Group #23776414 — subgroups

Subgroups — #23776414

Order 10112.

384 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19099299	C₂	1	2	yes	maximal, complemented
2	#19098946	C₂	1	2	yes	maximal, complemented
3	#19098776	C₂	1	4	yes	maximal, complemented
4	#19098760	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₇₉	C₂	1	2	yes	maximal, complemented
8	#493×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
10	C₂×#11929612	C₂²	1	4	yes	—
11	#14579148	—	2	2	no	—
12	#14579148	C₂²	1	4	yes	—
13	#14579142	—	2	4	no	—
14	C₄×Dic₁₅₈	C₂²	1	2	yes	—
15	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
16	D₄×Dic₇₉	—	2	2	no	—
17	D₄×Dic₇₉	C₂²	1	4	yes	—
18	#14579095	C₂²	1	1	yes	characteristic
19	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₇₉⋊C₈	C₂²	1	2	yes	—
21	#14579086	C₂²	1	2	yes	—
22	#14579082	C₂²	1	4	yes	—
23	C₈×Dic₇₉	C₂²	1	2	yes	—
24	C₂×D₈×C₇₉	C₂²	1	2	yes	—
25	C₂×D₈×C₇₉	C₂²	1	2	yes	—
26	#127×C₇₉	C₂²	1	2	yes	—
27	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
29	C₂×Dic₁₅₈	C₂³	1	2	yes	—
30	C₂²×Dic₇₉	—	2	4	no	—
31	Dic₃₁₆	—	2	2	no	—
32	#11929612	—	4	4	no	—
33	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
34	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
35	C₄⋊Dic₇₉	C₂³	1	2	yes	—
36	Dic₇₉⋊C₄	—	2	2	no	—
37	Dic₇₉⋊C₄	D₄	1	2	yes	—
38	C₂²⋊Dic₇₉	—	2	4	no	—
39	C₂²⋊Dic₇₉	—	2	4	no	—
40	C₂²⋊Dic₇₉	—	2	4	no	—
41	C₄×Dic₇₉	—	2	2	no	—
42	C₄×Dic₇₉	C₂³	1	2	yes	—
43	C₂×D₄×C₇₉	—	2	2	no	—
44	C₂×D₄×C₇₉	—	2	2	no	—
45	C₂×D₄×C₇₉	C₂³	1	4	yes	—
46	D₈×C₇₉	—	2	4	no	—
47	D₈×C₇₉	—	4	2	no	—
48	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	C₄²×C₇₉	C₂³	1	1	yes	characteristic
51	C₂³×C₇₉	—	2	4	no	—
52	D₄×C₇₉	—	2	4	no	—
53	D₄×C₇₉	—	2	4	no	—
54	D₄×C₇₉	—	4	4	no	—
55	D₄×C₇₉	—	8	2	no	—
56	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₇₉	—	2	2	no	—
60	C₂×Dic₇₉	—	2	2	no	—
61	C₂×Dic₇₉	—	8	4	no	—
62	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
63	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
64	Dic₁₅₈	—	2	2	no	—
65	Dic₁₅₈	—	2	2	no	—
66	Dic₁₅₈	—	2	2	no	—
67	C₈×C₇₉	—	2	1	no	—
68	C₈×C₇₉	#41	1	2	yes	—
69	C₇₉⋊C₈	—	2	2	no	—
70	C₂²×C₇₉	—	4	4	no	—
71	C₂²×C₇₉	—	4	4	no	—
72	C₂²×C₇₉	—	4	4	no	—
73	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	#135	1	2	yes	—
76	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
77	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
78	Dic₇₉	—	2	2	no	—
79	Dic₇₉	—	2	2	no	—
80	Dic₇₉	—	4	2	no	—
81	Dic₇₉	—	4	2	no	—
82	C₂×C₇₉	—	8	4	no	—
83	C₂×C₇₉	#550	1	1	yes	characteristic
84	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
85	C₂×C₇₉	#576	1	1	yes	characteristic
86	#3396	—	79	1	no	maximal, Sylow
87	C₇₉	#3396	1	1	yes	characteristic, complemented, Sylow
88	#559	—	79	2	no	—
89	#495	—	79	2	no	—
90	#493	D₇₉	1	1	yes	characteristic
91	#463	—	79	4	no	—
92	#460	—	79	2	no	—
93	C₈×Q₈	—	79	1	no	—
94	C₄×Q₁₆	—	79	1	no	—
95	C₄×D₈	—	79	2	no	—
96	C₂×Q₁₆	—	79	1	no	—
97	C₂×SD₁₆	—	79	4	no	—
98	C₂×D₈	C₂×D₇₉	1	2	yes	—
99	C₂×D₈	C₂×D₇₉	1	2	yes	—
100	#127	C₂×D₇₉	1	2	yes	—
101	#124	—	79	4	no	—
102	#124	—	158	2	no	—
103	#123	—	158	4	no	—
104	C₄×Q₈	—	79	1	no	—
105	C₄×Q₈	—	79	2	no	—
106	C₄×D₄	—	79	4	no	—
107	C₄×D₄	—	158	2	no	—
108	#107	—	79	1	no	—
109	C₄⋊C₈	—	79	2	no	—
110	C₄⋊C₈	—	79	1	no	—
111	#103	—	79	2	no	—
112	#102	—	79	4	no	—
113	C₄×C₈	—	79	2	no	—
114	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
115	C₂×Q₈	—	79	1	no	—
116	C₂×Q₈	—	79	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	C₂²×D₇₉	1	4	yes	—
120	C₂²×C₄	—	158	4	no	—
121	Q₁₆	—	158	2	no	—
122	SD₁₆	—	316	4	no	—
123	D₈	—	2	4	no	—
124	D₈	—	4	2	no	—
125	C₂×C₈	—	79	2	no	—
126	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
128	C₄⋊C₄	—	79	2	no	—
129	C₄⋊C₄	—	79	1	no	—
130	C₄⋊C₄	—	79	2	no	—
131	C₄⋊C₄	—	158	2	no	—
132	C₂²⋊C₄	—	158	4	no	—
133	C₂²⋊C₄	—	158	4	no	—
134	C₂²⋊C₄	—	158	4	no	—
135	C₄²	—	79	2	no	—
136	C₄²	—	158	2	no	—
137	C₄²	C₂²×D₇₉	1	1	yes	characteristic
138	C₂³	—	2	4	no	—
139	Q₈	—	158	2	no	—
140	Q₈	—	158	2	no	—
141	Q₈	—	158	2	no	—
142	D₄	—	2	4	no	—
143	D₄	—	2	4	no	—
144	D₄	—	4	4	no	—
145	D₄	—	8	2	no	—
146	C₂×C₄	—	79	2	no	—
147	C₂×C₄	—	79	2	no	—
148	C₂×C₄	—	632	4	no	—
149	C₂×C₄	—	158	2	no	—
150	C₂×C₄	—	158	2	no	—
151	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
152	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	158	2	no	—
156	C₈	#11929623	1	2	yes	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	316	2	no	—
163	C₄	—	316	2	no	—
164	C₄	—	158	2	no	—
165	C₄	—	158	2	no	—
166	C₄	#14579180	1	2	yes	—
167	C₄	C₂×#11929623	1	1	yes	characteristic
168	C₄	#14579206	1	1	yes	characteristic
169	C₂	—	8	4	no	—
170	C₂	#19099233	1	1	yes	characteristic
171	C₂	C₂×#14579180	1	1	yes	characteristic
172	C₂	#19099364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099299

C₂

yes

maximal, complemented

#19098946

C₂

yes

maximal, complemented

#19098776

C₂

yes

maximal, complemented

#19098760

C₂

yes

maximal, complemented

C₈×Dic₁₅₈

C₂

yes

maximal, characteristic, complemented

C₄×Dic₃₁₆

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₉

C₂

yes

maximal, complemented

#493×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929612

C₂²

yes

—

#14579148

—

#14579148

C₂²

yes

—

#14579142

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

D₄×Dic₇₉

—

D₄×Dic₇₉

C₂²

yes

—

#14579095

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

Dic₇₉⋊C₈

C₂²

yes

—

#14579086

C₂²

yes

—

#14579082

C₂²

yes

—

C₈×Dic₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

—

#127×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

#11929612

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

D₄

yes

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂³

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#559

—

#495

—

#493

D₇₉

yes

characteristic

#463

—

#460

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×D₈

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

C₂×D₇₉

yes

—

C₂×D₈

C₂×D₇₉

yes

—

100

#127

C₂×D₇₉

yes

—

101

#124

—

102

#124

—

158

—

103

#123

—

158

—

104

C₄×Q₈

—

105

C₄×Q₈

—

106

C₄×D₄

—

107

C₄×D₄

—

158

—

108

#107

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

112

#102

—

113

C₄×C₈

—

114

C₄×C₈

C₂×D₇₉

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

C₂²×D₇₉

yes

—

120

C₂²×C₄

—

158

—

121

Q₁₆

—

158

—

122

SD₁₆

—

316

—

123

D₈

—

124

D₈

—

125

C₂×C₈

—

126

C₂×C₈

C₂²×D₇₉

yes

characteristic

127

C₂×C₈

C₂²×D₇₉

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

158

—

132

C₂²⋊C₄

—

158

—

133

C₂²⋊C₄

—

158

—

134

C₂²⋊C₄

—

158

—

135

C₄²

—

136

C₄²

—

158

—

137

C₄²

C₂²×D₇₉

yes

characteristic

138

C₂³

—

139

Q₈

—

158

—

140

Q₈

—

158

—

141

Q₈

—

158

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

632

—

149

C₂×C₄

—

158

—

150

C₂×C₄

—

158

—

151

C₂×C₄

D₄×D₇₉

yes

characteristic

152

C₂×C₄

D₄×D₇₉

yes

characteristic

153

C₂×C₄

C₂³×D₇₉

yes

characteristic

154

C₈

—

155

C₈

—

158

—

156

C₈

#11929623

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂×D₄×D₇₉

yes

characteristic

161

C₄

—

162

C₄

—

316

—

163

C₄

—

316

—

164

C₄

—

158

—

165

C₄

—

158

—

166

C₄

#14579180

yes

—

167

C₄

C₂×#11929623

yes

characteristic

168

C₄

#14579206

yes

characteristic

169

C₂

—

170

C₂

#19099233

yes

characteristic

171

C₂

C₂×#14579180

yes

characteristic

172

C₂

#19099364

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting, prime radical	#493×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#493	D₇₉
hypocenter	C₇₉	#3396
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting, prime radical

#493×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #19099299	C₂	semidirect
2. #19098946	C₂	semidirect
3. #19098776	C₂	semidirect
4. #19098760	C₂	semidirect
5. C₈×Dic₁₅₈	C₂	semidirect
6. C₄×Dic₃₁₆	C₂	semidirect
7. D₈×Dic₇₉	C₂	semidirect
87. C₇₉	#3396	semidirect

H ≅ G/N

type

1. #19099299

C₂

semidirect

2. #19098946

C₂

semidirect

3. #19098776

C₂

semidirect

4. #19098760

C₂

semidirect

5. C₈×Dic₁₅₈

C₂

semidirect

6. C₄×Dic₃₁₆

C₂

semidirect

7. D₈×Dic₇₉

C₂

semidirect

87. C₇₉

#3396

semidirect

Subgroups — #23776414

Conjugacy classes of subgroups

Complements