Group #11771332 — subgroups

Subgroups — #11771332

Order 1152.

187 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 96 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	#10589327	C₂	1	1	yes	maximal, characteristic
2	C₃×#4461	C₂	1	1	yes	maximal, characteristic
3	C₃×#4455	C₂	1	1	yes	maximal, characteristic
4	#67511	—	3	1	no	maximal
5	#67442	—	3	1	no	maximal
6	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	2	yes	—
8	C₂×C₃×C₃⋊C₁₆	C₄	1	2	yes	—
9	C₃×C₄×C₃⋊C₈	C₄	1	2	yes	—
10	C₃×#346	—	3	1	no	—
11	C₃×#322	—	3	1	no	—
12	#4461	—	3	1	no	—
13	#4461	—	3	1	no	—
14	#4461	—	6	1	no	—
15	#4461	D₃	1	1	yes	characteristic
16	#4455	D₃	1	1	yes	characteristic
17	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₃⋊C₈	—	2	2	no	—
21	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
22	C₃×C₃⋊C₁₆	C₈	1	4	yes	complemented
23	#1313	—	9	1	no	Sylow
24	C₂×C₃×C₁₆	—	3	2	no	—
25	C₃×C₄×C₈	—	2	1	no	—
26	C₃×C₄×C₈	—	3	2	no	—
27	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
28	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃⋊C₁₆	—	3	2	no	—
30	C₂×C₃⋊C₁₆	—	3	2	no	—
31	C₂×C₃⋊C₁₆	—	6	2	no	—
32	C₂×C₃⋊C₁₆	Dic₃	1	2	yes	—
33	C₄×C₃⋊C₈	Dic₃	1	2	yes	—
34	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
35	C₂×C₃²×C₄	M₁₆	1	2	yes	—
36	C₃²×C₈	C₂×C₈	1	2	yes	—
37	C₃²×C₈	M₁₆	1	2	yes	—
38	C₃×C₃⋊C₈	—	4	2	no	—
39	#346	—	3	1	no	—
40	#346	—	9	1	no	—
41	#322	—	3	1	no	—
42	C₂×C₃×C₈	—	2	1	no	—
43	C₂×C₃×C₈	—	2	1	no	—
44	C₂×C₃×C₈	—	6	2	no	—
45	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
46	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
47	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
48	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
49	C₃×C₄²	—	2	1	no	—
50	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
51	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₃⋊C₈	—	2	2	no	—
53	C₃×C₁₆	—	3	4	no	—
54	C₃⋊C₁₆	—	6	4	no	—
55	C₃⋊C₁₆	—	6	2	no	—
56	C₃⋊C₁₆	—	6	2	no	—
57	C₃⋊C₁₆	C₃⋊C₈	1	4	yes	complemented
58	C₂²×C₃²	#97	1	1	yes	characteristic
59	C₃²×C₄	—	2	2	no	—
60	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
61	C₃²×C₄	#97	1	1	yes	characteristic
62	C₂×C₁₆	—	3	2	no	—
63	C₂×C₁₆	—	9	2	no	—
64	C₄×C₈	—	3	2	no	—
65	C₄×C₈	D₃²	1	1	yes	characteristic
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	—	2	1	no	—
68	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
69	C₂×C₃×C₄	#208	1	2	yes	—
70	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
71	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
72	C₃×C₈	—	2	2	no	—
73	C₃×C₈	—	2	2	no	—
74	C₃×C₈	—	12	2	no	—
75	C₃×C₈	D₃×C₈	1	2	yes	—
76	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
77	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
78	C₃×C₈	C₂×C₃⋊C₈	1	2	yes	—
79	C₃⋊C₈	—	4	2	no	—
80	C₂×C₃²	#322	1	1	yes	characteristic
81	C₂×C₃²	#347	1	2	yes	—
82	C₂×C₈	—	6	2	no	—
83	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
84	C₂×C₈	#624	1	1	yes	characteristic
85	C₄²	D₃×Dic₃	1	1	yes	characteristic
86	C₁₆	—	3	4	no	—
87	C₁₆	—	18	2	no	—
88	C₂²×C₃	—	2	1	no	—
89	C₂²×C₃	#803	1	1	yes	characteristic
90	C₂²×C₃	#815	1	1	yes	characteristic
91	C₃×C₄	—	2	2	no	—
92	C₃×C₄	—	2	2	no	—
93	C₃×C₄	—	2	1	no	—
94	C₃×C₄	—	2	1	no	—
95	C₃×C₄	—	4	2	no	—
96	C₃×C₄	C₄×C₃⋊C₈	1	1	yes	characteristic
97	C₃×C₄	C₈×Dic₃	1	1	yes	characteristic
98	C₃×C₄	#815	1	1	yes	characteristic
99	C₃×C₄	#815	1	1	yes	characteristic
100	C₃²	#1313	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄	#3711	1	2	yes	—
102	C₂×C₄	Dic₃²	1	1	yes	characteristic
103	C₈	—	12	2	no	—
104	C₈	D₃×C₃⋊C₈	1	2	yes	—
105	C₈	#3712	1	2	yes	—
106	C₂×C₃	—	2	1	no	—
107	C₂×C₃	—	2	2	no	—
108	C₂×C₃	#4454	1	1	yes	characteristic
109	C₂×C₃	#4455	1	1	yes	characteristic
110	C₂×C₃	#4466	1	2	yes	—
111	C₂×C₃	#4512	1	2	yes	—
112	C₂²	#63554	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	Dic₃×C₃⋊C₈	1	1	yes	characteristic
115	C₄	#63555	1	1	yes	characteristic
116	C₃	—	2	1	no	—
117	C₃	#67442	1	1	yes	characteristic, complemented
118	C₃	#67511	1	1	yes	characteristic, complemented
119	C₂	#10589078	1	1	yes	characteristic
120	C₂	#10589105	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10589327

C₂

yes

maximal, characteristic

C₃×#4461

C₂

yes

maximal, characteristic

C₃×#4455

C₂

yes

maximal, characteristic

—

—

C₃²×C₄×C₈

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₄

yes

—

C₂×C₃×C₃⋊C₁₆

C₄

yes

—

C₃×C₄×C₃⋊C₈

C₄

yes

—

—

—

—

—

—

D₃

yes

characteristic

#4455

D₃

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₃²×C₄²

C₂×C₄

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₃⋊(C₃⋊C₁₆)

—

C₃×C₃⋊C₁₆

C₈

yes

complemented

#1313

—

Sylow

C₂×C₃×C₁₆

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

C₃×C₄×C₈

C₂×D₃

yes

characteristic

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

Dic₃

yes

—

C₄×C₃⋊C₈

Dic₃

yes

—

C₂×C₃²×C₄

C₄²

yes

characteristic

C₂×C₃²×C₄

M₁₆

yes

—

C₃²×C₈

C₂×C₈

yes

—

C₃²×C₈

M₁₆

yes

—

C₃×C₃⋊C₈

—

#346

—

#346

—

#322

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₂×C₃×C₈

C₂×Dic₃

yes

characteristic

C₃×C₄²

—

C₃×C₄²

C₄×D₃

yes

characteristic

C₃×C₄²

C₂×Dic₃

yes

characteristic

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

C₃⋊C₈

yes

complemented

C₂²×C₃²

#97

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄×C₈

yes

characteristic

C₃²×C₄

#97

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

D₃²

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

ΓL(1,𝔽₂₅)

yes

—

C₂×C₃×C₄

#208

yes

—

C₂×C₃×C₄

C₄×Dic₃

yes

characteristic

C₂×C₃×C₄

C₄×Dic₃

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

D₃×C₈

yes

—

C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

C₃×C₈

C₂×C₃⋊C₈

yes

—

C₃⋊C₈

—

C₂×C₃²

#322

yes

characteristic

C₂×C₃²

#347

yes

—

C₂×C₈

—

C₂×C₈

D₃×Dic₃

yes

characteristic

C₂×C₈

#624

yes

characteristic

C₄²

D₃×Dic₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

yes

C₂²×C₃

yes

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×C₃⋊C₈

yes

characteristic

C₃×C₄

C₈×Dic₃

yes

C₃×C₄

yes

C₃×C₄

yes

100

C₃²

#1313

yes

characteristic, complemented, Sylow

101

C₂×C₄

#3711

yes

—

102

C₂×C₄

Dic₃²

yes

characteristic

103

C₈

—

104

C₈

D₃×C₃⋊C₈

yes

—

105

C₈

#3712

yes

—

106

C₂×C₃

—

107

C₂×C₃

—

108

C₂×C₃

#4454

yes

characteristic

109

C₂×C₃

#4455

yes

characteristic

110

C₂×C₃

#4466

yes

—

111

C₂×C₃

#4512

yes

—

112

C₂²

#63554

yes

characteristic

113

C₄

—

114

C₄

Dic₃×C₃⋊C₈

yes

characteristic

115

C₄

#63555

yes

characteristic

116

C₃

—

117

C₃

#67442

yes

characteristic, complemented

118

C₃

#67511

yes

characteristic, complemented

119

C₂

#10589078

yes

characteristic

120

C₂

#10589105

yes

—

	subgroup	G/H
center	C₂×C₄	Dic₃²
commutator	C₃²×C₄	C₄×C₈
Frattini, hypercenter	C₄×C₈	D₃²
Fitting, cosocle	C₃²×C₄×C₈	C₂²
socle, prime radical	C₂²×C₃²	#97
hypocenter	C₃²	#1313
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

Dic₃²

commutator

C₃²×C₄

C₄×C₈

Frattini, hypercenter

C₄×C₈

D₃²

Fitting, cosocle

C₃²×C₄×C₈

C₂²

C₂²×C₃²

C₃²

G ▷ C₃²×C₄ ▷ C₃²

C₁ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
22. C₃×C₃⋊C₁₆	C₈	semidirect
57. C₃⋊C₁₆	C₃⋊C₈	semidirect
100. C₃²	#1313	semidirect
117. C₃	#67442	semidirect
118. C₃	#67511	semidirect

H ≅ G/N

type

22. C₃×C₃⋊C₁₆

C₈

semidirect

57. C₃⋊C₁₆

C₃⋊C₈

semidirect

100. C₃²

#1313

semidirect

117. C₃

#67442

semidirect

118. C₃

#67511

semidirect

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

subgroup	index
#10589327	2
C₃×#4461	2
C₃×#4455	2
#67511	3
#67442	3

subgroup

index

Subgroups — #11771332

Conjugacy classes of subgroups

Complements

Maximal subgroups