Group #23763136 — subgroups

Subgroups — C₂×#19098332 #23763136

Order 10112.

432 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 205 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	C₂²×#11929608	C₂	1	2	yes	maximal
2	#19098332	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11929608	—	2	4	no	—
5	C₂×#11929608	—	2	2	no	—
6	C₂×#11929608	C₄	1	8	yes	complemented
7	C₂×#11929608	C₂²	1	4	yes	—
8	C₂²×C₇₉⋊C₈	—	2	2	no	—
9	C₂³×C₄×C₇₉	C₄	1	2	yes	—
10	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
12	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
15	#11929608	—	2	8	no	—
16	#11929608	—	2	4	no	—
17	#11929608	C₂×C₄	1	8	yes	complemented
18	C₂×C₇₉⋊C₈	—	2	8	no	—
19	C₂×C₇₉⋊C₈	—	2	4	no	—
20	C₂⁴×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₉	—	2	2	no	—
22	C₂²×C₄×C₇₉	—	2	2	no	—
23	C₂²×C₄×C₇₉	—	4	4	no	—
24	C₂²×C₄×C₇₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₇₉	D₄	1	2	yes	—
28	C₂²×C₄×C₇₉	D₄	1	2	yes	—
29	C₂²×C₄×C₇₉	D₄	1	2	yes	—
30	C₂²×C₄×C₇₉	Q₈	1	2	yes	—
31	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₉	—	2	4	no	—
33	C₄⋊C₄×C₇₉	—	2	4	no	—
34	C₂³×C₇₉	—	2	2	no	—
35	C₂³×C₇₉	—	2	2	no	—
36	C₂³×C₇₉	C₄²	1	2	yes	—
37	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂³×C₇₉	C₄⋊C₄	1	2	yes	—
40	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	—	2	4	no	—
42	C₂×C₄×C₇₉	—	4	4	no	—
43	C₂×C₄×C₇₉	—	4	4	no	—
44	C₂×C₄×C₇₉	—	4	4	no	—
45	C₂×C₄×C₇₉	C₄²	1	2	yes	—
46	C₂×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
47	C₂×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
48	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
50	C₂×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
51	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
56	C₇₉⋊C₈	—	2	8	no	—
57	C₂²×C₇₉	—	2	4	no	—
58	C₂²×C₇₉	—	2	4	no	—
59	C₂²×C₇₉	—	2	1	no	—
60	C₂²×C₇₉	—	2	1	no	—
61	C₂²×C₇₉	—	2	1	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	#95	1	2	yes	—
64	C₂²×C₇₉	#95	1	2	yes	—
65	C₂²×C₇₉	#100	1	2	yes	—
66	C₂²×C₇₉	#101	1	2	yes	—
67	C₂²×C₇₉	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₇₉	—	4	4	no	—
71	C₄×C₇₉	#95	1	4	yes	—
72	C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
73	C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₇₉	—	2	2	no	—
75	C₂×C₇₉	—	2	1	no	—
76	C₂×C₇₉	—	2	1	no	—
77	C₂×C₇₉	#343	1	4	yes	complemented
78	C₂×C₇₉	C₂×#95	1	1	yes	characteristic
79	C₂×C₇₉	C₂×#100	1	1	yes	characteristic
80	C₂×C₇₉	C₂×#101	1	1	yes	characteristic
81	C₂×#343	—	79	1	no	maximal, Sylow
82	C₇₉	C₂×#343	1	1	yes	characteristic, complemented, Sylow
83	C₂²×M₁₆	—	79	2	no	—
84	C₂²×C₄⋊C₄	D₇₉	1	1	yes	characteristic
85	#343	—	79	4	no	—
86	C₂³×C₄	Dic₇₉	1	2	yes	—
87	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
88	C₂×M₁₆	—	79	4	no	—
89	C₂×M₁₆	—	79	8	no	—
90	C₂×M₁₆	—	158	2	no	—
91	C₂×M₁₆	—	158	4	no	—
92	C₂²×C₈	—	158	2	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	Dic₇₉	1	2	yes	—
96	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
97	C₂⁴	C₂×Dic₇₉	1	1	yes	characteristic
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	4	4	no	—
101	C₂²×C₄	Dic₁₅₈	1	2	yes	—
102	C₂²×C₄	C₄×D₇₉	1	4	yes	—
103	C₂²×C₄	D₃₁₆	1	2	yes	—
104	C₂²×C₄	C₂×Dic₇₉	1	4	yes	—
105	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
107	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
108	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
109	M₁₆	—	79	8	no	—
110	M₁₆	—	158	8	no	—
111	M₁₆	—	158	4	no	—
112	C₂×C₈	—	158	8	no	—
113	C₂×C₈	—	158	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₂³	—	2	2	no	—
117	C₂³	—	2	2	no	—
118	C₂³	C₄×Dic₇₉	1	2	yes	—
119	C₂³	C₂²⋊Dic₇₉	1	1	yes	characteristic
120	C₂³	C₂²⋊Dic₇₉	1	1	yes	characteristic
121	C₂³	C₄⋊Dic₇₉	1	2	yes	—
122	C₂³	C₂²×Dic₇₉	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	C₄×Dic₇₉	1	2	yes	—
128	C₂×C₄	(C₂×D₇₉)⋊C₄	1	4	yes	—
129	C₂×C₄	(C₂×D₇₉)⋊C₄	1	4	yes	—
130	C₂×C₄	C₂²⋊Dic₇₉	1	2	yes	—
131	C₂×C₄	Dic₇₉⋊C₄	1	4	yes	—
132	C₂×C₄	Dic₇₉⋊C₄	1	4	yes	—
133	C₂×C₄	C₄⋊Dic₇₉	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
136	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
138	C₈	—	158	8	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	4	no	—
144	C₂²	—	2	4	no	—
145	C₂²	C₇₉⋊#95	1	2	yes	—
146	C₂²	C₇₉⋊#95	1	2	yes	—
147	C₂²	#14579078	1	2	yes	—
148	C₂²	#14579080	1	2	yes	—
149	C₂²	C₂×C₄×Dic₇₉	1	1	yes	characteristic
150	C₂²	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
151	C₂²	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
152	C₄	—	4	4	no	—
153	C₄	C₇₉⋊#95	1	4	yes	—
154	C₄	C₂×(C₂×D₇₉)⋊C₄	1	2	yes	—
155	C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#19098332	1	4	yes	direct product factor
160	C₂	C₂×C₇₉⋊#95	1	1	yes	characteristic
161	C₂	C₂×#14579078	1	1	yes	characteristic
162	C₂	C₂×#14579080	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11929608

C₂

yes

maximal

#19098332

C₂

yes

maximal, direct product factor

C₂²×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

—

C₂×#11929608

—

C₂×#11929608

C₄

yes

complemented

C₂×#11929608

C₂²

yes

—

C₂²×C₇₉⋊C₈

—

C₂³×C₄×C₇₉

C₄

yes

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₄

yes

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

#11929608

—

#11929608

—

#11929608

C₂×C₄

yes

complemented

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

Q₈

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₄²

yes

—

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₇₉

C₄⋊C₄

yes

—

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄²

yes

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#95

yes

—

C₂²×C₇₉

#95

yes

—

C₂²×C₇₉

#100

yes

—

C₂²×C₇₉

#101

yes

—

C₂²×C₇₉

C₂×C₄²

yes

characteristic

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#95

yes

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#343

yes

complemented

C₂×C₇₉

C₂×#95

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄⋊C₄

D₇₉

yes

characteristic

#343

—

C₂³×C₄

Dic₇₉

yes

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

158

—

C₂×M₁₆

—

158

—

C₂²×C₈

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₇₉

yes

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂⁴

C₂×Dic₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

Dic₁₅₈

yes

—

102

C₂²×C₄

C₄×D₇₉

yes

—

103

C₂²×C₄

D₃₁₆

yes

—

104

C₂²×C₄

C₂×Dic₇₉

yes

—

105

C₂²×C₄

C₂×Dic₇₉

yes

characteristic

106

C₂²×C₄

C₂²⋊D₇₉

yes

—

107

C₂²×C₄

C₂²⋊D₇₉

yes

—

108

C₂²×C₄

C₂²×D₇₉

yes

characteristic

109

M₁₆

—

110

M₁₆

—

158

—

111

M₁₆

—

158

—

112

C₂×C₈

—

158

—

113

C₂×C₈

—

158

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂³

—

117

C₂³

—

118

C₂³

C₄×Dic₇₉

yes

—

119

C₂³

C₂²⋊Dic₇₉

yes

characteristic

120

C₂³

C₂²⋊Dic₇₉

yes

characteristic

121

C₂³

C₄⋊Dic₇₉

yes

—

122

C₂³

C₂²×Dic₇₉

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₄×Dic₇₉

yes

—

128

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

129

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

130

C₂×C₄

C₂²⋊Dic₇₉

yes

—

131

C₂×C₄

Dic₇₉⋊C₄

yes

—

132

C₂×C₄

Dic₇₉⋊C₄

yes

—

133

C₂×C₄

C₄⋊Dic₇₉

yes

—

134

C₂×C₄

C₂×C₄×D₇₉

yes

—

135

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

136

C₂×C₄

C₂×D₃₁₆

yes

characteristic

137

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

138

C₈

—

158

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₇₉⋊#95

yes

—

146

C₂²

C₇₉⋊#95

yes

—

147

C₂²

#14579078

yes

—

148

C₂²

#14579080

yes

—

149

C₂²

C₂×C₄×Dic₇₉

yes

characteristic

150

C₂²

C₂×C₂²⋊Dic₇₉

yes

characteristic

151

C₂²

C₂×C₄⋊Dic₇₉

yes

characteristic

152

C₄

—

153

C₄

C₇₉⋊#95

yes

—

154

C₄

C₂×(C₂×D₇₉)⋊C₄

yes

—

155

C₄

C₂×Dic₇₉⋊C₄

yes

—

156

C₂

—

157

C₂

—

158

C₂

—

159

C₂

#19098332

yes

direct product factor

160

C₂

C₂×C₇₉⋊#95

yes

characteristic

161

C₂

C₂×#14579078

yes

characteristic

162

C₂

C₂×#14579080

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₇₉
commutator	C₂²×C₇₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂²×C₄⋊C₄×C₇₉	C₂
socle	C₂³×C₇₉	C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂×C₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₇₉
hypocenter	C₇₉	C₂×#343
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊Dic₇₉

commutator

C₂²×C₇₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂²×C₄⋊C₄×C₇₉

C₂

socle

C₂³×C₇₉

C₂²⋊C₄

prime radical

C₂⁴×C₇₉

C₂×C₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₇₉

hypocenter

C₇₉

C₂×#343

LCS

G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #19098332	C₂	direct (N × H)
6. C₂×#11929608	C₄	semidirect
17. #11929608	C₂×C₄	semidirect
77. C₂×C₇₉	#343	semidirect
82. C₇₉	C₂×#343	semidirect
159. C₂	#19098332	direct (N × H)

H ≅ G/N

type

2. #19098332

C₂

direct (N × H)

6. C₂×#11929608

C₄

semidirect

17. #11929608

C₂×C₄

semidirect

77. C₂×C₇₉

#343

semidirect

82. C₇₉

C₂×#343

semidirect

159. C₂

#19098332

direct (N × H)

Subgroups — C₂×#19098332 #23763136

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098332 #23763136

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098332 #23763136