Group #10694692 — subgroups

Subgroups — #10694692

Order 768.

558 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 127 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×#493	C₂	1	1	yes	maximal, characteristic
2	#71759	C₂	1	4	yes	maximal, complemented
3	#69791	C₂	1	1	yes	maximal, characteristic, complemented
4	#69726	C₂	1	1	yes	maximal, characteristic, complemented
5	#9780	—	3	1	no	maximal, Sylow
6	C₂²×C₃×D₈	—	2	2	no	—
7	C₂×C₃×#127	C₂²	1	2	yes	—
8	C₃×#493	—	2	2	no	—
9	C₃×#493	C₄	1	4	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#4925	C₂²	1	2	yes	—
12	#4924	C₂²	1	2	yes	—
13	#4552	—	2	2	no	—
14	#4544	—	2	8	no	—
15	#4454	—	2	2	no	—
16	C₂×#493	D₃	1	1	yes	characteristic
17	#1837	—	3	1	no	—
18	#1545	—	3	4	no	—
19	#1452	—	3	1	no	—
20	C₂²×C₃×D₄	—	2	2	no	—
21	C₂²×C₃×D₄	—	2	4	no	—
22	C₂×C₃×D₈	—	2	8	no	—
23	C₂×C₃×D₈	—	4	8	no	—
24	C₂²×C₃×C₈	D₄	1	2	yes	—
25	C₃×#127	—	2	4	no	—
26	C₃×#127	—	2	2	no	—
27	C₃×#127	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×#208	—	2	2	no	—
30	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
31	C₃×C₄×C₈	D₄	1	2	yes	—
32	#830	—	2	2	no	—
33	C₄⋊(C₃⋊C₈)	—	2	4	no	—
34	C₄⋊(C₃⋊C₈)	D₄	1	4	yes	—
35	C₄×C₃⋊C₈	—	2	2	no	—
36	C₂²×D₈	—	2	2	no	—
37	C₂×#127	C₂×D₃	1	2	yes	—
38	#493	—	2	2	no	—
39	#493	Dic₃	1	4	yes	—
40	#431	—	3	2	no	—
41	#423	—	3	2	no	—
42	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
43	#341	—	6	2	no	—
44	#331	—	6	8	no	—
45	#322	—	6	2	no	—
46	C₂⁴×C₃	—	4	2	no	—
47	C₂×C₃×D₄	—	2	4	no	—
48	C₂×C₃×D₄	—	2	8	no	—
49	C₂×C₃×D₄	—	4	4	no	—
50	C₂×C₃×D₄	—	4	2	no	—
51	C₂×C₃×D₄	—	4	8	no	—
52	C₂×C₃×D₄	—	8	8	no	—
53	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	#41	1	2	yes	—
55	C₃×D₈	—	4	16	no	—
56	C₂×C₃×C₈	—	2	4	no	—
57	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
58	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄²	C₂×D₄	1	2	yes	—
60	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
61	#208	—	4	4	no	—
62	C₂×C₃⋊C₈	—	2	4	no	—
63	C₂×C₃⋊C₈	—	4	2	no	—
64	C₂²×D₄	—	2	4	no	—
65	C₂²×D₄	—	2	2	no	—
66	C₂×D₈	—	2	8	no	—
67	C₂×D₈	—	4	8	no	—
68	C₂×M₁₆	—	6	2	no	—
69	C₂²×C₈	#67	1	2	yes	—
70	#127	—	2	4	no	—
71	#127	—	2	2	no	—
72	#127	C₂×Dic₃	1	4	yes	—
73	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
74	C₄⋊C₈	—	3	4	no	—
75	C₄⋊C₈	—	6	4	no	—
76	C₂²⋊C₈	—	6	2	no	—
77	C₄×C₈	—	6	2	no	—
78	C₄×C₈	C₂×Dic₃	1	2	yes	—
79	C₄×C₈	#67	1	2	yes	—
80	C₂³×C₃	—	4	4	no	—
81	C₂³×C₃	—	8	8	no	—
82	C₂³×C₃	—	8	4	no	—
83	C₂³×C₃	#124	1	1	yes	characteristic
84	C₃×D₄	—	4	8	no	—
85	C₃×D₄	—	8	8	no	—
86	C₃×D₄	—	8	8	no	—
87	C₃×D₄	—	8	4	no	—
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
91	C₂×C₃×C₄	#121	1	2	yes	—
92	C₂×C₃×C₄	#127	1	1	yes	characteristic
93	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
94	C₃×C₈	—	2	4	no	—
95	C₃⋊C₈	—	4	4	no	—
96	C₃⋊C₈	—	8	2	no	—
97	C₂⁴	—	4	2	no	—
98	C₂×D₄	—	2	8	no	—
99	C₂×D₄	—	2	4	no	—
100	C₂×D₄	—	4	8	no	—
101	C₂×D₄	—	4	4	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	—	8	8	no	—
104	C₂²×C₄	#237	1	2	yes	—
105	C₂²×C₄	C₂×#67	1	1	yes	characteristic
106	D₈	—	4	16	no	—
107	M₁₆	—	12	4	no	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	—	6	4	no	—
110	C₂×C₈	—	12	2	no	—
111	C₂×C₈	#217	1	4	yes	—
112	C₄²	D₃×D₄	1	2	yes	—
113	C₄²	C₂²×Dic₃	1	1	yes	characteristic
114	C₄²	C₂×#67	1	1	yes	characteristic
115	C₂²×C₃	—	2	2	no	—
116	C₂²×C₃	—	8	4	no	—
117	C₂²×C₃	—	8	8	no	—
118	C₂²×C₃	—	16	8	no	—
119	C₂²×C₃	#390	1	1	yes	characteristic
120	C₂²×C₃	#490	1	2	yes	—
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	#418	1	2	yes	—
123	C₃×C₄	#472	1	4	yes	—
124	C₃×C₄	#496	1	2	yes	—
125	C₂³	—	4	4	no	—
126	C₂³	—	8	8	no	—
127	C₂³	—	8	4	no	—
128	C₂³	#936	1	1	yes	characteristic
129	D₄	—	4	8	no	—
130	D₄	—	8	8	no	—
131	D₄	—	8	8	no	—
132	D₄	—	8	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	#932	1	4	yes	—
135	C₂×C₄	D₄×Dic₃	1	2	yes	—
136	C₂×C₄	#938	1	2	yes	—
137	C₂×C₄	#940	1	1	yes	characteristic
138	C₂×C₄	C₂×#217	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	12	4	no	—
141	C₈	—	24	2	no	—
142	C₂×C₃	—	2	2	no	—
143	C₂×C₃	—	16	4	no	—
144	C₂×C₃	#1961	1	1	yes	characteristic
145	C₂×C₃	#1978	1	2	yes	—
146	C₂²	—	2	2	no	—
147	C₂²	—	8	4	no	—
148	C₂²	—	8	8	no	—
149	C₂²	—	16	8	no	—
150	C₂²	#5071	1	2	yes	—
151	C₂²	#5221	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	#5074	1	2	yes	—
154	C₄	#5160	1	4	yes	—
155	C₄	#5215	1	2	yes	—
156	C₃	#9780	1	1	yes	characteristic, complemented, Sylow
157	C₂	—	2	2	no	—
158	C₂	—	16	4	no	—
159	C₂	#71765	1	1	yes	characteristic
160	C₂	#72041	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#493

C₂

yes

maximal, characteristic

#71759

C₂

yes

maximal, complemented

#69791

C₂

yes

maximal, characteristic, complemented

#69726

C₂

yes

maximal, characteristic, complemented

#9780

—

maximal, Sylow

C₂²×C₃×D₈

—

C₂×C₃×#127

C₂²

yes

—

C₃×#493

—

C₃×#493

C₄

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#4925

C₂²

yes

—

#4924

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

C₂²×C₃×D₄

—

C₂²×C₃×D₄

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₂²×C₃×C₈

D₄

yes

—

C₃×#127

—

C₃×#127

—

C₃×#127

C₂×C₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×#208

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#830

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

C₄×C₃⋊C₈

—

C₂²×D₈

—

C₂×#127

C₂×D₃

yes

—

#493

—

#493

Dic₃

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

C₂⁴×C₃

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₃×D₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

—

#127

—

#127

—

#127

C₂×Dic₃

yes

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Dic₃

yes

—

C₄×C₈

#67

yes

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

#124

yes

characteristic

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄×D₄

yes

—

C₂×C₃×C₄

#121

yes

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

#237

yes

—

105

C₂²×C₄

C₂×#67

yes

characteristic

106

D₈

—

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#217

yes

—

112

C₄²

D₃×D₄

yes

—

113

C₄²

C₂²×Dic₃

yes

characteristic

114

C₄²

C₂×#67

yes

characteristic

115

C₂²×C₃

—

116

C₂²×C₃

—

117

C₂²×C₃

—

118

C₂²×C₃

—

119

C₂²×C₃

#390

yes

characteristic

120

C₂²×C₃

#490

yes

—

121

C₃×C₄

—

122

C₃×C₄

#418

yes

—

123

C₃×C₄

#472

yes

—

124

C₃×C₄

#496

yes

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

#936

yes

characteristic

129

D₄

—

130

D₄

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

#932

yes

—

135

C₂×C₄

D₄×Dic₃

yes

—

136

C₂×C₄

#938

yes

—

137

C₂×C₄

#940

yes

characteristic

138

C₂×C₄

C₂×#217

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂×C₃

—

143

C₂×C₃

—

144

C₂×C₃

#1961

yes

characteristic

145

C₂×C₃

#1978

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#5071

yes

—

151

C₂²

#5221

yes

characteristic

152

C₄

—

153

C₄

#5074

yes

—

154

C₄

#5160

yes

—

155

C₄

#5215

yes

—

156

C₃

#9780

yes

characteristic, complemented, Sylow

157

C₂

—

158

C₂

—

159

C₂

#71765

yes

characteristic

160

C₂

#72041

yes

—

	subgroup	G/H
center	C₂²	#5221
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting, prime radical	C₂×C₃×#493	C₂
socle	C₂²×C₃	#390
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#493	D₃
hypocenter	C₃	#9780
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂²

#5221

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting, prime radical

C₂×C₃×#493

C₂

socle

C₂²×C₃

#390

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
2. #71759	C₂	semidirect
3. #69791	C₂	semidirect
4. #69726	C₂	semidirect
156. C₃	#9780	semidirect

H ≅ G/N

type

2. #71759

C₂

semidirect

3. #69791

C₂

semidirect

4. #69726

C₂

semidirect

156. C₃

#9780

semidirect

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

subgroup	index
C₂×C₃×#493	2
#71759	2
#69791	2
#69726	2
#9780	3

subgroup

index

C₂×C₃×#493

Subgroups — #10694692

Conjugacy classes of subgroups

Complements

Maximal subgroups