Group #23761900 — subgroups

Subgroups — #23761900

Order 10112.

240 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19099120	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098301	C₂	1	4	yes	maximal
4	#431×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#14579161	C₄	1	1	yes	characteristic, complemented
6	#14579161	C₄	1	1	yes	characteristic, complemented
7	#14579153	—	2	1	no	—
8	#14579140	—	2	1	no	—
9	#14579140	C₂²	1	2	yes	—
10	C₄×Dic₁₅₈	—	2	1	no	—
11	C₄×Dic₁₅₈	C₄	1	2	yes	complemented
12	#14579120	—	2	1	no	—
13	C₂×C₄⋊Dic₇₉	—	2	1	no	—
14	C₂×C₄×Dic₇₉	—	2	1	no	—
15	C₇₉⋊#95	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₉	—	2	1	no	—
18	C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
19	C₄×C₈×C₇₉	—	2	1	no	—
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
22	C₂²×Dic₇₉	—	2	2	no	—
23	C₄⋊Dic₇₉	—	2	4	no	—
24	C₄⋊Dic₇₉	—	2	1	no	—
25	C₄⋊Dic₇₉	—	2	1	no	—
26	C₄⋊Dic₇₉	—	2	1	no	—
27	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
28	Dic₇₉⋊C₄	—	2	2	no	—
29	Dic₇₉⋊C₄	—	2	2	no	—
30	C₂²⋊Dic₇₉	—	2	2	no	—
31	C₄×Dic₇₉	—	2	2	no	—
32	C₄×Dic₇₉	—	4	2	no	—
33	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₇₉	—	2	2	no	—
37	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	—	4	1	no	—
44	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
49	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
50	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₇₉	—	2	4	no	—
53	C₂×Dic₇₉	—	2	2	no	—
54	C₂×Dic₇₉	—	2	2	no	—
55	C₂×Dic₇₉	—	4	2	no	—
56	C₂×Dic₇₉	—	4	2	no	—
57	Dic₁₅₈	—	4	2	no	—
58	Dic₁₅₈	—	4	2	no	—
59	C₈×C₇₉	—	4	2	no	—
60	C₂²×C₇₉	—	2	1	no	—
61	C₂²×C₇₉	—	2	1	no	—
62	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₇₉	—	2	1	no	—
66	C₄×C₇₉	—	2	1	no	—
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	#103	1	2	yes	—
69	C₄×C₇₉	#103	1	2	yes	—
70	C₄×C₇₉	#104	1	2	yes	—
71	Dic₇₉	—	4	4	no	—
72	Dic₇₉	—	4	2	no	—
73	Dic₇₉	—	4	2	no	—
74	C₂×C₇₉	—	2	1	no	—
75	C₂×C₇₉	—	2	1	no	—
76	C₂×C₇₉	#410	1	1	yes	characteristic
77	C₂×C₇₉	C₂×#103	1	1	yes	characteristic
78	C₂×C₇₉	C₂×#104	1	1	yes	characteristic
79	#1503	—	79	1	no	maximal, Sylow
80	C₇₉	#1503	1	1	yes	characteristic, complemented, Sylow
81	#533	—	79	1	no	—
82	#431	D₇₉	1	1	yes	characteristic
83	C₄×C₄⋊C₄	—	79	1	no	—
84	#328	—	79	4	no	—
85	C₄⋊Q₈	—	79	1	no	—
86	C₄⋊Q₈	—	79	1	no	—
87	#125	—	158	1	no	—
88	C₂²⋊Q₈	—	79	2	no	—
89	C₂²⋊Q₈	—	158	1	no	—
90	C₄×Q₈	—	79	2	no	—
91	C₄×Q₈	—	158	1	no	—
92	#117	—	158	1	no	—
93	C₂×C₄⋊C₄	—	158	1	no	—
94	C₂×C₄²	—	158	1	no	—
95	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
96	C₄⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	C₂×D₇₉	1	2	yes	—
98	C₄×C₈	—	2	1	no	—
99	#95	—	79	2	no	—
100	C₂×Q₈	—	79	2	no	—
101	C₂×Q₈	—	158	1	no	—
102	C₂²×C₄	—	158	2	no	—
103	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
104	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₄⋊C₄	—	79	2	no	—
108	C₄⋊C₄	—	158	4	no	—
109	C₄⋊C₄	—	158	2	no	—
110	C₄⋊C₄	—	158	1	no	—
111	C₄⋊C₄	—	158	1	no	—
112	C₄⋊C₄	—	158	1	no	—
113	C₄⋊C₄	—	158	2	no	—
114	C₂²⋊C₄	—	158	2	no	—
115	C₄²	—	316	2	no	—
116	C₄²	—	158	2	no	—
117	C₄²	C₄×D₇₉	1	1	yes	characteristic
118	C₄²	C₄×D₇₉	1	1	yes	characteristic
119	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
120	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
121	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
122	Q₈	—	316	2	no	—
123	Q₈	—	316	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	316	2	no	—
127	C₂×C₄	—	316	2	no	—
128	C₂×C₄	—	158	4	no	—
129	C₂×C₄	—	158	2	no	—
130	C₂×C₄	—	158	2	no	—
131	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	#11929613	1	2	yes	—
136	C₂×C₄	Dic₃₁₆	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
143	C₂²	C₂×#11929613	1	1	yes	characteristic
144	C₂²	C₂×Dic₃₁₆	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	316	4	no	—
149	C₄	—	316	2	no	—
150	C₄	—	316	2	no	—
151	C₄	#14579086	1	2	yes	—
152	C₄	#14579086	1	2	yes	—
153	C₄	#14579087	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#19098536	1	1	yes	characteristic
157	C₂	C₂×#14579086	1	1	yes	characteristic
158	C₂	C₂×#14579087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099120

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₉

C₂

yes

maximal, characteristic

#19098301

C₂

yes

maximal

#431×C₇₉

C₂

yes

maximal, characteristic

#14579161

C₄

yes

characteristic, complemented

#14579161

C₄

yes

characteristic, complemented

#14579153

—

#14579140

—

#14579140

C₂²

yes

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

C₄

yes

complemented

#14579120

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄×Dic₇₉

—

C₇₉⋊#95

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×Q₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#103

yes

—

C₄×C₇₉

#103

yes

—

C₄×C₇₉

#104

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#533

—

#431

D₇₉

yes

characteristic

C₄×C₄⋊C₄

—

#328

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

158

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

C₄×Q₈

—

158

—

#117

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂×D₇₉

yes

—

C₄×C₈

—

#95

—

100

C₂×Q₈

—

101

C₂×Q₈

—

158

—

102

C₂²×C₄

—

158

—

103

C₂²×C₄

D₃₁₆

yes

characteristic

104

C₂²×C₄

D₃₁₆

yes

characteristic

105

C₂²×C₄

C₂²×D₇₉

yes

characteristic

106

C₂×C₈

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

158

—

109

C₄⋊C₄

—

158

—

110

C₄⋊C₄

—

158

—

111

C₄⋊C₄

—

158

—

112

C₄⋊C₄

—

158

—

113

C₄⋊C₄

—

158

—

114

C₂²⋊C₄

—

158

—

115

C₄²

—

316

—

116

C₄²

—

158

—

117

C₄²

C₄×D₇₉

yes

characteristic

118

C₄²

C₄×D₇₉

yes

characteristic

119

C₄²

C₂²⋊D₇₉

yes

characteristic

120

C₄²

C₂²⋊D₇₉

yes

characteristic

121

C₂³

C₂×D₃₁₆

yes

characteristic

122

Q₈

—

316

—

123

Q₈

—

316

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

316

—

127

C₂×C₄

—

316

—

128

C₂×C₄

—

158

—

129

C₂×C₄

—

158

—

130

C₂×C₄

—

158

—

131

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

132

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

135

C₂×C₄

#11929613

yes

—

136

C₂×C₄

Dic₃₁₆

yes

—

137

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

143

C₂²

C₂×#11929613

yes

characteristic

144

C₂²

C₂×Dic₃₁₆

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

316

—

149

C₄

—

316

—

150

C₄

—

316

—

151

C₄

#14579086

yes

—

152

C₄

#14579086

yes

—

153

C₄

#14579087

yes

—

154

C₂

—

155

C₂

—

156

C₂

#19098536

yes

characteristic

157

C₂

C₂×#14579086

yes

characteristic

158

C₂

C₂×#14579087

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#431×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#431	D₇₉
hypocenter	C₇₉	#1503
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₇₉)⋊C₄

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#431×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
5. #14579161	C₄	semidirect
6. #14579161	C₄	semidirect
11. C₄×Dic₁₅₈	C₄	semidirect
80. C₇₉	#1503	semidirect

H ≅ G/N

type

5. #14579161

C₄

semidirect

6. #14579161

C₄

semidirect

11. C₄×Dic₁₅₈

C₄

semidirect

80. C₇₉

#1503

semidirect

Subgroups — #23761900

Conjugacy classes of subgroups

Complements