Group #23764308 — subgroups

Subgroups — #23764308

Order 10112.

316 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×D₆₃₂	C₂	1	1	yes	maximal, characteristic
2	#19098569	C₂	1	2	yes	maximal, complemented
3	C₂×#14579077	C₂	1	2	yes	maximal, complemented
4	#19098518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#108×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₁₆	C₂²	1	2	yes	—
7	C₂×D₆₃₂	—	2	2	no	—
8	C₂×D₆₃₂	—	2	1	no	—
9	C₂×D₆₃₂	—	2	1	no	—
10	C₂×D₆₃₂	C₄	1	4	yes	complemented
11	C₂×#11929608	C₂²	1	1	yes	characteristic
12	#14579120	C₂²	1	1	yes	characteristic
13	#14579083	—	2	4	no	—
14	#14579077	—	2	4	no	—
15	#14579071	—	2	1	no	—
16	C₈×Dic₇₉	—	2	1	no	—
17	C₂×M₁₆×C₇₉	C₂²	1	2	yes	—
18	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#108×C₇₉	—	2	2	no	—
20	C₂³×D₇₉	—	2	2	no	—
21	C₂×D₃₁₆	—	2	2	no	—
22	C₂×D₃₁₆	—	2	2	no	—
23	C₂×D₃₁₆	—	2	2	no	—
24	C₂×D₃₁₆	—	2	2	no	—
25	C₂×D₃₁₆	—	2	2	no	—
26	C₂×D₃₁₆	C₂×C₄	1	4	yes	—
27	D₆₃₂	—	2	4	no	—
28	D₆₃₂	—	2	4	no	—
29	#11929608	—	2	1	no	—
30	#11929608	D₄	1	2	yes	—
31	C₂×C₇₉⋊C₈	—	2	1	no	—
32	C₄⋊Dic₇₉	D₄	1	2	yes	—
33	C₂²⋊Dic₇₉	—	2	1	no	—
34	C₄×Dic₇₉	—	2	1	no	—
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₇₉	—	2	2	no	—
37	M₁₆×C₇₉	—	2	2	no	—
38	C₂×C₈×C₇₉	—	2	2	no	—
39	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₇₉	D₄	1	2	yes	—
41	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
43	C₂³×C₇₉	#41	1	1	yes	characteristic
44	C₂²×D₇₉	—	2	4	no	—
45	C₂²×D₇₉	—	4	2	no	—
46	C₂²×D₇₉	—	4	2	no	—
47	C₂²×D₇₉	—	8	4	no	—
48	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
54	C₂×Dic₇₉	—	2	2	no	—
55	D₃₁₆	—	2	4	no	—
56	D₃₁₆	—	2	4	no	—
57	D₃₁₆	—	4	2	no	—
58	D₃₁₆	—	4	2	no	—
59	C₈×C₇₉	—	4	2	no	—
60	C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
61	C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
62	C₇₉⋊C₈	—	2	2	no	—
63	C₂²×C₇₉	—	2	2	no	—
64	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
65	C₂²×C₇₉	#121	1	1	yes	characteristic
66	C₂²×C₇₉	#124	1	1	yes	characteristic
67	C₂×D₇₉	—	4	4	no	—
68	C₂×D₇₉	—	8	4	no	—
69	C₂×D₇₉	—	8	4	no	—
70	C₂×D₇₉	—	8	4	no	—
71	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₇₉	#121	1	1	yes	characteristic
74	C₄×C₇₉	#127	1	1	yes	characteristic
75	Dic₇₉	—	4	2	no	—
76	C₂×C₇₉	—	2	1	no	—
77	C₂×C₇₉	—	2	1	no	—
78	C₂×C₇₉	#390	1	1	yes	characteristic
79	C₂×C₇₉	#472	1	2	yes	—
80	D₇₉	—	8	4	no	—
81	#1978	—	79	1	no	maximal, Sylow
82	C₇₉	#1978	1	1	yes	characteristic, complemented, Sylow
83	C₂²×D₈	—	79	1	no	—
84	C₂×#108	D₇₉	1	1	yes	characteristic, complemented
85	#418	—	79	2	no	—
86	C₂×#100	—	79	2	no	—
87	#405	—	79	1	no	—
88	C₂²×D₄	—	79	2	no	—
89	C₂×D₈	—	79	4	no	—
90	C₂×D₈	—	158	2	no	—
91	C₂×D₈	—	158	1	no	—
92	C₂×D₈	—	158	1	no	—
93	C₂×M₁₆	—	79	1	no	—
94	C₂×M₁₆	C₂×D₇₉	1	2	yes	—
95	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
96	#117	—	79	1	no	—
97	#108	—	2	2	no	—
98	#102	—	158	4	no	—
99	#100	—	158	4	no	—
100	#97	—	158	1	no	—
101	C₄×C₈	—	158	1	no	—
102	C₂⁴	—	158	2	no	—
103	C₂×D₄	—	79	4	no	—
104	C₂×D₄	—	158	2	no	—
105	C₂×D₄	—	158	2	no	—
106	C₂×D₄	—	158	2	no	—
107	C₂×D₄	—	158	2	no	—
108	C₂×D₄	—	158	2	no	—
109	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
110	D₈	—	158	4	no	—
111	D₈	—	158	4	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	79	2	no	—
115	M₁₆	—	158	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	158	1	no	—
118	C₂×C₈	C₄×D₇₉	1	2	yes	—
119	C₂×C₈	D₃₁₆	1	1	yes	characteristic
120	C₂×C₈	D₃₁₆	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
122	C₄⋊C₄	—	79	2	no	—
123	C₂²⋊C₄	—	158	1	no	—
124	C₄²	—	158	1	no	—
125	C₂³	—	316	2	no	—
126	C₂³	—	316	2	no	—
127	C₂³	—	632	4	no	—
128	C₂³	—	158	4	no	—
129	C₂³	#11929625	1	1	yes	characteristic
130	D₄	—	316	2	no	—
131	D₄	—	316	2	no	—
132	D₄	—	158	4	no	—
133	D₄	—	158	4	no	—
134	C₂×C₄	—	158	2	no	—
135	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
136	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
137	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
139	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
140	C₂×C₄	#11929625	1	1	yes	characteristic
141	C₈	—	4	2	no	—
142	C₈	—	158	2	no	—
143	C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
144	C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	316	4	no	—
147	C₂²	—	632	4	no	—
148	C₂²	—	632	4	no	—
149	C₂²	—	632	4	no	—
150	C₂²	#14579123	1	1	yes	characteristic
151	C₂²	#14579136	1	1	yes	characteristic
152	C₂²	#14579150	1	1	yes	characteristic
153	C₄	—	316	2	no	—
154	C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
155	C₄	#14579123	1	1	yes	characteristic
156	C₄	#14579136	1	1	yes	characteristic
157	C₄	#14579157	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	632	4	no	—
161	C₂	#19098464	1	1	yes	characteristic
162	C₂	#19098834	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₆₃₂

C₂

yes

maximal, characteristic

#19098569

C₂

yes

maximal, complemented

C₂×#14579077

C₂

yes

maximal, complemented

#19098518

C₂

yes

maximal, characteristic, complemented

C₂×#108×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₁₆

C₂²

yes

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

C₄

yes

complemented

C₂×#11929608

C₂²

yes

characteristic

#14579120

C₂²

yes

—

—

—

C₈×Dic₇₉

—

C₂×M₁₆×C₇₉

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#108×C₇₉

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂×C₄

yes

—

D₆₃₂

—

D₆₃₂

—

#11929608

—

#11929608

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

D₄

yes

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

#41

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊C₄

yes

—

C₈×C₇₉

C₂²⋊C₄

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₄×D₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#390

yes

characteristic

C₂×C₇₉

#472

yes

—

D₇₉

—

#1978

—

maximal, Sylow

C₇₉

#1978

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×#108

D₇₉

yes

characteristic, complemented

#418

—

C₂×#100

—

#405

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

—

C₂²×C₈

C₂×D₇₉

yes

—

—

—

158

—

#100

—

158

—

100

#97

—

158

—

101

C₄×C₈

—

158

—

102

C₂⁴

—

158

—

103

C₂×D₄

—

104

C₂×D₄

—

158

—

105

C₂×D₄

—

158

—

106

C₂×D₄

—

158

—

107

C₂×D₄

—

158

—

108

C₂×D₄

—

158

—

109

C₂²×C₄

C₂²×D₇₉

yes

characteristic

110

D₈

—

158

—

111

D₈

—

158

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

158

—

116

C₂×C₈

—

117

C₂×C₈

—

158

—

118

C₂×C₈

C₄×D₇₉

yes

—

119

C₂×C₈

D₃₁₆

yes

characteristic

120

C₂×C₈

D₃₁₆

yes

characteristic

121

C₂×C₈

C₂²⋊D₇₉

yes

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

158

—

124

C₄²

—

158

—

125

C₂³

—

316

—

126

C₂³

—

316

—

127

C₂³

—

632

—

128

C₂³

—

158

—

129

C₂³

#11929625

yes

characteristic

130

D₄

—

316

—

131

D₄

—

316

—

132

D₄

—

158

—

133

D₄

—

158

—

134

C₂×C₄

—

158

—

135

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

136

C₂×C₄

D₄×D₇₉

yes

characteristic

137

C₂×C₄

D₄×D₇₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

139

C₂×C₄

C₂×D₃₁₆

yes

characteristic

140

C₂×C₄

#11929625

yes

characteristic

141

C₈

—

142

C₈

—

158

—

143

C₈

(C₂×D₇₉)⋊C₄

yes

—

144

C₈

(C₂×D₇₉)⋊C₄

yes

—

145

C₂²

—

146

C₂²

—

316

—

147

C₂²

—

632

—

148

C₂²

—

632

—

149

C₂²

—

632

—

150

C₂²

#14579123

yes

characteristic

151

C₂²

#14579136

yes

characteristic

152

C₂²

#14579150

yes

characteristic

153

C₄

—

316

—

154

C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

155

C₄

#14579123

yes

characteristic

156

C₄

#14579136

yes

characteristic

157

C₄

#14579157

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

632

—

161

C₂

#19098464

yes

characteristic

162

C₂

#19098834

yes

—

	subgroup	G/H
center	C₂²	#14579136
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×#108×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	C₂²×D₆₃₂	C₂
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×#108	D₇₉
hypocenter	C₇₉	#1978
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14579136

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×#108×C₇₉

C₂

socle

C₂²×C₇₉

#121

prime radical

C₂²×D₆₃₂

C₂

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. #19098569	C₂	semidirect
3. C₂×#14579077	C₂	semidirect
4. #19098518	C₂	semidirect
5. C₂×#108×C₇₉	C₂	semidirect
10. C₂×D₆₃₂	C₄	semidirect
82. C₇₉	#1978	semidirect
84. C₂×#108	D₇₉	semidirect

H ≅ G/N

type

2. #19098569

C₂

semidirect

3. C₂×#14579077

C₂

semidirect

4. #19098518

C₂

semidirect

5. C₂×#108×C₇₉

C₂

semidirect

10. C₂×D₆₃₂

C₄

semidirect

82. C₇₉

#1978

semidirect

84. C₂×#108

D₇₉

semidirect

Subgroups — #23764308

Conjugacy classes of subgroups

Complements