Group #23774050 — subgroups

Subgroups — #23774050

Order 10112.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19099168	C₂	1	2	yes	maximal, complemented
2	#19098965	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098940	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098928	C₂	1	2	yes	maximal, complemented
5	#19098816	C₂	1	2	yes	maximal, complemented
6	#19098799	C₂	1	2	yes	maximal, complemented
7	#19098676	C₂	1	2	yes	maximal, complemented
8	C₄×D₆₃₂	C₂	1	2	yes	maximal, complemented
9	#405×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₆₃₂	C₂²	1	2	yes	—
11	C₂×D₆₃₂	C₂²	1	2	yes	—
12	#14579158	C₂²	1	2	yes	—
13	#14579153	C₂²	1	2	yes	—
14	#14579146	—	2	4	no	—
15	#14579135	C₂²	1	4	yes	—
16	#14579134	—	2	4	no	—
17	C₄×D₃₁₆	C₂²	1	4	yes	—
18	#14579095	C₂²	1	2	yes	—
19	#14579093	C₂²	1	2	yes	—
20	#14579083	C₂²	1	4	yes	—
21	#14579083	C₂²	1	4	yes	—
22	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#117×C₇₉	C₂²	1	1	yes	characteristic
25	#97×C₇₉	C₂²	1	1	yes	characteristic
26	#97×C₇₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×D₃₁₆	—	2	4	no	—
30	C₂×D₃₁₆	C₂³	1	4	yes	—
31	C₂×C₂²⋊D₇₉	—	2	4	no	—
32	C₂×C₄×D₇₉	—	2	4	no	—
33	D₆₃₂	—	2	4	no	—
34	D₆₃₂	—	2	4	no	—
35	C₄⋊Dic₇₉	C₂³	1	4	yes	—
36	Dic₇₉⋊C₄	—	2	4	no	—
37	(C₂×D₇₉)⋊C₄	—	2	4	no	—
38	(C₂×D₇₉)⋊C₄	—	2	4	no	—
39	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₇₉	—	2	2	no	—
41	C₂×C₈×C₇₉	—	2	2	no	—
42	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
47	C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
48	C₄²×C₇₉	C₂³	1	1	yes	characteristic
49	C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×D₇₉	—	2	4	no	—
52	C₂×C₄×C₇₉	—	2	1	no	—
53	C₂×C₄×C₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
58	C₂²⋊D₇₉	—	8	4	no	—
59	C₂×Dic₇₉	—	2	4	no	—
60	D₃₁₆	—	2	4	no	—
61	D₃₁₆	—	2	4	no	—
62	D₃₁₆	—	8	4	no	—
63	C₄×D₇₉	—	8	4	no	—
64	C₈×C₇₉	—	2	1	no	—
65	C₈×C₇₉	—	2	1	no	—
66	C₈×C₇₉	#41	1	2	yes	—
67	C₈×C₇₉	#41	1	2	yes	—
68	C₂²×C₇₉	—	2	1	no	—
69	C₂²×C₇₉	—	2	2	no	—
70	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
71	C₂×D₇₉	—	4	4	no	—
72	C₂×D₇₉	—	4	4	no	—
73	C₂×D₇₉	—	4	4	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	—	2	2	no	—
77	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
78	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
79	Dic₇₉	—	4	4	no	—
80	C₂×C₇₉	—	4	1	no	—
81	C₂×C₇₉	#532	1	1	yes	characteristic
82	C₂×C₇₉	#576	1	2	yes	—
83	D₇₉	—	8	4	no	—
84	#3176	—	79	1	no	maximal, Sylow
85	C₇₉	#3176	1	1	yes	characteristic, complemented, Sylow
86	#540	—	79	2	no	—
87	#499	—	79	1	no	—
88	#493	—	79	1	no	—
89	#490	—	79	2	no	—
90	#469	—	79	2	no	—
91	#466	—	79	2	no	—
92	#442	—	79	2	no	—
93	C₄×D₈	—	79	2	no	—
94	#405	D₇₉	1	1	yes	characteristic, complemented
95	C₂×D₈	—	79	2	no	—
96	C₂×D₈	—	79	2	no	—
97	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
99	#127	—	79	2	no	—
100	#125	—	79	2	no	—
101	#123	—	158	4	no	—
102	#121	—	79	4	no	—
103	#121	—	158	4	no	—
104	C₄×D₄	—	79	4	no	—
105	#117	C₂×D₇₉	1	1	yes	characteristic
106	#107	—	79	2	no	—
107	#106	—	79	2	no	—
108	#102	—	79	4	no	—
109	#102	—	79	4	no	—
110	#97	C₂×D₇₉	1	1	yes	characteristic
111	#97	C₂×D₇₉	1	1	yes	characteristic
112	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
113	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
114	C₂×D₄	—	79	4	no	—
115	C₂×D₄	—	158	4	no	—
116	C₂×D₄	—	158	4	no	—
117	C₂²×C₄	—	158	4	no	—
118	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
119	D₈	—	158	4	no	—
120	D₈	—	158	4	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
124	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
127	C₄⋊C₄	—	79	4	no	—
128	C₄⋊C₄	—	158	4	no	—
129	C₄⋊C₄	D₃₁₆	1	2	yes	—
130	C₂²⋊C₄	—	158	4	no	—
131	C₂²⋊C₄	—	158	4	no	—
132	C₂²⋊C₄	D₃₁₆	1	2	yes	—
133	C₄²	C₂²×D₇₉	1	1	yes	characteristic
134	C₄²	C₂²×D₇₉	1	1	yes	characteristic
135	C₂³	—	158	4	no	—
136	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
137	D₄	—	632	4	no	—
138	D₄	—	632	4	no	—
139	D₄	—	158	4	no	—
140	D₄	—	158	4	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	632	4	no	—
144	C₂×C₄	—	158	4	no	—
145	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
146	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
147	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
148	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	#11929624	1	2	yes	—
152	C₈	#11929624	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	316	4	no	—
156	C₂²	—	316	4	no	—
157	C₂²	—	316	4	no	—
158	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	1	no	—
162	C₄	—	316	4	no	—
163	C₄	C₂×#11929624	1	1	yes	characteristic
164	C₄	C₂×#11929624	1	1	yes	characteristic
165	C₂	—	4	1	no	—
166	C₂	—	632	4	no	—
167	C₂	#19099113	1	1	yes	characteristic
168	C₂	#19099368	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099168

C₂

yes

maximal, complemented

#19098965

C₂

yes

maximal, characteristic, complemented

#19098940

C₂

yes

maximal, characteristic, complemented

#19098928

C₂

yes

maximal, complemented

#19098816

C₂

yes

maximal, complemented

#19098799

C₂

yes

maximal, complemented

#19098676

C₂

yes

maximal, complemented

C₄×D₆₃₂

C₂

yes

maximal, complemented

#405×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₆₃₂

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

—

#14579158

C₂²

yes

—

#14579153

C₂²

yes

—

#14579146

—

#14579135

C₂²

yes

—

#14579134

—

C₄×D₃₁₆

C₂²

yes

—

#14579095

C₂²

yes

—

#14579093

C₂²

yes

—

#14579083

C₂²

yes

—

#14579083

C₂²

yes

—

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

#97×C₇₉

C₂²

yes

characteristic

#97×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

—

C₂²⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

—

C₈×C₇₉

#41

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

C₂×#41

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#532

yes

characteristic

C₂×C₇₉

#576

yes

—

D₇₉

—

#3176

—

maximal, Sylow

C₇₉

#3176

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

C₄×D₈

—

#405

D₇₉

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

#127

—

100

#125

—

101

#123

—

158

—

102

#121

—

103

#121

—

158

—

104

C₄×D₄

—

105

#117

C₂×D₇₉

yes

characteristic

106

#107

—

107

#106

—

108

#102

—

109

#102

—

110

#97

C₂×D₇₉

yes

characteristic

111

#97

C₂×D₇₉

yes

characteristic

112

C₄×C₈

C₂×D₇₉

yes

characteristic

113

C₄×C₈

C₂×D₇₉

yes

characteristic

114

C₂×D₄

—

115

C₂×D₄

—

158

—

116

C₂×D₄

—

158

—

117

C₂²×C₄

—

158

—

118

C₂²×C₄

C₂²×D₇₉

yes

characteristic

119

D₈

—

158

—

120

D₈

—

158

—

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

C₂²×D₇₉

yes

characteristic

124

C₂×C₈

C₂²×D₇₉

yes

characteristic

125

C₂×C₈

C₂²×D₇₉

yes

characteristic

126

C₂×C₈

C₂²×D₇₉

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

158

—

129

C₄⋊C₄

D₃₁₆

yes

—

130

C₂²⋊C₄

—

158

—

131

C₂²⋊C₄

—

158

—

132

C₂²⋊C₄

D₃₁₆

yes

—

133

C₄²

C₂²×D₇₉

yes

characteristic

134

C₄²

C₂²×D₇₉

yes

characteristic

135

C₂³

—

158

—

136

C₂³

C₂×D₃₁₆

yes

characteristic

137

D₄

—

632

—

138

D₄

—

632

—

139

D₄

—

158

—

140

D₄

—

158

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

632

—

144

C₂×C₄

—

158

—

145

C₂×C₄

C₂×D₃₁₆

yes

—

146

C₂×C₄

C₂×D₃₁₆

yes

—

147

C₂×C₄

C₂×D₃₁₆

yes

characteristic

148

C₂×C₄

C₂³×D₇₉

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

#11929624

yes

—

152

C₈

#11929624

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

316

—

156

C₂²

—

316

—

157

C₂²

—

316

—

158

C₂²

C₂²×D₃₁₆

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

316

—

163

C₄

C₂×#11929624

yes

characteristic

164

C₄

C₂×#11929624

yes

characteristic

165

C₂

—

166

C₂

—

632

—

167

C₂

#19099113

yes

characteristic

168

C₂

#19099368

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#405×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#405	D₇₉
hypocenter	C₇₉	#3176
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#405×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
1. #19099168	C₂	semidirect
2. #19098965	C₂	semidirect
3. #19098940	C₂	semidirect
4. #19098928	C₂	semidirect
5. #19098816	C₂	semidirect
6. #19098799	C₂	semidirect
7. #19098676	C₂	semidirect
8. C₄×D₆₃₂	C₂	semidirect
9. #405×C₇₉	C₂	semidirect
85. C₇₉	#3176	semidirect
94. #405	D₇₉	semidirect

H ≅ G/N

type

1. #19099168

C₂

semidirect

2. #19098965

C₂

semidirect

3. #19098940

C₂

semidirect

4. #19098928

C₂

semidirect

5. #19098816

C₂

semidirect

6. #19098799

C₂

semidirect

7. #19098676

C₂

semidirect

8. C₄×D₆₃₂

C₂

semidirect

9. #405×C₇₉

C₂

semidirect

85. C₇₉

#3176

semidirect

94. #405

D₇₉

semidirect

Subgroups — #23774050

Conjugacy classes of subgroups

Complements