Group #23763068 — subgroups

Subgroups — #23763068

Order 10112.

242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19098940	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098939	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098939	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098937	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098911	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098285	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929611	—	2	2	no	—
9	C₂×D₆₃₂	—	2	2	no	—
10	C₂×#11929609	—	2	2	no	—
11	C₂×#11929609	—	2	2	no	—
12	#14579158	C₂²	1	1	yes	characteristic
13	#14579158	C₂²	1	1	yes	characteristic
14	#14579081	—	2	2	no	—
15	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
18	#127×C₇₉	C₂²	1	1	yes	characteristic
19	#102×C₇₉	—	2	2	no	—
20	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×D₃₁₆	—	2	1	no	—
22	C₂×D₃₁₆	—	2	1	no	—
23	C₂×D₃₁₆	—	2	1	no	—
24	C₂×D₃₁₆	—	2	1	no	—
25	C₂×D₃₁₆	—	2	1	no	—
26	C₂×D₃₁₆	—	2	1	no	—
27	#11929611	—	4	4	no	—
28	D₆₃₂	—	4	4	no	—
29	#11929609	—	4	4	no	—
30	#11929609	—	4	4	no	—
31	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
32	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
33	C₂×Q₈×C₇₉	—	2	1	no	—
34	C₂×D₄×C₇₉	—	2	1	no	—
35	C₂×D₄×C₇₉	—	2	1	no	—
36	C₂×D₄×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	D₄	1	2	yes	—
38	C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	C₂³×C₇₉	—	4	1	no	—
42	C₂²×D₇₉	—	4	1	no	—
43	C₂²×D₇₉	—	4	1	no	—
44	Q₈×C₇₉	—	4	2	no	—
45	D₄×C₇₉	—	4	2	no	—
46	D₄×C₇₉	—	4	2	no	—
47	D₄×C₇₉	—	4	2	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
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	D₃₁₆	—	4	2	no	—
53	D₃₁₆	—	4	2	no	—
54	D₃₁₆	—	4	2	no	—
55	D₃₁₆	—	4	2	no	—
56	D₃₁₆	—	4	2	no	—
57	D₃₁₆	—	4	2	no	—
58	C₈×C₇₉	—	2	2	no	—
59	C₇₉⋊C₈	—	2	2	no	—
60	C₇₉⋊C₈	D₈	1	4	yes	complemented
61	C₂²×C₇₉	—	8	1	no	—
62	C₂²×C₇₉	—	8	1	no	—
63	C₂²×C₇₉	—	8	1	no	—
64	C₂²×C₇₉	#127	1	1	yes	characteristic
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	C₂×D₇₉	—	8	1	no	—
71	C₄×C₇₉	—	8	1	no	—
72	C₄×C₇₉	C₂×D₈	1	2	yes	—
73	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₇₉	—	16	1	no	—
76	C₂×C₇₉	#493	1	1	yes	characteristic
77	C₂×C₇₉	#496	1	1	yes	characteristic
78	C₂×C₇₉	#496	1	1	yes	characteristic
79	D₇₉	—	16	1	no	—
80	D₇₉	—	16	1	no	—
81	#1721	—	79	1	no	maximal, Sylow
82	C₇₉	#1721	1	1	yes	characteristic, complemented, Sylow
83	#493	—	79	1	no	—
84	#493	—	79	1	no	—
85	#493	—	79	1	no	—
86	#492	—	79	1	no	—
87	#486	—	79	1	no	—
88	#486	D₇₉	1	1	yes	characteristic, complemented
89	#322	—	79	1	no	—
90	C₂×SD₁₆	—	158	2	no	—
91	C₂×D₈	—	158	2	no	—
92	C₂×D₈	—	158	2	no	—
93	C₂×D₈	—	158	2	no	—
94	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
95	#127	—	79	1	no	—
96	#127	—	79	1	no	—
97	#127	C₂×D₇₉	1	1	yes	characteristic
98	#102	—	2	2	no	—
99	#102	—	158	2	no	—
100	C₄×C₈	—	79	1	no	—
101	C₄×C₈	—	79	1	no	—
102	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
103	C₂×Q₈	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	158	1	no	—
108	C₂×D₄	—	158	1	no	—
109	C₂×D₄	—	158	1	no	—
110	C₂×D₄	—	158	1	no	—
111	C₂×D₄	—	158	1	no	—
112	C₂×D₄	—	158	1	no	—
113	SD₁₆	—	316	4	no	—
114	D₈	—	316	4	no	—
115	D₈	—	316	4	no	—
116	D₈	—	316	4	no	—
117	C₂×C₈	—	79	2	no	—
118	C₂×C₈	—	79	2	no	—
119	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄²	C₂²×D₇₉	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	316	1	no	—
125	C₂³	—	316	1	no	—
126	Q₈	—	4	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	316	2	no	—
131	D₄	—	316	2	no	—
132	D₄	—	316	2	no	—
133	D₄	—	316	2	no	—
134	D₄	—	316	2	no	—
135	D₄	—	316	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	79	4	no	—
142	C₈	—	158	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	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	—	632	1	no	—
152	C₂²	#14579157	1	1	yes	characteristic
153	C₄	—	8	1	no	—
154	C₄	D₈×D₇₉	1	2	yes	—
155	C₄	#14579186	1	2	yes	—
156	C₄	#14579188	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	1264	1	no	—
159	C₂	—	1264	1	no	—
160	C₂	#19098938	1	1	yes	characteristic
161	C₂	#19098949	1	1	yes	characteristic
162	C₂	#19098953	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098940

C₂

yes

maximal, characteristic, complemented

#19098939

C₂

yes

maximal, characteristic, complemented

#19098939

C₂

yes

maximal, characteristic, complemented

#19098937

C₂

yes

maximal, characteristic, complemented

#19098911

C₂

yes

maximal, characteristic, complemented

#19098285

C₂

yes

maximal, characteristic, complemented

#486×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929611

—

C₂×D₆₃₂

—

C₂×#11929609

—

C₂×#11929609

—

#14579158

C₂²

yes

characteristic

#14579158

C₂²

yes

characteristic

#14579081

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#127×C₇₉

C₂²

yes

characteristic

#102×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929611

—

D₆₃₂

—

#11929609

—

#11929609

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

D₈

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#127

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

—

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

D₇₉

—

#1721

—

maximal, Sylow

C₇₉

#1721

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₉

yes

characteristic, complemented

#322

—

C₂×SD₁₆

—

158

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

—

—

C₂×D₇₉

yes

characteristic

#102

—

#102

—

158

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₂×D₇₉

yes

characteristic

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

158

—

108

C₂×D₄

—

158

—

109

C₂×D₄

—

158

—

110

C₂×D₄

—

158

—

111

C₂×D₄

—

158

—

112

C₂×D₄

—

158

—

113

SD₁₆

—

316

—

114

D₈

—

316

—

115

D₈

—

316

—

116

D₈

—

316

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂²⋊D₇₉

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

C₂²×D₇₉

yes

characteristic

123

C₂³

—

124

C₂³

—

316

—

125

C₂³

—

316

—

126

Q₈

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

316

—

131

D₄

—

316

—

132

D₄

—

316

—

133

D₄

—

316

—

134

D₄

—

316

—

135

D₄

—

316

—

136

C₂×C₄

—

137

C₂×C₄

D₄×D₇₉

yes

characteristic

138

C₂×C₄

D₄×D₇₉

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

158

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

632

—

147

C₂²

—

632

—

148

C₂²

—

632

—

149

C₂²

—

632

—

150

C₂²

—

632

—

151

C₂²

—

632

—

152

C₂²

#14579157

yes

characteristic

153

C₄

—

154

C₄

D₈×D₇₉

yes

—

155

C₄

#14579186

yes

—

156

C₄

#14579188

yes

—

157

C₂

—

158

C₂

—

1264

—

159

C₂

—

1264

—

160

C₂

#19098938

yes

characteristic

161

C₂

#19098949

yes

characteristic

162

C₂

#19098953

yes

characteristic

	subgroup	G/H
center	C₂²	#14579157
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#486×C₇₉	C₂
socle	C₂²×C₇₉	#127
hypercenter	#486	D₇₉
hypocenter	C₇₉	#1721
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14579157

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#486×C₇₉

C₂

C₂²×C₇₉

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #19098940	C₂	semidirect
2. #19098939	C₂	semidirect
3. #19098939	C₂	semidirect
4. #19098937	C₂	semidirect
5. #19098911	C₂	semidirect
6. #19098285	C₂	semidirect
7. #486×C₇₉	C₂	semidirect
60. C₇₉⋊C₈	D₈	semidirect
82. C₇₉	#1721	semidirect
88. #486	D₇₉	semidirect

H ≅ G/N

type

1. #19098940

C₂

semidirect

2. #19098939

C₂

semidirect

3. #19098939

C₂

semidirect

4. #19098937

C₂

semidirect

5. #19098911

C₂

semidirect

6. #19098285

C₂

semidirect

7. #486×C₇₉

C₂

semidirect

60. C₇₉⋊C₈

D₈

semidirect

82. C₇₉

#1721

semidirect

88. #486

D₇₉

semidirect

Subgroups — #23763068

Conjugacy classes of subgroups

Complements