Group #11772385 — subgroups

Subgroups — #11772385

Order 1152.

234 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10593771	C₂	1	1	yes	maximal, characteristic
2	C₂×#63764	C₂	1	1	yes	maximal, characteristic
3	C₃⋊D₃×#108	C₂	1	1	yes	maximal, characteristic
4	#64283	—	2	1	no	—
5	M₁₆×C₃⋊D₃	—	2	1	no	—
6	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₂×#3708	—	2	1	no	—
8	#63771	C₄	1	2	yes	—
9	#63770	—	2	1	no	—
10	#63764	—	2	1	no	—
11	#63764	C₄	1	2	yes	—
12	C₃²×#108	C₄	1	1	yes	characteristic
13	#63649	—	2	1	no	—
14	#63646	C₄	1	1	yes	characteristic
15	D₃×#108	—	6	2	no	—
16	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
17	#3792	—	4	1	no	—
18	#3788	—	2	2	no	—
19	#3787	—	4	1	no	—
20	C₃²×M₁₆	—	2	1	no	—
21	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
22	#3748	—	2	1	no	—
23	C₂×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
24	#3743	—	4	1	no	—
25	C₈×C₃⋊D₃	—	4	1	no	—
26	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
27	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
28	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
29	C₈×C₃⋊D₃	Q₈	1	1	yes	characteristic
30	#3708	—	2	2	no	—
31	#1381	—	9	1	no	maximal, Sylow
32	D₃×M₁₆	—	6	4	no	—
33	C₂×D₃×C₈	—	6	2	no	—
34	C₃×#108	—	2	2	no	—
35	#822	—	6	4	no	—
36	#819	—	6	2	no	—
37	C₂²×C₃⋊D₃	Q₁₆	1	1	yes	characteristic
38	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₃⋊Dic₃	SD₁₆	1	1	yes	characteristic
40	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₃⋊D₃	D₈	1	1	yes	characteristic
43	C₄×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
44	#622	—	4	2	no	—
45	C₃²×C₈	—	4	1	no	—
46	C₃²×C₈	C₄²	1	1	yes	characteristic
47	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₃⋊(C₃⋊C₈)	—	4	1	no	—
49	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
50	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₆	—	9	1	no	—
52	C₂×#108	—	9	1	no	—
53	C₂×#108	—	9	1	no	—
54	C₂×C₄×D₃	—	6	2	no	—
55	C₃×M₁₆	—	2	4	no	—
56	C₂×C₃×C₈	—	2	2	no	—
57	#208	—	6	4	no	—
58	C₂×C₃⋊C₈	—	6	2	no	—
59	ΓL(1,𝔽₂₅)	—	12	4	no	—
60	D₃×C₈	—	6	2	no	—
61	D₃×C₈	—	6	2	no	—
62	D₃×C₈	—	6	2	no	—
63	D₃×C₈	—	6	2	no	—
64	D₃×C₈	—	12	4	no	—
65	C₂²×C₃²	#103	1	1	yes	characteristic
66	C₂×C₃⋊D₃	—	2	2	no	—
67	C₂×C₃⋊D₃	#103	1	1	yes	characteristic
68	C₂×C₃⋊D₃	#107	1	1	yes	characteristic
69	C₃²×C₄	#95	1	1	yes	characteristic
70	C₃²×C₄	#102	1	1	yes	characteristic
71	C₃⋊Dic₃	#102	1	1	yes	characteristic
72	C₃⋊Dic₃	#106	1	1	yes	characteristic
73	C₂×M₁₆	—	18	1	no	—
74	C₂×M₁₆	—	18	1	no	—
75	C₂²×C₈	—	9	1	no	—
76	C₂×C₁₆	—	9	2	no	—
77	C₂×C₁₆	—	18	1	no	—
78	C₂×C₁₆	—	18	1	no	—
79	#108	—	9	2	no	—
80	#108	—	9	1	no	—
81	#108	—	18	1	no	—
82	#108	—	18	1	no	—
83	#108	#157	1	1	yes	characteristic
84	C₂²×D₃	—	6	2	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×Dic₃	—	6	2	no	—
87	C₄×D₃	—	6	2	no	—
88	C₄×D₃	—	6	2	no	—
89	C₄×D₃	—	6	2	no	—
90	C₄×D₃	—	6	2	no	—
91	C₃×C₈	—	2	2	no	—
92	C₃×C₈	—	2	2	no	—
93	C₃×C₈	—	4	4	no	—
94	C₃⋊C₈	—	6	2	no	—
95	C₃⋊C₈	—	6	2	no	—
96	C₃⋊C₈	—	12	4	no	—
97	C₂×C₃²	—	2	1	no	—
98	C₂×C₃²	#340	1	1	yes	characteristic
99	C₃⋊D₃	—	2	1	no	—
100	C₃⋊D₃	#368	1	2	yes	—
101	C₂²×C₄	—	9	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	18	2	no	—
104	M₁₆	—	18	1	no	—
105	M₁₆	—	36	1	no	—
106	M₁₆	—	36	1	no	—
107	C₂×C₈	—	9	1	no	—
108	C₂×C₈	—	9	1	no	—
109	C₂×C₈	—	9	1	no	—
110	C₂×C₈	—	9	1	no	—
111	C₂×C₈	—	9	1	no	—
112	C₂×C₈	—	36	1	no	—
113	C₂×C₈	—	36	1	no	—
114	C₂×C₈	C₂×#157	1	1	yes	characteristic
115	C₁₆	—	18	2	no	—
116	C₂²×C₃	—	2	2	no	—
117	C₂×D₃	—	6	2	no	—
118	C₂×D₃	—	6	2	no	—
119	C₂×D₃	—	12	4	no	—
120	C₃×C₄	—	2	2	no	—
121	C₃×C₄	—	2	2	no	—
122	Dic₃	—	6	2	no	—
123	Dic₃	—	6	2	no	—
124	C₃²	#1381	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	9	1	no	—
126	C₂×C₄	—	9	1	no	—
127	C₂×C₄	—	9	1	no	—
128	C₂×C₄	—	9	1	no	—
129	C₂×C₄	—	9	1	no	—
130	C₂×C₄	—	9	1	no	—
131	C₂×C₄	C₂²⋊#157	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	9	1	no	—
134	C₈	—	9	1	no	—
135	C₈	—	36	2	no	—
136	C₈	—	36	1	no	—
137	C₈	C₄×#157	1	1	yes	characteristic
138	C₈	C₄⋊#157	1	1	yes	characteristic
139	C₂×C₃	—	2	2	no	—
140	C₂×C₃	—	4	2	no	—
141	D₃	—	6	4	no	—
142	D₃	—	12	2	no	—
143	C₂²	—	9	1	no	—
144	C₂²	—	9	1	no	—
145	C₂²	—	18	2	no	—
146	C₂²	#63784	1	1	yes	characteristic
147	C₄	—	9	1	no	—
148	C₄	—	9	1	no	—
149	C₄	#63781	1	1	yes	characteristic
150	C₄	#63782	1	1	yes	characteristic
151	C₃	—	2	2	no	—
152	C₂	—	2	1	no	—
153	C₂	—	9	2	no	—
154	C₂	—	18	1	no	—
155	C₂	#10590163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593771

C₂

yes

maximal, characteristic

C₂×#63764

C₂

yes

maximal, characteristic

C₃⋊D₃×#108

C₂

yes

maximal, characteristic

#64283

—

M₁₆×C₃⋊D₃

—

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

C₂×#3708

—

#63771

C₄

yes

—

#63770

—

#63764

—

#63764

C₄

yes

—

C₃²×#108

C₄

yes

characteristic

#63649

—

#63646

C₄

yes

characteristic

D₃×#108

—

C₂×C₄×C₃⋊D₃

D₄

yes

—

—

—

C₃²×M₁₆

—

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

#3748

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

#3743

—

C₈×C₃⋊D₃

—

C₈×C₃⋊D₃

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

D₄

yes

characteristic

C₈×C₃⋊D₃

Q₈

yes

—

—

D₃×M₁₆

—

C₂×D₃×C₈

—

C₃×#108

—

#822

—

#819

—

C₂²×C₃⋊D₃

Q₁₆

yes

characteristic

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

C₄⋊C₄

yes

characteristic

C₄×C₃⋊D₃

D₈

yes

characteristic

C₄×C₃⋊D₃

SD₁₆

yes

characteristic

#622

—

C₃²×C₈

—

C₃²×C₈

C₄²

yes

characteristic

C₃²×C₈

C₄⋊C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

C₂²⋊C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

C₄⋊C₄

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

C₂×C₄×D₃

—

C₃×M₁₆

—

C₂×C₃×C₈

—

#208

—

C₂×C₃⋊C₈

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

#103

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

yes

C₂×C₃⋊D₃

yes

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

yes

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#340

yes

characteristic

C₃⋊D₃

—

100

C₃⋊D₃

#368

yes

—

101

C₂²×C₄

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂×#157

yes

characteristic

115

C₁₆

—

116

C₂²×C₃

—

117

C₂×D₃

—

118

C₂×D₃

—

119

C₂×D₃

—

120

C₃×C₄

—

121

C₃×C₄

—

122

Dic₃

—

123

Dic₃

—

124

C₃²

#1381

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊#157

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

C₄×#157

yes

characteristic

138

C₈

C₄⋊#157

yes

characteristic

139

C₂×C₃

—

140

C₂×C₃

—

141

D₃

—

142

D₃

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#63784

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#63781

yes

characteristic

150

C₄

#63782

yes

characteristic

151

C₃

—

152

C₂

—

153

C₂

—

154

C₂

—

155

C₂

#10590163

yes

characteristic

	subgroup	G/H
center	C₄	#63782
commutator	C₃²×C₈	C₄²
Frattini	C₂×C₈	C₂×#157
Fitting	C₃²×#108	C₄
socle	C₂×C₃²	#340
prime radical	C₂²×C₃⋊D₃	Q₁₆
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypercenter	#108	#157
hypocenter	C₃²	#1381
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

subgroup

G/H

center

C₄

#63782

commutator

C₃²×C₈

C₄²

Frattini

C₂×C₈

C₂×#157

Fitting

C₃²×#108

C₄

socle

C₂×C₃²

#340

prime radical

C₂²×C₃⋊D₃

Q₁₆

cosocle

C₂×C₈×C₃⋊D₃

C₂²

C₃²

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

N	H ≅ G/N	type
124. C₃²	#1381	semidirect

H ≅ G/N

type

124. C₃²

#1381

semidirect

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

subgroup	index
#10593771	2
C₂×#63764	2
C₃⋊D₃×#108	2
#1381	9

subgroup

index

#10593771

C₂×#63764

C₃⋊D₃×#108

#1381

Subgroups — #11772385

Conjugacy classes of subgroups

Complements

Maximal subgroups