Group #11711531 — subgroups

Subgroups — #11711531

Order 768.

179 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 43 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	C₄×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#73119	C₂	1	2	yes	maximal
3	#12792	—	3	1	no	maximal, Sylow
4	C₂×#131⋊C₃	C₄	1	1	yes	characteristic
5	C₂×#131⋊C₃	C₂²	1	1	yes	characteristic
6	C₄×#41⋊C₃	C₄	1	1	yes	characteristic
7	#4628	—	2	2	no	—
8	#4463	—	4	1	no	maximal
9	C₄×#131	D₃	1	1	yes	characteristic
10	C₂×#364	—	3	1	no	—
11	C₂×#364	—	3	1	no	—
12	#2117	—	3	2	no	—
13	#1378	—	3	2	no	—
14	C₂×#41⋊C₃	C₂×C₄	1	1	yes	characteristic
15	#131⋊C₃	C₈	1	2	yes	—
16	#131⋊C₃	D₄	1	1	yes	characteristic
17	#131⋊C₃	Q₈	1	1	yes	characteristic
18	C₄×SL(2,𝔽₃)	—	2	1	no	—
19	C₃×C₄×C₈	—	4	1	no	—
20	#812	—	4	2	no	—
21	C₂×#131	Dic₃	1	1	yes	characteristic
22	C₂×#131	C₂×D₃	1	1	yes	characteristic
23	C₄×#41	Dic₃	1	1	yes	characteristic
24	C₂×M₃₂	—	3	2	no	—
25	C₂×M₃₂	—	3	2	no	—
26	#405	—	3	1	no	—
27	#405	—	3	1	no	—
28	C₄×M₁₆	—	3	1	no	—
29	C₂×C₄×C₈	—	3	1	no	—
30	#364	—	6	1	no	—
31	#364	—	6	1	no	—
32	#364	—	6	1	no	—
33	#364	—	6	1	no	—
34	#350	—	6	2	no	—
35	#350	—	6	2	no	—
36	#41⋊C₃	C₄⋊C₄	1	1	yes	characteristic
37	#41⋊C₃	C₂×C₈	1	1	yes	characteristic
38	C₂×SL(2,𝔽₃)	M₁₆	1	1	yes	characteristic
39	C₂×C₃×C₈	—	4	1	no	—
40	C₂×C₃×C₈	—	4	1	no	—
41	C₃×C₄²	—	4	1	no	—
42	C₃⋊C₁₆	—	8	2	no	—
43	C₂×#41	C₂×Dic₃	1	1	yes	characteristic
44	#131	—	3	2	no	—
45	#131	—	3	1	no	—
46	#131	—	3	1	no	—
47	#131	C₃⋊C₈	1	2	yes	—
48	#131	Dic₆	1	1	yes	characteristic
49	#131	D₁₂	1	1	yes	characteristic
50	C₂×M₁₆	—	3	1	no	—
51	C₂×M₁₆	—	3	1	no	—
52	C₂²×C₈	—	3	1	no	—
53	C₂²×C₈	—	3	1	no	—
54	C₄×Q₈	—	2	1	no	—
55	C₄×D₄	—	6	1	no	—
56	#117	—	3	1	no	—
57	C₂×C₄²	—	3	1	no	—
58	M₃₂	—	6	2	no	—
59	M₃₂	—	6	2	no	—
60	M₃₂	—	6	2	no	—
61	M₃₂	—	6	2	no	—
62	C₂×C₁₆	—	6	2	no	—
63	C₂×C₁₆	—	6	2	no	—
64	#97	—	6	1	no	—
65	C₄×C₈	—	3	1	no	—
66	C₄×C₈	—	3	1	no	—
67	C₄×C₈	—	3	1	no	—
68	C₄×C₈	S₄	1	1	yes	characteristic
69	C₂×C₃×C₄	—	4	1	no	—
70	C₂×C₃×C₄	—	8	1	no	—
71	SL(2,𝔽₃)	C₄⋊C₈	1	1	yes	characteristic
72	C₃×C₈	—	4	2	no	—
73	C₃×C₈	—	4	1	no	—
74	C₃×C₈	—	4	1	no	—
75	#41	—	3	1	no	—
76	#41	—	3	1	no	—
77	#41	C₂×C₃⋊C₈	1	1	yes	characteristic
78	#41	C₄⋊Dic₃	1	1	yes	characteristic
79	C₂×Q₈	#208	1	1	yes	characteristic
80	C₂×D₄	—	3	1	no	—
81	C₂²×C₄	—	3	1	no	—
82	C₂²×C₄	—	6	1	no	—
83	M₁₆	—	3	2	no	—
84	M₁₆	—	3	1	no	—
85	M₁₆	—	3	1	no	—
86	M₁₆	—	6	1	no	—
87	M₁₆	—	6	1	no	—
88	C₂×C₈	—	3	1	no	—
89	C₂×C₈	—	3	1	no	—
90	C₂×C₈	—	3	2	no	—
91	C₂×C₈	—	3	2	no	—
92	C₂×C₈	—	3	1	no	—
93	C₂×C₈	—	3	1	no	—
94	C₂×C₈	—	3	1	no	—
95	C₂×C₈	—	3	1	no	—
96	C₂×C₈	SL(2,ℤ/4)	1	1	yes	characteristic
97	C₂×C₈	C₂×S₄	1	1	yes	characteristic
98	C₄⋊C₄	—	6	1	no	—
99	C₂²⋊C₄	—	3	2	no	—
100	C₄²	—	3	1	no	—
101	C₄²	—	6	1	no	—
102	C₄²	SL(2,ℤ/4)	1	1	yes	characteristic
103	C₁₆	—	12	2	no	—
104	C₁₆	—	12	2	no	—
105	C₂²×C₃	—	4	1	no	—
106	C₃×C₄	—	4	1	no	—
107	C₃×C₄	—	4	1	no	—
108	C₃×C₄	—	8	2	no	—
109	C₂³	—	3	1	no	—
110	Q₈	—	3	1	no	—
111	Q₈	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
112	D₄	—	3	1	no	—
113	D₄	—	3	1	no	—
114	D₄	—	6	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	3	1	no	—
117	C₂×C₄	—	3	1	no	—
118	C₂×C₄	—	3	1	no	—
119	C₂×C₄	—	3	1	no	—
120	C₂×C₄	—	3	1	no	—
121	C₂×C₄	—	6	1	no	—
122	C₂×C₄	—	12	2	no	—
123	C₂×C₄	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
124	C₈	—	3	2	no	—
125	C₈	—	3	1	no	—
126	C₈	—	3	1	no	—
127	C₈	A₄⋊C₈	1	2	yes	—
128	C₈	A₄⋊Q₈	1	1	yes	characteristic
129	C₈	C₄⋊S₄	1	1	yes	characteristic
130	C₂×C₃	—	4	1	no	—
131	C₂×C₃	—	8	1	no	—
132	C₂²	—	3	1	no	—
133	C₂²	—	3	1	no	—
134	C₂²	—	12	1	no	—
135	C₂²	#5409	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	3	1	no	—
138	C₄	—	3	1	no	—
139	C₄	—	12	1	no	—
140	C₄	C₂×A₄⋊C₈	1	1	yes	characteristic
141	C₄	C₄⋊SL(2,ℤ/4)	1	1	yes	characteristic
142	C₃	—	4	1	no	Sylow
143	C₂	—	2	1	no	—
144	C₂	—	6	1	no	—
145	C₂	—	6	1	no	—
146	C₂	C₄⋊(A₄⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#131⋊C₃

C₂

yes

maximal, characteristic

#73119

C₂

yes

maximal

#12792

—

maximal, Sylow

C₂×#131⋊C₃

C₄

yes

characteristic

C₂×#131⋊C₃

C₂²

yes

characteristic

C₄×#41⋊C₃

C₄

yes

—

—

D₃

yes

—

—

—

—

C₂×#41⋊C₃

C₂×C₄

yes

characteristic

#131⋊C₃

C₈

yes

—

#131⋊C₃

D₄

yes

characteristic

#131⋊C₃

Q₈

yes

characteristic

C₄×SL(2,𝔽₃)

—

C₃×C₄×C₈

—

#812

—

C₂×#131

Dic₃

yes

characteristic

C₂×#131

C₂×D₃

yes

characteristic

C₄×#41

Dic₃

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

#405

—

#405

—

C₄×M₁₆

—

C₂×C₄×C₈

—

—

—

—

—

—

—

C₄⋊C₄

yes

characteristic

#41⋊C₃

C₂×C₈

yes

characteristic

C₂×SL(2,𝔽₃)

M₁₆

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃⋊C₁₆

—

C₂×#41

C₂×Dic₃

yes

—

—

—

C₃⋊C₈

yes

—

#131

Dic₆

yes

characteristic

#131

D₁₂

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

S₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

SL(2,𝔽₃)

C₄⋊C₈

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

#41

—

#41

—

#41

C₂×C₃⋊C₈

yes

characteristic

#41

C₄⋊Dic₃

yes

characteristic

C₂×Q₈

#208

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SL(2,ℤ/4)

yes

characteristic

C₂×C₈

C₂×S₄

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

SL(2,ℤ/4)

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₂²×C₃

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₂³

—

110

Q₈

—

111

Q₈

C₄⋊(C₃⋊C₈)

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×SL(2,ℤ/4)

yes

characteristic

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

A₄⋊C₈

yes

—

128

C₈

A₄⋊Q₈

yes

characteristic

129

C₈

C₄⋊S₄

yes

characteristic

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#5409

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

C₂×A₄⋊C₈

yes

characteristic

141

C₄

C₄⋊SL(2,ℤ/4)

yes

characteristic

142

C₃

—

Sylow

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

C₄⋊(A₄⋊C₈)

yes

characteristic

	subgroup	G/H
center	C₈	C₄⋊S₄
commutator	#41⋊C₃	C₂×C₈
Frattini	C₂×C₈	C₂×S₄
Fitting	C₄×#131	D₃
socle	C₂	C₄⋊(A₄⋊C₈)
prime radical	C₂×#41⋊C₃	C₂×C₄
cosocle	C₂×#131⋊C₃	C₂²
hypercenter	C₄×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₄⋊C₈
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄⋊S₄

commutator

#41⋊C₃

C₂×C₈

Frattini

C₂×C₈

C₂×S₄

Fitting

C₄×#131

D₃

socle

C₂

C₄⋊(A₄⋊C₈)

prime radical

C₂×#41⋊C₃

C₂×C₄

cosocle

C₂×#131⋊C₃

C₂²

hypercenter

C₄×C₈

S₄

hypocenter

SL(2,𝔽₃)

C₄⋊C₈

LCS

G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

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

subgroup	index
C₄×#131⋊C₃	2
#73119	2
#12792	3
#4463	4

subgroup

index

C₄×#131⋊C₃

#73119

#12792

#4463

Subgroups — #11711531

Conjugacy classes of subgroups

Maximal subgroups