Group #11786328 — subgroups

Subgroups — #11786328

Order 1152.

636 conjugacy classes of subgroups; only the 233 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 233 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×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×#63649	C₂	1	4	yes	maximal
3	#10592623	C₂	1	1	yes	maximal, characteristic
4	C₂×#63630	C₂	1	1	yes	maximal, characteristic
5	#70519	—	3	4	no	maximal
6	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#3748	C₂²	1	2	yes	—
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₃²×C₄⋊C₈	—	2	2	no	—
11	#63649	—	2	8	no	—
12	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
13	#63630	C₄	1	4	yes	—
14	C₃×#423	D₃	1	4	yes	—
15	C₂×#822	—	3	16	no	—
16	#4924	—	3	4	no	—
17	C₂×#803	—	3	4	no	—
18	C₂²×C₃²×C₄	Q₈	1	2	yes	—
19	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
20	C₃²×M₁₆	—	2	4	no	—
21	C₂×C₃²×C₈	—	2	2	no	—
22	C₃²×C₄²	C₂×C₄	1	2	yes	—
23	C₃²×C₄²	D₄	1	2	yes	—
24	#3748	—	2	4	no	—
25	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
26	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
27	C₂×C₃⋊(C₃⋊C₈)	D₄	1	4	yes	—
28	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	4	yes	—
29	#1849	—	9	1	no	Sylow
30	C₂×C₃×M₁₆	C₂×D₃	1	8	yes	—
31	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
32	C₂×#208	—	3	8	no	—
33	C₂²×C₃⋊C₈	—	3	8	no	—
34	C₃×C₄⋊C₈	—	2	8	no	—
35	#822	—	6	32	no	—
36	C₄⋊(C₃⋊C₈)	—	6	8	no	—
37	#803	—	3	16	no	—
38	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₃²×C₄	—	4	2	no	—
40	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
41	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
43	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
45	C₃²×C₈	—	4	2	no	—
46	C₃⋊(C₃⋊C₈)	—	4	2	no	—
47	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	8	yes	—
48	C₂×#108	—	9	4	no	—
49	#423	—	9	1	no	—
50	#423	C₃⋊D₃	1	1	yes	characteristic
51	C₂×#97	—	9	1	no	—
52	C₂²×C₃×C₄	Dic₆	1	8	yes	—
53	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
54	C₃×M₁₆	—	2	16	no	—
55	C₂×C₃×C₈	—	2	8	no	—
56	C₃×C₄²	C₄×D₃	1	8	yes	—
57	C₃×C₄²	#67	1	8	yes	—
58	#208	—	6	16	no	—
59	C₂×C₃⋊C₈	—	3	16	no	—
60	C₂×C₃⋊C₈	—	3	16	no	—
61	C₂×C₃⋊C₈	—	3	16	no	—
62	C₂×C₃⋊C₈	—	6	8	no	—
63	C₂²×C₃²	—	2	2	no	—
64	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
66	C₃²×C₄	—	2	4	no	—
67	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
68	C₃²×C₄	#127	1	1	yes	characteristic
69	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×M₁₆	—	9	2	no	—
71	C₂×M₁₆	C₂×C₃⋊D₃	1	2	yes	—
72	C₂²×C₈	—	9	2	no	—
73	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
74	#108	—	18	8	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	18	2	no	—
77	#97	—	9	4	no	—
78	C₂³×C₃	C₂×Dic₆	1	4	yes	—
79	C₂×C₃×C₄	—	4	8	no	—
80	C₂×C₃×C₄	Dic₃⋊C₄	1	16	yes	—
81	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
82	C₂×C₃×C₄	D₃×D₄	1	8	yes	—
83	C₂×C₃×C₄	D₃×Q₈	1	8	yes	—
84	C₂×C₃×C₄	C₂×#67	1	4	yes	—
85	C₃×C₈	—	4	8	no	—
86	C₃⋊C₈	—	3	32	no	—
87	C₃⋊C₈	—	12	8	no	—
88	C₂×C₃²	—	2	2	no	—
89	C₂×C₃²	#384	1	1	yes	characteristic
90	C₂×C₃²	#430	1	2	yes	—
91	C₂²×C₄	#634	1	2	yes	—
92	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
93	M₁₆	—	2	4	no	—
94	M₁₆	—	18	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	9	4	no	—
97	C₂×C₈	—	9	4	no	—
98	C₂×C₈	—	9	4	no	—
99	C₂×C₈	—	18	2	no	—
100	C₄²	C₄×C₃⋊D₃	1	2	yes	—
101	C₄²	#638	1	2	yes	—
102	C₂²×C₃	—	2	8	no	—
103	C₂²×C₃	#888	1	8	yes	—
104	C₂²×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
105	C₃×C₄	—	2	16	no	—
106	C₃×C₄	D₃×C₄⋊C₄	1	8	yes	—
107	C₃×C₄	#940	1	4	yes	—
108	C₃×C₄	#944	1	4	yes	—
109	C₃²	#1849	1	1	yes	characteristic, complemented, Sylow
110	C₂³	C₂×#634	1	1	yes	characteristic
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	#3750	1	4	yes	—
113	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
114	C₂×C₄	D₄×C₃⋊D₃	1	2	yes	—
115	C₂×C₄	Q₈×C₃⋊D₃	1	2	yes	—
116	C₂×C₄	C₂×#638	1	1	yes	characteristic
117	C₈	—	4	2	no	—
118	C₈	—	9	8	no	—
119	C₈	—	36	2	no	—
120	C₂×C₃	—	2	8	no	—
121	C₂×C₃	#4975	1	4	yes	—
122	C₂×C₃	#5124	1	8	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	#64097	1	2	yes	—
125	C₂²	C₂×#3750	1	1	yes	characteristic
126	C₄	—	2	4	no	—
127	C₄	C₄⋊C₄×C₃⋊D₃	1	2	yes	—
128	C₄	#64149	1	1	yes	characteristic
129	C₄	#64153	1	1	yes	characteristic
130	C₃	#70519	1	4	yes	complemented
131	C₂	—	2	2	no	—
132	C₂	#10592674	1	1	yes	characteristic
133	C₂	#10592823	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#423

C₂

yes

maximal, characteristic

C₂×#63649

C₂

yes

maximal

#10592623

C₂

yes

maximal, characteristic

C₂×#63630

C₂

yes

maximal, characteristic

#70519

—

maximal

C₂×C₃²×M₁₆

C₂²

yes

—

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₂×#3748

C₂²

yes

—

C₂²×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃²×C₄⋊C₈

—

#63649

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

#63630

C₄

yes

—

C₃×#423

D₃

yes

—

C₂×#822

—

#4924

—

C₂×#803

—

C₂²×C₃²×C₄

Q₈

yes

—

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

—

C₃²×C₄²

C₂×C₄

yes

—

C₃²×C₄²

D₄

yes

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

Q₈

yes

—

#1849

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×#208

—

C₂²×C₃⋊C₈

—

C₃×C₄⋊C₈

—

#822

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₂³×C₃²

C₂×Q₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×Q₈

yes

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

C₄⋊C₄

yes

—

C₂×#108

—

#423

—

#423

C₃⋊D₃

yes

characteristic

C₂×#97

—

C₂²×C₃×C₄

Dic₆

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄²

C₄×D₃

yes

—

C₃×C₄²

#67

yes

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃²

C₄⋊Q₈

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×C₄⋊C₄

yes

—

C₃²×C₄

#127

yes

characteristic

C₃²×C₄

C₄⋊Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×C₃⋊D₃

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×C₃⋊D₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₂³×C₃

C₂×Dic₆

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

D₃×Q₈

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#384

yes

characteristic

C₂×C₃²

#430

yes

—

C₂²×C₄

#634

yes

—

C₂²×C₄

C₂²×C₃⋊D₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄²

C₄×C₃⋊D₃

yes

—

101

C₄²

#638

yes

—

102

C₂²×C₃

—

103

C₂²×C₃

#888

yes

—

104

C₂²×C₃

C₂×Dic₃⋊C₄

yes

—

105

C₃×C₄

—

106

C₃×C₄

D₃×C₄⋊C₄

yes

—

107

C₃×C₄

#940

yes

—

108

C₃×C₄

#944

yes

—

109

C₃²

#1849

yes

characteristic, complemented, Sylow

110

C₂³

C₂×#634

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

#3750

yes

—

113

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

114

C₂×C₄

D₄×C₃⋊D₃

yes

—

115

C₂×C₄

Q₈×C₃⋊D₃

yes

—

116

C₂×C₄

C₂×#638

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂×C₃

—

121

C₂×C₃

#4975

yes

—

122

C₂×C₃

#5124

yes

—

123

C₂²

—

124

C₂²

#64097

yes

—

125

C₂²

C₂×#3750

yes

characteristic

126

C₄

—

127

C₄

C₄⋊C₄×C₃⋊D₃

yes

—

128

C₄

#64149

yes

characteristic

129

C₄

#64153

yes

characteristic

130

C₃

#70519

yes

complemented

131

C₂

—

132

C₂

#10592674

yes

characteristic

133

C₂

#10592823

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#638
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#423	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#423	C₃⋊D₃
hypocenter	C₃²	#1849
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×#638

commutator

C₂×C₃²×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#423

C₂

socle

C₂²×C₃²

C₂×C₄⋊C₄

prime radical

C₂³×C₃²

C₂×Q₈

cosocle

C₂²×C₃²×C₄

C₂³

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
109. C₃²	#1849	semidirect
130. C₃	#70519	semidirect

H ≅ G/N

type

109. C₃²

#1849

semidirect

130. C₃

#70519

semidirect

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

subgroup	index
C₃²×#423	2
C₂×#63649	2
#10592623	2
C₂×#63630	2
#70519	3

subgroup

index

C₃²×#423

Subgroups — #11786328

Conjugacy classes of subgroups

Complements

Maximal subgroups