃×C₄	#213	1	2	yes	—
107	C₂×C₃×C₄	#213	1	2	yes	—
108	C₂×C₃×C₄	C₂×#67	1	2	yes	—
109	C₂×Dic₃	—	4	2	no	—
110	C₂×Dic₃	—	4	2	no	—
111	Dic₆	—	2	4	no	—
112	Dic₆	—	4	2	no	—
113	C₃×C₈	—	2	2	no	—
114	C₃×C₈	—	2	2	no	—
115	C₃×C₈	—	2	1	no	—
116	C₃×C₈	—	2	1	no	—
117	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
118	C₃×C₈	#235	1	2	yes	—
119	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
120	C₂×C₃²	#510	1	2	yes	—
121	C₂×Q₈	—	6	2	no	—
122	Q₁₆	—	3	4	no	—
123	Q₁₆	—	6	2	no	—
124	C₂×C₈	#625	1	1	yes	characteristic
125	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
126	C₄⋊C₄	—	6	2	no	—
127	C₄⋊C₄	—	12	2	no	—
128	C₄²	—	12	2	no	—
129	C₄²	#625	1	1	yes	characteristic
130	C₁₆	—	9	2	no	—
131	C₁₆	—	18	1	no	—
132	C₂²×C₃	—	2	1	no	—
133	C₂²×C₃	C₂×#213	1	2	yes	—
134	C₃×C₄	—	2	2	no	—
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	2	1	no	—
137	C₃×C₄	—	2	1	no	—
138	C₃×C₄	—	4	2	no	—
139	C₃×C₄	—	12	4	no	—
140	C₃×C₄	—	24	2	no	—
141	C₃×C₄	C₄×#67	1	2	yes	—
142	C₃×C₄	#950	1	2	yes	—
143	Dic₃	—	4	4	no	—
144	Dic₃	—	8	2	no	—
145	C₃²	#2176	1	1	yes	characteristic, complemented, Sylow
146	Q₈	—	6	4	no	—
147	Q₈	—	12	2	no	—
148	C₂×C₄	—	12	2	no	—
149	C₂×C₄	—	12	2	no	—
150	C₂×C₄	#3713	1	2	yes	—
151	C₂×C₄	C₂×#625	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	#3798	1	1	yes	characteristic
154	C₈	C₄×D₃²	1	1	yes	characteristic
155	C₂×C₃	—	2	1	no	—
156	C₂×C₃	—	2	2	no	—
157	C₂×C₃	C₄×#213	1	2	yes	—
158	C₂×C₃	#5179	1	2	yes	—
159	C₂×C₃	#5195	1	2	yes	—
160	C₂²	C₂×#3713	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	12	4	no	—
163	C₄	—	24	2	no	—
164	C₄	#63822	1	1	yes	characteristic
165	C₄	C₄×#625	1	1	yes	characteristic
166	C₃	—	2	1	no	—
167	C₃	#70909	1	2	yes	complemented
168	C₂	C₄×#3713	1	1	yes	characteristic
169	C₂	#10590636	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×Dic₁₂

C₂

yes

maximal

C₂×#63550

C₂

yes

maximal, characteristic

#10589327

C₂

yes

maximal, characteristic

#10589142

C₂

yes

maximal, characteristic

#10589117

C₂

yes

maximal

#70909

—

maximal

C₂×C₃×Dic₁₂

C₂²

yes

—

C₃×C₄×Dic₆

—

C₃²×C₄×C₈

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₂²

yes

characteristic

C₃×#818

C₂²

yes

—

C₃×#816

—

#63550

C₄

yes

complemented

C₃×C₄×Q₁₆

—

C₂×#829

—

C₄×Dic₁₂

D₃

yes

—

—

—

—

—

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

C₂³

yes

characteristic

C₃²×C₄²

D₄

yes

characteristic

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₃×C₄×Dic₃

—

C₃×Dic₁₂

—

C₃×Dic₁₂

C₂×C₄

yes

—

C₃⋊(C₃⋊C₁₆)

—

C₃⋊(C₃⋊C₁₆)

C₂×C₄

yes

—

#2176

—

Sylow

C₂×C₃×Q₁₆

—

C₃×C₄×Q₈

—

C₂×Dic₁₂

C₂×D₃

yes

—

C₄×Dic₆

—

C₃×#107

—

C₃×#103

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₃

yes

—

#829

—

#818

C₂×D₃

yes

—

#816

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃²×C₄

D₈

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃²×C₈

—

C₃²×C₈

C₂²×C₄

yes

characteristic

C₃²×C₈

#41

yes

characteristic

C₂×Q₃₂

—

C₄×Q₁₆

—

#367

—

#358

—

#346

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

#67

yes

—

C₂×C₃×C₈

C₂²×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

#67

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

Dic₁₂

—

Dic₁₂

C₄×D₃

yes

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

C₂×D₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄×D₄

yes

characteristic

C₃²×C₄

#135

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

D₃²

yes

characteristic

100

C₃×Q₈

—

101

C₃×Q₈

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

#213

yes

—

107

C₂×C₃×C₄

#213

yes

—

108

C₂×C₃×C₄

C₂×#67

yes

—

109

C₂×Dic₃

—

110

C₂×Dic₃

—

111

Dic₆

—

112

Dic₆

—

113

C₃×C₈

—

114

C₃×C₈

—

115

C₃×C₈

—

116

C₃×C₈

—

117

C₃×C₈

C₂×C₄×D₃

yes

—

118

C₃×C₈

#235

yes

—

119

C₂×C₃²

C₄×D₈

yes

characteristic

120

C₂×C₃²

#510

yes

—

121

C₂×Q₈

—

122

Q₁₆

—

123

Q₁₆

—

124

C₂×C₈

#625

yes

characteristic

125

C₂×C₈

C₂×D₃²

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

#625

yes

characteristic

130

C₁₆

—

131

C₁₆

—

132

C₂²×C₃

—

133

C₂²×C₃

C₂×#213

yes

—

134

C₃×C₄

—

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

—

138

C₃×C₄

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

C₄×#67

yes

—

142

C₃×C₄

#950

yes

—

143

Dic₃

—

144

Dic₃

—

145

C₃²

#2176

yes

characteristic, complemented, Sylow

146

Q₈

—

147

Q₈

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#3713

yes

—

151

C₂×C₄

C₂×#625

yes

characteristic

152

C₈

—

153

C₈

#3798

yes

characteristic

154

C₈

C₄×D₃²

yes

characteristic

155

C₂×C₃

—

156

C₂×C₃

—

157

C₂×C₃

C₄×#213

yes

—

158

C₂×C₃

#5179

yes

—

159

C₂×C₃

#5195

yes

—

160

C₂²

C₂×#3713

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

#63822

yes

characteristic

165

C₄

C₄×#625

yes

characteristic

166

C₃

—

167

C₃

#70909

yes

complemented

168

C₂

C₄×#3713

yes

characteristic

169

C₂

#10590636

yes

—

	subgroup	G/H
center	C₂²	C₂×#3713
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle, prime radical	C₂²×C₃²	C₂×D₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#2176
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×#3713

commutator

C₃²×C₈

C₂²×C₄

Frattini

C₂×C₈

C₂×D₃²

Fitting

C₃²×C₄×C₈

C₂²

socle, prime radical

C₂²×C₃²

C₂×D₈

cosocle

C₂×C₃²×C₈

C₂³

hypercenter

C₄×C₈

D₃²

hypocenter

C₃²

#2176

LCS

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
14. #63550	C₄	semidirect
145. C₃²	#2176	semidirect
167. C₃	#70909	semidirect

H ≅ G/N

type

14. #63550

C₄

semidirect

145. C₃²

#2176

semidirect

167. C₃

#70909

semidirect

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

subgroup	index
C₃×C₄×Dic₁₂	2
C₂×#63550	2
#10589327	2
#10589142	2
#10589117	2
#70909	3

subgroup

index

C₃×C₄×Dic₁₂

Subgroups — #11797792

Conjugacy classes of subgroups

Complements

Maximal subgroups