Group #11771752 — subgroups

Subgroups — #11771752

Order 1152.

330 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 83 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	#10592834	C₂	1	1	yes	maximal, characteristic
2	C₃²×#349	C₂	1	1	yes	maximal, characteristic, complemented
3	#10589426	C₂	1	1	yes	maximal, characteristic, complemented
4	#67535	—	3	4	no	maximal
5	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
6	#64139	—	2	1	no	—
7	C₂×#3748	—	2	1	no	—
8	C₂×#3745	C₄	1	1	yes	characteristic
9	C₃²×M₃₂	—	2	1	no	—
10	#63648	—	2	1	no	—
11	#63644	C₄	1	1	yes	characteristic
12	#5135	—	3	4	no	—
13	C₃×#349	D₃	1	4	yes	complemented
14	C₃⋊#338	—	3	4	no	—
15	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
16	C₂×#638	—	4	1	no	—
17	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
18	C₃²×M₁₆	—	4	1	no	—
19	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
21	#3752	—	4	1	no	—
22	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
23	#3748	—	4	1	no	—
24	#3748	—	4	1	no	—
25	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
26	#3745	—	4	1	no	—
27	C₃²×C₁₆	—	4	1	no	—
28	#1341	—	9	1	no	Sylow
29	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
30	#930	—	6	4	no	—
31	C₂×#208	—	6	4	no	—
32	C₂×D₂₄	—	3	4	no	—
33	C₃×M₃₂	—	2	4	no	—
34	#821	—	6	4	no	—
35	#817	—	3	4	no	—
36	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
37	C₂²×C₃⋊D₃	—	4	1	no	—
38	C₂×C₃²×C₄	—	2	1	no	—
39	C₂×C₃²×C₄	—	2	1	no	—
40	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
42	#638	—	8	2	no	—
43	C₂×C₃⋊Dic₃	—	4	1	no	—
44	C₄⋊(C₃⋊D₃)	—	4	1	no	—
45	C₄⋊(C₃⋊D₃)	—	4	1	no	—
46	C₃²×C₈	—	2	1	no	—
47	C₃²×C₈	—	2	1	no	—
48	C₃⋊(C₃⋊C₈)	—	4	2	no	—
49	#469	—	9	1	no	—
50	#349	C₃⋊D₃	1	1	yes	characteristic, complemented
51	#338	—	9	1	no	—
52	C₂²×C₃×C₄	D₁₂	1	4	yes	—
53	C₂×#67	—	12	4	no	—
54	C₂×D₁₂	—	6	4	no	—
55	C₃×M₁₆	—	4	4	no	—
56	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
57	C₂×C₃×C₈	#67	1	4	yes	—
58	(C₂×D₃)⋊C₄	—	12	4	no	—
59	C₄⋊Dic₃	—	6	4	no	—
60	#208	—	12	4	no	—
61	#208	—	12	4	no	—
62	C₂×C₃⋊C₈	—	6	8	no	—
63	D₂₄	—	12	4	no	—
64	C₃×C₁₆	—	4	4	no	—
65	C₂²×C₃²	—	2	1	no	—
66	C₂²×C₃²	—	4	1	no	—
67	C₂²×C₃²	#102	1	1	yes	characteristic
68	C₂×C₃⋊D₃	—	8	1	no	—
69	C₂×C₃⋊D₃	—	8	1	no	—
70	C₂×C₃⋊D₃	—	8	1	no	—
71	C₃²×C₄	—	2	1	no	—
72	C₃²×C₄	#99	1	1	yes	characteristic
73	C₃²×C₄	#104	1	1	yes	characteristic
74	C₃⋊Dic₃	—	8	1	no	—
75	C₂×D₈	—	9	1	no	—
76	C₂×M₁₆	—	18	1	no	—
77	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
78	#121	—	18	1	no	—
79	M₃₂	—	2	1	no	—
80	#106	—	9	1	no	—
81	#102	—	18	1	no	—
82	C₂³×C₃	#204	1	4	yes	—
83	C₂²×D₃	—	12	4	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	D₂₄	1	4	yes	—
87	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
88	#67	—	24	8	no	—
89	C₂×Dic₃	—	12	4	no	—
90	D₁₂	—	12	4	no	—
91	D₁₂	—	12	4	no	—
92	C₃×C₈	—	2	4	no	—
93	C₃×C₈	—	2	4	no	—
94	C₃⋊C₈	—	12	8	no	—
95	C₂×C₃²	—	2	1	no	—
96	C₂×C₃²	—	4	1	no	—
97	C₂×C₃²	#327	1	1	yes	characteristic
98	C₃⋊D₃	—	16	1	no	—
99	C₂×D₄	—	18	1	no	—
100	C₂×D₄	—	36	1	no	—
101	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
102	D₈	—	36	1	no	—
103	M₁₆	—	4	1	no	—
104	M₁₆	—	36	1	no	—
105	M₁₆	—	36	1	no	—
106	C₂×C₈	—	18	2	no	—
107	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
108	C₂×C₈	#638	1	1	yes	characteristic
109	C₄⋊C₄	—	18	1	no	—
110	C₂²⋊C₄	—	36	1	no	—
111	C₁₆	—	4	1	no	—
112	C₂²×C₃	—	2	4	no	—
113	C₂²×C₃	—	4	4	no	—
114	C₂²×C₃	#821	1	4	yes	—
115	C₂×D₃	—	24	4	no	—
116	C₂×D₃	—	24	4	no	—
117	C₂×D₃	—	24	4	no	—
118	C₃×C₄	—	2	4	no	—
119	C₃×C₄	#806	1	4	yes	—
120	C₃×C₄	#825	1	4	yes	—
121	Dic₃	—	24	4	no	—
122	C₃²	#1341	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	36	1	no	—
124	C₂³	#3744	1	1	yes	characteristic
125	D₄	—	36	1	no	—
126	D₄	—	36	1	no	—
127	D₄	—	72	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	36	1	no	—
131	C₂×C₄	#3745	1	1	yes	characteristic
132	C₂×C₄	#3752	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	36	2	no	—
136	C₂×C₃	—	2	4	no	—
137	C₂×C₃	—	4	4	no	—
138	C₂×C₃	#4470	1	4	yes	—
139	D₃	—	48	4	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	—	72	1	no	—
143	C₂²	—	72	1	no	—
144	C₂²	—	72	1	no	—
145	C₂²	#63648	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	72	1	no	—
148	C₄	#63633	1	1	yes	characteristic
149	C₄	#63652	1	1	yes	characteristic
150	C₃	#67535	1	4	yes	complemented
151	C₂	—	2	1	no	—
152	C₂	—	4	1	no	—
153	C₂	—	144	1	no	—
154	C₂	#10589336	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592834

C₂

yes

maximal, characteristic

C₃²×#349

C₂

yes

maximal, characteristic, complemented

#10589426

C₂

yes

maximal, characteristic, complemented

#67535

—

maximal

C₂×C₃²×M₁₆

C₂²

yes

—

—

C₄

yes

characteristic

C₃²×M₃₂

—

#63648

—

#63644

C₄

yes

characteristic

#5135

—

C₃×#349

D₃

yes

complemented

C₃⋊#338

—

C₂²×C₃²×C₄

D₄

yes

characteristic

C₂×#638

—

C₂×C₄⋊(C₃⋊D₃)

—

C₃²×M₁₆

—

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

#3752

—

C₄⋊(C₃⋊Dic₃)

—

#3748

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

#3745

—

C₃²×C₁₆

—

#1341

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

#930

—

C₂×#208

—

C₂×D₂₄

—

C₃×M₃₂

—

#821

—

#817

—

C₂³×C₃²

SD₁₆

yes

characteristic

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

D₈

yes

characteristic

#638

—

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₃²×C₈

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#469

—

#349

C₃⋊D₃

yes

characteristic, complemented

#338

—

C₂²×C₃×C₄

D₁₂

yes

—

C₂×#67

—

C₂×D₁₂

—

C₃×M₁₆

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

#67

yes

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

D₂₄

—

C₃×C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#102

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₃⋊D₃

yes

characteristic

#121

—

M₃₂

—

#106

—

#102

—

C₂³×C₃

#204

yes

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₂₄

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

#67

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#327

yes

characteristic

C₃⋊D₃

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂²×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

102

D₈

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

C₄×C₃⋊D₃

yes

characteristic

108

C₂×C₈

#638

yes

characteristic

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₁₆

—

112

C₂²×C₃

—

113

C₂²×C₃

—

114

C₂²×C₃

#821

yes

—

115

C₂×D₃

—

116

C₂×D₃

—

117

C₂×D₃

—

118

C₃×C₄

—

119

C₃×C₄

#806

yes

—

120

C₃×C₄

#825

yes

—

121

Dic₃

—

122

C₃²

#1341

yes

characteristic, complemented, Sylow

123

C₂³

—

124

C₂³

#3744

yes

characteristic

125

D₄

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#3745

yes

characteristic

132

C₂×C₄

#3752

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₂×C₃

—

137

C₂×C₃

—

138

C₂×C₃

#4470

yes

—

139

D₃

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#63648

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

#63633

yes

characteristic

149

C₄

#63652

yes

characteristic

150

C₃

#67535

yes

complemented

151

C₂

—

152

C₂

—

153

C₂

—

144

—

154

C₂

#10589336

yes

characteristic

	subgroup	G/H
center	C₂	#10589336
commutator	C₂×C₃²×C₈	C₂×C₄
Frattini	C₂×M₁₆	C₂×C₃⋊D₃
Fitting	C₃²×#349	C₂
socle	C₂×C₃²	#327
prime radical	#10592834	C₂
cosocle	C₂×C₃²×M₁₆	C₂²
hypercenter	#349	C₃⋊D₃
hypocenter	C₃²	#1341
LCS	G ▷ C₂×C₃²×C₈ ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₂

#10589336

commutator

C₂×C₃²×C₈

C₂×C₄

Frattini

C₂×M₁₆

C₂×C₃⋊D₃

Fitting

C₃²×#349

C₂

C₂×C₃²

C₂

C₂×C₃²×M₁₆

C₂²

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₈ ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
2. C₃²×#349	C₂	semidirect
3. #10589426	C₂	semidirect
13. C₃×#349	D₃	semidirect
50. #349	C₃⋊D₃	semidirect
122. C₃²	#1341	semidirect
150. C₃	#67535	semidirect

H ≅ G/N

type

2. C₃²×#349

C₂

semidirect

3. #10589426

C₂

semidirect

13. C₃×#349

D₃

semidirect

50. #349

C₃⋊D₃

semidirect

122. C₃²

#1341

semidirect

150. C₃

#67535

semidirect

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

subgroup	index
#10592834	2
C₃²×#349	2
#10589426	2
#67535	3

subgroup

index

#10592834

C₃²×#349

#10589426

#67535

Subgroups — #11771752

Conjugacy classes of subgroups

Complements

Maximal subgroups