Group #10711548 — subgroups

Subgroups — #10711548

Order 768.

342 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₃×C₄²×C₈	C₂	1	1	yes	maximal, characteristic
2	#69743	C₂	1	2	yes	maximal
3	#67658	C₂	1	4	yes	maximal
4	#12883	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₃×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₄³	C₄	1	1	yes	characteristic
10	C₂×#812	C₂²	1	4	yes	—
11	C₄⋊(C₃⋊C₁₆)	—	2	4	no	—
12	#2150	—	3	2	no	—
13	C₄²×C₈	D₃	1	1	yes	characteristic
14	#1376	—	3	4	no	—
15	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄²	—	2	2	no	—
19	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₃×C₄×C₈	—	2	4	no	—
22	C₃×C₄×C₈	—	2	4	no	—
23	C₃×C₄×C₈	C₈	1	2	yes	—
24	C₃×C₄×C₈	C₈	1	2	yes	—
25	C₃×C₄×C₈	D₄	1	2	yes	—
26	C₃×C₄×C₈	Q₈	1	2	yes	—
27	#812	—	2	8	no	—
28	C₂×C₃⋊C₁₆	—	2	4	no	—
29	C₂×M₃₂	—	3	4	no	—
30	C₂×C₄×C₈	Dic₃	1	2	yes	—
31	C₂×C₄×C₈	Dic₃	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
33	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
34	C₄³	Dic₃	1	1	yes	characteristic
35	C₄⋊C₁₆	—	6	4	no	—
36	C₂²×C₃×C₄	—	2	2	no	—
37	C₂²×C₃×C₄	M₁₆	1	2	yes	—
38	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₃×C₈	—	2	2	no	—
40	C₂×C₃×C₈	—	2	2	no	—
41	C₂×C₃×C₈	—	2	2	no	—
42	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₃×C₈	C₂×C₈	1	4	yes	—
44	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₃×C₈	#41	1	4	yes	—
47	C₃×C₄²	—	2	8	no	—
48	C₃×C₄²	—	2	2	no	—
49	C₃×C₄²	C₄⋊C₄	1	2	yes	—
50	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
51	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
52	C₃×C₄²	M₁₆	1	4	yes	—
53	C₃⋊C₁₆	—	4	4	no	—
54	C₂²×C₈	C₂×Dic₃	1	2	yes	—
55	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
56	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
57	C₂×C₄²	—	2	2	no	—
58	C₂×C₄²	C₂×Dic₃	1	2	yes	—
59	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
60	M₃₂	—	6	8	no	—
61	C₂×C₁₆	—	6	4	no	—
62	C₄×C₈	—	2	4	no	—
63	C₄×C₈	—	2	4	no	—
64	C₄×C₈	C₃⋊C₈	1	2	yes	—
65	C₄×C₈	C₃⋊C₈	1	2	yes	—
66	C₄×C₈	Dic₆	1	2	yes	—
67	C₄×C₈	D₁₂	1	2	yes	—
68	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₃×C₄	—	2	8	no	—
70	C₂×C₃×C₄	—	2	4	no	—
71	C₂×C₃×C₄	—	2	4	no	—
72	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
73	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
74	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄	#117	1	2	yes	—
76	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
78	C₃×C₈	—	2	2	no	—
79	C₃×C₈	—	2	4	no	—
80	C₃×C₈	#124	1	2	yes	—
81	C₃×C₈	#125	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	#208	1	2	yes	—
84	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	C₂×C₃⋊C₈	1	4	yes	—
89	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
90	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
91	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
92	C₂×C₈	#235	1	4	yes	—
93	C₄²	—	2	8	no	—
94	C₄²	—	2	2	no	—
95	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
96	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
97	C₄²	#208	1	4	yes	—
98	C₄²	C₄⋊Dic₃	1	2	yes	—
99	C₁₆	—	12	4	no	—
100	C₂²×C₃	—	2	2	no	—
101	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₃	#432	1	2	yes	—
103	C₃×C₄	—	2	8	no	—
104	C₃×C₄	—	2	2	no	—
105	C₃×C₄	#364	1	4	yes	—
106	C₃×C₄	#382	1	1	yes	characteristic
107	C₃×C₄	#423	1	1	yes	characteristic
108	C₃×C₄	#431	1	2	yes	—
109	C₂³	C₂×#208	1	1	yes	characteristic
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
114	C₂×C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
115	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
116	C₂×C₄	C₂×#208	1	2	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
118	C₂×C₄	#926	1	2	yes	—
119	C₈	—	2	2	no	—
120	C₈	—	2	4	no	—
121	C₈	#870	1	2	yes	—
122	C₈	#875	1	2	yes	—
123	C₂×C₃	—	2	2	no	—
124	C₂×C₃	#1831	1	1	yes	characteristic
125	C₂×C₃	C₂×#364	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
128	C₂²	#4926	1	2	yes	—
129	C₄	—	2	8	no	—
130	C₄	—	2	2	no	—
131	C₄	#4463	1	4	yes	—
132	C₄	#4924	1	1	yes	characteristic
133	C₄	#4925	1	2	yes	—
134	C₄	#4936	1	1	yes	characteristic
135	C₃	#12883	1	1	yes	characteristic, complemented, Sylow
136	C₂	—	2	2	no	—
137	C₂	#69392	1	1	yes	characteristic
138	C₂	C₂×#4463	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄²×C₈

C₂

yes

maximal, characteristic

#69743

C₂

yes

maximal

#67658

C₂

yes

maximal

#12883

—

maximal, Sylow

C₂×C₃×C₄×C₈

C₄

yes

—

C₂×C₃×C₄×C₈

C₄

yes

characteristic

C₂×C₃×C₄×C₈

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄³

C₄

yes

characteristic

C₂×#812

C₂²

yes

—

C₄⋊(C₃⋊C₁₆)

—

#2150

—

C₄²×C₈

D₃

yes

characteristic

#1376

—

C₂²×C₃×C₈

C₂×C₄

yes

characteristic

C₂²×C₃×C₈

C₂×C₄

yes

—

C₂²×C₃×C₈

C₂³

yes

characteristic

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂×C₄

yes

—

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₈

yes

—

C₃×C₄×C₈

C₈

yes

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

Q₈

yes

—

#812

—

C₂×C₃⋊C₁₆

—

C₂×M₃₂

—

C₂×C₄×C₈

Dic₃

yes

—

C₂×C₄×C₈

Dic₃

yes

characteristic

C₂×C₄×C₈

C₂×D₃

yes

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

C₄³

Dic₃

yes

characteristic

C₄⋊C₁₆

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

M₁₆

yes

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄⋊C₄

yes

—

C₂×C₃×C₈

C₂×C₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×Q₈

yes

characteristic

C₂×C₃×C₈

#41

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₄⋊C₄

yes

—

C₃×C₄²

C₂×C₈

yes

characteristic

C₃×C₄²

C₂×C₈

yes

characteristic

C₃×C₄²

M₁₆

yes

—

C₃⋊C₁₆

—

C₂²×C₈

C₂×Dic₃

yes

—

C₂²×C₈

C₂×Dic₃

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₃

yes

—

C₂×C₄²

C₂×Dic₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₃⋊C₈

yes

—

C₄×C₈

C₃⋊C₈

yes

—

C₄×C₈

Dic₆

yes

—

C₄×C₈

D₁₂

yes

—

C₂³×C₃

C₂×M₁₆

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₈

yes

—

C₂×C₃×C₄

C₄⋊C₈

yes

—

C₂×C₃×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

#117

yes

—

C₂×C₃×C₄

C₂²×C₈

yes

characteristic

C₂×C₃×C₄

C₂×M₁₆

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

#124

yes

—

C₃×C₈

#125

yes

—

C₂²×C₄

—

C₂²×C₄

#208

yes

—

C₂²×C₄

C₂²×Dic₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊C₈

yes

—

C₂×C₈

C₄⋊Dic₃

yes

—

C₂×C₈

C₂×Dic₆

yes

characteristic

C₂×C₈

C₂×D₁₂

yes

characteristic

C₂×C₈

#235

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₃⋊C₈

yes

characteristic

C₄²

C₂×C₃⋊C₈

yes

characteristic

C₄²

#208

yes

—

C₄²

C₄⋊Dic₃

yes

—

C₁₆

—

100

C₂²×C₃

—

101

C₂²×C₃

C₂×C₄⋊C₈

yes

characteristic

102

C₂²×C₃

#432

yes

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

#364

yes

—

106

C₃×C₄

#382

yes

characteristic

107

C₃×C₄

#423

yes

characteristic

108

C₃×C₄

#431

yes

—

109

C₂³

C₂×#208

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄⋊(C₃⋊C₈)

yes

—

114

C₂×C₄

C₄⋊(C₃⋊C₈)

yes

—

115

C₂×C₄

C₂²×C₃⋊C₈

yes

characteristic

116

C₂×C₄

C₂×#208

yes

—

117

C₂×C₄

C₂×C₄⋊Dic₃

yes

characteristic

118

C₂×C₄

#926

yes

—

119

C₈

—

120

C₈

—

121

C₈

#870

yes

—

122

C₈

#875

yes

—

123

C₂×C₃

—

124

C₂×C₃

#1831

yes

characteristic

125

C₂×C₃

C₂×#364

yes

—

126

C₂²

—

127

C₂²

C₂×C₄⋊(C₃⋊C₈)

yes

characteristic

128

C₂²

#4926

yes

—

129

C₄

—

130

C₄

—

131

C₄

#4463

yes

—

132

C₄

#4924

yes

characteristic

133

C₄

#4925

yes

—

134

C₄

#4936

yes

characteristic

135

C₃

#12883

yes

characteristic, complemented, Sylow

136

C₂

—

137

C₂

#69392

yes

characteristic

138

C₂

C₂×#4463

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×C₄²×C₈	C₂
socle	C₂²×C₃	C₂×C₄⋊C₈
prime radical	C₂³×C₃	C₂×M₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₄²×C₈	D₃
hypocenter	C₃	#12883
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₂×C₈

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₃

Fitting

C₃×C₄²×C₈

C₂

socle

C₂²×C₃

C₂×C₄⋊C₈

prime radical

C₂³×C₃

C₂×M₁₆

cosocle

C₂²×C₃×C₈

C₂³

hypercenter

C₄²×C₈

D₃

hypocenter

C₃

#12883

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
135. C₃	#12883	semidirect

H ≅ G/N

type

135. C₃

#12883

semidirect

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

subgroup	index
C₃×C₄²×C₈	2
#69743	2
#67658	2
#12883	3

subgroup

index

C₃×C₄²×C₈

#69743

#67658

#12883

Subgroups — #10711548

Conjugacy classes of subgroups

Complements

Maximal subgroups