Group #23762242 — subgroups

Subgroups — #23762242

Order 10112.

166 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 63 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	#19098687	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11929613	C₂	1	1	yes	maximal, characteristic
3	#19098640	C₂	1	1	yes	maximal, characteristic
4	#19098293	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098290	C₂	1	1	yes	maximal, characteristic
6	#19098285	C₂	1	1	yes	maximal, characteristic, complemented
7	#334×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929613	C₄	1	1	yes	characteristic
9	C₂×#11929607	—	2	1	no	—
10	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
11	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
12	#14579093	C₄	1	1	yes	characteristic
13	Dic₇₉⋊C₈	—	2	1	no	—
14	#14579086	C₄	1	1	yes	characteristic
15	#14579083	C₄	1	1	yes	characteristic
16	(C₂×D₇₉)⋊C₈	—	2	1	no	—
17	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₇₉	—	2	1	no	—
20	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
24	C₂×D₃₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₇₉	—	2	1	no	—
26	#11929613	—	2	2	no	—
27	#11929607	—	4	2	no	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
31	C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
33	Dic₇₉⋊C₄	—	2	1	no	—
34	(C₂×D₇₉)⋊C₄	—	2	1	no	—
35	C₄×Dic₇₉	—	2	1	no	—
36	C₂×C₈×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	C₂²×D₇₉	—	2	1	no	—
42	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
45	C₂×Dic₇₉	—	2	1	no	—
46	C₂×Dic₇₉	—	2	1	no	—
47	C₂×Dic₇₉	—	2	1	no	—
48	D₃₁₆	—	2	1	no	—
49	D₃₁₆	—	2	1	no	—
50	C₄×D₇₉	—	4	1	no	—
51	C₄×D₇₉	—	4	1	no	—
52	Dic₁₅₈	—	2	1	no	—
53	Dic₁₅₈	M₁₆	1	2	yes	complemented
54	C₈×C₇₉	—	2	1	no	—
55	C₈×C₇₉	—	2	1	no	—
56	C₈×C₇₉	—	4	1	no	—
57	C₈×C₇₉	—	4	1	no	—
58	C₇₉⋊C₈	—	2	1	no	—
59	C₇₉⋊C₈	—	4	1	no	—
60	C₇₉⋊C₈	SD₁₆	1	2	yes	complemented
61	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
62	C₂×D₇₉	—	4	1	no	—
63	C₂×D₇₉	—	4	1	no	—
64	C₂×D₇₉	—	4	1	no	—
65	C₄×C₇₉	—	2	1	no	—
66	C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₇₉	#131	1	1	yes	characteristic
68	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₉	#135	1	1	yes	characteristic
70	Dic₇₉	—	2	2	no	—
71	Dic₇₉	—	4	1	no	—
72	Dic₇₉	—	4	1	no	—
73	C₂×C₇₉	#435	1	1	yes	characteristic
74	C₂×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
75	C₂×C₇₉	#444	1	1	yes	characteristic
76	D₇₉	—	8	1	no	—
77	#1583	—	79	1	no	maximal, Sylow
78	C₇₉	#1583	1	1	yes	characteristic, complemented, Sylow
79	C₈×Q₈	—	79	1	no	—
80	C₄×SD₁₆	—	79	1	no	—
81	#436	—	79	1	no	—
82	#334	D₇₉	1	1	yes	characteristic, complemented
83	Q₈⋊C₈	—	79	1	no	—
84	D₄⋊C₈	—	79	1	no	—
85	#322	—	79	1	no	—
86	C₂×SD₁₆	—	79	1	no	—
87	C₂×M₁₆	—	158	1	no	—
88	C₄×Q₈	—	79	1	no	—
89	C₄×D₄	—	79	1	no	—
90	#106	—	79	1	no	—
91	C₄⋊C₈	—	158	1	no	—
92	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
94	#103	—	79	1	no	—
95	#102	—	79	1	no	—
96	C₂²⋊C₈	—	158	1	no	—
97	C₄×C₈	—	79	1	no	—
98	C₄×C₈	—	79	1	no	—
99	C₄×C₈	—	158	1	no	—
100	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
101	C₂×Q₈	—	79	1	no	—
102	C₂×D₄	—	79	1	no	—
103	C₂²×C₄	—	158	1	no	—
104	SD₁₆	—	158	2	no	—
105	M₁₆	—	316	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	79	1	no	—
109	C₂×C₈	—	79	1	no	—
110	C₂×C₈	—	158	1	no	—
111	C₂×C₈	C₄×D₇₉	1	1	yes	characteristic
112	C₂×C₈	C₄×D₇₉	1	1	yes	characteristic
113	C₄⋊C₄	—	79	1	no	—
114	C₄⋊C₄	—	79	1	no	—
115	C₄⋊C₄	—	158	1	no	—
116	C₂²⋊C₄	—	158	1	no	—
117	C₄²	—	158	1	no	—
118	C₄²	C₂²×D₇₉	1	1	yes	characteristic
119	C₂³	—	158	1	no	—
120	Q₈	—	79	2	no	—
121	Q₈	—	158	1	no	—
122	D₄	—	158	1	no	—
123	D₄	—	158	1	no	—
124	C₂×C₄	—	316	1	no	—
125	C₂×C₄	—	316	1	no	—
126	C₂×C₄	—	158	1	no	—
127	C₂×C₄	—	158	1	no	—
128	C₂×C₄	—	158	1	no	—
129	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
130	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
131	C₂×C₄	#11929625	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	316	1	no	—
137	C₈	—	79	2	no	—
138	C₈	—	158	1	no	—
139	C₂²	—	316	1	no	—
140	C₂²	—	316	1	no	—
141	C₂²	—	316	1	no	—
142	C₂²	#14579123	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	316	1	no	—
145	C₄	—	316	1	no	—
146	C₄	—	158	2	no	—
147	C₄	M₁₆×D₇₉	1	1	yes	characteristic
148	C₄	#14579167	1	1	yes	characteristic
149	C₄	SD₁₆×D₇₉	1	1	yes	characteristic
150	C₄	#14579182	1	1	yes	characteristic
151	C₂	—	632	1	no	—
152	C₂	#19098633	1	1	yes	characteristic
153	C₂	#19098650	1	1	yes	characteristic
154	C₂	#19098684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098687

C₂

yes

maximal, characteristic, complemented

C₄×#11929613

C₂

yes

maximal, characteristic

#19098640

C₂

yes

maximal, characteristic

#19098293

C₂

yes

maximal, characteristic, complemented

#19098290

C₂

yes

maximal, characteristic

#19098285

C₂

yes

maximal, characteristic, complemented

#334×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929613

C₄

yes

characteristic

C₂×#11929607

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

C₄×D₃₁₆

C₂²

yes

characteristic

#14579093

C₄

yes

characteristic

Dic₇₉⋊C₈

—

#14579086

C₄

yes

characteristic

#14579083

C₄

yes

characteristic

(C₂×D₇₉)⋊C₈

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

C₂×C₄

yes

characteristic

C₂×D₃₁₆

C₂×C₄

yes

characteristic

C₂×C₄×D₇₉

—

#11929613

—

#11929607

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂×C₄

yes

characteristic

C₄⋊Dic₇₉

C₂×C₄

yes

characteristic

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

M₁₆

yes

complemented

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₇₉

C₄×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×M₁₆

yes

characteristic

C₄×C₇₉

#131

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₄×C₇₉

#135

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#435

yes

characteristic

C₂×C₇₉

C₄×SD₁₆

yes

characteristic

C₂×C₇₉

#444

yes

characteristic

D₇₉

—

#1583

—

maximal, Sylow

C₇₉

#1583

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×SD₁₆

—

#436

—

#334

D₇₉

yes

characteristic, complemented

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×SD₁₆

—

C₂×M₁₆

—

158

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

158

—

C₄⋊C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₈

C₂×D₇₉

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

158

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

158

—

100

C₄×C₈

C₂×D₇₉

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

158

—

104

SD₁₆

—

158

—

105

M₁₆

—

316

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

158

—

111

C₂×C₈

C₄×D₇₉

yes

characteristic

112

C₂×C₈

C₄×D₇₉

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

158

—

116

C₂²⋊C₄

—

158

—

117

C₄²

—

158

—

118

C₄²

C₂²×D₇₉

yes

characteristic

119

C₂³

—

158

—

120

Q₈

—

121

Q₈

—

158

—

122

D₄

—

158

—

123

D₄

—

158

—

124

C₂×C₄

—

316

—

125

C₂×C₄

—

316

—

126

C₂×C₄

—

158

—

127

C₂×C₄

—

158

—

128

C₂×C₄

—

158

—

129

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

130

C₂×C₄

D₄×D₇₉

yes

characteristic

131

C₂×C₄

#11929625

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

316

—

137

C₈

—

138

C₈

—

158

—

139

C₂²

—

316

—

140

C₂²

—

316

—

141

C₂²

—

316

—

142

C₂²

#14579123

yes

characteristic

143

C₄

—

144

C₄

—

316

—

145

C₄

—

316

—

146

C₄

—

158

—

147

C₄

M₁₆×D₇₉

yes

characteristic

148

C₄

#14579167

yes

characteristic

149

C₄

SD₁₆×D₇₉

yes

characteristic

150

C₄

#14579182

yes

characteristic

151

C₂

—

632

—

152

C₂

#19098633

yes

characteristic

153

C₂

#19098650

yes

characteristic

154

C₂

#19098684

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	#334×C₇₉	C₂
socle	C₂²×C₇₉	C₄×D₄
prime radical	C₂×D₃₁₆	C₂×C₄
cosocle	C₄²×C₇₉	C₂³
hypercenter	#334	D₇₉
hypocenter	C₇₉	#1583
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

#334×C₇₉

C₂

socle

C₂²×C₇₉

C₄×D₄

prime radical

C₂×D₃₁₆

C₂×C₄

cosocle

C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #19098687	C₂	semidirect
4. #19098293	C₂	semidirect
6. #19098285	C₂	semidirect
7. #334×C₇₉	C₂	semidirect
53. Dic₁₅₈	M₁₆	semidirect
60. C₇₉⋊C₈	SD₁₆	semidirect
78. C₇₉	#1583	semidirect
82. #334	D₇₉	semidirect

H ≅ G/N

type

1. #19098687

C₂

semidirect

4. #19098293

C₂

semidirect

6. #19098285

C₂

semidirect

7. #334×C₇₉

C₂

semidirect

53. Dic₁₅₈

M₁₆

semidirect

60. C₇₉⋊C₈

SD₁₆

semidirect

78. C₇₉

#1583

semidirect

82. #334

D₇₉

semidirect

Subgroups — #23762242

Conjugacy classes of subgroups

Complements