Group #11772197 — subgroups

Subgroups — #11772197

Order 1152.

143 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 143 conjugacy classes of subgroups — too many to draw. Only its 70 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	#10589329	C₂	1	1	yes	maximal, characteristic
2	C₃×#4463	C₂	1	1	yes	maximal, characteristic
3	C₃×#4461	C₂	1	1	yes	maximal, characteristic
4	#67567	—	3	1	no	maximal
5	#67514	—	3	1	no	maximal
6	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
7	#63639	—	2	1	no	—
8	C₃×#812	—	2	1	no	—
9	C₂×C₃×C₃⋊C₁₆	C₄	1	1	yes	characteristic
10	C₂×C₃×C₃⋊C₁₆	C₄	1	1	yes	characteristic
11	C₃×#364	—	3	1	no	—
12	C₃×#346	—	3	1	no	—
13	#4463	—	3	1	no	—
14	#4463	—	3	1	no	—
15	#4463	—	6	1	no	—
16	#4463	D₃	1	1	yes	characteristic
17	#4461	D₃	1	1	yes	characteristic
18	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
20	C₃²×C₄²	Q₈	1	1	yes	characteristic
21	C₃⋊(C₃⋊C₁₆)	—	4	1	no	—
22	C₃×C₃⋊C₁₆	—	2	1	no	—
23	C₃×C₃⋊C₁₆	—	4	1	no	—
24	C₃×C₃⋊C₁₆	C₈	1	2	yes	—
25	#1369	—	9	1	no	Sylow
26	C₃×M₃₂	—	6	1	no	—
27	C₂×C₃×C₁₆	—	3	1	no	—
28	C₂×C₃×C₁₆	—	3	1	no	—
29	C₃×C₄×C₈	—	2	1	no	—
30	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
31	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
32	#812	—	2	1	no	—
33	#812	—	6	2	no	—
34	#812	—	6	1	no	—
35	#812	—	6	1	no	—
36	C₂×C₃⋊C₁₆	Dic₃	1	1	yes	characteristic
37	C₂×C₃⋊C₁₆	Dic₃	1	1	yes	characteristic
38	C₂×C₃²×C₄	—	2	1	no	—
39	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
40	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
41	C₃²×C₈	M₁₆	1	1	yes	characteristic
42	C₃²×C₈	SD₁₆	1	1	yes	characteristic
43	C₃²×C₈	SD₁₆	1	1	yes	characteristic
44	#364	—	3	1	no	—
45	#364	—	9	1	no	—
46	#346	—	3	1	no	—
47	C₂×C₃×C₈	—	2	1	no	—
48	C₂×C₃×C₈	—	2	1	no	—
49	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
50	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
51	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₃×C₈	#67	1	1	yes	characteristic
53	C₃×C₄²	—	2	1	no	—
54	C₃×C₄²	Dic₆	1	1	yes	characteristic
55	C₃×C₄²	Dic₆	1	1	yes	characteristic
56	C₃×C₁₆	—	3	2	no	—
57	C₃×C₁₆	—	6	1	no	—
58	C₃×C₁₆	—	12	1	no	—
59	C₃⋊C₁₆	—	2	1	no	—
60	C₃⋊C₁₆	—	4	1	no	—
61	C₃⋊C₁₆	—	12	2	no	—
62	C₃⋊C₁₆	—	12	1	no	—
63	C₃⋊C₁₆	—	12	1	no	—
64	C₃⋊C₁₆	C₃⋊C₈	1	2	yes	—
65	C₂²×C₃²	#108	1	1	yes	characteristic
66	C₃²×C₄	—	4	1	no	—
67	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
68	C₃²×C₄	#106	1	1	yes	characteristic
69	M₃₂	—	6	1	no	—
70	M₃₂	—	18	1	no	—
71	C₂×C₁₆	—	3	1	no	—
72	C₂×C₁₆	—	3	1	no	—
73	C₄×C₈	D₃²	1	1	yes	characteristic
74	C₂×C₃×C₄	—	2	1	no	—
75	C₂×C₃×C₄	—	2	1	no	—
76	C₂×C₃×C₄	—	2	1	no	—
77	C₂×C₃×C₄	—	4	1	no	—
78	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
80	C₃×C₈	—	2	1	no	—
81	C₃×C₈	—	2	1	no	—
82	C₃×C₈	—	2	1	no	—
83	C₃×C₈	—	2	1	no	—
84	C₃×C₈	D₃×C₈	1	1	yes	characteristic
85	C₃×C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
86	C₃×C₈	#204	1	1	yes	characteristic
87	C₃×C₈	#204	1	1	yes	characteristic
88	C₃×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
89	C₃×C₈	#208	1	1	yes	characteristic
90	C₃×C₈	#214	1	1	yes	characteristic
91	C₃×C₈	#215	1	1	yes	characteristic
92	C₂×C₃²	—	2	1	no	—
93	C₂×C₃²	#334	1	1	yes	characteristic
94	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
95	C₂×C₈	#626	1	1	yes	characteristic
96	C₄²	#627	1	1	yes	characteristic
97	C₁₆	—	3	2	no	—
98	C₁₆	—	6	1	no	—
99	C₁₆	—	12	1	no	—
100	C₁₆	—	36	1	no	—
101	C₂²×C₃	—	2	1	no	—
102	C₂²×C₃	#819	1	1	yes	characteristic
103	C₂²×C₃	#822	1	1	yes	characteristic
104	C₃×C₄	—	2	1	no	—
105	C₃×C₄	—	2	1	no	—
106	C₃×C₄	—	4	1	no	—
107	C₃×C₄	—	4	1	no	—
108	C₃×C₄	—	4	2	no	—
109	C₃×C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
110	C₃×C₄	#808	1	1	yes	characteristic
111	C₃×C₄	Dic₃⋊C₈	1	1	yes	characteristic
112	C₃×C₄	#817	1	1	yes	characteristic
113	C₃²	#1369	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
116	C₈	D₃×C₃⋊C₈	1	1	yes	characteristic
117	C₈	#3711	1	1	yes	characteristic
118	C₈	#3716	1	1	yes	characteristic
119	C₈	#3717	1	1	yes	characteristic
120	C₂×C₃	—	2	1	no	—
121	C₂×C₃	—	2	1	no	—
122	C₂×C₃	—	2	1	no	—
123	C₂×C₃	—	4	1	no	—
124	C₂×C₃	#4457	1	1	yes	characteristic
125	C₂×C₃	#4480	1	1	yes	characteristic
126	C₂²	#63577	1	1	yes	characteristic
127	C₄	—	4	1	no	—
128	C₄	Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic
129	C₄	#63573	1	1	yes	characteristic
130	C₃	—	2	1	no	—
131	C₃	#67514	1	1	yes	characteristic, complemented
132	C₃	#67567	1	1	yes	characteristic, complemented
133	C₂	—	2	1	no	—
134	C₂	#10589097	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10589329

C₂

yes

maximal, characteristic

C₃×#4463

C₂

yes

maximal, characteristic

C₃×#4461

C₂

yes

maximal, characteristic

—

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63639

—

C₃×#812

—

C₂×C₃×C₃⋊C₁₆

C₄

yes

characteristic

C₂×C₃×C₃⋊C₁₆

C₄

yes

—

—

—

—

—

D₃

yes

characteristic

#4461

D₃

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₃²×C₄²

Q₈

yes

characteristic

C₃⋊(C₃⋊C₁₆)

—

C₃×C₃⋊C₁₆

—

C₃×C₃⋊C₁₆

—

C₃×C₃⋊C₁₆

C₈

yes

—

#1369

—

Sylow

C₃×M₃₂

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

C₃×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂×C₃⋊C₁₆

Dic₃

yes

characteristic

C₂×C₃⋊C₁₆

Dic₃

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₃²×C₈

C₂×C₈

yes

characteristic

C₃²×C₈

M₁₆

yes

characteristic

C₃²×C₈

SD₁₆

yes

characteristic

C₃²×C₈

SD₁₆

yes

—

—

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₂×C₃×C₈

D₁₂

yes

characteristic

C₂×C₃×C₈

C₂×Dic₃

yes

characteristic

C₂×C₃×C₈

#67

yes

characteristic

C₃×C₄²

—

C₃×C₄²

Dic₆

yes

characteristic

C₃×C₄²

Dic₆

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

C₃⋊C₈

yes

—

C₂²×C₃²

#108

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄⋊C₈

yes

characteristic

C₃²×C₄

#106

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

D₃²

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊Dic₃

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

D₃×C₈

yes

characteristic

C₃×C₈

ΓL(1,𝔽₂₅)

yes

C₃×C₈

yes

C₃×C₈

yes

C₃×C₈

C₂×C₃⋊C₈

yes

C₃×C₈

yes

C₃×C₈

yes

C₃×C₈

yes

C₂×C₃²

—

C₂×C₃²

#334

yes

characteristic

C₂×C₈

D₃×Dic₃

yes

C₂×C₈

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

100

C₁₆

—

101

C₂²×C₃

—

102

C₂²×C₃

#819

yes

characteristic

103

C₂²×C₃

#822

yes

characteristic

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

C₄⋊(C₃⋊C₈)

yes

characteristic

110

C₃×C₄

#808

yes

characteristic

111

C₃×C₄

Dic₃⋊C₈

yes

characteristic

112

C₃×C₄

#817

yes

characteristic

113

C₃²

#1369

yes

characteristic, complemented, Sylow

114

C₂×C₄

—

115

C₂×C₄

Dic₃⋊Dic₃

yes

characteristic

116

C₈

D₃×C₃⋊C₈

yes

characteristic

117

C₈

#3711

yes

characteristic

118

C₈

#3716

yes

characteristic

119

C₈

#3717

yes

characteristic

120

C₂×C₃

—

121

C₂×C₃

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

#4457

yes

characteristic

125

C₂×C₃

#4480

yes

characteristic

126

C₂²

#63577

yes

characteristic

127

C₄

—

128

C₄

Dic₃⋊(C₃⋊C₈)

yes

characteristic

129

C₄

#63573

yes

characteristic

130

C₃

—

131

C₃

#67514

yes

characteristic, complemented

132

C₃

#67567

yes

characteristic, complemented

133

C₂

—

134

C₂

#10589097

yes

characteristic

	subgroup	G/H
center	C₄	#63573
commutator	C₃²×C₈	C₂×C₈
Frattini, hypercenter	C₄×C₈	D₃²
Fitting, cosocle	C₃²×C₄×C₈	C₂²
socle	C₂×C₃²	#334
prime radical	C₂²×C₃²	#108
hypocenter	C₃²	#1369
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₄

#63573

commutator

C₃²×C₈

C₂×C₈

Frattini, hypercenter

C₄×C₈

D₃²

Fitting, cosocle

C₃²×C₄×C₈

C₂²

C₂×C₃²

C₂²×C₃²

C₃²

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
113. C₃²	#1369	semidirect
131. C₃	#67514	semidirect
132. C₃	#67567	semidirect

H ≅ G/N

type

113. C₃²

#1369

semidirect

131. C₃

#67514

semidirect

132. C₃

#67567

semidirect

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

subgroup	index
#10589329	2
C₃×#4463	2
C₃×#4461	2
#67567	3
#67514	3

subgroup

index

Subgroups — #11772197

Conjugacy classes of subgroups

Complements

Maximal subgroups