Group #23773789 — subgroups

Subgroups — #23773789

Order 10112.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14579149	C₂	1	1	yes	maximal, characteristic
2	#19098925	C₂	1	4	yes	maximal, complemented
3	#19098920	C₂	1	4	yes	maximal, complemented
4	#19098571	C₂	1	2	yes	maximal, complemented
5	#19098567	C₂	1	2	yes	maximal
6	C₄×#11929608	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂²×D₃₁₆	C₂²	1	1	yes	characteristic
10	C₂×#11929608	C₂²	1	2	yes	—
11	#14579149	—	2	2	no	—
12	#14579149	C₂²	1	2	yes	—
13	#14579149	C₂²	1	2	yes	—
14	C₂×(C₂×D₇₉)⋊C₄	—	2	2	no	—
15	#14579084	C₂²	1	8	yes	complemented
16	#14579081	C₂²	1	8	yes	—
17	#14579072	C₂²	1	2	yes	complemented
18	C₄×C₇₉⋊C₈	C₂²	1	2	yes	complemented
19	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	complemented
20	#125×C₇₉	C₂²	1	2	yes	complemented
21	C₂²⋊Q₈×C₇₉	—	2	2	no	—
22	C₄×Q₈×C₇₉	—	2	2	no	—
23	#117×C₇₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
25	C₂³×D₇₉	—	2	1	no	—
26	C₂×Dic₁₅₈	—	2	1	no	—
27	C₂×Dic₁₅₈	—	2	4	no	—
28	C₂×Dic₁₅₈	C₂³	1	2	yes	—
29	C₂×D₃₁₆	—	2	1	no	—
30	C₂×D₃₁₆	—	2	4	no	—
31	C₂×D₃₁₆	C₂³	1	2	yes	—
32	C₂²×Dic₇₉	—	2	1	no	—
33	#11929608	—	2	4	no	—
34	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
35	(C₂×D₇₉)⋊C₄	—	2	4	no	—
36	(C₂×D₇₉)⋊C₄	—	2	4	no	—
37	C₂×Q₈×C₇₉	—	2	2	no	—
38	C₂²×C₄×C₇₉	D₄	1	2	yes	—
39	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₇₉	—	2	4	no	—
41	C₄⋊C₄×C₇₉	—	2	2	no	—
42	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₇₉	—	2	2	no	—
44	C₄²×C₇₉	—	2	2	no	—
45	C₄²×C₇₉	D₄	1	2	yes	—
46	C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₇₉	—	2	2	no	—
50	C₂²×D₇₉	—	4	4	no	—
51	C₂²×D₇₉	—	4	2	no	—
52	Q₈×C₇₉	—	4	4	no	—
53	C₂×C₄×C₇₉	—	2	4	no	—
54	C₂×C₄×C₇₉	—	4	2	no	—
55	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#41	1	4	yes	—
59	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₇₉	—	2	2	no	—
61	C₂×Dic₇₉	—	4	2	no	—
62	D₃₁₆	—	2	4	no	—
63	D₃₁₆	—	4	2	no	—
64	Dic₁₅₈	—	2	4	no	—
65	Dic₁₅₈	—	4	2	no	—
66	C₇₉⋊C₈	—	2	4	no	—
67	C₂²×C₇₉	—	2	2	no	—
68	C₂²×C₇₉	#124	1	2	yes	—
69	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₇₉	—	4	2	no	—
71	C₂×D₇₉	—	4	4	no	—
72	C₂×D₇₉	—	8	4	no	—
73	C₄×C₇₉	—	2	2	no	—
74	C₄×C₇₉	—	2	2	no	—
75	C₄×C₇₉	—	4	4	no	—
76	C₄×C₇₉	#124	1	2	yes	—
77	C₄×C₇₉	C₂×#41	1	2	yes	—
78	Dic₇₉	—	4	2	no	—
79	C₂×C₇₉	—	2	2	no	—
80	C₂×C₇₉	C₂×#124	1	1	yes	characteristic
81	C₂×C₇₉	#575	1	2	yes	—
82	D₇₉	—	8	2	no	—
83	#3140	—	79	1	no	maximal, Sylow
84	C₇₉	#3140	1	1	yes	characteristic, complemented, Sylow
85	#533	D₇₉	1	1	yes	characteristic, complemented
86	C₂×#124	—	79	1	no	—
87	#489	—	79	4	no	—
88	#488	—	79	4	no	—
89	#419	—	79	2	no	—
90	#418	—	79	2	no	—
91	C₄×M₁₆	—	79	1	no	—
92	C₂²×Q₈	—	79	1	no	—
93	C₂²×D₄	—	79	1	no	—
94	C₂×M₁₆	—	79	2	no	—
95	C₄⋊Q₈	C₂×D₇₉	1	2	yes	complemented
96	#125	C₂×D₇₉	1	2	yes	complemented
97	#124	—	79	2	no	—
98	#124	—	79	2	no	—
99	#124	—	158	2	no	—
100	C₂²⋊Q₈	—	2	2	no	—
101	C₄×Q₈	—	2	2	no	—
102	#117	C₂×D₇₉	1	2	yes	—
103	C₂×C₂²⋊C₄	—	158	2	no	—
104	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
105	#103	—	79	8	no	—
106	#102	—	79	8	no	—
107	#97	—	79	2	no	—
108	C₄×C₈	—	79	2	no	—
109	C₂⁴	—	158	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	79	2	no	—
112	C₂×Q₈	—	158	4	no	—
113	C₂×Q₈	—	158	1	no	—
114	C₂×D₄	—	79	2	no	—
115	C₂×D₄	—	158	1	no	—
116	C₂×D₄	—	158	4	no	—
117	C₂²×C₄	—	158	1	no	—
118	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
119	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
120	M₁₆	—	158	4	no	—
121	C₂×C₈	—	79	4	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	158	4	no	—
127	C₂²⋊C₄	—	158	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂²⋊D₇₉	1	2	yes	—
130	C₄²	C₂²×D₇₉	1	1	yes	characteristic
131	C₄²	C₂²×D₇₉	1	1	yes	characteristic
132	C₂³	—	316	4	no	—
133	C₂³	—	316	2	no	—
134	C₂³	—	158	2	no	—
135	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
136	Q₈	—	4	4	no	—
137	Q₈	—	316	2	no	—
138	Q₈	—	158	4	no	—
139	D₄	—	316	2	no	—
140	D₄	—	158	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	316	2	no	—
144	C₂×C₄	—	158	2	no	—
145	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
148	C₂×C₄	#11929625	1	4	yes	—
149	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
150	C₈	—	158	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	316	2	no	—
153	C₂²	—	316	4	no	—
154	C₂²	—	632	4	no	—
155	C₂²	#14579150	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	316	2	no	—
161	C₄	#14579150	1	2	yes	—
162	C₄	C₂×#11929625	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	632	2	no	—
165	C₂	C₂×#14579150	1	1	yes	characteristic
166	C₂	#19099361	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579149

C₂

yes

maximal, characteristic

#19098925

C₂

yes

maximal, complemented

#19098920

C₂

yes

maximal, complemented

#19098571

C₂

yes

maximal, complemented

#19098567

C₂

yes

maximal

C₄×#11929608

C₂

yes

maximal, characteristic, complemented

#533×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₂²×D₃₁₆

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

—

#14579149

—

#14579149

C₂²

yes

—

#14579149

C₂²

yes

—

C₂×(C₂×D₇₉)⋊C₄

—

#14579084

C₂²

yes

complemented

#14579081

C₂²

yes

—

#14579072

C₂²

yes

complemented

C₄×C₇₉⋊C₈

C₂²

yes

complemented

C₄⋊Q₈×C₇₉

C₂²

yes

complemented

#125×C₇₉

C₂²

yes

complemented

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂³×D₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂²×Dic₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

C₂³

yes

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#124

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#124

yes

—

C₄×C₇₉

C₂×#41

yes

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#124

yes

characteristic

C₂×C₇₉

#575

yes

—

D₇₉

—

#3140

—

maximal, Sylow

C₇₉

#3140

yes

characteristic, complemented, Sylow

#533

D₇₉

yes

characteristic, complemented

—

—

—

—

—

C₄×M₁₆

—

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₇₉

yes

complemented

#125

C₂×D₇₉

yes

—

—

—

158

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

C₂×D₇₉

yes

—

103

C₂×C₂²⋊C₄

—

158

—

104

C₂×C₄²

C₂×D₇₉

yes

characteristic

105

#103

—

106

#102

—

107

#97

—

108

C₄×C₈

—

109

C₂⁴

—

158

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

158

—

113

C₂×Q₈

—

158

—

114

C₂×D₄

—

115

C₂×D₄

—

158

—

116

C₂×D₄

—

158

—

117

C₂²×C₄

—

158

—

118

C₂²×C₄

C₂²⋊D₇₉

yes

—

119

C₂²×C₄

C₂²×D₇₉

yes

characteristic

120

M₁₆

—

158

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²×D₇₉

yes

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

158

—

127

C₂²⋊C₄

—

158

—

128

C₄²

—

129

C₄²

C₂²⋊D₇₉

yes

—

130

C₄²

C₂²×D₇₉

yes

characteristic

131

C₄²

C₂²×D₇₉

yes

characteristic

132

C₂³

—

316

—

133

C₂³

—

316

—

134

C₂³

—

158

—

135

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

136

Q₈

—

137

Q₈

—

316

—

138

Q₈

—

158

—

139

D₄

—

316

—

140

D₄

—

158

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

316

—

144

C₂×C₄

—

158

—

145

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

146

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

148

C₂×C₄

#11929625

yes

—

149

C₂×C₄

C₂³×D₇₉

yes

characteristic

150

C₈

—

158

—

151

C₂²

—

152

C₂²

—

316

—

153

C₂²

—

316

—

154

C₂²

—

632

—

155

C₂²

#14579150

yes

—

156

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

316

—

161

C₄

#14579150

yes

—

162

C₄

C₂×#11929625

yes

—

163

C₂

—

164

C₂

—

632

—

165

C₂

C₂×#14579150

yes

characteristic

166

C₂

#19099361

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#533×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂²×D₃₁₆	C₂²
hypercenter	#533	D₇₉
hypocenter	C₇₉	#3140
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#533×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂²×D₃₁₆

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #19098925	C₂	semidirect
3. #19098920	C₂	semidirect
4. #19098571	C₂	semidirect
6. C₄×#11929608	C₂	semidirect
7. #533×C₇₉	C₂	semidirect
15. #14579084	C₂²	semidirect
17. #14579072	C₂²	semidirect
18. C₄×C₇₉⋊C₈	C₂²	semidirect
19. C₄⋊Q₈×C₇₉	C₂²	semidirect
20. #125×C₇₉	C₂²	semidirect
84. C₇₉	#3140	semidirect
85. #533	D₇₉	semidirect
95. C₄⋊Q₈	C₂×D₇₉	semidirect
96. #125	C₂×D₇₉	semidirect

H ≅ G/N

type

2. #19098925

C₂

semidirect

3. #19098920

C₂

semidirect

4. #19098571

C₂

semidirect

6. C₄×#11929608

C₂

semidirect

7. #533×C₇₉

C₂

semidirect

15. #14579084

C₂²

semidirect

17. #14579072

C₂²

semidirect

18. C₄×C₇₉⋊C₈

C₂²

semidirect

19. C₄⋊Q₈×C₇₉

C₂²

semidirect

20. #125×C₇₉

C₂²

semidirect

84. C₇₉

#3140

semidirect

85. #533

D₇₉

semidirect

95. C₄⋊Q₈

C₂×D₇₉

semidirect

96. #125

C₂×D₇₉

semidirect

Subgroups — #23773789

Conjugacy classes of subgroups

Complements