Group #11782502 — subgroups

Subgroups — #11782502

Order 1152.

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

normal subgroups only

This group has 570 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	#10592393	C₂	1	2	yes	maximal
2	#10592392	C₂	1	1	yes	maximal, characteristic
3	#10592380	C₂	1	2	yes	maximal, complemented
4	#10592379	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	#68059	—	3	4	no	maximal
7	C₂×#3744	—	2	2	no	—
8	#64083	C₂²	1	1	yes	characteristic
9	#64078	C₂²	1	2	yes	—
10	#64078	C₂²	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	2	yes	—
13	#63648	—	2	4	no	—
14	#63644	—	2	2	no	—
15	#63643	—	2	4	no	—
16	#4694	—	3	8	no	—
17	#4693	—	3	4	no	—
18	#4681	—	3	8	no	—
19	#4680	—	3	4	no	—
20	C₃×C₈²	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	#3744	—	4	4	no	—
32	#1708	—	9	1	no	Sylow
33	C₂×#204	—	6	8	no	—
34	#874	—	3	4	no	—
35	#869	—	3	8	no	—
36	#869	—	3	4	no	—
37	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
38	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
39	#821	—	6	16	no	—
40	#817	—	6	8	no	—
41	#816	—	6	16	no	—
42	C₂²×C₃⋊D₃	—	4	1	no	—
43	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
45	C₂×C₃⋊Dic₃	—	4	1	no	—
46	C₂×C₃⋊Dic₃	—	4	2	no	—
47	C₄⋊(C₃⋊D₃)	—	4	2	no	—
48	C₄⋊(C₃⋊D₃)	—	4	4	no	—
49	#634	—	4	2	no	—
50	#634	—	4	4	no	—
51	C₃²×C₈	—	2	4	no	—
52	C₃²×C₈	SD₁₆	1	4	yes	—
53	#498	—	9	1	no	—
54	#492	—	9	1	no	—
55	#487	—	9	2	no	—
56	#486	—	9	2	no	—
57	C₈²	C₃⋊D₃	1	1	yes	characteristic, complemented
58	C₂×D₁₂	—	6	8	no	—
59	C₂×D₁₂	—	6	4	no	—
60	C₂×Dic₆	—	6	8	no	—
61	C₂×Dic₆	—	6	4	no	—
62	C₂×C₃×C₈	D₁₂	1	16	yes	—
63	C₂×C₃×C₈	D₁₂	1	8	yes	—
64	C₃×C₄²	C₂²×D₃	1	4	yes	—
65	C₄⋊Dic₃	—	6	8	no	—
66	C₄⋊Dic₃	—	6	8	no	—
67	C₄⋊Dic₃	—	6	8	no	—
68	#204	—	12	16	no	—
69	C₂²×C₃²	#127	1	1	yes	characteristic
70	C₂×C₃⋊D₃	—	8	1	no	—
71	C₂×C₃⋊D₃	—	8	2	no	—
72	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
73	C₃²×C₄	#135	1	4	yes	—
74	C₃⋊Dic₃	—	8	1	no	—
75	C₃⋊Dic₃	—	8	2	no	—
76	C₂×SD₁₆	—	18	2	no	—
77	C₄⋊Q₈	—	9	2	no	—
78	C₄⋊Q₈	—	9	1	no	—
79	#127	—	9	1	no	—
80	#106	—	18	2	no	—
81	#103	—	18	4	no	—
82	#102	—	18	4	no	—
83	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
84	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
85	C₂²×D₃	—	12	4	no	—
86	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
87	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
88	C₂×Dic₃	—	12	4	no	—
89	C₂×Dic₃	—	12	8	no	—
90	D₁₂	—	12	8	no	—
91	D₁₂	—	12	16	no	—
92	Dic₆	—	12	8	no	—
93	Dic₆	—	12	16	no	—
94	C₃×C₈	—	2	16	no	—
95	C₃×C₈	#204	1	16	yes	—
96	C₂×C₃²	#492	1	1	yes	characteristic
97	C₂×C₃²	#495	1	2	yes	—
98	C₃⋊D₃	—	16	1	no	—
99	C₂×Q₈	—	18	2	no	—
100	C₂×Q₈	—	18	1	no	—
101	C₂×D₄	—	18	2	no	—
102	C₂×D₄	—	18	1	no	—
103	SD₁₆	—	36	4	no	—
104	C₂×C₈	C₄⋊(C₃⋊D₃)	1	4	yes	—
105	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
106	C₄⋊C₄	—	18	2	no	—
107	C₄⋊C₄	—	18	2	no	—
108	C₄⋊C₄	—	18	2	no	—
109	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
110	C₂²×C₃	#874	1	4	yes	—
111	C₂×D₃	—	24	4	no	—
112	C₂×D₃	—	24	8	no	—
113	C₃×C₄	C₂×#204	1	8	yes	—
114	C₃×C₄	#904	1	16	yes	—
115	Dic₃	—	24	4	no	—
116	Dic₃	—	24	8	no	—
117	C₃²	#1708	1	1	yes	characteristic, complemented, Sylow
118	C₂³	—	36	1	no	—
119	Q₈	—	36	2	no	—
120	Q₈	—	36	4	no	—
121	D₄	—	36	2	no	—
122	D₄	—	36	4	no	—
123	C₂×C₄	—	36	1	no	—
124	C₂×C₄	—	36	2	no	—
125	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
127	C₈	—	2	4	no	—
128	C₈	#3744	1	4	yes	—
129	C₂×C₃	#4693	1	4	yes	—
130	C₂×C₃	#4696	1	8	yes	—
131	D₃	—	48	4	no	—
132	C₂²	—	72	1	no	—
133	C₂²	—	72	2	no	—
134	C₂²	#64083	1	1	yes	characteristic
135	C₄	—	72	1	no	—
136	C₄	—	72	2	no	—
137	C₄	C₂×#3744	1	2	yes	—
138	C₄	#64113	1	4	yes	—
139	C₃	#68059	1	4	yes	complemented
140	C₂	—	144	1	no	—
141	C₂	#10592392	1	1	yes	characteristic
142	C₂	#10592395	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10592393

C₂

yes

maximal

#10592392

C₂

yes

maximal, characteristic

#10592380

C₂

yes

maximal, complemented

#10592379

C₂

yes

maximal, characteristic, complemented

C₃²×C₈²

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

characteristic

#64078

C₂²

yes

—

#64078

C₂²

yes

characteristic

C₃²×C₄×C₈

C₂²

yes

characteristic

C₃²×C₄×C₈

C₂²

yes

—

—

—

—

—

—

—

—

C₃×C₈²

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)

—

—

—

—

—

—

—

C₃×C₄×C₈

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

#821

—

#817

—

#816

—

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

SD₁₆

yes

—

—

—

—

—

C₈²

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

—

#204

—

C₂²×C₃²

#127

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

C₂×SD₁₆

yes

—

C₃²×C₄

#135

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

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

#204

yes

—

C₂×C₃²

#492

yes

characteristic

C₂×C₃²

#495

yes

—

C₃⋊D₃

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

SD₁₆

—

104

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

—

105

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

C₂²×C₃⋊D₃

yes

characteristic

110

C₂²×C₃

#874

yes

—

111

C₂×D₃

—

112

C₂×D₃

—

113

C₃×C₄

C₂×#204

yes

—

114

C₃×C₄

#904

yes

—

115

Dic₃

—

116

Dic₃

—

117

C₃²

#1708

yes

characteristic, complemented, Sylow

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

126

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

—

127

C₈

—

128

C₈

#3744

yes

—

129

C₂×C₃

#4693

yes

—

130

C₂×C₃

#4696

yes

—

131

D₃

—

132

C₂²

—

133

C₂²

—

134

C₂²

#64083

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

C₂×#3744

yes

—

138

C₄

#64113

yes

—

139

C₃

#68059

yes

complemented

140

C₂

—

144

—

141

C₂

#10592392

yes

characteristic

142

C₂

#10592395

yes

—

	subgroup	G/H
center	C₂²	#64083
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×C₈²	C₂
socle	C₂²×C₃²	#127
prime radical	#64083	C₂²
hypercenter	C₈²	C₃⋊D₃
hypocenter	C₃²	#1708
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#64083

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×C₈²

C₂

C₂²×C₃²

C₂²

C₈²

C₃⋊D₃

hypocenter

C₃²

#1708

LCS

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
3. #10592380	C₂	semidirect
4. #10592379	C₂	semidirect
5. C₃²×C₈²	C₂	semidirect
20. C₃×C₈²	D₃	semidirect
57. C₈²	C₃⋊D₃	semidirect
117. C₃²	#1708	semidirect
139. C₃	#68059	semidirect

H ≅ G/N

type

3. #10592380

C₂

semidirect

4. #10592379

C₂

semidirect

5. C₃²×C₈²

C₂

semidirect

20. C₃×C₈²

D₃

semidirect

57. C₈²

C₃⋊D₃

semidirect

117. C₃²

#1708

semidirect

139. C₃

#68059

semidirect

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

subgroup	index
#10592393	2
#10592392	2
#10592380	2
#10592379	2
C₃²×C₈²	2
#68059	3

subgroup

index

C₃²×C₈²

Subgroups — #11782502

Conjugacy classes of subgroups

Complements

Maximal subgroups