Group #10711536 — subgroups

Subgroups — #10711536

Order 768.

322 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#69749	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#67804	C₂	1	4	yes	maximal
5	#12880	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₁₆	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	C₂²×C₃⋊C₁₆	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	C₂×C₄⋊C₁₆	—	3	1	no	—
16	C₂×C₄⋊C₁₆	D₃	1	1	yes	characteristic
17	#1726	—	3	1	no	—
18	#1374	—	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	C₂×C₃×C₁₆	—	2	2	no	—
25	C₂×C₃×C₁₆	—	2	4	no	—
26	C₃×C₄×C₈	D₄	1	2	yes	—
27	C₃×C₄×C₈	Q₈	1	2	yes	—
28	C₃⋊#95	C₈	1	2	yes	—
29	C₃⋊#95	C₈	1	2	yes	—
30	#815	—	2	4	no	—
31	C₈×Dic₃	—	2	2	no	—
32	C₈×Dic₃	—	2	2	no	—
33	C₂×C₃⋊C₁₆	—	2	2	no	—
34	C₂×C₃⋊C₁₆	—	2	4	no	—
35	#803	—	2	2	no	—
36	C₂²×C₁₆	—	3	2	no	—
37	C₂²×C₁₆	C₂×D₃	1	2	yes	—
38	C₂×#97	—	3	2	no	—
39	C₂×#97	—	3	1	no	—
40	C₂×C₄×C₈	—	3	2	no	—
41	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
42	#376	—	3	1	no	—
43	C₄⋊C₁₆	—	2	2	no	—
44	C₄⋊C₁₆	—	6	2	no	—
45	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₃	C₂×C₈	1	4	yes	—
49	C₂×C₃×C₈	—	2	2	no	—
50	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₃×C₈	#41	1	2	yes	—
54	C₂×C₃×C₈	#41	1	2	yes	—
55	C₃×C₄²	C₄⋊C₄	1	2	yes	—
56	C₃×C₄²	M₁₆	1	2	yes	—
57	C₄×Dic₃	—	2	2	no	—
58	C₄×Dic₃	—	2	2	no	—
59	C₂×C₃⋊C₈	—	2	2	no	—
60	C₂×C₃⋊C₈	—	2	2	no	—
61	C₂×C₃⋊C₈	M₁₆	1	4	yes	—
62	C₃×C₁₆	—	4	2	no	—
63	C₃⋊C₁₆	—	4	2	no	—
64	C₂²×C₈	—	3	2	no	—
65	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
66	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₄²	—	3	2	no	—
68	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
69	C₂×C₁₆	—	2	4	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	6	4	no	—
72	C₂×C₁₆	—	6	2	no	—
73	#97	—	6	4	no	—
74	#97	—	6	2	no	—
75	C₄×C₈	—	6	2	no	—
76	C₄×C₈	—	6	2	no	—
77	C₄×C₈	Dic₆	1	2	yes	—
78	C₄×C₈	#67	1	2	yes	—
79	#95	—	3	2	no	—
80	#95	—	3	2	no	—
81	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
85	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
86	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄	#117	1	2	yes	—
88	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
90	C₂×Dic₃	—	2	4	no	—
91	C₂×Dic₃	—	2	4	no	—
92	C₂×Dic₃	—	2	4	no	—
93	C₃×C₈	—	2	2	no	—
94	C₃×C₈	#124	1	1	yes	characteristic
95	C₃×C₈	#124	1	1	yes	characteristic
96	C₃×C₈	#125	1	2	yes	—
97	C₃⋊C₈	—	2	4	no	—
98	C₂²×C₄	—	3	4	no	—
99	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
100	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
101	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	3	4	no	—
104	C₂×C₈	—	6	2	no	—
105	C₂×C₈	—	6	2	no	—
106	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
107	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
108	C₂×C₈	#237	1	2	yes	—
109	C₂×C₈	#239	1	2	yes	—
110	C₂×C₈	C₂×#67	1	1	yes	characteristic
111	C₄²	—	6	2	no	—
112	C₄²	—	6	2	no	—
113	C₄²	ΓL(1,𝔽₂₅)	1	2	yes	—
114	C₄²	Dic₃⋊C₄	1	2	yes	—
115	C₁₆	—	4	2	no	—
116	C₁₆	—	12	2	no	—
117	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₃	#431	1	2	yes	—
119	C₂²×C₃	#504	1	2	yes	—
120	C₂²×C₃	#504	1	2	yes	—
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	—	2	2	no	—
123	C₃×C₄	#382	1	1	yes	characteristic
124	C₃×C₄	#423	1	1	yes	characteristic
125	C₃×C₄	#432	1	2	yes	—
126	Dic₃	—	4	4	no	—
127	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	6	4	no	—
131	C₂×C₄	—	6	4	no	—
132	C₂×C₄	—	6	4	no	—
133	C₂×C₄	Dic₃⋊C₈	1	2	yes	—
134	C₂×C₄	Dic₃⋊C₈	1	2	yes	—
135	C₂×C₄	#892	1	2	yes	—
136	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
137	C₂×C₄	D₃×M₁₆	1	2	yes	—
138	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	6	4	no	—
141	C₈	#890	1	2	yes	—
142	C₈	#936	1	1	yes	characteristic
143	C₈	#947	1	1	yes	characteristic
144	C₂×C₃	#1831	1	1	yes	characteristic
145	C₂×C₃	#2151	1	1	yes	characteristic
146	C₂×C₃	#2151	1	1	yes	characteristic
147	C₂×C₃	#2164	1	4	yes	—
148	C₂²	#4833	1	2	yes	—
149	C₂²	#4901	1	2	yes	—
150	C₂²	#4907	1	2	yes	—
151	C₂²	C₂×Dic₃⋊C₈	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	12	4	no	—
155	C₄	#4835	1	2	yes	—
156	C₄	#4977	1	1	yes	characteristic
157	C₄	#5118	1	1	yes	characteristic
158	C₃	#12880	1	1	yes	characteristic, complemented, Sylow
159	C₂	#69276	1	4	yes	—
160	C₂	#70443	1	1	yes	characteristic
161	C₂	#71401	1	1	yes	characteristic
162	C₂	#71421	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#69749

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊C₁₆)

C₂

yes

maximal, characteristic

#67804

C₂

yes

maximal

#12880

—

maximal, Sylow

C₂²×C₃×C₁₆

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#815

C₄

yes

—

C₂×C₈×Dic₃

C₂²

yes

—

C₂²×C₃⋊C₁₆

C₂²

yes

—

#4932

C₄

yes

characteristic

C₂×#803

C₄

yes

characteristic

C₃×C₄⋊C₁₆

—

C₄⋊(C₃⋊C₁₆)

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

D₃

yes

characteristic

#1726

—

#1374

—

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

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

Q₈

yes

—

C₃⋊#95

C₈

yes

—

C₃⋊#95

C₈

yes

—

#815

—

C₈×Dic₃

—

C₈×Dic₃

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

#803

—

C₂²×C₁₆

—

C₂²×C₁₆

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

C₂×C₈

yes

characteristic

C₂²×C₃×C₄

C₂×C₈

yes

characteristic

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×Dic₃

C₂×C₈

yes

—

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

M₁₆

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

M₁₆

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

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₆

yes

—

C₄×C₈

#67

yes

—

#95

—

#95

—

C₂³×C₃

C₂²×C₈

yes

characteristic

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

C₂×M₁₆

yes

characteristic

C₂×C₃×C₄

C₂×M₁₆

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₃×C₈

yes

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

D₃×C₈

yes

characteristic

100

C₂²×C₄

D₃×C₈

yes

characteristic

101

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

Dic₃⋊C₄

yes

—

107

C₂×C₈

C₂×Dic₆

yes

characteristic

108

C₂×C₈

#237

yes

—

109

C₂×C₈

#239

yes

—

110

C₂×C₈

C₂×#67

yes

characteristic

111

C₄²

—

112

C₄²

—

113

C₄²

ΓL(1,𝔽₂₅)

yes

—

114

C₄²

Dic₃⋊C₄

yes

—

115

C₁₆

—

116

C₁₆

—

117

C₂²×C₃

C₂×C₄⋊C₈

yes

characteristic

118

C₂²×C₃

#431

yes

—

119

C₂²×C₃

#504

yes

—

120

C₂²×C₃

#504

yes

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

#382

yes

characteristic

124

C₃×C₄

#423

yes

characteristic

125

C₃×C₄

#432

yes

—

126

Dic₃

—

127

C₂³

C₂×D₃×C₈

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

Dic₃⋊C₈

yes

—

134

C₂×C₄

Dic₃⋊C₈

yes

—

135

C₂×C₄

#892

yes

—

136

C₂×C₄

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

137

C₂×C₄

D₃×M₁₆

yes

—

138

C₂×C₄

C₂×Dic₃⋊C₄

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

#890

yes

—

142

C₈

#936

yes

characteristic

143

C₈

#947

yes

characteristic

144

C₂×C₃

#1831

yes

characteristic

145

C₂×C₃

#2151

yes

characteristic

146

C₂×C₃

#2151

yes

characteristic

147

C₂×C₃

#2164

yes

—

148

C₂²

#4833

yes

—

149

C₂²

#4901

yes

—

150

C₂²

#4907

yes

—

151

C₂²

C₂×Dic₃⋊C₈

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#4835

yes

—

156

C₄

#4977

yes

characteristic

157

C₄

#5118

yes

characteristic

158

C₃

#12880

yes

characteristic, complemented, Sylow

159

C₂

#69276

yes

—

160

C₂

#70443

yes

characteristic

161

C₂

#71401

yes

characteristic

162

C₂

#71421

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₃
commutator, socle, prime radical	C₂³×C₃	C₂²×C₈
Fitting	C₂×C₃×C₄⋊C₁₆	C₂
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₃
hypocenter	C₃	#12880
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₃

commutator, socle, prime radical

C₂³×C₃

C₂²×C₈

Fitting

C₂×C₃×C₄⋊C₁₆

C₂

cosocle

C₂²×C₃×C₈

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
158. C₃	#12880	semidirect

H ≅ G/N

type

158. C₃

#12880

semidirect

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

subgroup	index
C₂×C₃×C₄⋊C₁₆	2
#69749	2
C₂×C₄⋊(C₃⋊C₁₆)	2
#67804	2
#12880	3

subgroup

index

C₂×C₃×C₄⋊C₁₆

#69749

C₂×C₄⋊(C₃⋊C₁₆)

#67804

#12880

Subgroups — #10711536

Conjugacy classes of subgroups

Complements

Maximal subgroups