Group #23764134 — subgroups

Subgroups — #23764134

Order 10112.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14579095	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098462	C₂	1	2	yes	maximal
4	#19098325	C₂	1	2	yes	maximal
5	C₂×#107×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇₉	—	2	2	no	—
8	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
9	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
10	#14579095	—	2	2	no	—
11	C₈×Dic₇₉	—	2	2	no	—
12	C₇₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
15	#107×C₇₉	C₄	1	4	yes	complemented
16	C₂²×Dic₇₉	—	2	2	no	—
17	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
18	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
19	C₂×C₇₉⋊C₈	—	2	1	no	—
20	C₂×C₇₉⋊C₈	—	2	1	no	—
21	C₂×C₇₉⋊C₈	Q₈	1	2	yes	—
22	C₄⋊Dic₇₉	—	2	2	no	—
23	C₄⋊Dic₇₉	—	2	2	no	—
24	C₄⋊Dic₇₉	—	4	4	no	—
25	C₄×Dic₇₉	—	2	1	no	—
26	C₄×Dic₇₉	—	2	1	no	—
27	C₂²×C₄×C₇₉	—	2	2	no	—
28	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₇₉	D₄	1	2	yes	—
31	C₂×C₈×C₇₉	Q₈	1	2	yes	—
32	C₄⋊C₄×C₇₉	—	2	2	no	—
33	C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
34	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₉	—	2	4	no	—
36	C₂×C₄×C₇₉	—	4	2	no	—
37	C₂×C₄×C₇₉	—	4	2	no	—
38	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
44	C₂×Dic₇₉	—	2	1	no	—
45	C₂×Dic₇₉	—	2	1	no	—
46	C₂×Dic₇₉	—	2	1	no	—
47	C₂×Dic₇₉	—	2	1	no	—
48	C₂×Dic₇₉	—	4	2	no	—
49	C₂×Dic₇₉	—	4	2	no	—
50	C₂×Dic₇₉	—	4	2	no	—
51	C₂×Dic₇₉	D₈	1	2	yes	—
52	C₂×Dic₇₉	Q₁₆	1	2	yes	—
53	C₈×C₇₉	C₄⋊C₄	1	4	yes	complemented
54	C₇₉⋊C₈	—	2	2	no	—
55	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
56	C₂²×C₇₉	#121	1	1	yes	characteristic
57	C₂²×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₇₉	#135	1	1	yes	characteristic
61	C₂²×C₇₉	#135	1	1	yes	characteristic
62	C₄×C₇₉	—	4	4	no	—
63	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
66	C₄×C₇₉	#125	1	1	yes	characteristic
67	Dic₇₉	—	2	2	no	—
68	Dic₇₉	—	2	2	no	—
69	Dic₇₉	—	8	2	no	—
70	C₂×C₇₉	#389	1	1	yes	characteristic
71	C₂×C₇₉	C₄×D₈	1	1	yes	characteristic
72	C₂×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
73	C₂×C₇₉	#466	1	1	yes	characteristic
74	C₂×C₇₉	#467	1	1	yes	characteristic
75	C₂×C₇₉	#499	1	1	yes	characteristic
76	C₂×C₇₉	#500	1	1	yes	characteristic
77	#1942	—	79	1	no	maximal, Sylow
78	C₇₉	#1942	1	1	yes	characteristic, complemented, Sylow
79	C₂×#107	—	79	1	no	—
80	C₂×#107	D₇₉	1	1	yes	characteristic
81	C₂×C₄×C₈	—	79	1	no	—
82	#389	—	79	2	no	—
83	#340	—	79	2	no	—
84	C₂²×C₈	—	79	1	no	—
85	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	79	2	no	—
87	C₂×C₄⋊C₄	—	158	2	no	—
88	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
89	C₂×C₄²	—	79	1	no	—
90	#107	—	158	2	no	—
91	#107	Dic₇₉	1	4	yes	complemented
92	C₄×C₈	—	158	2	no	—
93	#95	—	158	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	79	1	no	—
96	C₂²×C₄	—	79	1	no	—
97	C₂²×C₄	—	158	2	no	—
98	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
99	C₂×C₈	—	79	2	no	—
100	C₂×C₈	—	158	1	no	—
101	C₂×C₈	—	158	1	no	—
102	C₂×C₈	Dic₁₅₈	1	2	yes	—
103	C₂×C₈	D₃₁₆	1	2	yes	—
104	C₂×C₈	C₂×Dic₇₉	1	2	yes	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	316	4	no	—
107	C₄⋊C₄	—	158	2	no	—
108	C₄⋊C₄	—	158	2	no	—
109	C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
110	C₄²	—	158	1	no	—
111	C₄²	—	158	1	no	—
112	C₂³	D₄×D₇₉	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	316	2	no	—
117	C₂×C₄	—	316	2	no	—
118	C₂×C₄	—	316	2	no	—
119	C₂×C₄	—	79	2	no	—
120	C₂×C₄	—	79	2	no	—
121	C₂×C₄	—	158	1	no	—
122	C₂×C₄	—	158	1	no	—
123	C₂×C₄	—	158	1	no	—
124	C₂×C₄	—	158	1	no	—
125	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
126	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
127	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
128	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
129	C₂×C₄	#11929623	1	1	yes	characteristic
130	C₂×C₄	#11929625	1	1	yes	characteristic
131	C₈	—	158	2	no	—
132	C₈	C₄⋊Dic₇₉	1	4	yes	complemented
133	C₂²	D₄×Dic₇₉	1	1	yes	characteristic
134	C₂²	#14579136	1	1	yes	characteristic
135	C₂²	#14579160	1	1	yes	characteristic
136	C₂²	D₈×D₇₉	1	1	yes	characteristic
137	C₂²	Q₁₆×D₇₉	1	1	yes	characteristic
138	C₂²	#14579180	1	1	yes	characteristic
139	C₂²	#14579184	1	1	yes	characteristic
140	C₄	—	4	4	no	—
141	C₄	—	632	2	no	—
142	C₄	—	158	2	no	—
143	C₄	—	158	2	no	—
144	C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
145	C₄	Q₈×Dic₇₉	1	1	yes	characteristic
146	C₄	#14579141	1	1	yes	characteristic
147	C₄	#14579152	1	1	yes	characteristic
148	C₂	#19098462	1	1	yes	characteristic
149	C₂	D₈×Dic₇₉	1	1	yes	characteristic
150	C₂	Q₁₆×Dic₇₉	1	1	yes	characteristic
151	C₂	#19098800	1	1	yes	characteristic
152	C₂	#19098805	1	1	yes	characteristic
153	C₂	#19098964	1	1	yes	characteristic
154	C₂	#19098967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579095

C₂

yes

maximal, characteristic

C₂×C₈×Dic₇₉

C₂

yes

maximal, characteristic

#19098462

C₂

yes

maximal

#19098325

C₂

yes

maximal

C₂×#107×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579095

—

C₈×Dic₇₉

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

#107×C₇₉

C₄

yes

complemented

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

characteristic

C₂²×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

Q₈

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₈

yes

—

C₂×Dic₇₉

Q₁₆

yes

—

C₈×C₇₉

C₄⋊C₄

yes

complemented

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×D₄

yes

characteristic

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

C₂×Q₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

C₄×Q₈

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

#125

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#389

yes

characteristic

C₂×C₇₉

C₄×D₈

yes

characteristic

C₂×C₇₉

C₄×Q₁₆

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#107

D₇₉

yes

characteristic

C₂×C₄×C₈

—

#389

—

#340

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

—

#107

—

158

—

#107

Dic₇₉

yes

complemented

C₄×C₈

—

158

—

#95

—

158

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

158

—

101

C₂×C₈

—

158

—

102

C₂×C₈

Dic₁₅₈

yes

—

103

C₂×C₈

D₃₁₆

yes

—

104

C₂×C₈

C₂×Dic₇₉

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

316

—

107

C₄⋊C₄

—

158

—

108

C₄⋊C₄

—

158

—

109

C₄⋊C₄

C₂×Dic₇₉

yes

—

110

C₄²

—

158

—

111

C₄²

—

158

—

112

C₂³

D₄×D₇₉

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

316

—

117

C₂×C₄

—

316

—

118

C₂×C₄

—

316

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

158

—

122

C₂×C₄

—

158

—

123

C₂×C₄

—

158

—

124

C₂×C₄

—

158

—

125

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

126

C₂×C₄

C₂×D₃₁₆

yes

characteristic

127

C₂×C₄

Q₈×D₇₉

yes

characteristic

128

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

129

C₂×C₄

#11929623

yes

characteristic

130

C₂×C₄

#11929625

yes

characteristic

131

C₈

—

158

—

132

C₈

C₄⋊Dic₇₉

yes

complemented

133

C₂²

D₄×Dic₇₉

yes

characteristic

134

C₂²

#14579136

yes

characteristic

135

C₂²

#14579160

yes

characteristic

136

C₂²

D₈×D₇₉

yes

characteristic

137

C₂²

Q₁₆×D₇₉

yes

characteristic

138

C₂²

#14579180

yes

characteristic

139

C₂²

#14579184

yes

characteristic

140

C₄

—

141

C₄

—

632

—

142

C₄

—

158

—

143

C₄

—

158

—

144

C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

145

C₄

Q₈×Dic₇₉

yes

characteristic

146

C₄

#14579141

yes

characteristic

147

C₄

#14579152

yes

characteristic

148

C₂

#19098462

yes

characteristic

149

C₂

D₈×Dic₇₉

yes

characteristic

150

C₂

Q₁₆×Dic₇₉

yes

characteristic

151

C₂

#19098800

yes

characteristic

152

C₂

#19098805

yes

characteristic

153

C₂

#19098964

yes

characteristic

154

C₂

#19098967

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×#107×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×#107	D₇₉
hypocenter	C₇₉	#1942
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×#107×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
15. #107×C₇₉	C₄	semidirect
53. C₈×C₇₉	C₄⋊C₄	semidirect
78. C₇₉	#1942	semidirect
91. #107	Dic₇₉	semidirect
132. C₈	C₄⋊Dic₇₉	semidirect

H ≅ G/N

type

15. #107×C₇₉

C₄

semidirect

53. C₈×C₇₉

C₄⋊C₄

semidirect

78. C₇₉

#1942

semidirect

91. #107

Dic₇₉

semidirect

132. C₈

C₄⋊Dic₇₉

semidirect

Subgroups — #23764134

Conjugacy classes of subgroups

Complements