Group #10709484 — subgroups

Subgroups — #10709484

Order 768.

218 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 71 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	#69295	C₂	1	2	yes	maximal
2	#68065	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₁₂	C₂	1	1	yes	maximal, characteristic
4	C₃×#1370	C₂	1	1	yes	maximal, characteristic
5	#67564	C₂	1	2	yes	maximal
6	#12446	—	3	1	no	maximal, Sylow
7	#4699	—	2	1	no	—
8	C₄×Dic₁₂	C₂²	1	1	yes	characteristic
9	#4682	—	2	1	no	—
10	#4682	C₂²	1	2	yes	—
11	C₈×Dic₆	—	2	1	no	—
12	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₃×C₈²	C₂²	1	1	yes	characteristic
14	#4505	—	2	2	no	—
15	#4503	—	2	2	no	—
16	#4458	C₂²	1	1	yes	characteristic
17	#4456	—	2	1	no	—
18	#2225	—	3	2	no	—
19	#1713	—	3	1	no	—
20	C₈×Q₁₆	—	3	1	no	—
21	#1370	D₃	1	1	yes	characteristic
22	#1364	—	3	2	no	—
23	C₂×Dic₁₂	—	4	1	no	—
24	C₂×Dic₁₂	—	4	1	no	—
25	C₂×Dic₁₂	D₄	1	1	yes	characteristic
26	#869	—	2	2	no	—
27	C₄×Dic₆	—	2	1	no	—
28	C₂×C₃×C₁₆	—	2	2	no	—
29	C₃×C₄×C₈	D₄	1	1	yes	characteristic
30	C₃×C₄×C₈	D₄	1	1	yes	characteristic
31	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
32	#818	—	2	2	no	—
33	#818	—	2	2	no	—
34	#818	—	4	2	no	—
35	#818	D₄	1	1	yes	characteristic
36	#816	—	2	1	no	—
37	Dic₃⋊C₈	—	4	1	no	—
38	C₈×Dic₃	—	4	1	no	—
39	C₄⋊(C₃⋊C₈)	—	2	1	no	—
40	#500	—	3	2	no	—
41	#500	—	6	1	no	—
42	#494	—	6	1	no	—
43	C₈×Q₈	—	6	1	no	—
44	C₄×Q₁₆	—	3	1	no	—
45	#367	—	6	2	no	—
46	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
47	#358	—	6	2	no	—
48	#335	—	3	1	no	—
49	Q₈⋊C₈	—	6	1	no	—
50	C₈²	C₂×D₃	1	1	yes	characteristic
51	C₂×Dic₆	—	2	1	no	—
52	C₂×Dic₆	—	4	2	no	—
53	C₂×C₃×C₈	—	2	1	no	—
54	C₂×C₃×C₈	D₈	1	1	yes	characteristic
55	C₂×C₃×C₈	D₈	1	1	yes	characteristic
56	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₈	#41	1	1	yes	characteristic
58	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₄⋊Dic₃	—	2	1	no	—
60	C₄⋊Dic₃	—	4	2	no	—
61	C₄⋊Dic₃	—	4	2	no	—
62	Dic₃⋊C₄	—	4	1	no	—
63	C₄×Dic₃	—	4	1	no	—
64	C₂×C₃⋊C₈	—	4	1	no	—
65	Dic₁₂	—	2	1	no	—
66	Dic₁₂	—	8	2	no	—
67	Dic₁₂	—	8	2	no	—
68	Dic₁₂	Q₁₆	1	2	yes	complemented
69	C₃×C₁₆	—	4	2	no	—
70	C₂×Q₁₆	—	3	1	no	—
71	C₂×Q₁₆	—	12	1	no	—
72	C₂×Q₁₆	—	12	1	no	—
73	C₄⋊Q₈	—	6	2	no	—
74	C₄×Q₈	—	6	1	no	—
75	C₂×C₁₆	—	2	2	no	—
76	#107	—	3	1	no	—
77	#107	—	6	2	no	—
78	#107	—	6	2	no	—
79	#107	—	12	2	no	—
80	C₄⋊C₈	—	6	1	no	—
81	C₄⋊C₈	—	12	1	no	—
82	#103	—	6	1	no	—
83	C₄×C₈	—	12	1	no	—
84	C₄×C₈	D₁₂	1	1	yes	characteristic
85	C₄×C₈	D₁₂	1	1	yes	characteristic
86	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃×C₄	#121	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
89	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×Dic₃	—	4	1	no	—
91	C₂×Dic₃	—	4	1	no	—
92	C₂×Dic₃	—	8	2	no	—
93	Dic₆	—	2	2	no	—
94	Dic₆	—	4	1	no	—
95	Dic₆	—	8	2	no	—
96	Dic₆	—	8	2	no	—
97	C₃×C₈	—	2	2	no	—
98	C₃×C₈	—	2	1	no	—
99	C₃×C₈	—	2	1	no	—
100	C₃×C₈	SD₃₂	1	2	yes	—
101	C₃×C₈	C₂×Q₁₆	1	1	yes	characteristic
102	C₃×C₈	#135	1	1	yes	characteristic
103	C₃⋊C₈	—	8	1	no	—
104	C₂×Q₈	—	6	1	no	—
105	C₂×Q₈	—	12	2	no	—
106	Q₁₆	—	3	2	no	—
107	Q₁₆	—	6	1	no	—
108	Q₁₆	—	24	2	no	—
109	Q₁₆	—	24	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	12	1	no	—
112	C₂×C₈	D₂₄	1	1	yes	characteristic
113	C₂×C₈	D₂₄	1	1	yes	characteristic
114	C₂×C₈	D₃×D₄	1	1	yes	characteristic
115	C₂×C₈	#239	1	1	yes	characteristic
116	C₄⋊C₄	—	6	1	no	—
117	C₄⋊C₄	—	12	1	no	—
118	C₄⋊C₄	—	12	2	no	—
119	C₄⋊C₄	—	12	2	no	—
120	C₄²	—	12	1	no	—
121	C₄²	C₂×D₁₂	1	1	yes	characteristic
122	C₁₆	—	4	2	no	—
123	C₂²×C₃	#459	1	1	yes	characteristic
124	C₃×C₄	—	2	1	no	—
125	C₃×C₄	#467	1	1	yes	characteristic
126	C₃×C₄	#473	1	1	yes	characteristic
127	C₃×C₄	C₂×SD₃₂	1	1	yes	characteristic
128	C₃×C₄	#509	1	1	yes	characteristic
129	Dic₃	—	4	2	no	—
130	Dic₃	—	8	1	no	—
131	Dic₃	—	16	2	no	—
132	Q₈	—	6	2	no	—
133	Q₈	—	12	1	no	—
134	Q₈	—	24	2	no	—
135	Q₈	—	24	2	no	—
136	C₂×C₄	—	12	1	no	—
137	C₂×C₄	—	12	1	no	—
138	C₂×C₄	—	24	2	no	—
139	C₂×C₄	#895	1	1	yes	characteristic
140	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
141	C₂×C₄	#908	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	24	1	no	—
146	C₈	#800	1	2	yes	—
147	C₈	#912	1	1	yes	characteristic
148	C₈	D₃×Q₁₆	1	1	yes	characteristic
149	C₂×C₃	#1669	1	1	yes	characteristic
150	C₂×C₃	#2207	1	1	yes	characteristic
151	C₂×C₃	#2214	1	1	yes	characteristic
152	C₂²	#4843	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	12	2	no	—
155	C₄	—	24	1	no	—
156	C₄	—	48	2	no	—
157	C₄	#4887	1	1	yes	characteristic
158	C₄	#4896	1	1	yes	characteristic
159	C₄	C₂×#800	1	1	yes	characteristic
160	C₄	#4908	1	1	yes	characteristic
161	C₃	#12446	1	1	yes	characteristic, complemented, Sylow
162	C₂	#68577	1	1	yes	characteristic
163	C₂	#69292	1	1	yes	characteristic
164	C₂	#69306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#69295

C₂

yes

maximal

#68065

C₂

yes

maximal, characteristic

C₈×Dic₁₂

C₂

yes

maximal, characteristic

C₃×#1370

C₂

yes

maximal, characteristic

#67564

C₂

yes

—

—

C₄×Dic₁₂

C₂²

yes

characteristic

#4682

—

#4682

C₂²

yes

—

C₈×Dic₆

—

C₃×C₄⋊C₁₆

C₂²

yes

—

C₃×C₈²

C₂²

yes

—

—

C₂²

yes

—

—

—

C₈×Q₁₆

—

#1370

D₃

yes

characteristic

#1364

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

D₄

yes

characteristic

#869

—

C₄×Dic₆

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

C₂³

yes

—

—

—

D₄

yes

characteristic

#816

—

Dic₃⋊C₈

—

C₈×Dic₃

—

C₄⋊(C₃⋊C₈)

—

#500

—

#500

—

#494

—

C₈×Q₈

—

C₄×Q₁₆

—

#367

—

C₄⋊C₁₆

C₂×D₃

yes

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₃

yes

characteristic

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₈

yes

characteristic

C₂×C₃×C₈

D₈

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

#41

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

Q₁₆

yes

complemented

C₃×C₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

C₂²×D₃

yes

characteristic

C₂×C₃×C₄

#121

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

100

C₃×C₈

SD₃₂

yes

—

101

C₃×C₈

C₂×Q₁₆

yes

characteristic

102

C₃×C₈

#135

yes

characteristic

103

C₃⋊C₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

Q₁₆

—

107

Q₁₆

—

108

Q₁₆

—

109

Q₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

D₂₄

yes

characteristic

113

C₂×C₈

D₂₄

yes

characteristic

114

C₂×C₈

D₃×D₄

yes

characteristic

115

C₂×C₈

#239

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₄²

C₂×D₁₂

yes

characteristic

122

C₁₆

—

123

C₂²×C₃

#459

yes

characteristic

124

C₃×C₄

—

125

C₃×C₄

#467

yes

characteristic

126

C₃×C₄

#473

yes

characteristic

127

C₃×C₄

C₂×SD₃₂

yes

characteristic

128

C₃×C₄

#509

yes

characteristic

129

Dic₃

—

130

Dic₃

—

131

Dic₃

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#895

yes

characteristic

140

C₂×C₄

C₂×D₂₄

yes

characteristic

141

C₂×C₄

#908

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#800

yes

—

147

C₈

#912

yes

characteristic

148

C₈

D₃×Q₁₆

yes

characteristic

149

C₂×C₃

#1669

yes

characteristic

150

C₂×C₃

#2207

yes

characteristic

151

C₂×C₃

#2214

yes

characteristic

152

C₂²

#4843

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#4887

yes

characteristic

158

C₄

#4896

yes

characteristic

159

C₄

C₂×#800

yes

characteristic

160

C₄

#4908

yes

characteristic

161

C₃

#12446

yes

characteristic, complemented, Sylow

162

C₂

#68577

yes

characteristic

163

C₂

#69292

yes

characteristic

164

C₂

#69306

yes

characteristic

	subgroup	G/H
center	C₂²	#4843
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1370	C₂
socle, prime radical	C₂²×C₃	#459
hypercenter	#1370	D₃
hypocenter	C₃	#12446
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#4843

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
68. Dic₁₂	Q₁₆	semidirect
161. C₃	#12446	semidirect

H ≅ G/N

type

68. Dic₁₂

Q₁₆

semidirect

161. C₃

#12446

semidirect

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

subgroup	index
#69295	2
#68065	2
C₈×Dic₁₂	2
C₃×#1370	2
#67564	2
#12446	3

subgroup

index

C₈×Dic₁₂

Subgroups — #10709484

Conjugacy classes of subgroups

Complements

Maximal subgroups