Group #11782476 — subgroups

Subgroups — #11782476

Order 1152.

540 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 169 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	C₃²×#500	C₂	1	1	yes	maximal, characteristic
2	#10592779	C₂	1	2	yes	maximal
3	#10592778	C₂	1	2	yes	maximal
4	#10592398	C₂	1	1	yes	maximal, characteristic
5	C₈×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#71360	—	3	4	no	maximal
7	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#3746	—	2	2	no	—
9	#64078	C₂²	1	2	yes	—
10	C₃²×#107	—	2	2	no	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#63662	—	2	4	no	—
13	#63635	—	2	4	no	—
14	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₃×#500	D₃	1	4	yes	—
17	#5080	—	3	8	no	—
18	#5079	—	3	8	no	—
19	#4699	—	3	4	no	—
20	C₈×C₃⋊C₈	—	3	4	no	—
21	C₂×C₃²×Q₈	—	2	2	no	—
22	C₂×#634	—	2	2	no	—
23	C₂×C₃²×C₈	D₄	1	2	yes	—
24	C₃²×C₄⋊C₄	—	2	2	no	—
25	C₃²×C₄⋊C₄	—	2	2	no	—
26	C₃²×C₄²	C₂³	1	1	yes	characteristic
27	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
28	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
29	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
30	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
31	#3746	—	4	4	no	—
32	#1707	—	9	1	no	Sylow
33	C₃×C₄⋊Q₈	C₂×D₃	1	8	yes	—
34	C₂×Dic₁₂	—	6	8	no	—
35	#869	—	3	8	no	—
36	C₃×#107	—	2	8	no	—
37	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
38	#835	—	6	16	no	—
39	#808	—	6	16	no	—
40	C₄×C₃⋊C₈	—	3	4	no	—
41	C₄×C₃⋊C₈	—	3	4	no	—
42	C₃²×Q₈	—	4	4	no	—
43	C₂×C₃²×C₄	—	4	2	no	—
44	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃⋊Dic₃	—	4	2	no	—
48	#634	—	4	4	no	—
49	C₃²×C₈	SD₁₆	1	4	yes	—
50	C₃⋊(C₃⋊C₈)	—	2	2	no	—
51	C₃⋊(C₃⋊C₈)	Q₁₆	1	4	yes	complemented
52	#500	C₃⋊D₃	1	1	yes	characteristic
53	#498	—	9	2	no	—
54	#494	—	9	1	no	—
55	#487	—	9	2	no	—
56	C₈²	—	9	1	no	—
57	C₂×C₃×Q₈	—	2	8	no	—
58	C₂×Dic₆	—	6	8	no	—
59	C₂×C₃×C₈	#67	1	8	yes	—
60	C₃×C₄⋊C₄	—	2	8	no	—
61	C₃×C₄⋊C₄	—	2	8	no	—
62	C₃×C₄²	C₂²×D₃	1	4	yes	—
63	C₄⋊Dic₃	—	6	8	no	—
64	C₄⋊Dic₃	—	6	8	no	—
65	C₂×C₃⋊C₈	—	3	8	no	—
66	C₂×C₃⋊C₈	—	3	8	no	—
67	Dic₁₂	—	12	16	no	—
68	C₂²×C₃²	#127	1	1	yes	characteristic
69	C₃²×C₄	—	8	2	no	—
70	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
71	C₃²×C₄	C₂×Q₁₆	1	2	yes	—
72	C₃²×C₄	#135	1	2	yes	—
73	C₃⋊Dic₃	—	8	2	no	—
74	C₂×Q₁₆	—	18	2	no	—
75	C₄⋊Q₈	—	9	2	no	—
76	C₄⋊Q₈	C₂×C₃⋊D₃	1	2	yes	—
77	#107	—	2	2	no	—
78	#106	—	18	4	no	—
79	#103	—	18	4	no	—
80	C₄×C₈	—	9	1	no	—
81	C₄×C₈	—	9	1	no	—
82	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
83	C₃×Q₈	—	4	16	no	—
84	C₂×C₃×C₄	—	4	8	no	—
85	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
86	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
87	C₂×C₃×C₄	C₂×#67	1	4	yes	—
88	C₂×Dic₃	—	12	8	no	—
89	Dic₆	—	12	16	no	—
90	C₃×C₈	#215	1	16	yes	—
91	C₃⋊C₈	—	3	16	no	—
92	C₃⋊C₈	—	6	8	no	—
93	C₂×C₃²	#492	1	1	yes	characteristic
94	C₂×C₃²	#494	1	1	yes	characteristic
95	C₂×C₃²	#495	1	1	yes	characteristic
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	18	2	no	—
98	Q₁₆	—	36	4	no	—
99	C₂×C₈	—	9	2	no	—
100	C₂×C₈	—	9	2	no	—
101	C₂×C₈	#638	1	2	yes	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	18	2	no	—
105	C₄⋊C₄	—	18	2	no	—
106	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
107	C₂²×C₃	#940	1	4	yes	—
108	C₃×C₄	—	8	8	no	—
109	C₃×C₄	#912	1	8	yes	—
110	C₃×C₄	D₃×Q₁₆	1	8	yes	—
111	C₃×C₄	C₂×#215	1	8	yes	—
112	Dic₃	—	24	8	no	—
113	C₃²	#1707	1	1	yes	characteristic, complemented, Sylow
114	Q₈	—	4	4	no	—
115	Q₈	—	36	4	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	36	2	no	—
118	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
119	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
120	C₂×C₄	C₂×#638	1	1	yes	characteristic
121	C₈	—	9	4	no	—
122	C₈	—	18	2	no	—
123	C₈	#3755	1	4	yes	—
124	C₂×C₃	#5085	1	4	yes	—
125	C₂×C₃	#5155	1	4	yes	—
126	C₂×C₃	#5183	1	4	yes	—
127	C₂²	#64149	1	1	yes	characteristic
128	C₄	—	8	2	no	—
129	C₄	—	72	2	no	—
130	C₄	#64121	1	2	yes	—
131	C₄	Q₁₆×C₃⋊D₃	1	2	yes	—
132	C₄	C₂×#3755	1	2	yes	—
133	C₃	#71360	1	4	yes	complemented
134	C₂	#10592784	1	1	yes	characteristic
135	C₂	#10592854	1	1	yes	characteristic
136	C₂	#10592882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#500

C₂

yes

maximal, characteristic

#10592779

C₂

yes

maximal

#10592778

C₂

yes

maximal

#10592398

C₂

yes

maximal, characteristic

C₈×C₃⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#71360

—

maximal

C₃²×C₄⋊Q₈

C₂²

yes

—

C₂×#3746

—

#64078

C₂²

yes

—

C₃²×#107

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63662

—

#63635

—

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₃×#500

D₃

yes

—

#5080

—

#5079

—

#4699

—

C₈×C₃⋊C₈

—

C₂×C₃²×Q₈

—

C₂×#634

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂³

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

#3746

—

#1707

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₂×Dic₁₂

—

#869

—

C₃×#107

—

C₃×C₄×C₈

C₂×D₃

yes

—

#835

—

#808

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₃²×C₈

SD₁₆

yes

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

Q₁₆

yes

complemented

#500

C₃⋊D₃

yes

—

—

—

C₈²

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

#127

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂×SD₁₆

yes

—

C₃²×C₄

C₂×Q₁₆

yes

—

C₃²×C₄

#135

yes

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₃⋊D₃

yes

—

#107

—

#106

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

#215

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

#638

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂²×C₃⋊D₃

yes

characteristic

107

C₂²×C₃

#940

yes

—

108

C₃×C₄

—

109

C₃×C₄

#912

yes

—

110

C₃×C₄

D₃×Q₁₆

yes

—

111

C₃×C₄

C₂×#215

yes

—

112

Dic₃

—

113

C₃²

#1707

yes

characteristic, complemented, Sylow

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

D₄×C₃⋊D₃

yes

characteristic

119

C₂×C₄

D₄×C₃⋊D₃

yes

characteristic

120

C₂×C₄

C₂×#638

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

#3755

yes

—

124

C₂×C₃

#5085

yes

—

125

C₂×C₃

#5155

yes

—

126

C₂×C₃

#5183

yes

—

127

C₂²

#64149

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#64121

yes

—

131

C₄

Q₁₆×C₃⋊D₃

yes

—

132

C₄

C₂×#3755

yes

—

133

C₃

#71360

yes

complemented

134

C₂

#10592784

yes

characteristic

135

C₂

#10592854

yes

characteristic

136

C₂

#10592882

yes

characteristic

	subgroup	G/H
center	C₂²	#64149
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#500	C₂
socle, prime radical	C₂²×C₃²	#127
hypercenter	#500	C₃⋊D₃
hypocenter	C₃²	#1707
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#64149

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#500

C₂

C₂²×C₃²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
51. C₃⋊(C₃⋊C₈)	Q₁₆	semidirect
113. C₃²	#1707	semidirect
133. C₃	#71360	semidirect

H ≅ G/N

type

51. C₃⋊(C₃⋊C₈)

Q₁₆

semidirect

113. C₃²

#1707

semidirect

133. C₃

#71360

semidirect

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

subgroup	index
C₃²×#500	2
#10592779	2
#10592778	2
#10592398	2
C₈×C₃⋊(C₃⋊C₈)	2
#71360	3

subgroup

index

C₃²×#500

C₈×C₃⋊(C₃⋊C₈)

Subgroups — #11782476

Conjugacy classes of subgroups

Complements

Maximal subgroups