Group #11782936 — subgroups

Subgroups — #11782936

Order 1152.

594 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 185 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	#10592398	C₂	1	1	yes	maximal, characteristic, complemented
2	#10592393	C₂	1	2	yes	maximal
3	#10592392	C₂	1	1	yes	maximal, characteristic
4	#10592381	C₂	1	2	yes	maximal, complemented
5	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#68098	—	3	4	no	maximal
7	C₂×#3746	—	2	2	no	—
8	C₂×#3744	—	2	2	no	—
9	#64083	C₂²	1	1	yes	characteristic
10	#64078	C₂²	1	1	yes	characteristic
11	#64078	C₂²	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	2	yes	—
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	#63648	—	2	4	no	—
15	#63645	—	2	4	no	—
16	#4699	—	3	4	no	—
17	#4694	—	3	8	no	—
18	#4693	—	3	4	no	—
19	#4682	—	3	8	no	—
20	C₃×#322	D₃	1	4	yes	complemented
21	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
22	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
23	C₂×#634	—	2	1	no	—
24	C₂×#634	—	2	2	no	—
25	C₂×C₃²×C₈	D₄	1	4	yes	—
26	C₂×C₃²×C₈	D₄	1	2	yes	—
27	C₃²×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
29	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
30	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
31	#3746	—	4	4	no	—
32	#3744	—	4	4	no	—
33	#1722	—	9	1	no	Sylow
34	C₂×Dic₁₂	—	6	8	no	—
35	C₂×#204	—	6	8	no	—
36	#874	—	3	4	no	—
37	#869	—	3	4	no	—
38	#869	—	3	8	no	—
39	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
40	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
41	#821	—	6	16	no	—
42	#818	—	6	16	no	—
43	C₂²×C₃⋊D₃	—	4	1	no	—
44	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
46	C₂×C₃⋊Dic₃	—	4	1	no	—
47	C₂×C₃⋊Dic₃	—	4	2	no	—
48	C₄⋊(C₃⋊D₃)	—	4	2	no	—
49	C₄⋊(C₃⋊D₃)	—	4	4	no	—
50	#634	—	4	2	no	—
51	#634	—	4	4	no	—
52	C₃²×C₈	—	2	4	no	—
53	C₃²×C₈	D₈	1	4	yes	—
54	#500	—	9	2	no	—
55	#494	—	9	1	no	—
56	#492	—	9	1	no	—
57	#486	—	9	2	no	—
58	#322	C₃⋊D₃	1	1	yes	characteristic, complemented
59	C₂×D₁₂	—	6	8	no	—
60	C₂×D₁₂	—	6	4	no	—
61	C₂×Dic₆	—	6	4	no	—
62	C₂×Dic₆	—	6	8	no	—
63	C₂×C₃×C₈	D₁₂	1	16	yes	—
64	C₂×C₃×C₈	D₁₂	1	8	yes	—
65	C₃×C₄²	C₂²×D₃	1	4	yes	—
66	C₄⋊Dic₃	—	6	8	no	—
67	C₄⋊Dic₃	—	6	8	no	—
68	C₄⋊Dic₃	—	6	8	no	—
69	Dic₁₂	—	12	16	no	—
70	#204	—	12	16	no	—
71	C₂²×C₃²	#127	1	1	yes	characteristic
72	C₂×C₃⋊D₃	—	8	1	no	—
73	C₂×C₃⋊D₃	—	8	2	no	—
74	C₃²×C₄	C₂×D₈	1	2	yes	—
75	C₃²×C₄	#137	1	4	yes	—
76	C₃⋊Dic₃	—	8	1	no	—
77	C₃⋊Dic₃	—	8	2	no	—
78	C₂×Q₁₆	—	18	2	no	—
79	C₂×SD₁₆	—	18	2	no	—
80	C₄⋊Q₈	—	9	2	no	—
81	C₄⋊Q₈	—	9	1	no	—
82	#127	—	9	1	no	—
83	#107	—	18	4	no	—
84	#102	—	18	4	no	—
85	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
86	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
87	C₂²×D₃	—	12	4	no	—
88	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
89	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
90	C₂×Dic₃	—	12	4	no	—
91	C₂×Dic₃	—	12	8	no	—
92	D₁₂	—	12	8	no	—
93	D₁₂	—	12	16	no	—
94	Dic₆	—	12	8	no	—
95	Dic₆	—	12	16	no	—
96	C₃×C₈	—	2	16	no	—
97	C₃×C₈	D₂₄	1	16	yes	—
98	C₂×C₃²	#493	1	1	yes	characteristic
99	C₂×C₃²	#497	1	2	yes	—
100	C₃⋊D₃	—	16	1	no	—
101	C₂×Q₈	—	18	1	no	—
102	C₂×Q₈	—	18	2	no	—
103	C₂×D₄	—	18	2	no	—
104	C₂×D₄	—	18	1	no	—
105	Q₁₆	—	36	4	no	—
106	SD₁₆	—	36	4	no	—
107	C₂×C₈	C₄⋊(C₃⋊D₃)	1	4	yes	—
108	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
109	C₄⋊C₄	—	18	2	no	—
110	C₄⋊C₄	—	18	2	no	—
111	C₄⋊C₄	—	18	2	no	—
112	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
113	C₂²×C₃	#874	1	4	yes	—
114	C₂×D₃	—	24	4	no	—
115	C₂×D₃	—	24	8	no	—
116	C₃×C₄	C₂×D₂₄	1	8	yes	—
117	C₃×C₄	#909	1	16	yes	—
118	Dic₃	—	24	4	no	—
119	Dic₃	—	24	8	no	—
120	C₃²	#1722	1	1	yes	characteristic, complemented, Sylow
121	C₂³	—	36	1	no	—
122	Q₈	—	36	2	no	—
123	Q₈	—	36	4	no	—
124	D₄	—	36	2	no	—
125	D₄	—	36	4	no	—
126	C₂×C₄	—	36	2	no	—
127	C₂×C₄	—	36	1	no	—
128	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
130	C₈	—	2	4	no	—
131	C₈	#3745	1	4	yes	—
132	C₂×C₃	#4695	1	4	yes	—
133	C₂×C₃	#4715	1	8	yes	—
134	D₃	—	48	4	no	—
135	C₂²	—	72	1	no	—
136	C₂²	—	72	2	no	—
137	C₂²	#64083	1	1	yes	characteristic
138	C₄	—	72	1	no	—
139	C₄	—	72	2	no	—
140	C₄	C₂×#3745	1	2	yes	—
141	C₄	#64118	1	4	yes	—
142	C₃	#68098	1	4	yes	complemented
143	C₂	—	144	1	no	—
144	C₂	#10592394	1	1	yes	characteristic
145	C₂	#10592414	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10592398

C₂

yes

maximal, characteristic, complemented

#10592393

C₂

yes

maximal

#10592392

C₂

yes

maximal, characteristic

#10592381

C₂

yes

maximal, complemented

C₃²×#322

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#64078

C₂²

yes

characteristic

#64078

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

—

—

—

—

—

—

D₃

yes

complemented

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×#634

—

C₂×#634

—

C₂×C₃²×C₈

D₄

yes

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄²

C₂³

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

—

—

—

C₂×Dic₁₂

—

—

—

—

—

C₃×C₄×C₈

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

#821

—

#818

—

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

#634

—

#634

—

C₃²×C₈

—

C₃²×C₈

D₈

yes

—

—

—

—

—

C₃⋊D₃

yes

characteristic, complemented

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

D₁₂

yes

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

#204

—

C₂²×C₃²

#127

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

C₂×D₈

yes

—

C₃²×C₄

#137

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

#107

—

#102

—

C₄×C₈

C₂×C₃⋊D₃

yes

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

C₂×D₁₂

yes

—

C₂×C₃×C₄

C₂×D₁₂

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

D₂₄

yes

—

C₂×C₃²

#493

yes

characteristic

C₂×C₃²

#497

yes

—

100

C₃⋊D₃

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

Q₁₆

—

106

SD₁₆

—

107

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

—

108

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

C₂²×C₃⋊D₃

yes

characteristic

113

C₂²×C₃

#874

yes

—

114

C₂×D₃

—

115

C₂×D₃

—

116

C₃×C₄

C₂×D₂₄

yes

—

117

C₃×C₄

#909

yes

—

118

Dic₃

—

119

Dic₃

—

120

C₃²

#1722

yes

characteristic, complemented, Sylow

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

129

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

—

130

C₈

—

131

C₈

#3745

yes

—

132

C₂×C₃

#4695

yes

—

133

C₂×C₃

#4715

yes

—

134

D₃

—

135

C₂²

—

136

C₂²

—

137

C₂²

#64083

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

C₂×#3745

yes

—

141

C₄

#64118

yes

—

142

C₃

#68098

yes

complemented

143

C₂

—

144

—

144

C₂

#10592394

yes

characteristic

145

C₂

#10592414

yes

—

	subgroup	G/H
center	C₂²	#64083
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#322	C₂
socle	C₂²×C₃²	#127
prime radical	#64083	C₂²
hypercenter	#322	C₃⋊D₃
hypocenter	C₃²	#1722
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#64083

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#322

C₂

C₂²×C₃²

C₂²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #10592398	C₂	semidirect
4. #10592381	C₂	semidirect
5. C₃²×#322	C₂	semidirect
20. C₃×#322	D₃	semidirect
58. #322	C₃⋊D₃	semidirect
120. C₃²	#1722	semidirect
142. C₃	#68098	semidirect

H ≅ G/N

type

1. #10592398

C₂

semidirect

4. #10592381

C₂

semidirect

5. C₃²×#322

C₂

semidirect

20. C₃×#322

D₃

semidirect

58. #322

C₃⋊D₃

semidirect

120. C₃²

#1722

semidirect

142. C₃

#68098

semidirect

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

subgroup	index
#10592398	2
#10592393	2
#10592392	2
#10592381	2
C₃²×#322	2
#68098	3

subgroup

index

C₃²×#322

Subgroups — #11782936

Conjugacy classes of subgroups

Complements

Maximal subgroups