Group #10711550 — subgroups

Subgroups — #10711550

Order 768.

302 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2150	C₂	1	1	yes	maximal, characteristic
2	#69749	C₂	1	1	yes	maximal, characteristic
3	#69743	C₂	1	1	yes	maximal, characteristic
4	#67810	C₂	1	4	yes	maximal
5	#12884	—	3	1	no	maximal, Sylow
6	C₂×C₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×#815	C₄	1	2	yes	—
9	C₂×C₈×Dic₃	C₂²	1	2	yes	—
10	C₂×#812	C₂²	1	2	yes	—
11	#4932	C₄	1	1	yes	characteristic
12	C₂×#803	C₄	1	1	yes	characteristic
13	C₃×C₄⋊C₁₆	—	2	2	no	—
14	C₄⋊(C₃⋊C₁₆)	—	2	2	no	—
15	#2150	—	3	1	no	—
16	#2150	D₃	1	1	yes	characteristic
17	#1726	—	3	1	no	—
18	#1376	—	3	4	no	—
19	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₃	C₂×C₄	1	2	yes	—
23	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
24	C₃×M₃₂	—	2	4	no	—
25	C₂×C₃×C₁₆	—	2	2	no	—
26	C₃×C₄×C₈	D₄	1	2	yes	—
27	C₃×C₄×C₈	Q₈	1	2	yes	—
28	C₃⋊#95	—	2	2	no	—
29	#815	—	2	4	no	—
30	C₈×Dic₃	—	2	2	no	—
31	C₈×Dic₃	—	2	2	no	—
32	#812	—	2	4	no	—
33	C₂×C₃⋊C₁₆	—	2	2	no	—
34	#803	C₈	1	4	yes	—
35	C₂×M₃₂	—	3	2	no	—
36	C₂×M₃₂	C₂×D₃	1	2	yes	—
37	C₂×#97	—	3	1	no	—
38	C₂×#97	—	3	2	no	—
39	C₂×C₄×C₈	—	3	2	no	—
40	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	#376	—	3	1	no	—
42	C₄⋊C₁₆	—	2	2	no	—
43	C₄⋊C₁₆	—	6	2	no	—
44	C₂²×C₃×C₄	M₁₆	1	2	yes	—
45	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₃	—	2	2	no	—
47	C₂×C₃×C₈	—	2	2	no	—
48	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃×C₈	#41	1	2	yes	—
52	C₂×C₃×C₈	#41	1	2	yes	—
53	C₃×C₄²	C₄⋊C₄	1	2	yes	—
54	C₃×C₄²	C₂×C₈	1	2	yes	—
55	C₄×Dic₃	—	2	2	no	—
56	C₄×Dic₃	M₁₆	1	4	yes	—
57	C₂×C₃⋊C₈	—	2	2	no	—
58	C₂×C₃⋊C₈	—	2	2	no	—
59	C₂×C₃⋊C₈	C₂×C₈	1	4	yes	—
60	C₃×C₁₆	—	4	2	no	—
61	C₃⋊C₁₆	—	4	2	no	—
62	C₂²×C₈	—	3	2	no	—
63	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
64	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
65	C₂×C₄²	—	3	2	no	—
66	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
67	M₃₂	—	2	4	no	—
68	M₃₂	—	6	4	no	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	—	6	2	no	—
71	#97	—	3	4	no	—
72	#97	—	6	4	no	—
73	C₄×C₈	—	6	2	no	—
74	C₄×C₈	—	6	2	no	—
75	C₄×C₈	Dic₆	1	2	yes	—
76	C₄×C₈	#67	1	2	yes	—
77	#95	—	6	2	no	—
78	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₃×C₄	—	4	2	no	—
80	C₂×C₃×C₄	C₄⋊C₈	1	4	yes	—
81	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄	#117	1	2	yes	—
83	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
85	C₂×Dic₃	—	2	4	no	—
86	C₂×Dic₃	—	4	4	no	—
87	C₃×C₈	—	2	2	no	—
88	C₃×C₈	#124	1	1	yes	characteristic
89	C₃×C₈	#124	1	1	yes	characteristic
90	C₃×C₈	#125	1	2	yes	—
91	C₃⋊C₈	—	2	4	no	—
92	C₂²×C₄	—	6	2	no	—
93	C₂²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
94	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	3	4	no	—
97	C₂×C₈	—	6	2	no	—
98	C₂×C₈	—	6	2	no	—
99	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
100	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
101	C₂×C₈	#237	1	2	yes	—
102	C₂×C₈	#239	1	2	yes	—
103	C₂×C₈	C₂×#67	1	1	yes	characteristic
104	C₄²	—	3	4	no	—
105	C₄²	—	6	2	no	—
106	C₄²	D₃×C₈	1	2	yes	—
107	C₄²	Dic₃⋊C₄	1	2	yes	—
108	C₁₆	—	4	2	no	—
109	C₁₆	—	12	2	no	—
110	C₂²×C₃	—	2	2	no	—
111	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₃	#432	1	2	yes	—
113	C₃×C₄	—	2	4	no	—
114	C₃×C₄	#382	1	1	yes	characteristic
115	C₃×C₄	#423	1	1	yes	characteristic
116	C₃×C₄	#431	1	2	yes	—
117	Dic₃	—	4	4	no	—
118	C₂³	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	6	4	no	—
121	C₂×C₄	—	12	4	no	—
122	C₂×C₄	Dic₃⋊C₈	1	4	yes	—
123	C₂×C₄	#892	1	2	yes	—
124	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
125	C₂×C₄	D₃×M₁₆	1	2	yes	—
126	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	6	4	no	—
129	C₈	#890	1	2	yes	—
130	C₈	#936	1	1	yes	characteristic
131	C₈	#947	1	1	yes	characteristic
132	C₂×C₃	—	2	2	no	—
133	C₂×C₃	#1831	1	1	yes	characteristic
134	C₂×C₃	#2153	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	#4835	1	2	yes	—
137	C₂²	C₂×Dic₃⋊C₈	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	12	4	no	—
140	C₄	#4833	1	2	yes	—
141	C₄	#4977	1	1	yes	characteristic
142	C₄	#5118	1	1	yes	characteristic
143	C₃	#12884	1	1	yes	characteristic, complemented, Sylow
144	C₂	—	2	2	no	—
145	C₂	#70443	1	1	yes	characteristic
146	C₂	#71423	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2150

C₂

yes

maximal, characteristic

#69749

C₂

yes

maximal, characteristic

#69743

C₂

yes

maximal, characteristic

#67810

C₂

yes

maximal

#12884

—

maximal, Sylow

C₂×C₃×M₃₂

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#815

C₄

yes

—

C₂×C₈×Dic₃

C₂²

yes

—

C₂×#812

C₂²

yes

—

#4932

C₄

yes

characteristic

C₂×#803

C₄

yes

characteristic

C₃×C₄⋊C₁₆

—

C₄⋊(C₃⋊C₁₆)

—

#2150

—

#2150

D₃

yes

characteristic

#1726

—

#1376

—

C₂²×C₃×C₈

C₂×C₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

characteristic

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₂×C₄×Dic₃

C₂×C₄

yes

—

C₂²×C₃⋊C₈

C₂×C₄

yes

—

C₃×M₃₂

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

Q₈

yes

—

C₃⋊#95

—

#815

—

C₈×Dic₃

—

C₈×Dic₃

—

#812

—

C₂×C₃⋊C₁₆

—

#803

C₈

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²×C₃×C₄

M₁₆

yes

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄⋊C₄

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×Q₈

yes

characteristic

C₂×C₃×C₈

#41

yes

—

C₂×C₃×C₈

#41

yes

—

C₃×C₄²

C₄⋊C₄

yes

—

C₃×C₄²

C₂×C₈

yes

—

C₄×Dic₃

—

C₄×Dic₃

M₁₆

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×C₈

yes

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₄×D₃

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₆

yes

—

C₄×C₈

#67

yes

—

#95

—

C₂³×C₃

C₂×M₁₆

yes

characteristic

C₂×C₃×C₄

—

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

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₃×C₈

yes

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

ΓL(1,𝔽₂₅)

yes

—

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₃⋊C₄

yes

—

100

C₂×C₈

C₂×Dic₆

yes

characteristic

101

C₂×C₈

#237

yes

—

102

C₂×C₈

#239

yes

—

103

C₂×C₈

C₂×#67

yes

characteristic

104

C₄²

—

105

C₄²

—

106

C₄²

D₃×C₈

yes

—

107

C₄²

Dic₃⋊C₄

yes

—

108

C₁₆

—

109

C₁₆

—

110

C₂²×C₃

—

111

C₂²×C₃

C₂×C₄⋊C₈

yes

characteristic

112

C₂²×C₃

#432

yes

—

113

C₃×C₄

—

114

C₃×C₄

#382

yes

characteristic

115

C₃×C₄

#423

yes

characteristic

116

C₃×C₄

#431

yes

—

117

Dic₃

—

118

C₂³

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

Dic₃⋊C₈

yes

—

123

C₂×C₄

#892

yes

—

124

C₂×C₄

C₂×D₃×C₈

yes

characteristic

125

C₂×C₄

D₃×M₁₆

yes

—

126

C₂×C₄

C₂×Dic₃⋊C₄

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

#890

yes

—

130

C₈

#936

yes

characteristic

131

C₈

#947

yes

characteristic

132

C₂×C₃

—

133

C₂×C₃

#1831

yes

characteristic

134

C₂×C₃

#2153

yes

—

135

C₂²

—

136

C₂²

#4835

yes

—

137

C₂²

C₂×Dic₃⋊C₈

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

#4833

yes

—

141

C₄

#4977

yes

characteristic

142

C₄

#5118

yes

characteristic

143

C₃

#12884

yes

characteristic, complemented, Sylow

144

C₂

—

145

C₂

#70443

yes

characteristic

146

C₂

#71423

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#2150	C₂
socle	C₂²×C₃	C₂×C₄⋊C₈
prime radical	C₂³×C₃	C₂×M₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#2150	D₃
hypocenter	C₃	#12884
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₃

Fitting

C₃×#2150

C₂

socle

C₂²×C₃

C₂×C₄⋊C₈

prime radical

C₂³×C₃

C₂×M₁₆

cosocle

C₂²×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
143. C₃	#12884	semidirect

H ≅ G/N

type

143. C₃

#12884

semidirect

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

subgroup	index
C₃×#2150	2
#69749	2
#69743	2
#67810	2
#12884	3

subgroup

index

Subgroups — #10711550

Conjugacy classes of subgroups

Complements

Maximal subgroups