Group #23764286 — subgroups

Subgroups — #23764286

Order 10112.

300 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14579183	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579087	C₂	1	2	yes	maximal, complemented
3	#19098531	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#108×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11929624	C₂²	1	2	yes	—
7	#14579183	—	2	2	no	—
8	#14579183	—	2	2	no	—
9	C₂×Dic₃₁₆	C₄	1	1	yes	characteristic, complemented
10	C₂×#11929613	C₄	1	2	yes	complemented
11	C₂×D₆₃₂	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
14	#14579087	—	2	4	no	—
15	(C₂×D₇₉)⋊C₈	—	2	4	no	—
16	C₈×Dic₇₉	—	2	2	no	—
17	C₂×M₁₆×C₇₉	C₂²	1	2	yes	—
18	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#108×C₇₉	—	2	2	no	—
20	#11929624	—	2	2	no	—
21	#11929624	—	2	2	no	—
22	#11929624	—	2	2	no	—
23	#11929624	—	2	2	no	—
24	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
25	C₂×D₃₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₂²⋊D₇₉	—	2	2	no	—
27	C₂²×Dic₇₉	—	2	1	no	—
28	C₂×C₄×D₇₉	—	2	2	no	—
29	Dic₃₁₆	—	2	2	no	—
30	#11929613	—	2	4	no	—
31	D₆₃₂	—	2	2	no	—
32	C₂×C₇₉⋊C₈	—	2	1	no	—
33	C₂×C₇₉⋊C₈	—	2	1	no	—
34	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
35	C₄×Dic₇₉	—	2	1	no	—
36	C₄×Dic₇₉	D₄	1	2	yes	—
37	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	M₁₆×C₇₉	—	2	2	no	—
39	M₁₆×C₇₉	—	2	2	no	—
40	C₂×C₈×C₇₉	—	2	2	no	—
41	C₂×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	D₄	1	2	yes	—
44	C₂×C₈×C₇₉	D₄	1	2	yes	—
45	C₂³×C₇₉	#41	1	1	yes	characteristic
46	C₂²×D₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
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	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
53	C₂²⋊D₇₉	—	4	2	no	—
54	C₂²⋊D₇₉	—	4	2	no	—
55	C₂×Dic₇₉	—	2	4	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	—	2	2	no	—
58	D₃₁₆	—	2	2	no	—
59	D₃₁₆	—	2	2	no	—
60	C₄×D₇₉	—	4	2	no	—
61	C₄×D₇₉	—	4	2	no	—
62	Dic₁₅₈	—	2	2	no	—
63	Dic₁₅₈	—	2	2	no	—
64	C₈×C₇₉	—	4	2	no	—
65	C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
66	C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
67	C₇₉⋊C₈	—	2	2	no	—
68	C₂²×C₇₉	—	2	2	no	—
69	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
70	C₂²×C₇₉	#121	1	1	yes	characteristic
71	C₂²×C₇₉	#124	1	1	yes	characteristic
72	C₂×D₇₉	—	4	2	no	—
73	C₂×D₇₉	—	4	4	no	—
74	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
76	C₄×C₇₉	#121	1	1	yes	characteristic
77	C₄×C₇₉	#127	1	1	yes	characteristic
78	Dic₇₉	—	2	4	no	—
79	Dic₇₉	—	4	2	no	—
80	C₂×C₇₉	—	2	1	no	—
81	C₂×C₇₉	—	2	1	no	—
82	C₂×C₇₉	#390	1	1	yes	characteristic
83	C₂×C₇₉	#443	1	2	yes	—
84	D₇₉	—	8	2	no	—
85	#1974	—	79	1	no	maximal, Sylow
86	C₇₉	#1974	1	1	yes	characteristic, complemented, Sylow
87	C₂×#135	—	79	1	no	—
88	C₂×#108	D₇₉	1	1	yes	characteristic, complemented
89	C₂×#104	—	79	2	no	—
90	#408	—	79	2	no	—
91	C₂×C₄×C₈	—	79	1	no	—
92	C₂×#41	—	79	2	no	—
93	#135	—	158	2	no	—
94	#135	—	158	2	no	—
95	C₂×Q₁₆	—	79	1	no	—
96	C₂×SD₁₆	—	79	2	no	—
97	C₂×D₈	—	79	1	no	—
98	C₂×M₁₆	C₂×D₇₉	1	2	yes	—
99	C₂²×C₈	—	79	1	no	—
100	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
101	C₂×C₄²	—	79	1	no	—
102	#108	—	2	2	no	—
103	#104	—	158	4	no	—
104	C₂²⋊C₈	—	158	4	no	—
105	C₄×C₈	—	158	2	no	—
106	#41	—	158	2	no	—
107	#41	—	158	2	no	—
108	#41	—	158	2	no	—
109	#41	—	158	2	no	—
110	C₂×Q₈	—	79	2	no	—
111	C₂×D₄	—	79	2	no	—
112	C₂×D₄	—	158	2	no	—
113	C₂²×C₄	—	158	2	no	—
114	C₂²×C₄	—	158	1	no	—
115	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
116	Q₁₆	—	158	2	no	—
117	SD₁₆	—	158	4	no	—
118	D₈	—	158	2	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	79	2	no	—
123	C₂×C₈	—	158	1	no	—
124	C₂×C₈	—	158	1	no	—
125	C₂×C₈	C₄×D₇₉	1	1	yes	characteristic
126	C₂×C₈	C₄×D₇₉	1	1	yes	characteristic
127	C₂×C₈	D₃₁₆	1	2	yes	—
128	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
129	C₄²	—	79	2	no	—
130	C₄²	—	158	1	no	—
131	C₂³	—	158	2	no	—
132	C₂³	#11929625	1	1	yes	characteristic
133	Q₈	—	158	2	no	—
134	Q₈	—	158	2	no	—
135	D₄	—	316	2	no	—
136	D₄	—	316	2	no	—
137	D₄	—	158	2	no	—
138	D₄	—	158	2	no	—
139	C₂×C₄	—	316	2	no	—
140	C₂×C₄	—	316	2	no	—
141	C₂×C₄	—	158	2	no	—
142	C₂×C₄	—	158	4	no	—
143	C₂×C₄	—	158	2	no	—
144	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
145	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
146	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
148	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
149	C₂×C₄	#11929625	1	1	yes	characteristic
150	C₈	—	4	2	no	—
151	C₈	—	158	2	no	—
152	C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
153	C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	316	4	no	—
156	C₂²	—	316	2	no	—
157	C₂²	#14579123	1	1	yes	characteristic
158	C₂²	#14579136	1	1	yes	characteristic
159	C₂²	#14579150	1	1	yes	characteristic
160	C₄	—	316	2	no	—
161	C₄	—	158	4	no	—
162	C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
163	C₄	#14579123	1	1	yes	characteristic
164	C₄	#14579136	1	1	yes	characteristic
165	C₄	#14579157	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	632	2	no	—
169	C₂	#19098464	1	1	yes	characteristic
170	C₂	#19098679	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579183

C₂

yes

maximal, characteristic

C₂×#14579087

C₂

yes

maximal, complemented

#19098531

C₂

yes

maximal, complemented

C₂×C₈×Dic₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#108×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

C₂²

yes

—

#14579183

—

#14579183

—

C₂×Dic₃₁₆

C₄

yes

characteristic, complemented

C₂×#11929613

C₄

yes

complemented

C₂×D₆₃₂

C₄

yes

characteristic, complemented

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579087

—

(C₂×D₇₉)⋊C₈

—

C₈×Dic₇₉

—

C₂×M₁₆×C₇₉

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#108×C₇₉

—

—

—

—

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×D₃₁₆

C₂×C₄

yes

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

—

Dic₃₁₆

—

#11929613

—

D₆₃₂

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂³×C₇₉

#41

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊C₄

yes

—

C₈×C₇₉

C₂²⋊C₄

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₄×D₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#390

yes

characteristic

C₂×C₇₉

#443

yes

—

D₇₉

—

#1974

—

maximal, Sylow

C₇₉

#1974

yes

characteristic, complemented, Sylow

C₂×#135

—

C₂×#108

D₇₉

yes

characteristic, complemented

C₂×#104

—

#408

—

C₂×C₄×C₈

—

C₂×#41

—

#135

—

158

—

#135

—

158

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

C₂×D₇₉

yes

—

C₂²×C₈

—

100

C₂²×C₈

C₂×D₇₉

yes

characteristic

101

C₂×C₄²

—

102

#108

—

103

#104

—

158

—

104

C₂²⋊C₈

—

158

—

105

C₄×C₈

—

158

—

106

#41

—

158

—

107

#41

—

158

—

108

#41

—

158

—

109

#41

—

158

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

158

—

113

C₂²×C₄

—

158

—

114

C₂²×C₄

—

158

—

115

C₂²×C₄

C₂²×D₇₉

yes

characteristic

116

Q₁₆

—

158

—

117

SD₁₆

—

158

—

118

D₈

—

158

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

158

—

124

C₂×C₈

—

158

—

125

C₂×C₈

C₄×D₇₉

yes

characteristic

126

C₂×C₈

C₄×D₇₉

yes

characteristic

127

C₂×C₈

D₃₁₆

yes

—

128

C₂×C₈

C₂²⋊D₇₉

yes

—

129

C₄²

—

130

C₄²

—

158

—

131

C₂³

—

158

—

132

C₂³

#11929625

yes

characteristic

133

Q₈

—

158

—

134

Q₈

—

158

—

135

D₄

—

316

—

136

D₄

—

316

—

137

D₄

—

158

—

138

D₄

—

158

—

139

C₂×C₄

—

316

—

140

C₂×C₄

—

316

—

141

C₂×C₄

—

158

—

142

C₂×C₄

—

158

—

143

C₂×C₄

—

158

—

144

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

145

C₂×C₄

D₄×D₇₉

yes

characteristic

146

C₂×C₄

D₄×D₇₉

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

148

C₂×C₄

C₂×D₃₁₆

yes

characteristic

149

C₂×C₄

#11929625

yes

characteristic

150

C₈

—

151

C₈

—

158

—

152

C₈

(C₂×D₇₉)⋊C₄

yes

—

153

C₈

(C₂×D₇₉)⋊C₄

yes

—

154

C₂²

—

155

C₂²

—

316

—

156

C₂²

—

316

—

157

C₂²

#14579123

yes

characteristic

158

C₂²

#14579136

yes

characteristic

159

C₂²

#14579150

yes

characteristic

160

C₄

—

316

—

161

C₄

—

158

—

162

C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

163

C₄

#14579123

yes

characteristic

164

C₄

#14579136

yes

characteristic

165

C₄

#14579157

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

632

—

169

C₂

#19098464

yes

characteristic

170

C₂

#19098679

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×#108×C₇₉	C₂
socle	C₂²×C₇₉	C₄×D₄
prime radical	C₂×#14579183	C₂
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×#108	D₇₉
hypocenter	C₇₉	#1974
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×#108×C₇₉

C₂

socle

C₂²×C₇₉

C₄×D₄

prime radical

C₂×#14579183

C₂

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. C₂×#14579087	C₂	semidirect
3. #19098531	C₂	semidirect
4. C₂×C₈×Dic₇₉	C₂	semidirect
5. C₂×#108×C₇₉	C₂	semidirect
9. C₂×Dic₃₁₆	C₄	semidirect
10. C₂×#11929613	C₄	semidirect
11. C₂×D₆₃₂	C₄	semidirect
86. C₇₉	#1974	semidirect
88. C₂×#108	D₇₉	semidirect

H ≅ G/N

type

2. C₂×#14579087

C₂

semidirect

3. #19098531

C₂

semidirect

4. C₂×C₈×Dic₇₉

C₂

semidirect

5. C₂×#108×C₇₉

C₂

semidirect

9. C₂×Dic₃₁₆

C₄

semidirect

10. C₂×#11929613

C₄

semidirect

11. C₂×D₆₃₂

C₄

semidirect

86. C₇₉

#1974

semidirect

88. C₂×#108

D₇₉

semidirect

Subgroups — #23764286

Conjugacy classes of subgroups

Complements