Group #23764828 — subgroups

Subgroups — #23764828

Order 10112.

256 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14579095	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098423	C₂	1	2	yes	maximal
3	#19098326	C₂	1	2	yes	maximal, complemented
4	#19098326	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
9	#14579095	—	2	2	no	—
10	C₇₉⋊#95	—	2	2	no	—
11	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₇₉	—	2	2	no	—
15	C₂²×Dic₇₉	—	2	2	no	—
16	C₂²×Dic₇₉	—	2	2	no	—
17	C₄⋊Dic₇₉	—	2	2	no	—
18	C₄⋊Dic₇₉	—	2	2	no	—
19	C₄⋊Dic₇₉	—	2	2	no	—
20	C₄⋊Dic₇₉	—	2	2	no	—
21	C₂²⋊Dic₇₉	—	4	4	no	—
22	C₂⁴×C₇₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₉	—	2	1	no	—
24	C₂²×C₄×C₇₉	—	2	1	no	—
25	C₂²×C₄×C₇₉	—	2	2	no	—
26	C₂²×C₄×C₇₉	—	2	2	no	—
27	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₇₉	—	2	1	no	—
30	C₂×C₈×C₇₉	—	2	2	no	—
31	C₂×C₈×C₇₉	—	2	2	no	—
32	C₂×C₈×C₇₉	D₄	1	2	yes	—
33	C₂³×C₇₉	—	2	1	no	—
34	C₂³×C₇₉	—	2	2	no	—
35	C₂³×C₇₉	—	2	1	no	—
36	C₂³×C₇₉	—	2	2	no	—
37	C₂³×C₇₉	Q₁₆	1	2	yes	—
38	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	—	2	2	no	—
41	C₂×C₄×C₇₉	—	4	1	no	—
42	C₂×C₄×C₇₉	—	4	1	no	—
43	C₂×C₄×C₇₉	—	4	2	no	—
44	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#41	1	2	yes	—
46	C₂×C₄×C₇₉	#41	1	2	yes	—
47	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
48	C₂×Dic₇₉	—	4	2	no	—
49	C₂×Dic₇₉	—	4	2	no	—
50	C₂×Dic₇₉	—	4	2	no	—
51	C₂×Dic₇₉	—	4	2	no	—
52	C₂×Dic₇₉	—	4	2	no	—
53	C₂×Dic₇₉	—	4	2	no	—
54	C₈×C₇₉	—	2	2	no	—
55	C₈×C₇₉	—	4	1	no	—
56	C₂²×C₇₉	—	2	2	no	—
57	C₂²×C₇₉	—	2	4	no	—
58	C₂²×C₇₉	—	4	1	no	—
59	C₂²×C₇₉	—	4	1	no	—
60	C₂²×C₇₉	—	4	2	no	—
61	C₂²×C₇₉	#121	1	1	yes	characteristic
62	C₂²×C₇₉	#123	1	2	yes	—
63	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₇₉	#135	1	1	yes	characteristic
65	C₂²×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
66	C₄×C₇₉	—	4	1	no	—
67	C₄×C₇₉	#123	1	1	yes	characteristic
68	C₄×C₇₉	#124	1	1	yes	characteristic
69	C₄×C₇₉	#126	1	2	yes	—
70	Dic₇₉	—	8	2	no	—
71	Dic₇₉	—	8	2	no	—
72	C₂×C₇₉	—	4	2	no	—
73	C₂×C₇₉	#397	1	1	yes	characteristic
74	C₂×C₇₉	#467	1	1	yes	characteristic
75	C₂×C₇₉	#469	1	1	yes	characteristic
76	C₂×C₇₉	#482	1	2	yes	—
77	C₂×C₇₉	#484	1	2	yes	—
78	#2076	—	79	1	no	maximal, Sylow
79	C₇₉	#2076	1	1	yes	characteristic, complemented, Sylow
80	C₂×#107	—	79	1	no	—
81	C₂×C₂²⋊C₈	D₇₉	1	1	yes	characteristic
82	#380	—	79	2	no	—
83	#340	—	79	1	no	—
84	#340	—	79	2	no	—
85	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	79	2	no	—
89	C₂×C₄⋊C₄	—	79	2	no	—
90	C₂×C₂²⋊C₄	—	158	2	no	—
91	#107	—	158	2	no	—
92	C₂²⋊C₈	—	2	2	no	—
93	#95	—	158	2	no	—
94	C₂⁴	D₃₁₆	1	1	yes	characteristic
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	158	2	no	—
100	C₂²×C₄	—	158	2	no	—
101	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
107	C₄⋊C₄	—	158	2	no	—
108	C₄⋊C₄	—	158	2	no	—
109	C₄⋊C₄	—	158	2	no	—
110	C₄⋊C₄	—	158	2	no	—
111	C₂²⋊C₄	—	316	4	no	—
112	C₂³	—	2	1	no	—
113	C₂³	—	2	2	no	—
114	C₂³	—	2	1	no	—
115	C₂³	—	2	2	no	—
116	C₂³	Dic₃₁₆	1	2	yes	—
117	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	316	2	no	—
124	C₂×C₄	—	316	2	no	—
125	C₂×C₄	—	316	2	no	—
126	C₂×C₄	—	316	2	no	—
127	C₂×C₄	—	316	2	no	—
128	C₂×C₄	—	316	2	no	—
129	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂×C₄	#11929623	1	2	yes	—
131	C₂×C₄	#11929623	1	2	yes	—
132	C₂×C₄	#11929624	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	1	no	—
135	C₂²	—	2	4	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	#14579135	1	1	yes	characteristic
141	C₂²	#14579145	1	2	yes	—
142	C₂²	C₂×Dic₃₁₆	1	1	yes	characteristic
143	C₂²	#14579183	1	1	yes	characteristic
144	C₂²	#14579189	1	2	yes	—
145	C₄	—	4	1	no	—
146	C₄	—	632	2	no	—
147	C₄	—	632	2	no	—
148	C₄	#14579142	1	1	yes	characteristic
149	C₄	#14579148	1	1	yes	characteristic
150	C₄	#14579154	1	2	yes	—
151	C₂	—	4	2	no	—
152	C₂	#19098488	1	1	yes	characteristic
153	C₂	#19098804	1	1	yes	characteristic
154	C₂	#19098816	1	1	yes	characteristic
155	C₂	#19098888	1	2	yes	—
156	C₂	#19098899	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579095

C₂

yes

maximal, characteristic, complemented

#19098423

C₂

yes

maximal

#19098326

C₂

yes

maximal, complemented

#19098326

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

—

#14579095

—

C₇₉⋊#95

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²⋊C₈×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₂⁴×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

Q₁₆

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

#123

yes

—

C₂²×C₇₉

C₂×Q₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#484

yes

—

#2076

—

maximal, Sylow

C₇₉

#2076

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×C₂²⋊C₈

D₇₉

yes

—

—

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

158

—

#107

—

158

—

C₂²⋊C₈

—

#95

—

158

—

C₂⁴

D₃₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

158

—

100

C₂²×C₄

—

158

—

101

C₂²×C₄

D₃₁₆

yes

characteristic

102

C₂²×C₄

C₂²×D₇₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²⋊D₇₉

yes

—

107

C₄⋊C₄

—

158

—

108

C₄⋊C₄

—

158

—

109

C₄⋊C₄

—

158

—

110

C₄⋊C₄

—

158

—

111

C₂²⋊C₄

—

316

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

Dic₃₁₆

yes

—

117

C₂³

C₂×D₃₁₆

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

316

—

124

C₂×C₄

—

316

—

125

C₂×C₄

—

316

—

126

C₂×C₄

—

316

—

127

C₂×C₄

—

316

—

128

C₂×C₄

—

316

—

129

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂×C₄

#11929623

yes

—

131

C₂×C₄

#11929623

yes

—

132

C₂×C₄

#11929624

yes

characteristic

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#14579135

yes

characteristic

141

C₂²

#14579145

yes

—

142

C₂²

C₂×Dic₃₁₆

yes

characteristic

143

C₂²

#14579183

yes

characteristic

144

C₂²

#14579189

yes

—

145

C₄

—

146

C₄

—

632

—

147

C₄

—

632

—

148

C₄

#14579142

yes

characteristic

149

C₄

#14579148

yes

characteristic

150

C₄

#14579154

yes

—

151

C₂

—

152

C₂

#19098488

yes

characteristic

153

C₂

#19098804

yes

characteristic

154

C₂

#19098816

yes

characteristic

155

C₂

#19098888

yes

—

156

C₂

#19098899

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×C₂²⋊C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂⁴×C₇₉	D₄
hypercenter	C₂×C₂²⋊C₈	D₇₉
hypocenter	C₇₉	#2076
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×C₂²⋊C₈×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

prime radical

C₂⁴×C₇₉

D₄

hypercenter

C₂×C₂²⋊C₈

D₇₉

hypocenter

C₇₉

#2076

LCS

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×#14579095	C₂	semidirect
3. #19098326	C₂	semidirect
4. #19098326	C₂	semidirect
79. C₇₉	#2076	semidirect

H ≅ G/N

type

1. C₂×#14579095

C₂

semidirect

3. #19098326

C₂

semidirect

4. #19098326

C₂

semidirect

79. C₇₉

#2076

semidirect

Subgroups — #23764828

Conjugacy classes of subgroups

Complements