Group #10711553 — subgroups

Subgroups — #10711553

Order 768.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2151	C₂	1	1	yes	maximal, characteristic
2	#69895	C₂	1	1	yes	maximal, characteristic
3	#69885	C₂	1	1	yes	maximal, characteristic
4	#67814	C₂	1	2	yes	maximal
5	#67805	C₂	1	2	yes	maximal
6	#12885	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₃×#405	C₂²	1	1	yes	characteristic
10	#5101	C₂²	1	2	yes	—
11	C₂×#812	C₂²	1	2	yes	—
12	C₃×C₄⋊C₁₆	—	2	2	no	—
13	C₃⋊#338	C₄	1	1	yes	characteristic
14	#4557	C₄	1	1	yes	characteristic
15	#4531	C₄	1	2	yes	—
16	#4463	—	2	2	no	—
17	#2153	—	3	1	no	—
18	#2151	D₃	1	1	yes	characteristic
19	#1747	—	3	1	no	—
20	#1379	—	3	2	no	—
21	#1378	—	3	2	no	—
22	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₃×#117	C₂×C₄	1	1	yes	characteristic
25	#926	C₂×C₄	1	2	yes	—
26	C₂×#208	C₂×C₄	1	2	yes	—
27	C₃×M₃₂	—	2	2	no	—
28	C₂×C₃×C₁₆	—	2	1	no	—
29	C₂×C₃×C₁₆	—	2	1	no	—
30	C₂×C₃×C₁₆	—	2	2	no	—
31	C₃×#97	Q₈	1	2	yes	—
32	C₃×C₄×C₈	D₄	1	2	yes	—
33	#815	—	2	2	no	—
34	C₈×Dic₃	—	2	2	no	—
35	#812	—	2	2	no	—
36	#812	—	2	2	no	—
37	C₂×C₃⋊C₁₆	—	2	2	no	—
38	C₂×M₃₂	—	3	2	no	—
39	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
41	#405	—	3	2	no	—
42	#405	C₂×D₃	1	1	yes	characteristic
43	#364	—	6	2	no	—
44	C₄⋊C₁₆	—	2	2	no	—
45	#338	—	3	1	no	—
46	H(3,ℤ/4)	—	3	1	no	—
47	H(3,ℤ/4)	—	3	2	no	—
48	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₃×M₁₆	—	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×C₄⋊C₄	—	2	1	no	—
56	C₃×C₂²⋊C₄	—	2	1	no	—
57	C₃×C₄²	C₄⋊C₄	1	2	yes	—
58	#217	—	2	2	no	—
59	C₄⋊Dic₃	—	2	2	no	—
60	C₄×Dic₃	—	2	2	no	—
61	#208	—	2	2	no	—
62	#208	—	2	2	no	—
63	C₂×C₃⋊C₈	—	2	2	no	—
64	C₃×C₁₆	—	4	1	no	—
65	C₃×C₁₆	—	4	1	no	—
66	C₃⋊C₁₆	—	4	2	no	—
67	C₂×M₁₆	—	3	2	no	—
68	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
69	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
70	#117	—	3	2	no	—
71	#117	C₄×D₃	1	1	yes	characteristic
72	M₃₂	—	2	2	no	—
73	M₃₂	—	6	2	no	—
74	M₃₂	—	6	2	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	6	2	no	—
79	#97	—	6	2	no	—
80	#97	Dic₆	1	2	yes	—
81	C₄×C₈	—	6	2	no	—
82	C₄×C₈	#67	1	2	yes	—
83	C₂³×C₃	#131	1	1	yes	characteristic
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄	#117	1	2	yes	—
87	C₂×C₃×C₄	#131	1	1	yes	characteristic
88	C₂×C₃×C₄	#131	1	2	yes	—
89	C₂×Dic₃	—	4	2	no	—
90	C₃×C₈	—	2	2	no	—
91	C₃×C₈	#124	1	1	yes	characteristic
92	C₃×C₈	#124	1	1	yes	characteristic
93	C₃×C₈	#125	1	2	yes	—
94	C₃⋊C₈	—	4	2	no	—
95	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	6	2	no	—
98	M₁₆	—	6	2	no	—
99	C₂×C₈	—	6	2	no	—
100	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
101	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
102	C₂×C₈	#237	1	2	yes	—
103	C₂×C₈	#239	1	2	yes	—
104	C₂×C₈	C₂×#67	1	1	yes	characteristic
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	6	2	no	—
107	C₂²⋊C₄	—	2	1	no	—
108	C₂²⋊C₄	—	6	2	no	—
109	C₄²	—	6	2	no	—
110	C₄²	Dic₃⋊C₄	1	2	yes	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	4	1	no	—
113	C₁₆	—	12	2	no	—
114	C₂²×C₃	—	4	1	no	—
115	C₂²×C₃	#424	1	1	yes	characteristic
116	C₂²×C₃	#433	1	2	yes	—
117	C₃×C₄	—	4	2	no	—
118	C₃×C₄	#382	1	1	yes	characteristic
119	C₃×C₄	#424	1	1	yes	characteristic
120	C₃×C₄	#433	1	2	yes	—
121	Dic₃	—	8	2	no	—
122	C₂³	#901	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	12	2	no	—
125	C₂×C₄	#892	1	2	yes	—
126	C₂×C₄	#901	1	1	yes	characteristic
127	C₂×C₄	#907	1	2	yes	—
128	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	12	2	no	—
131	C₈	#890	1	2	yes	—
132	C₈	#936	1	1	yes	characteristic
133	C₈	#947	1	1	yes	characteristic
134	C₂×C₃	—	2	2	no	—
135	C₂×C₃	—	2	1	no	—
136	C₂×C₃	#1832	1	1	yes	characteristic
137	C₂²	—	4	1	no	—
138	C₂²	#4840	1	2	yes	—
139	C₂²	#5102	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	24	2	no	—
142	C₄	#4840	1	2	yes	—
143	C₄	#4977	1	1	yes	characteristic
144	C₄	#5119	1	1	yes	characteristic
145	C₃	#12885	1	1	yes	characteristic, complemented, Sylow
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#70444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2151

C₂

yes

maximal, characteristic

#69895

C₂

yes

maximal, characteristic

#69885

C₂

yes

maximal, characteristic

#67814

C₂

yes

maximal

#67805

C₂

yes

maximal

#12885

—

maximal, Sylow

C₂×C₃×M₃₂

C₂²

yes

characteristic

C₂²×C₃×C₁₆

C₂²

yes

characteristic

C₃×#405

C₂²

yes

characteristic

#5101

C₂²

yes

—

C₂×#812

C₂²

yes

—

C₃×C₄⋊C₁₆

—

C₃⋊#338

C₄

yes

characteristic

#4557

C₄

yes

characteristic

#4531

C₄

yes

—

#4463

—

#2153

—

#2151

D₃

yes

—

—

—

C₂×C₃×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

characteristic

C₃×#117

C₂×C₄

yes

characteristic

#926

C₂×C₄

yes

—

C₂×#208

C₂×C₄

yes

—

C₃×M₃₂

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₃×#97

Q₈

yes

—

C₃×C₄×C₈

D₄

yes

—

#815

—

C₈×Dic₃

—

#812

—

#812

—

C₂×C₃⋊C₁₆

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₃

yes

characteristic

C₂²×C₁₆

C₂×D₃

yes

characteristic

#405

—

#405

C₂×D₃

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₃×M₁₆

—

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×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

C₄⋊C₄

yes

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₃

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

characteristic

#117

—

#117

C₄×D₃

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

Dic₆

yes

—

C₄×C₈

—

C₄×C₈

#67

yes

—

C₂³×C₃

#131

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

#117

yes

—

C₂×C₃×C₄

#131

yes

characteristic

C₂×C₃×C₄

#131

yes

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₃×C₈

yes

—

C₃⋊C₈

—

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

Dic₃⋊C₄

yes

—

101

C₂×C₈

C₂×Dic₆

yes

characteristic

102

C₂×C₈

#237

yes

—

103

C₂×C₈

#239

yes

—

104

C₂×C₈

C₂×#67

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

Dic₃⋊C₄

yes

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₂²×C₃

—

115

C₂²×C₃

#424

yes

characteristic

116

C₂²×C₃

#433

yes

—

117

C₃×C₄

—

118

C₃×C₄

#382

yes

characteristic

119

C₃×C₄

#424

yes

characteristic

120

C₃×C₄

#433

yes

—

121

Dic₃

—

122

C₂³

#901

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#892

yes

—

126

C₂×C₄

#901

yes

characteristic

127

C₂×C₄

#907

yes

—

128

C₂×C₄

C₂×Dic₃⋊C₄

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

#890

yes

—

132

C₈

#936

yes

characteristic

133

C₈

#947

yes

characteristic

134

C₂×C₃

—

135

C₂×C₃

—

136

C₂×C₃

#1832

yes

characteristic

137

C₂²

—

138

C₂²

#4840

yes

—

139

C₂²

#5102

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#4840

yes

—

143

C₄

#4977

yes

characteristic

144

C₄

#5119

yes

characteristic

145

C₃

#12885

yes

characteristic, complemented, Sylow

146

C₂

—

147

C₂

—

148

C₂

#70444

yes

characteristic

	subgroup	G/H
center	C₈	#947
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#2151	C₂
socle	C₂×C₃	#1832
prime radical	C₂³×C₃	#131
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#2151	D₃
hypocenter	C₃	#12885
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#947

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₃

C₂

C₂×C₃

C₂³×C₃

C₂²×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

N	H ≅ G/N	type
145. C₃	#12885	semidirect

H ≅ G/N

type

145. C₃

#12885

semidirect

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

subgroup	index
C₃×#2151	2
#69895	2
#69885	2
#67814	2
#67805	2
#12885	3

subgroup

index

Subgroups — #10711553

Conjugacy classes of subgroups

Complements

Maximal subgroups