Group #11771514 — subgroups

Subgroups — #11771514

Order 1152.

270 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 85 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	#10592818	C₂	1	1	yes	maximal, characteristic
2	C₃²×#349	C₂	1	1	yes	maximal, characteristic
3	#10589419	C₂	1	1	yes	maximal, characteristic
4	#67533	—	3	4	no	maximal
5	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
6	#64135	C₄	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
8	C₃²×M₃₂	—	2	1	no	—
9	#63641	—	2	1	no	—
10	#63641	—	2	1	no	—
11	#63639	—	2	1	no	—
12	#5119	—	3	4	no	—
13	C₃×#349	D₃	1	4	yes	—
14	#4553	—	3	4	no	—
15	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₃²×M₁₆	—	4	1	no	—
17	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
19	#3757	C₈	1	2	yes	—
20	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
21	C₄×C₃⋊Dic₃	—	2	1	no	—
22	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
23	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
24	C₃²×C₁₆	—	4	1	no	—
25	C₃⋊(C₃⋊C₁₆)	—	4	1	no	—
26	#1328	—	9	1	no	Sylow
27	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
28	#926	—	3	4	no	—
29	C₂²×C₃⋊C₈	—	3	4	no	—
30	C₃×M₃₂	—	2	4	no	—
31	Dic₃⋊C₈	—	6	4	no	—
32	Dic₃⋊C₈	—	6	4	no	—
33	#812	—	6	4	no	—
34	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
35	C₂×C₃²×C₄	—	2	1	no	—
36	C₂×C₃²×C₄	—	2	1	no	—
37	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
39	C₂×C₃⋊Dic₃	—	2	2	no	—
40	C₃²×C₈	—	2	1	no	—
41	C₃²×C₈	—	2	1	no	—
42	C₃⋊(C₃⋊C₈)	—	4	1	no	—
43	#424	—	9	1	no	—
44	#349	—	9	1	no	—
45	#349	C₃⋊D₃	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
47	C₃×M₁₆	—	4	4	no	—
48	C₂×C₃×C₈	D₁₂	1	4	yes	—
49	C₂×C₃×C₈	#67	1	4	yes	—
50	#217	—	3	8	no	—
51	C₄⋊Dic₃	—	6	4	no	—
52	C₄×Dic₃	—	6	4	no	—
53	C₂×C₃⋊C₈	—	6	4	no	—
54	C₂×C₃⋊C₈	—	12	4	no	—
55	C₃×C₁₆	—	4	4	no	—
56	C₃⋊C₁₆	—	12	4	no	—
57	C₂²×C₃²	—	2	1	no	—
58	C₂²×C₃²	—	4	1	no	—
59	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
60	C₃²×C₄	—	2	1	no	—
61	C₃²×C₄	#99	1	1	yes	characteristic
62	C₃²×C₄	#101	1	1	yes	characteristic
63	C₃⋊Dic₃	—	4	2	no	—
64	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
65	C₂²×C₈	—	9	1	no	—
66	#117	—	9	1	no	—
67	M₃₂	—	2	1	no	—
68	M₃₂	—	18	1	no	—
69	C₄⋊C₈	—	18	1	no	—
70	C₄⋊C₈	—	18	1	no	—
71	C₂³×C₃	D₃×C₈	1	4	yes	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
75	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
76	C₂×Dic₃	—	6	8	no	—
77	C₃×C₈	—	2	4	no	—
78	C₃×C₈	—	2	4	no	—
79	C₃⋊C₈	—	12	4	no	—
80	C₂×C₃²	—	2	1	no	—
81	C₂×C₃²	—	4	1	no	—
82	C₂×C₃²	#324	1	1	yes	characteristic
83	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
84	M₁₆	—	4	1	no	—
85	C₂×C₈	—	18	1	no	—
86	C₂×C₈	—	36	1	no	—
87	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
88	C₂×C₈	#638	1	1	yes	characteristic
89	C₄⋊C₄	—	18	1	no	—
90	C₂²⋊C₄	—	9	2	no	—
91	C₄²	—	18	1	no	—
92	C₁₆	—	4	1	no	—
93	C₁₆	—	36	1	no	—
94	C₂²×C₃	—	2	4	no	—
95	C₂²×C₃	—	4	4	no	—
96	C₂²×C₃	(C₂×D₃)⋊C₈	1	4	yes	—
97	C₃×C₄	—	2	4	no	—
98	C₃×C₄	#806	1	4	yes	—
99	C₃×C₄	#824	1	4	yes	—
100	Dic₃	—	12	8	no	—
101	C₃²	#1328	1	1	yes	characteristic, complemented, Sylow
102	C₂³	C₈×C₃⋊D₃	1	1	yes	characteristic
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	18	2	no	—
106	C₂×C₄	#3743	1	1	yes	characteristic
107	C₂×C₄	#3752	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	36	1	no	—
111	C₂×C₃	—	2	4	no	—
112	C₂×C₃	—	4	4	no	—
113	C₂×C₃	#4469	1	4	yes	—
114	C₂²	—	2	1	no	—
115	C₂²	—	4	1	no	—
116	C₂²	#63647	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	36	2	no	—
119	C₄	#63633	1	1	yes	characteristic
120	C₄	#63651	1	1	yes	characteristic
121	C₃	#67533	1	4	yes	complemented
122	C₂	—	2	1	no	—
123	C₂	—	4	1	no	—
124	C₂	#10589335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592818

C₂

yes

maximal, characteristic

C₃²×#349

C₂

yes

maximal, characteristic

#10589419

C₂

yes

maximal, characteristic

#67533

—

maximal

C₂×C₃²×M₁₆

C₂²

yes

characteristic

#64135

C₄

yes

characteristic

C₂²×C₃⋊(C₃⋊C₈)

C₄

yes

characteristic

C₃²×M₃₂

—

—

—

—

—

D₃

yes

—

#4553

—

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

#3757

C₈

yes

—

C₄⋊(C₃⋊Dic₃)

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

#1328

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

#926

—

C₂²×C₃⋊C₈

—

C₃×M₃₂

—

Dic₃⋊C₈

—

Dic₃⋊C₈

—

#812

—

C₂³×C₃²

C₂×C₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

M₁₆

yes

characteristic

C₂×C₃⋊Dic₃

—

C₃²×C₈

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#424

—

#349

—

#349

C₃⋊D₃

yes

characteristic

C₂²×C₃×C₄

C₄×D₃

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

#67

yes

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂²⋊C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₂×M₁₆

C₂×C₃⋊D₃

yes

characteristic

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂³×C₃

D₃×C₈

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

ΓL(1,𝔽₂₅)

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#324

yes

characteristic

C₂²×C₄

C₄×C₃⋊D₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₈

#638

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

(C₂×D₃)⋊C₈

yes

—

C₃×C₄

—

C₃×C₄

#806

yes

—

C₃×C₄

#824

yes

—

100

Dic₃

—

101

C₃²

#1328

yes

characteristic, complemented, Sylow

102

C₂³

C₈×C₃⋊D₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#3743

yes

characteristic

107

C₂×C₄

#3752

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂×C₃

#4469

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

#63647

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#63633

yes

characteristic

120

C₄

#63651

yes

characteristic

121

C₃

#67533

yes

complemented

122

C₂

—

123

C₂

—

124

C₂

#10589335

yes

characteristic

	subgroup	G/H
center	C₄	#63633
commutator, prime radical	C₂³×C₃²	C₂×C₈
Frattini	C₂×M₁₆	C₂×C₃⋊D₃
Fitting	C₃²×#349	C₂
socle	C₂×C₃²	#324
cosocle	C₂×C₃²×M₁₆	C₂²
hypercenter	#349	C₃⋊D₃
hypocenter	C₃²	#1328
LCS	G ▷ C₂³×C₃² ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₄

#63633

commutator, prime radical

C₂³×C₃²

C₂×C₈

Frattini

C₂×M₁₆

C₂×C₃⋊D₃

Fitting

C₃²×#349

C₂

socle

C₂×C₃²

#324

cosocle

C₂×C₃²×M₁₆

C₂²

C₃⋊D₃

C₃²

G ▷ C₂³×C₃² ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
101. C₃²	#1328	semidirect
121. C₃	#67533	semidirect

H ≅ G/N

type

101. C₃²

#1328

semidirect

121. C₃

#67533

semidirect

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

subgroup	index
#10592818	2
C₃²×#349	2
#10589419	2
#67533	3

subgroup

index

#10592818

C₃²×#349

#10589419

#67533

Subgroups — #11771514

Conjugacy classes of subgroups

Complements

Maximal subgroups