Group #11772028 — subgroups

Subgroups — #11772028

Order 1152.

306 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 95 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	#10592392	C₂	1	1	yes	maximal, characteristic
2	C₃²×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	#10589329	C₂	1	1	yes	maximal, characteristic, complemented
4	#67570	—	3	4	no	maximal
5	C₂×#3744	—	2	1	no	—
6	C₂×#3744	—	2	1	no	—
7	#64083	C₄	1	1	yes	characteristic
8	#64078	C₄	1	1	yes	characteristic
9	C₃²×M₃₂	—	2	1	no	—
10	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#63639	—	2	1	no	—
12	#4693	—	3	4	no	—
13	C₃×#364	D₃	1	4	yes	complemented
14	#4463	—	3	4	no	—
15	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
16	C₂×C₄⋊(C₃⋊D₃)	—	4	1	no	—
17	C₂×#634	—	2	1	no	—
18	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
19	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
20	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
22	#3744	—	4	1	no	—
23	#3744	—	4	1	no	—
24	#3744	—	4	1	no	—
25	#3744	—	4	1	no	—
26	C₃²×C₁₆	—	4	1	no	—
27	C₃⋊(C₃⋊C₁₆)	—	4	1	no	—
28	#1359	—	9	1	no	Sylow
29	C₂×#204	—	6	4	no	—
30	C₂×#204	—	6	4	no	—
31	#874	—	3	4	no	—
32	#869	—	3	4	no	—
33	C₃×M₃₂	—	2	4	no	—
34	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
35	#812	—	6	4	no	—
36	C₂²×C₃⋊D₃	—	4	1	no	—
37	C₂×C₃²×C₄	—	2	1	no	—
38	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₃⋊Dic₃	—	4	1	no	—
40	C₄⋊(C₃⋊D₃)	—	4	1	no	—
41	C₄⋊(C₃⋊D₃)	—	4	1	no	—
42	C₄⋊(C₃⋊D₃)	—	8	2	no	—
43	#634	—	4	1	no	—
44	#634	—	4	1	no	—
45	C₃²×C₈	D₈	1	1	yes	characteristic
46	C₃²×C₈	D₈	1	1	yes	characteristic
47	C₃²×C₈	SD₁₆	1	1	yes	characteristic
48	C₃²×C₈	SD₁₆	1	1	yes	characteristic
49	#492	—	9	1	no	—
50	#364	—	9	1	no	—
51	#364	C₃⋊D₃	1	1	yes	characteristic, complemented
52	C₂×D₁₂	—	6	4	no	—
53	C₂×D₁₂	—	12	4	no	—
54	C₂×Dic₆	—	6	4	no	—
55	C₂×C₃×C₈	D₁₂	1	4	yes	—
56	C₂×C₃×C₈	#67	1	4	yes	—
57	C₃×C₄²	C₄×D₃	1	4	yes	—
58	C₄⋊Dic₃	—	12	4	no	—
59	#204	—	12	4	no	—
60	#204	—	12	4	no	—
61	#204	—	12	4	no	—
62	#204	—	12	4	no	—
63	C₃×C₁₆	—	4	4	no	—
64	C₃⋊C₁₆	—	12	4	no	—
65	C₂²×C₃²	#100	1	1	yes	characteristic
66	C₂×C₃⋊D₃	—	8	2	no	—
67	C₂×C₃⋊D₃	—	8	1	no	—
68	C₃²×C₄	—	2	2	no	—
69	C₃²×C₄	#102	1	1	yes	characteristic
70	C₃²×C₄	#102	1	1	yes	characteristic
71	C₃⋊Dic₃	—	8	1	no	—
72	C₂×SD₁₆	—	18	1	no	—
73	C₂×SD₁₆	—	18	1	no	—
74	C₄⋊Q₈	—	9	1	no	—
75	#127	—	9	1	no	—
76	M₃₂	—	2	1	no	—
77	M₃₂	—	18	1	no	—
78	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
79	C₂²×D₃	—	12	4	no	—
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
82	C₂×Dic₃	—	12	4	no	—
83	D₁₂	—	12	4	no	—
84	D₁₂	—	12	4	no	—
85	D₁₂	—	24	8	no	—
86	Dic₆	—	12	4	no	—
87	Dic₆	—	12	4	no	—
88	C₃×C₈	#204	1	4	yes	—
89	C₃×C₈	D₂₄	1	4	yes	—
90	C₃×C₈	#213	1	4	yes	—
91	C₃×C₈	#215	1	4	yes	—
92	C₂×C₃²	—	2	1	no	—
93	C₂×C₃²	#331	1	1	yes	characteristic
94	C₃⋊D₃	—	16	1	no	—
95	C₂×Q₈	—	18	1	no	—
96	C₂×D₄	—	18	1	no	—
97	C₂×D₄	—	36	1	no	—
98	SD₁₆	—	36	1	no	—
99	SD₁₆	—	36	1	no	—
100	SD₁₆	—	36	1	no	—
101	SD₁₆	—	36	1	no	—
102	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
103	C₂×C₈	#638	1	1	yes	characteristic
104	C₄⋊C₄	—	36	1	no	—
105	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
106	C₁₆	—	4	1	no	—
107	C₁₆	—	36	1	no	—
108	C₂²×C₃	#823	1	4	yes	—
109	C₂×D₃	—	24	8	no	—
110	C₂×D₃	—	24	4	no	—
111	C₃×C₄	—	2	8	no	—
112	C₃×C₄	#809	1	4	yes	—
113	C₃×C₄	#821	1	4	yes	—
114	Dic₃	—	24	4	no	—
115	C₃²	#1359	1	1	yes	characteristic, complemented, Sylow
116	C₂³	—	36	1	no	—
117	Q₈	—	36	1	no	—
118	Q₈	—	36	1	no	—
119	D₄	—	36	1	no	—
120	D₄	—	36	1	no	—
121	D₄	—	72	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	36	1	no	—
124	C₂×C₄	#3752	1	1	yes	characteristic
125	C₈	#3744	1	1	yes	characteristic
126	C₈	#3745	1	1	yes	characteristic
127	C₈	#3753	1	1	yes	characteristic
128	C₈	#3755	1	1	yes	characteristic
129	C₂×C₃	—	2	4	no	—
130	C₂×C₃	#4485	1	4	yes	—
131	D₃	—	48	4	no	—
132	C₂²	—	72	2	no	—
133	C₂²	—	72	1	no	—
134	C₂²	#63650	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	72	1	no	—
137	C₄	#63636	1	1	yes	characteristic
138	C₄	#63648	1	1	yes	characteristic
139	C₃	#67570	1	4	yes	complemented
140	C₂	—	2	1	no	—
141	C₂	—	144	1	no	—
142	C₂	#10589351	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592392

C₂

yes

maximal, characteristic

C₃²×#364

C₂

yes

maximal, characteristic, complemented

#10589329

C₂

yes

maximal, characteristic, complemented

—

—

—

C₄

yes

characteristic

#64078

C₄

yes

characteristic

C₃²×M₃₂

—

C₃²×C₄×C₈

C₂²

yes

—

—

D₃

yes

complemented

#4463

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×#634

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₃²×C₄²

C₂×C₄

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

—

—

—

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

—

—

—

—

—

C₃×M₃₂

—

C₃×C₄×C₈

C₂×D₃

yes

—

#812

—

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

#634

—

#634

—

C₃²×C₈

D₈

yes

characteristic

C₃²×C₈

D₈

yes

characteristic

C₃²×C₈

SD₁₆

yes

characteristic

C₃²×C₈

SD₁₆

yes

—

—

C₃⋊D₃

yes

characteristic, complemented

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄²

C₄×D₃

yes

—

C₄⋊Dic₃

—

—

—

—

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

#100

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#127

—

M₃₂

—

M₃₂

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

C₃×C₈

#204

yes

—

C₃×C₈

D₂₄

yes

—

C₃×C₈

#213

yes

—

C₃×C₈

#215

yes

—

C₂×C₃²

—

C₂×C₃²

#331

yes

characteristic

C₃⋊D₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

100

SD₁₆

—

101

SD₁₆

—

102

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

characteristic

103

C₂×C₈

#638

yes

characteristic

104

C₄⋊C₄

—

105

C₄²

C₄×C₃⋊D₃

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₂²×C₃

#823

yes

—

109

C₂×D₃

—

110

C₂×D₃

—

111

C₃×C₄

—

112

C₃×C₄

#809

yes

—

113

C₃×C₄

#821

yes

—

114

Dic₃

—

115

C₃²

#1359

yes

characteristic, complemented, Sylow

116

C₂³

—

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#3752

yes

characteristic

125

C₈

#3744

yes

characteristic

126

C₈

#3745

yes

characteristic

127

C₈

#3753

yes

characteristic

128

C₈

#3755

yes

characteristic

129

C₂×C₃

—

130

C₂×C₃

#4485

yes

—

131

D₃

—

132

C₂²

—

133

C₂²

—

134

C₂²

#63650

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

#63636

yes

characteristic

138

C₄

#63648

yes

characteristic

139

C₃

#67570

yes

complemented

140

C₂

—

141

C₂

—

144

—

142

C₂

#10589351

yes

characteristic

	subgroup	G/H
center	C₂	#10589351
commutator	C₃²×C₄²	C₂×C₄
Frattini	C₄×C₈	C₂×C₃⋊D₃
Fitting	C₃²×#364	C₂
socle	C₂×C₃²	#331
prime radical	#64083	C₄
cosocle	C₃²×C₄×C₈	C₂²
hypercenter	#364	C₃⋊D₃
hypocenter	C₃²	#1359
LCS	G ▷ C₃²×C₄² ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#10589351

commutator

C₃²×C₄²

C₂×C₄

Frattini

C₄×C₈

C₂×C₃⋊D₃

Fitting

C₃²×#364

C₂

C₂×C₃²

C₄

C₃²×C₄×C₈

C₂²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
2. C₃²×#364	C₂	semidirect
3. #10589329	C₂	semidirect
13. C₃×#364	D₃	semidirect
51. #364	C₃⋊D₃	semidirect
115. C₃²	#1359	semidirect
139. C₃	#67570	semidirect

H ≅ G/N

type

2. C₃²×#364

C₂

semidirect

3. #10589329

C₂

semidirect

13. C₃×#364

D₃

semidirect

51. #364

C₃⋊D₃

semidirect

115. C₃²

#1359

semidirect

139. C₃

#67570

semidirect

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

subgroup	index
#10592392	2
C₃²×#364	2
#10589329	2
#67570	3

subgroup

index

#10592392

C₃²×#364

#10589329

#67570

Subgroups — #11772028

Conjugacy classes of subgroups

Complements

Maximal subgroups