Group #10709961 — subgroups

Subgroups — #10709961

Order 768.

318 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 91 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	#69061	C₂	1	2	yes	maximal, complemented
2	#69058	C₂	1	2	yes	maximal, complemented
3	#68063	C₂	1	1	yes	maximal, characteristic, complemented
4	#68062	C₂	1	1	yes	maximal, characteristic
5	C₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	#12529	—	3	1	no	maximal, Sylow
7	C₂×#800	—	2	4	no	—
8	#4699	C₂²	1	1	yes	characteristic
9	#4695	—	2	1	no	—
10	#4695	—	2	1	no	—
11	#4695	C₂²	1	2	yes	—
12	#4682	C₂²	1	1	yes	characteristic
13	#4681	—	2	1	no	—
14	#4680	—	2	1	no	—
15	C₃×C₄×C₁₆	C₂²	1	2	yes	—
16	C₃×C₈²	C₂²	1	1	yes	characteristic
17	#4509	—	2	4	no	—
18	#2248	—	3	2	no	—
19	#2240	—	3	2	no	—
20	#1712	—	3	1	no	—
21	#1707	—	3	1	no	—
22	#1312	D₃	1	1	yes	characteristic, complemented
23	C₂×Dic₁₂	—	2	2	no	—
24	C₂×D₂₄	—	2	2	no	—
25	C₂×D₂₄	—	2	2	no	—
26	C₂×D₂₄	—	4	2	no	—
27	C₂×D₂₄	—	4	2	no	—
28	#874	—	2	2	no	—
29	#869	—	2	1	no	—
30	#869	—	2	1	no	—
31	C₂×C₃×C₁₆	D₄	1	4	yes	—
32	C₃×C₄×C₈	D₄	1	1	yes	characteristic
33	C₃×C₄×C₈	D₄	1	1	yes	characteristic
34	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
35	#818	—	2	2	no	—
36	#817	—	4	2	no	—
37	#816	—	4	2	no	—
38	#800	—	4	8	no	—
39	C₂×SD₃₂	—	6	4	no	—
40	#500	—	3	1	no	—
41	#498	—	6	1	no	—
42	#494	—	3	1	no	—
43	#493	—	3	2	no	—
44	#493	—	6	1	no	—
45	#493	—	6	1	no	—
46	#487	—	6	1	no	—
47	#357	—	6	4	no	—
48	C₄×C₁₆	C₂×D₃	1	2	yes	—
49	C₈²	C₂×D₃	1	1	yes	characteristic
50	C₂×D₁₂	—	4	2	no	—
51	C₂×D₁₂	—	4	2	no	—
52	C₂×D₁₂	—	4	2	no	—
53	C₂×Dic₆	—	4	1	no	—
54	C₂×Dic₆	—	4	1	no	—
55	C₂×C₃×C₈	D₈	1	2	yes	—
56	C₂×C₃×C₈	D₈	1	2	yes	—
57	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
58	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₄⋊Dic₃	—	4	1	no	—
60	C₄⋊Dic₃	—	4	1	no	—
61	C₄⋊Dic₃	—	4	1	no	—
62	C₄⋊Dic₃	—	4	1	no	—
63	Dic₁₂	—	4	4	no	—
64	D₂₄	—	4	4	no	—
65	D₂₄	—	4	4	no	—
66	D₂₄	—	8	4	no	—
67	D₂₄	—	8	4	no	—
68	C₃×C₁₆	—	2	4	no	—
69	C₂×Q₁₆	—	6	2	no	—
70	C₂×D₈	—	6	2	no	—
71	C₂×D₈	—	6	2	no	—
72	C₂×D₈	—	12	2	no	—
73	C₂×D₈	—	12	2	no	—
74	C₄⋊Q₈	—	6	1	no	—
75	C₄⋊Q₈	—	6	1	no	—
76	#127	—	6	2	no	—
77	SD₃₂	—	12	8	no	—
78	C₂×C₁₆	D₁₂	1	4	yes	—
79	#107	—	6	2	no	—
80	#106	—	12	2	no	—
81	#103	—	12	2	no	—
82	C₄×C₈	D₁₂	1	1	yes	characteristic
83	C₄×C₈	D₁₂	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
85	C₂²×D₃	—	8	2	no	—
86	C₂×C₃×C₄	#127	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
89	C₂×Dic₃	—	8	1	no	—
90	C₂×Dic₃	—	8	1	no	—
91	D₁₂	—	8	2	no	—
92	D₁₂	—	8	2	no	—
93	D₁₂	—	8	4	no	—
94	D₁₂	—	8	4	no	—
95	Dic₆	—	8	2	no	—
96	Dic₆	—	8	2	no	—
97	C₃×C₈	—	2	2	no	—
98	C₃×C₈	SD₃₂	1	4	yes	complemented
99	C₃×C₈	AGL(1,ℤ/8)	1	4	yes	—
100	C₂×Q₈	—	12	1	no	—
101	C₂×Q₈	—	12	1	no	—
102	C₂×D₄	—	12	2	no	—
103	C₂×D₄	—	12	2	no	—
104	C₂×D₄	—	12	2	no	—
105	Q₁₆	—	12	4	no	—
106	D₈	—	12	4	no	—
107	D₈	—	12	4	no	—
108	D₈	—	24	4	no	—
109	D₈	—	24	4	no	—
110	C₂×C₈	D₂₄	1	2	yes	—
111	C₂×C₈	D₂₄	1	2	yes	—
112	C₂×C₈	C₂×D₁₂	1	2	yes	—
113	C₄⋊C₄	—	12	1	no	—
114	C₄⋊C₄	—	12	1	no	—
115	C₄⋊C₄	—	12	1	no	—
116	C₄⋊C₄	—	12	1	no	—
117	C₄²	C₂×D₁₂	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	C₂²×C₃	#493	1	1	yes	characteristic
120	C₂×D₃	—	16	2	no	—
121	C₂×D₃	—	16	2	no	—
122	C₂×D₃	—	16	2	no	—
123	C₃×C₄	#496	1	2	yes	—
124	C₃×C₄	C₂×SD₃₂	1	2	yes	—
125	C₃×C₄	#509	1	2	yes	—
126	Dic₃	—	16	1	no	—
127	Dic₃	—	16	1	no	—
128	C₂³	—	24	2	no	—
129	Q₈	—	24	2	no	—
130	Q₈	—	24	2	no	—
131	D₄	—	24	4	no	—
132	D₄	—	24	4	no	—
133	D₄	—	24	2	no	—
134	D₄	—	24	2	no	—
135	C₂×C₄	—	24	1	no	—
136	C₂×C₄	—	24	1	no	—
137	C₂×C₄	#874	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
139	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	#800	1	4	yes	complemented
142	C₈	#908	1	4	yes	—
143	C₂×C₃	#1721	1	1	yes	characteristic
144	C₂×C₃	#2248	1	1	yes	characteristic
145	C₂×C₃	#2250	1	1	yes	characteristic
146	D₃	—	32	2	no	—
147	C₂²	—	48	2	no	—
148	C₂²	—	48	2	no	—
149	C₂²	—	48	2	no	—
150	C₂²	#4695	1	1	yes	characteristic
151	C₄	—	48	1	no	—
152	C₄	—	48	1	no	—
153	C₄	#4712	1	2	yes	—
154	C₄	C₂×#800	1	2	yes	—
155	C₄	#4908	1	2	yes	—
156	C₃	#12529	1	1	yes	characteristic, complemented, Sylow
157	C₂	—	96	2	no	—
158	C₂	#68094	1	1	yes	characteristic
159	C₂	#69061	1	1	yes	characteristic
160	C₂	#69081	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#69061

C₂

yes

maximal, complemented

#69058

C₂

yes

maximal, complemented

#68063

C₂

yes

maximal, characteristic, complemented

#68062

C₂

yes

maximal, characteristic

C₃×#1312

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

—

C₂²

yes

—

#4682

C₂²

yes

characteristic

#4681

—

#4680

—

C₃×C₄×C₁₆

C₂²

yes

—

C₃×C₈²

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂×Dic₁₂

—

C₂×D₂₄

—

C₂×D₂₄

—

C₂×D₂₄

—

C₂×D₂₄

—

#874

—

#869

—

#869

—

C₂×C₃×C₁₆

D₄

yes

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

C₂³

yes

—

—

—

—

C₂×SD₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₃

yes

—

C₈²

C₂×D₃

yes

characteristic

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

D₂₄

—

D₂₄

—

D₂₄

—

D₂₄

—

C₃×C₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

D₁₂

yes

—

#107

—

#106

—

#103

—

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

C₂²×D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

SD₃₂

yes

complemented

C₃×C₈

AGL(1,ℤ/8)

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

Q₁₆

—

106

D₈

—

107

D₈

—

108

D₈

—

109

D₈

—

110

C₂×C₈

D₂₄

yes

—

111

C₂×C₈

D₂₄

yes

—

112

C₂×C₈

C₂×D₁₂

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

C₂×D₁₂

yes

characteristic

118

C₁₆

—

119

C₂²×C₃

#493

yes

characteristic

120

C₂×D₃

—

121

C₂×D₃

—

122

C₂×D₃

—

123

C₃×C₄

#496

yes

—

124

C₃×C₄

C₂×SD₃₂

yes

—

125

C₃×C₄

#509

yes

—

126

Dic₃

—

127

Dic₃

—

128

C₂³

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#874

yes

characteristic

138

C₂×C₄

C₂×D₂₄

yes

characteristic

139

C₂×C₄

C₂×D₂₄

yes

characteristic

140

C₈

—

141

C₈

#800

yes

complemented

142

C₈

#908

yes

—

143

C₂×C₃

#1721

yes

characteristic

144

C₂×C₃

#2248

yes

characteristic

145

C₂×C₃

#2250

yes

characteristic

146

D₃

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#4695

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

#4712

yes

—

154

C₄

C₂×#800

yes

—

155

C₄

#4908

yes

—

156

C₃

#12529

yes

characteristic, complemented, Sylow

157

C₂

—

158

C₂

#68094

yes

characteristic

159

C₂

#69061

yes

characteristic

160

C₂

#69081

yes

characteristic

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1312	C₂
socle	C₂²×C₃	#493
prime radical	#68062	C₂
hypercenter	#1312	D₃
hypocenter	C₃	#12529
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#4695

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #69061	C₂	semidirect
2. #69058	C₂	semidirect
3. #68063	C₂	semidirect
5. C₃×#1312	C₂	semidirect
22. #1312	D₃	semidirect
98. C₃×C₈	SD₃₂	semidirect
141. C₈	#800	semidirect
156. C₃	#12529	semidirect

H ≅ G/N

type

1. #69061

C₂

semidirect

2. #69058

C₂

semidirect

3. #68063

C₂

semidirect

5. C₃×#1312

C₂

semidirect

22. #1312

D₃

semidirect

98. C₃×C₈

SD₃₂

semidirect

141. C₈

#800

semidirect

156. C₃

#12529

semidirect

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

subgroup	index
#69061	2
#69058	2
#68063	2
#68062	2
C₃×#1312	2
#12529	3

subgroup

index

Subgroups — #10709961

Conjugacy classes of subgroups

Complements

Maximal subgroups