Group #23776386 — subgroups

Subgroups — #23776386

Order 10112.

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

normal subgroups only

This group has 360 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	#19099283	C₂	1	2	yes	maximal, complemented
2	#19098946	C₂	1	2	yes	maximal
3	#19098943	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098784	C₂	1	4	yes	maximal, complemented
5	#19098768	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098657	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₁₆	C₂²	1	2	yes	—
10	C₂×Dic₃₁₆	C₂²	1	2	yes	—
11	C₂×#11929612	C₂²	1	4	yes	—
12	#14579161	C₂²	1	2	yes	—
13	#14579154	—	2	4	no	—
14	#14579148	C₂²	1	4	yes	—
15	#14579140	—	2	2	no	—
16	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
17	#14579126	C₂²	1	4	yes	—
18	#14579120	—	2	2	no	—
19	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₇₉⋊C₈	C₂²	1	2	yes	—
21	#14579084	C₂²	1	4	yes	—
22	C₈×Dic₇₉	C₂²	1	2	yes	—
23	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₇₉	C₂²	1	2	yes	—
25	#107×C₇₉	C₂²	1	1	yes	characteristic
26	#102×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	4	yes	—
30	Dic₃₁₆	—	2	4	no	—
31	Dic₃₁₆	—	4	2	no	—
32	#11929612	—	4	4	no	—
33	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
34	C₄⋊Dic₇₉	—	2	2	no	—
35	C₄⋊Dic₇₉	—	2	2	no	—
36	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
37	Dic₇₉⋊C₄	—	2	4	no	—
38	Dic₇₉⋊C₄	D₄	1	2	yes	—
39	C₂²⋊Dic₇₉	—	2	4	no	—
40	C₂²⋊Dic₇₉	—	2	4	no	—
41	C₄×Dic₇₉	—	2	4	no	—
42	C₄×Dic₇₉	C₂³	1	2	yes	—
43	C₂×D₄×C₇₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₇₉	—	2	2	no	—
45	D₈×C₇₉	—	2	2	no	—
46	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
49	C₂²⋊C₄×C₇₉	—	2	2	no	—
50	C₄²×C₇₉	C₂³	1	1	yes	characteristic
51	C₂³×C₇₉	—	2	2	no	—
52	D₄×C₇₉	—	2	2	no	—
53	D₄×C₇₉	—	2	2	no	—
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	—	4	4	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	4	no	—
60	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
61	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
62	Dic₁₅₈	—	2	2	no	—
63	Dic₁₅₈	—	2	4	no	—
64	Dic₁₅₈	—	2	4	no	—
65	C₈×C₇₉	—	2	1	no	—
66	C₈×C₇₉	#41	1	2	yes	—
67	C₇₉⋊C₈	—	2	2	no	—
68	C₂²×C₇₉	—	4	2	no	—
69	C₂²×C₇₉	—	4	2	no	—
70	C₂²×C₇₉	—	4	2	no	—
71	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	—	4	2	no	—
74	C₄×C₇₉	#135	1	2	yes	—
75	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
76	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
77	Dic₇₉	—	2	2	no	—
78	Dic₇₉	—	2	2	no	—
79	Dic₇₉	—	4	4	no	—
80	C₂×C₇₉	—	8	2	no	—
81	C₂×C₇₉	#550	1	1	yes	characteristic
82	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
83	C₂×C₇₉	#578	1	1	yes	characteristic
84	#3393	—	79	1	no	maximal, Sylow
85	C₇₉	#3393	1	1	yes	characteristic, complemented, Sylow
86	#556	—	79	2	no	—
87	#495	—	79	2	no	—
88	#494	—	79	1	no	—
89	#464	—	79	4	no	—
90	#461	—	79	2	no	—
91	C₈×Q₈	—	79	1	no	—
92	C₄×Q₁₆	—	79	2	no	—
93	C₄×D₈	D₇₉	1	1	yes	characteristic
94	C₂×Q₁₆	—	79	2	no	—
95	C₂×Q₁₆	—	79	2	no	—
96	C₂×SD₁₆	—	79	4	no	—
97	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
98	C₄⋊Q₈	—	79	2	no	—
99	#126	—	158	4	no	—
100	#124	—	79	4	no	—
101	C₂²⋊Q₈	—	158	2	no	—
102	C₄×Q₈	—	79	1	no	—
103	C₄×Q₈	—	79	4	no	—
104	C₄×D₄	C₂×D₇₉	1	2	yes	—
105	#117	—	158	2	no	—
106	#107	C₂×D₇₉	1	1	yes	characteristic
107	C₄⋊C₈	—	79	2	no	—
108	C₄⋊C₈	—	79	1	no	—
109	#103	—	79	4	no	—
110	#102	C₂×D₇₉	1	2	yes	—
111	C₄×C₈	—	79	2	no	—
112	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
113	C₂×Q₈	—	79	1	no	—
114	C₂×Q₈	—	79	4	no	—
115	C₂×D₄	C₂²×D₇₉	1	2	yes	—
116	C₂²×C₄	—	2	2	no	—
117	Q₁₆	—	316	2	no	—
118	Q₁₆	—	158	4	no	—
119	SD₁₆	—	316	4	no	—
120	D₈	—	2	2	no	—
121	C₂×C₈	—	79	2	no	—
122	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
124	C₄⋊C₄	—	79	2	no	—
125	C₄⋊C₄	—	79	1	no	—
126	C₄⋊C₄	—	158	4	no	—
127	C₄⋊C₄	—	158	2	no	—
128	C₄⋊C₄	—	158	2	no	—
129	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
130	C₂²⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	158	4	no	—
132	C₂²⋊C₄	—	158	4	no	—
133	C₄²	—	79	2	no	—
134	C₄²	—	158	4	no	—
135	C₄²	C₂²×D₇₉	1	1	yes	characteristic
136	C₂³	—	2	2	no	—
137	Q₈	—	158	4	no	—
138	Q₈	—	158	4	no	—
139	Q₈	—	158	2	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	79	2	no	—
145	C₂×C₄	—	79	2	no	—
146	C₂×C₄	—	158	4	no	—
147	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
148	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	158	2	no	—
152	C₈	#11929624	1	2	yes	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	2	no	—
156	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	316	4	no	—
160	C₄	—	158	2	no	—
161	C₄	—	158	2	no	—
162	C₄	#14579180	1	2	yes	—
163	C₄	C₂×#11929624	1	1	yes	characteristic
164	C₄	#14579207	1	1	yes	characteristic
165	C₂	—	8	2	no	—
166	C₂	#19099234	1	1	yes	characteristic
167	C₂	C₂×#14579180	1	1	yes	characteristic
168	C₂	#19099377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099283

C₂

yes

maximal, complemented

#19098946

C₂

yes

maximal

#19098943

C₂

yes

maximal, characteristic, complemented

#19098784

C₂

yes

maximal, complemented

#19098768

C₂

yes

maximal, complemented

C₈×Dic₁₅₈

C₂

yes

maximal, characteristic, complemented

#19098657

C₂

yes

maximal, complemented

C₄×D₈×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

—

C₂×Dic₃₁₆

C₂²

yes

—

C₂×#11929612

C₂²

yes

—

#14579161

C₂²

yes

—

#14579154

—

#14579148

C₂²

yes

—

#14579140

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

#14579126

C₂²

yes

—

#14579120

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

Dic₇₉⋊C₈

C₂²

yes

—

#14579084

C₂²

yes

—

C₈×Dic₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₄×D₄×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

characteristic

#102×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×Dic₁₅₈

C₂³

yes

—

Dic₃₁₆

—

Dic₃₁₆

—

#11929612

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

D₄

yes

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂³

yes

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×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×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

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×D₈

D₇₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

—

#126

—

158

—

100

#124

—

101

C₂²⋊Q₈

—

158

—

102

C₄×Q₈

—

103

C₄×Q₈

—

104

C₄×D₄

C₂×D₇₉

yes

—

105

#117

—

158

—

106

#107

C₂×D₇₉

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

#103

—

110

#102

C₂×D₇₉

yes

—

111

C₄×C₈

—

112

C₄×C₈

C₂×D₇₉

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×D₄

C₂²×D₇₉

yes

—

116

C₂²×C₄

—

117

Q₁₆

—

316

—

118

Q₁₆

—

158

—

119

SD₁₆

—

316

—

120

D₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂²×D₇₉

yes

characteristic

123

C₂×C₈

C₂²×D₇₉

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

158

—

127

C₄⋊C₄

—

158

—

128

C₄⋊C₄

—

158

—

129

C₄⋊C₄

C₂²×D₇₉

yes

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

158

—

132

C₂²⋊C₄

—

158

—

133

C₄²

—

134

C₄²

—

158

—

135

C₄²

C₂²×D₇₉

yes

characteristic

136

C₂³

—

137

Q₈

—

158

—

138

Q₈

—

158

—

139

Q₈

—

158

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

158

—

147

C₂×C₄

D₄×D₇₉

yes

characteristic

148

C₂×C₄

D₄×D₇₉

yes

characteristic

149

C₂×C₄

C₂³×D₇₉

yes

characteristic

150

C₈

—

151

C₈

—

158

—

152

C₈

#11929624

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₂×D₄×D₇₉

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

316

—

160

C₄

—

158

—

161

C₄

—

158

—

162

C₄

#14579180

yes

—

163

C₄

C₂×#11929624

yes

characteristic

164

C₄

#14579207

yes

characteristic

165

C₂

—

166

C₂

#19099234

yes

characteristic

167

C₂

C₂×#14579180

yes

characteristic

168

C₂

#19099377

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₄×D₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×D₈×C₇₉	C₂²
hypercenter	C₄×D₈	D₇₉
hypocenter	C₇₉	#3393
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₄×D₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂×D₈×C₇₉

C₂²

hypercenter

C₄×D₈

D₇₉

hypocenter

C₇₉

#3393

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #19099283	C₂	semidirect
3. #19098943	C₂	semidirect
4. #19098784	C₂	semidirect
5. #19098768	C₂	semidirect
6. C₈×Dic₁₅₈	C₂	semidirect
7. #19098657	C₂	semidirect
85. C₇₉	#3393	semidirect

H ≅ G/N

type

1. #19099283

C₂

semidirect

3. #19098943

C₂

semidirect

4. #19098784

C₂

semidirect

5. #19098768

C₂

semidirect

6. C₈×Dic₁₅₈

C₂

semidirect

7. #19098657

C₂

semidirect

85. C₇₉

#3393

semidirect

Subgroups — #23776386

Conjugacy classes of subgroups

Complements