Group #23765434 — subgroups

Subgroups — #23765434

Order 10112.

186 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 49 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×D₁₂₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579102	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098938	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098645	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098363	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
9	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
10	C₂×#11929609	—	2	1	no	—
11	#14579157	—	2	1	no	—
12	#14579123	—	2	1	no	—
13	D₁₂₆₄	—	2	2	no	—
14	#14579102	—	2	2	no	—
15	C₂×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14579081	—	2	1	no	—
17	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
20	#107×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×D₃₁₆	—	2	1	no	—
22	C₂×D₃₁₆	—	2	1	no	—
23	C₂×C₂²⋊D₇₉	—	4	1	no	—
24	C₂×C₂²⋊D₇₉	—	4	1	no	—
25	C₂×C₄×D₇₉	—	4	1	no	—
26	D₆₃₂	—	2	1	no	—
27	D₆₃₂	—	2	1	no	—
28	D₆₃₂	—	2	1	no	—
29	D₆₃₂	D₄	1	2	yes	complemented
30	#11929609	—	4	2	no	—
31	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
32	(C₂×D₇₉)⋊C₄	—	4	1	no	—
33	C₄×Dic₇₉	D₄	1	1	yes	characteristic
34	C₇₉⋊C₁₆	—	2	1	no	—
35	C₂×D₄×C₇₉	—	2	1	no	—
36	D₈×C₇₉	—	2	1	no	—
37	D₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₁₆×C₇₉	—	2	1	no	—
41	C₂³×C₇₉	—	4	1	no	—
42	C₂²×D₇₉	—	4	1	no	—
43	C₂²×D₇₉	—	4	1	no	—
44	D₄×C₇₉	—	4	1	no	—
45	D₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	—	4	1	no	—
47	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₇₉	—	8	2	no	—
49	C₂²⋊D₇₉	—	8	2	no	—
50	C₂×Dic₇₉	D₈	1	1	yes	characteristic
51	C₂×Dic₇₉	D₈	1	1	yes	characteristic
52	D₃₁₆	—	2	2	no	—
53	D₃₁₆	—	4	1	no	—
54	D₃₁₆	—	4	1	no	—
55	D₃₁₆	—	4	1	no	—
56	C₄×D₇₉	—	8	2	no	—
57	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₈×C₇₉	#41	1	1	yes	characteristic
59	C₇₉⋊C₈	—	2	1	no	—
60	C₂²×C₇₉	—	8	1	no	—
61	C₂²×C₇₉	—	8	1	no	—
62	C₂²×C₇₉	—	8	1	no	—
63	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
64	C₂×D₇₉	—	4	2	no	—
65	C₂×D₇₉	—	8	1	no	—
66	C₂×D₇₉	—	8	1	no	—
67	C₂×D₇₉	—	8	1	no	—
68	C₂×D₇₉	—	8	1	no	—
69	C₂×D₇₉	—	8	1	no	—
70	C₄×C₇₉	—	8	1	no	—
71	C₄×C₇₉	#121	1	1	yes	characteristic
72	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	Dic₇₉	—	2	1	no	—
74	Dic₇₉	D₁₆	1	2	yes	complemented
75	C₂×C₇₉	—	16	1	no	—
76	C₂×C₇₉	#459	1	1	yes	characteristic
77	C₂×C₇₉	C₂×D₁₆	1	1	yes	characteristic
78	C₂×C₇₉	#509	1	1	yes	characteristic
79	D₇₉	—	8	2	no	—
80	D₇₉	—	16	1	no	—
81	#2206	—	79	1	no	maximal, Sylow
82	C₇₉	#2206	1	1	yes	characteristic, complemented, Sylow
83	C₂×D₁₆	—	79	1	no	—
84	C₂×D₁₆	—	79	1	no	—
85	#493	—	79	1	no	—
86	C₄×D₈	—	79	1	no	—
87	C₄⋊C₁₆	—	79	1	no	—
88	#357	—	79	1	no	—
89	#357	D₇₉	1	1	yes	characteristic, complemented
90	C₂×D₈	—	79	1	no	—
91	C₂×D₈	—	79	1	no	—
92	C₂×D₈	—	158	1	no	—
93	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
94	#127	—	158	1	no	—
95	C₄×D₄	—	158	1	no	—
96	D₁₆	—	158	2	no	—
97	D₁₆	—	158	2	no	—
98	C₂×C₁₆	—	79	1	no	—
99	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
100	#107	C₂×D₇₉	1	1	yes	characteristic
101	#102	—	158	1	no	—
102	C₄×C₈	—	79	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	316	1	no	—
105	C₂×D₄	—	316	1	no	—
106	C₂×D₄	—	158	1	no	—
107	C₂×D₄	—	158	1	no	—
108	C₂²×C₄	—	316	1	no	—
109	D₈	—	2	1	no	—
110	D₈	—	2	1	no	—
111	D₈	—	316	2	no	—
112	D₈	—	79	2	no	—
113	D₈	—	158	1	no	—
114	D₈	—	158	1	no	—
115	D₈	—	158	1	no	—
116	C₂×C₈	—	79	1	no	—
117	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
118	C₄⋊C₄	—	2	1	no	—
119	C₂²⋊C₄	—	316	1	no	—
120	C₄²	—	79	1	no	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	158	1	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	316	1	no	—
125	C₂³	—	316	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	316	1	no	—
129	D₄	—	316	1	no	—
130	D₄	—	316	1	no	—
131	D₄	—	632	2	no	—
132	D₄	—	632	2	no	—
133	D₄	—	158	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	79	1	no	—
136	C₂×C₄	—	79	1	no	—
137	C₂×C₄	—	632	2	no	—
138	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
139	C₈	—	158	1	no	—
140	C₈	D₄×D₇₉	1	1	yes	characteristic
141	C₈	#11929624	1	1	yes	characteristic
142	C₂²	—	8	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	316	2	no	—
146	C₂²	—	632	1	no	—
147	C₂²	—	632	1	no	—
148	C₂²	—	632	1	no	—
149	C₂²	—	632	1	no	—
150	C₂²	—	632	1	no	—
151	C₂²	D₈×D₇₉	1	1	yes	characteristic
152	C₄	—	8	1	no	—
153	C₄	—	79	2	no	—
154	C₄	—	158	1	no	—
155	C₄	#14579134	1	1	yes	characteristic
156	C₄	#14579188	1	1	yes	characteristic
157	C₂	—	16	1	no	—
158	C₂	—	632	2	no	—
159	C₂	—	1264	1	no	—
160	C₂	#19098757	1	1	yes	characteristic
161	C₂	D₁₆×D₇₉	1	1	yes	characteristic
162	C₂	#19099000	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₂₆₄

C₂

yes

maximal, characteristic, complemented

C₂×#14579102

C₂

yes

maximal, characteristic, complemented

#19098938

C₂

yes

maximal, characteristic, complemented

#19098645

C₂

yes

maximal, characteristic, complemented

Dic₇₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#19098363

C₂

yes

maximal, characteristic, complemented

#357×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×D₆₃₂

C₂²

yes

—

—

—

D₁₂₆₄

—

#14579102

—

C₂×C₇₉⋊C₁₆

C₂²

yes

characteristic

#14579081

—

C₈×Dic₇₉

C₂²

yes

characteristic

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

D₄

yes

complemented

#11929609

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

D₄

yes

characteristic

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

D₈

yes

characteristic

C₂×Dic₇₉

D₈

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₈×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

D₁₆

yes

complemented

C₂×C₇₉

—

C₂×C₇₉

#459

yes

characteristic

C₂×C₇₉

C₂×D₁₆

yes

characteristic

C₂×C₇₉

#509

yes

characteristic

D₇₉

—

D₇₉

—

#2206

—

maximal, Sylow

C₇₉

#2206

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

—

#493

—

C₄×D₈

—

C₄⋊C₁₆

—

#357

—

#357

D₇₉

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

158

—

C₂×D₈

C₂×D₇₉

yes

characteristic

#127

—

158

—

C₄×D₄

—

158

—

D₁₆

—

158

—

D₁₆

—

158

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₉

yes

characteristic

100

#107

C₂×D₇₉

yes

characteristic

101

#102

—

158

—

102

C₄×C₈

—

103

C₂×D₄

—

104

C₂×D₄

—

316

—

105

C₂×D₄

—

316

—

106

C₂×D₄

—

158

—

107

C₂×D₄

—

158

—

108

C₂²×C₄

—

316

—

109

D₈

—

110

D₈

—

111

D₈

—

316

—

112

D₈

—

113

D₈

—

158

—

114

D₈

—

158

—

115

D₈

—

158

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₇₉

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

316

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

158

—

123

C₂³

—

124

C₂³

—

316

—

125

C₂³

—

316

—

126

D₄

—

127

D₄

—

128

D₄

—

316

—

129

D₄

—

316

—

130

D₄

—

316

—

131

D₄

—

632

—

132

D₄

—

632

—

133

D₄

—

158

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

632

—

138

C₂×C₄

D₄×D₇₉

yes

characteristic

139

C₈

—

158

—

140

C₈

D₄×D₇₉

yes

characteristic

141

C₈

#11929624

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

316

—

146

C₂²

—

632

—

147

C₂²

—

632

—

148

C₂²

—

632

—

149

C₂²

—

632

—

150

C₂²

—

632

—

151

C₂²

D₈×D₇₉

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

158

—

155

C₄

#14579134

yes

characteristic

156

C₄

#14579188

yes

characteristic

157

C₂

—

158

C₂

—

632

—

159

C₂

—

1264

—

160

C₂

#19098757

yes

characteristic

161

C₂

D₁₆×D₇₉

yes

characteristic

162

C₂

#19099000

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#357×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
hypercenter	#357	D₇₉
hypocenter	C₇₉	#2206
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₉

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#357×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×D₁₂₆₄	C₂	semidirect
2. C₂×#14579102	C₂	semidirect
3. #19098938	C₂	semidirect
4. #19098645	C₂	semidirect
5. Dic₇₉⋊C₁₆	C₂	semidirect
6. #19098363	C₂	semidirect
7. #357×C₇₉	C₂	semidirect
29. D₆₃₂	D₄	semidirect
74. Dic₇₉	D₁₆	semidirect
82. C₇₉	#2206	semidirect
89. #357	D₇₉	semidirect

H ≅ G/N

type

1. C₂×D₁₂₆₄

C₂

semidirect

2. C₂×#14579102

C₂

semidirect

3. #19098938

C₂

semidirect

4. #19098645

C₂

semidirect

5. Dic₇₉⋊C₁₆

C₂

semidirect

6. #19098363

C₂

semidirect

7. #357×C₇₉

C₂

semidirect

29. D₆₃₂

D₄

semidirect

74. Dic₇₉

D₁₆

semidirect

82. C₇₉

#2206

semidirect

89. #357

D₇₉

semidirect

Subgroups — #23765434

Conjugacy classes of subgroups

Complements