Group #23773166 — subgroups

Subgroups — C₂×#19098849 #23773166

Order 10112.

416 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14579161	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
3	#19098849	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14579093	C₂	1	2	yes	maximal
5	C₂×#14579086	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₅₈	—	2	1	no	—
8	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	#14579161	—	2	2	no	—
10	#14579161	C₂²	1	4	yes	—
11	C₄×Dic₁₅₈	—	2	2	no	—
12	C₄×Dic₁₅₈	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₇₉	—	2	1	no	—
14	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
16	C₂×Dic₇₉⋊C₄	—	2	1	no	—
17	C₂×C₄×Dic₇₉	—	2	1	no	—
18	#14579093	C₂²	1	8	yes	—
19	#14579086	C₂²	1	8	yes	—
20	C₂²×C₈×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₉	C₂²	1	4	yes	—
23	C₂×Dic₁₅₈	—	2	1	no	—
24	C₂×Dic₁₅₈	—	2	4	no	—
25	C₂×Dic₁₅₈	—	2	2	no	—
26	C₂×Dic₁₅₈	—	4	4	no	—
27	C₂×Dic₁₅₈	Q₈	1	4	yes	complemented
28	C₂×Dic₁₅₈	C₂³	1	2	yes	—
29	C₂²×Dic₇₉	—	2	1	no	—
30	C₂²×Dic₇₉	—	2	2	no	—
31	C₂²×Dic₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	—	2	4	no	—
33	C₄⋊Dic₇₉	—	2	1	no	—
34	C₄⋊Dic₇₉	—	2	2	no	—
35	C₄⋊Dic₇₉	C₂³	1	4	yes	—
36	C₄⋊Dic₇₉	C₂³	1	2	yes	—
37	Dic₇₉⋊C₄	—	2	4	no	—
38	C₄×Dic₇₉	—	2	4	no	—
39	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₇₉	—	2	4	no	—
43	C₂×C₈×C₇₉	C₂³	1	4	yes	—
44	C₄²×C₇₉	D₄	1	2	yes	—
45	C₄²×C₇₉	C₂³	1	2	yes	—
46	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	SD₁₆	1	4	yes	—
49	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₇₉	#41	1	2	yes	—
54	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₉	—	2	4	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	—	2	2	no	—
58	C₂×Dic₇₉	—	2	4	no	—
59	C₂×Dic₇₉	—	4	4	no	—
60	C₂×Dic₇₉	—	4	2	no	—
61	Dic₁₅₈	—	2	4	no	—
62	Dic₁₅₈	—	2	2	no	—
63	Dic₁₅₈	—	4	8	no	—
64	Dic₁₅₈	C₂×Q₈	1	4	yes	complemented
65	C₈×C₇₉	—	2	4	no	—
66	C₂²×C₇₉	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₇₉	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₇₉	#137	1	2	yes	—
69	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₇₉	—	2	2	no	—
71	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
73	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
74	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
75	Dic₇₉	—	2	4	no	—
76	Dic₇₉	—	4	4	no	—
77	Dic₇₉	—	4	2	no	—
78	C₂×C₇₉	#475	1	4	yes	complemented
79	C₂×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
81	C₂×C₇₉	C₂×#137	1	1	yes	characteristic
82	C₂×#475	—	79	1	no	maximal, Sylow
83	C₇₉	C₂×#475	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄⋊Q₈	—	79	1	no	—
85	C₂×C₄×Q₈	—	79	1	no	—
86	#475	—	79	8	no	—
87	C₂×#106	—	79	2	no	—
88	C₂×C₄⋊C₈	D₇₉	1	1	yes	characteristic
89	C₂×#103	—	79	2	no	—
90	C₂²×Q₈	—	79	1	no	—
91	C₂²×Q₈	—	158	1	no	—
92	C₂²×C₈	C₂×D₇₉	1	2	yes	—
93	C₄⋊Q₈	—	79	4	no	—
94	C₄⋊Q₈	—	158	2	no	—
95	C₄×Q₈	—	79	4	no	—
96	C₄×Q₈	—	158	2	no	—
97	C₂×C₄⋊C₄	—	79	1	no	—
98	C₂×C₄⋊C₄	—	79	2	no	—
99	C₂×C₄⋊C₄	—	158	1	no	—
100	C₂×C₄⋊C₄	—	158	1	no	—
101	C₂×C₄²	—	158	1	no	—
102	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
103	#106	—	79	8	no	—
104	C₄⋊C₈	C₂×D₇₉	1	4	yes	—
105	#103	—	79	8	no	—
106	C₂×Q₈	—	316	4	no	—
107	C₂×Q₈	—	79	4	no	—
108	C₂×Q₈	—	79	2	no	—
109	C₂×Q₈	—	158	4	no	—
110	C₂×Q₈	—	158	1	no	—
111	C₂×Q₈	—	158	2	no	—
112	C₂²×C₄	—	158	1	no	—
113	C₂²×C₄	—	158	2	no	—
114	C₂²×C₄	—	158	1	no	—
115	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
116	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	C₂²×D₇₉	1	4	yes	—
120	C₄⋊C₄	—	79	4	no	—
121	C₄⋊C₄	—	79	2	no	—
122	C₄⋊C₄	—	158	4	no	—
123	C₄⋊C₄	—	158	1	no	—
124	C₄⋊C₄	—	158	2	no	—
125	C₄⋊C₄	—	158	4	no	—
126	C₄²	—	158	4	no	—
127	C₄²	D₃₁₆	1	2	yes	—
128	C₄²	C₂²×D₇₉	1	2	yes	—
129	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
130	Q₈	—	316	8	no	—
131	Q₈	—	79	4	no	—
132	Q₈	—	158	2	no	—
133	Q₈	—	158	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	316	4	no	—
136	C₂×C₄	—	316	2	no	—
137	C₂×C₄	—	158	2	no	—
138	C₂×C₄	—	158	4	no	—
139	C₂×C₄	—	158	4	no	—
140	C₂×C₄	—	158	2	no	—
141	C₂×C₄	#11929613	1	4	yes	—
142	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
143	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
144	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
145	C₂×C₄	Q₈×D₇₉	1	2	yes	—
146	C₂×C₄	#11929623	1	2	yes	—
147	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₂²	#14579141	1	2	yes	—
150	C₂²	C₂×#11929613	1	2	yes	—
151	C₂²	#14579194	1	2	yes	—
152	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	316	4	no	—
155	C₄	—	316	2	no	—
156	C₄	—	158	4	no	—
157	C₄	#14579141	1	2	yes	—
158	C₄	C₂×#11929613	1	4	yes	—
159	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
160	C₄	C₂×#11929623	1	1	yes	characteristic
161	C₂	#19098849	1	4	yes	direct product factor
162	C₂	C₂×#14579141	1	1	yes	characteristic
163	C₂	C₂²×#11929613	1	1	yes	characteristic
164	C₂	C₂×#14579194	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₅₈

C₂

yes

maximal, characteristic

#19098849

C₂

yes

maximal, direct product factor

C₂×#14579093

C₂

yes

maximal

C₂×#14579086

C₂

yes

maximal

C₂×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

C₂²×Dic₁₅₈

C₂²

yes

characteristic

#14579161

—

#14579161

C₂²

yes

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

—

#14579093

C₂²

yes

—

#14579086

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

Q₈

yes

complemented

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

C₂×Q₈

yes

complemented

C₈×C₇₉

—

C₂²×C₇₉

C₂²⋊Q₈

yes

—

C₂²×C₇₉

C₂×SD₁₆

yes

—

C₂²×C₇₉

#137

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

C₂×SD₁₆

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

C₂×#41

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#475

yes

complemented

C₂×C₇₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₇₉

C₂²×SD₁₆

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

#475

—

C₂×#106

—

C₂×C₄⋊C₈

D₇₉

yes

characteristic

C₂×#103

—

C₂²×Q₈

—

C₂²×Q₈

—

158

—

C₂²×C₈

C₂×D₇₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄×Q₈

—

C₄×Q₈

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

100

C₂×C₄⋊C₄

—

158

—

101

C₂×C₄²

—

158

—

102

C₂×C₄²

C₂×D₇₉

yes

characteristic

103

#106

—

104

C₄⋊C₈

C₂×D₇₉

yes

—

105

#103

—

106

C₂×Q₈

—

316

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

158

—

110

C₂×Q₈

—

158

—

111

C₂×Q₈

—

158

—

112

C₂²×C₄

—

158

—

113

C₂²×C₄

—

158

—

114

C₂²×C₄

—

158

—

115

C₂²×C₄

D₃₁₆

yes

characteristic

116

C₂²×C₄

D₃₁₆

yes

characteristic

117

C₂²×C₄

C₂²×D₇₉

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

C₂²×D₇₉

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

158

—

123

C₄⋊C₄

—

158

—

124

C₄⋊C₄

—

158

—

125

C₄⋊C₄

—

158

—

126

C₄²

—

158

—

127

C₄²

D₃₁₆

yes

—

128

C₄²

C₂²×D₇₉

yes

—

129

C₂³

C₂×D₃₁₆

yes

characteristic

130

Q₈

—

316

—

131

Q₈

—

132

Q₈

—

158

—

133

Q₈

—

158

—

134

C₂×C₄

—

135

C₂×C₄

—

316

—

136

C₂×C₄

—

316

—

137

C₂×C₄

—

158

—

138

C₂×C₄

—

158

—

139

C₂×C₄

—

158

—

140

C₂×C₄

—

158

—

141

C₂×C₄

#11929613

yes

—

142

C₂×C₄

C₂×D₃₁₆

yes

—

143

C₂×C₄

C₂×D₃₁₆

yes

characteristic

144

C₂×C₄

C₂×D₃₁₆

yes

—

145

C₂×C₄

Q₈×D₇₉

yes

—

146

C₂×C₄

#11929623

yes

—

147

C₂×C₄

C₂³×D₇₉

yes

characteristic

148

C₈

—

149

C₂²

#14579141

yes

—

150

C₂²

C₂×#11929613

yes

—

151

C₂²

#14579194

yes

—

152

C₂²

C₂²×D₃₁₆

yes

characteristic

153

C₄

—

154

C₄

—

316

—

155

C₄

—

316

—

156

C₄

—

158

—

157

C₄

#14579141

yes

—

158

C₄

C₂×#11929613

yes

—

159

C₄

C₂×Q₈×D₇₉

yes

characteristic

160

C₄

C₂×#11929623

yes

characteristic

161

C₂

#19098849

yes

direct product factor

162

C₂

C₂×#14579141

yes

characteristic

163

C₂

C₂²×#11929613

yes

characteristic

164

C₂

C₂×#14579194

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₉
hypocenter	C₇₉	C₂×#475
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×C₄⋊C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₉

hypocenter

C₇₉

C₂×#475

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #19098849	C₂	direct (N × H)
27. C₂×Dic₁₅₈	Q₈	semidirect
64. Dic₁₅₈	C₂×Q₈	semidirect
78. C₂×C₇₉	#475	semidirect
83. C₇₉	C₂×#475	semidirect
161. C₂	#19098849	direct (N × H)

H ≅ G/N

type

3. #19098849

C₂

direct (N × H)

27. C₂×Dic₁₅₈

Q₈

semidirect

64. Dic₁₅₈

C₂×Q₈

semidirect

78. C₂×C₇₉

#475

semidirect

83. C₇₉

C₂×#475

semidirect

161. C₂

#19098849

direct (N × H)

Subgroups — C₂×#19098849 #23773166

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098849 #23773166

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098849 #23773166