Group #23764862 — subgroups

Subgroups — #23764862

Order 10112.

204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19098611	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098567	C₂	1	1	yes	maximal, characteristic
3	#19098567	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579077	C₂	1	1	yes	maximal, characteristic
5	#19098333	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098333	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,ℤ/4)×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₁₆	C₂²	1	1	yes	characteristic
9	C₂×#11929608	C₂²	1	1	yes	characteristic
10	C₂×#11929608	C₂²	1	1	yes	characteristic
11	C₂×#11929608	C₂²	1	1	yes	characteristic
12	#14579096	—	2	2	no	—
13	#14579081	—	2	2	no	—
14	#14579081	—	2	2	no	—
15	#14579077	—	2	2	no	—
16	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	#117×C₇₉	C₂²	1	1	yes	characteristic
18	#117×C₇₉	C₂²	1	1	yes	characteristic
19	C₂³×D₇₉	—	2	1	no	—
20	C₂×D₃₁₆	—	2	1	no	—
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	D₄	1	2	yes	—
26	#11929608	—	2	1	no	—
27	#11929608	—	2	1	no	—
28	#11929608	—	2	1	no	—
29	#11929608	—	2	1	no	—
30	#11929608	—	2	1	no	—
31	#11929608	—	2	1	no	—
32	C₂×C₇₉⋊C₈	—	2	1	no	—
33	C₂×C₇₉⋊C₈	—	2	1	no	—
34	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₇₉	—	2	1	no	—
37	M₁₆×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₄⋊C₄×C₇₉	—	2	1	no	—
41	C₂²⋊C₄×C₇₉	—	2	1	no	—
42	C₂²⋊C₄×C₇₉	—	2	1	no	—
43	C₄²×C₇₉	—	2	1	no	—
44	C₄²×C₇₉	—	2	1	no	—
45	C₂³×C₇₉	#41	1	1	yes	characteristic
46	C₂²×D₇₉	—	2	2	no	—
47	C₂²×D₇₉	—	4	1	no	—
48	C₂²×D₇₉	—	4	1	no	—
49	C₂²×D₇₉	—	8	2	no	—
50	C₂×C₄×C₇₉	—	4	1	no	—
51	C₂×C₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
58	D₃₁₆	—	2	2	no	—
59	D₃₁₆	—	2	2	no	—
60	D₃₁₆	—	4	1	no	—
61	D₃₁₆	—	4	1	no	—
62	C₈×C₇₉	—	4	1	no	—
63	C₇₉⋊C₈	—	2	2	no	—
64	C₇₉⋊C₈	—	4	1	no	—
65	C₇₉⋊C₈	—	4	1	no	—
66	C₂²×C₇₉	—	4	1	no	—
67	C₂²×C₇₉	#123	1	1	yes	characteristic
68	C₂²×C₇₉	#124	1	1	yes	characteristic
69	C₂²×C₇₉	#126	1	1	yes	characteristic
70	C₂×D₇₉	—	4	2	no	—
71	C₂×D₇₉	—	8	2	no	—
72	C₂×D₇₉	—	8	2	no	—
73	C₂×D₇₉	—	8	2	no	—
74	C₄×C₇₉	—	8	1	no	—
75	C₄×C₇₉	—	8	1	no	—
76	C₄×C₇₉	#121	1	1	yes	characteristic
77	C₄×C₇₉	#123	1	1	yes	characteristic
78	C₄×C₇₉	#123	1	1	yes	characteristic
79	C₄×C₇₉	#126	1	1	yes	characteristic
80	C₂×C₇₉	—	2	1	no	—
81	C₂×C₇₉	—	2	1	no	—
82	C₂×C₇₉	—	2	1	no	—
83	C₂×C₇₉	#397	1	1	yes	characteristic
84	D₇₉	—	8	2	no	—
85	#2081	—	79	1	no	maximal, Sylow
86	C₇₉	#2081	1	1	yes	characteristic, complemented, Sylow
87	#430	—	79	1	no	—
88	#418	—	79	1	no	—
89	#418	—	79	1	no	—
90	C₂×#100	—	79	1	no	—
91	#344	—	79	1	no	—
92	#344	—	79	1	no	—
93	H(3,ℤ/4)	D₇₉	1	1	yes	characteristic, complemented
94	C₂²×D₄	—	79	1	no	—
95	C₂×M₁₆	—	79	1	no	—
96	C₂×M₁₆	—	79	1	no	—
97	C₂×M₁₆	—	79	1	no	—
98	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
99	#117	C₂×D₇₉	1	1	yes	characteristic
100	#117	C₂×D₇₉	1	1	yes	characteristic
101	#108	—	158	2	no	—
102	#102	—	158	2	no	—
103	#102	—	158	2	no	—
104	#100	—	158	2	no	—
105	C₂⁴	—	158	1	no	—
106	C₂×D₄	—	79	2	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	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	158	1	no	—
116	M₁₆	—	158	1	no	—
117	M₁₆	—	158	1	no	—
118	M₁₆	—	158	1	no	—
119	M₁₆	—	158	1	no	—
120	M₁₆	—	158	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	79	2	no	—
123	C₂×C₈	—	158	1	no	—
124	C₂×C₈	—	158	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	2	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₂³	—	316	1	no	—
132	C₂³	—	316	1	no	—
133	C₂³	—	632	2	no	—
134	C₂³	—	158	2	no	—
135	C₂³	#11929623	1	1	yes	characteristic
136	D₄	—	316	1	no	—
137	D₄	—	316	1	no	—
138	D₄	—	158	2	no	—
139	D₄	—	158	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
144	C₂×C₄	#11929624	1	1	yes	characteristic
145	C₂×C₄	#11929624	1	1	yes	characteristic
146	C₂×C₄	#11929624	1	1	yes	characteristic
147	C₂×C₄	#11929625	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	316	1	no	—
150	C₈	—	316	1	no	—
151	C₈	—	158	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	316	2	no	—
154	C₂²	—	632	2	no	—
155	C₂²	—	632	2	no	—
156	C₂²	—	632	2	no	—
157	C₂²	#14579143	1	1	yes	characteristic
158	C₂²	#14579147	1	1	yes	characteristic
159	C₂²	#14579156	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	8	1	no	—
162	C₄	#14579134	1	1	yes	characteristic
163	C₄	#14579146	1	1	yes	characteristic
164	C₄	#14579146	1	1	yes	characteristic
165	C₄	#14579155	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	632	2	no	—
170	C₂	#19098492	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098611

C₂

yes

maximal, characteristic, complemented

#19098567

C₂

yes

maximal, characteristic

#19098567

C₂

yes

maximal, characteristic

C₂×#14579077

C₂

yes

maximal, characteristic

#19098333

C₂

yes

maximal, characteristic, complemented

#19098333

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₁₆

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

—

—

—

—

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

D₄

yes

—

—

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

#41

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#397

yes

characteristic

D₇₉

—

#2081

—

maximal, Sylow

C₇₉

#2081

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

characteristic

#117

C₂×D₇₉

yes

characteristic

100

#117

C₂×D₇₉

yes

characteristic

101

#108

—

158

—

102

#102

—

158

—

103

#102

—

158

—

104

#100

—

158

—

105

C₂⁴

—

158

—

106

C₂×D₄

—

107

C₂×D₄

—

158

—

108

C₂×D₄

—

158

—

109

C₂×D₄

—

158

—

110

C₂×D₄

—

158

—

111

C₂×D₄

—

158

—

112

C₂²×C₄

C₂²×D₇₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

158

—

116

M₁₆

—

158

—

117

M₁₆

—

158

—

118

M₁₆

—

158

—

119

M₁₆

—

158

—

120

M₁₆

—

158

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

158

—

124

C₂×C₈

—

158

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₂³

—

316

—

132

C₂³

—

316

—

133

C₂³

—

632

—

134

C₂³

—

158

—

135

C₂³

#11929623

yes

characteristic

136

D₄

—

316

—

137

D₄

—

316

—

138

D₄

—

158

—

139

D₄

—

158

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

D₄×D₇₉

yes

characteristic

143

C₂×C₄

D₄×D₇₉

yes

characteristic

144

C₂×C₄

#11929624

yes

characteristic

145

C₂×C₄

#11929624

yes

characteristic

146

C₂×C₄

#11929624

yes

characteristic

147

C₂×C₄

#11929625

yes

characteristic

148

C₈

—

149

C₈

—

316

—

150

C₈

—

316

—

151

C₈

—

158

—

152

C₂²

—

153

C₂²

—

316

—

154

C₂²

—

632

—

155

C₂²

—

632

—

156

C₂²

—

632

—

157

C₂²

#14579143

yes

characteristic

158

C₂²

#14579147

yes

characteristic

159

C₂²

#14579156

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

#14579134

yes

characteristic

163

C₄

#14579146

yes

characteristic

164

C₄

#14579146

yes

characteristic

165

C₄

#14579155

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

632

—

170

C₂

#19098492

yes

characteristic

	subgroup	G/H
center	C₂	#19098492
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	H(3,ℤ/4)×C₇₉	C₂
socle	C₂×C₇₉	#397
prime radical	C₂²×D₃₁₆	C₂²
hypercenter	H(3,ℤ/4)	D₇₉
hypocenter	C₇₉	#2081
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₂

#19098492

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

H(3,ℤ/4)×C₇₉

C₂

socle

C₂×C₇₉

#397

prime radical

C₂²×D₃₁₆

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #19098611	C₂	semidirect
5. #19098333	C₂	semidirect
6. #19098333	C₂	semidirect
7. H(3,ℤ/4)×C₇₉	C₂	semidirect
86. C₇₉	#2081	semidirect
93. H(3,ℤ/4)	D₇₉	semidirect

H ≅ G/N

type

1. #19098611

C₂

semidirect

5. #19098333

C₂

semidirect

6. #19098333

C₂

semidirect

7. H(3,ℤ/4)×C₇₉

C₂

semidirect

86. C₇₉

#2081

semidirect

93. H(3,ℤ/4)

D₇₉

semidirect

Subgroups — #23764862

Conjugacy classes of subgroups

Complements