Group #23764133 — subgroups

Subgroups — #23764133

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×#14579095	C₂	1	1	yes	maximal, characteristic
3	#19098461	C₂	1	2	yes	maximal
4	#19098326	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
8	C₂×Dic₇₉⋊C₄	—	2	2	no	—
9	#14579095	—	2	2	no	—
10	#14579095	C₄	1	4	yes	complemented
11	C₇₉⋊#95	—	2	2	no	—
12	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₇₉	—	2	2	no	—
16	C₂²×Dic₇₉	—	2	2	no	—
17	C₂²×Dic₇₉	—	2	2	no	—
18	C₄⋊Dic₇₉	—	2	2	no	—
19	C₄⋊Dic₇₉	—	2	2	no	—
20	C₄⋊Dic₇₉	—	2	2	no	—
21	C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
22	Dic₇₉⋊C₄	—	4	4	no	—
23	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₇₉	—	2	1	no	—
27	C₂×C₈×C₇₉	—	2	1	no	—
28	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₉	D₄	1	2	yes	—
30	C₂×C₈×C₇₉	Q₈	1	2	yes	—
31	C₂×C₈×C₇₉	Q₈	1	2	yes	—
32	C₄²×C₇₉	—	2	1	no	—
33	C₄²×C₇₉	—	2	1	no	—
34	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₉	—	2	1	no	—
36	C₂×C₄×C₇₉	—	2	1	no	—
37	C₂×C₄×C₇₉	—	2	1	no	—
38	C₂×C₄×C₇₉	—	2	1	no	—
39	C₂×C₄×C₇₉	D₈	1	2	yes	—
40	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
41	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
47	C₂×Dic₇₉	—	2	4	no	—
48	C₂×Dic₇₉	—	4	2	no	—
49	C₂×Dic₇₉	—	4	2	no	—
50	C₂×Dic₇₉	—	4	2	no	—
51	C₂×Dic₇₉	—	4	2	no	—
52	C₂×Dic₇₉	—	4	2	no	—
53	C₈×C₇₉	—	2	2	no	—
54	C₈×C₇₉	C₄⋊C₄	1	4	yes	complemented
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	—	2	2	no	—
63	C₄×C₇₉	—	2	2	no	—
64	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
66	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₇₉	#125	1	1	yes	characteristic
68	Dic₇₉	—	4	4	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	—	79	1	no	—
81	C₂×C₄×C₈	D₇₉	1	1	yes	characteristic
82	#389	—	79	2	no	—
83	#340	—	79	2	no	—
84	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	79	2	no	—
87	C₂×C₄⋊C₄	—	79	2	no	—
88	C₂×C₄⋊C₄	—	158	2	no	—
89	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
90	#107	—	79	4	no	—
91	#107	—	158	2	no	—
92	C₄×C₈	—	2	2	no	—
93	#95	—	158	2	no	—
94	C₂²×C₄	—	158	2	no	—
95	C₂²×C₄	—	158	2	no	—
96	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
97	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	Dic₁₅₈	1	2	yes	—
102	C₂×C₈	Dic₁₅₈	1	2	yes	—
103	C₂×C₈	C₄×D₇₉	1	2	yes	—
104	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
105	C₄⋊C₄	—	316	4	no	—
106	C₄⋊C₄	—	79	4	no	—
107	C₄⋊C₄	—	158	2	no	—
108	C₄⋊C₄	—	158	2	no	—
109	C₄⋊C₄	—	158	2	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	316	2	no	—
118	C₂×C₄	—	316	2	no	—
119	C₂×C₄	—	316	2	no	—
120	C₂×C₄	—	316	2	no	—
121	C₂×C₄	—	316	2	no	—
122	C₂×C₄	—	158	4	no	—
123	C₂×C₄	D₆₃₂	1	2	yes	—
124	C₂×C₄	Dic₃₁₆	1	2	yes	—
125	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
127	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
128	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
129	C₂×C₄	#11929624	1	1	yes	characteristic
130	C₂×C₄	#11929624	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	Dic₇₉⋊C₄	1	4	yes	complemented
133	C₂²	C₄×D₃₁₆	1	1	yes	characteristic
134	C₂²	#14579135	1	1	yes	characteristic
135	C₂²	#14579161	1	1	yes	characteristic
136	C₂²	C₂×D₆₃₂	1	1	yes	characteristic
137	C₂²	C₂×Dic₃₁₆	1	1	yes	characteristic
138	C₂²	#14579183	1	1	yes	characteristic
139	C₂²	#14579183	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	316	4	no	—
143	C₄	—	632	2	no	—
144	C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
145	C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
146	C₄	#14579140	1	1	yes	characteristic
147	C₄	#14579153	1	1	yes	characteristic
148	C₂	#19098461	1	1	yes	characteristic
149	C₂	C₄×D₆₃₂	1	1	yes	characteristic
150	C₂	C₄×Dic₃₁₆	1	1	yes	characteristic
151	C₂	#19098799	1	1	yes	characteristic
152	C₂	#19098804	1	1	yes	characteristic
153	C₂	#19098965	1	1	yes	characteristic
154	C₂	#19098968	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579095

C₂

yes

maximal, characteristic

C₂×#14579095

C₂

yes

maximal, characteristic

#19098461

C₂

yes

maximal

#19098326

C₂

yes

maximal

C₂×C₄×C₈×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

—

#14579095

—

#14579095

C₄

yes

complemented

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

Q₈

yes

—

C₂×C₈×C₇₉

Q₈

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

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

—

C₈×C₇₉

—

C₈×C₇₉

C₄⋊C₄

yes

complemented

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

—

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

—

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

—

C₂×C₄×C₈

D₇₉

yes

characteristic

#389

—

#340

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#107

—

#107

—

158

—

C₄×C₈

—

#95

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

D₃₁₆

yes

characteristic

C₂²×C₄

D₃₁₆

yes

characteristic

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Dic₁₅₈

yes

—

102

C₂×C₈

Dic₁₅₈

yes

—

103

C₂×C₈

C₄×D₇₉

yes

—

104

C₂×C₈

C₂²⋊D₇₉

yes

—

105

C₄⋊C₄

—

316

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

158

—

108

C₄⋊C₄

—

158

—

109

C₄⋊C₄

—

158

—

110

C₄²

—

111

C₄²

—

112

C₂³

C₂×D₃₁₆

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

316

—

118

C₂×C₄

—

316

—

119

C₂×C₄

—

316

—

120

C₂×C₄

—

316

—

121

C₂×C₄

—

316

—

122

C₂×C₄

—

158

—

123

C₂×C₄

D₆₃₂

yes

—

124

C₂×C₄

Dic₃₁₆

yes

—

125

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

127

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

128

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

129

C₂×C₄

#11929624

yes

characteristic

130

C₂×C₄

#11929624

yes

characteristic

131

C₈

—

132

C₈

Dic₇₉⋊C₄

yes

complemented

133

C₂²

C₄×D₃₁₆

yes

characteristic

134

C₂²

#14579135

yes

characteristic

135

C₂²

#14579161

yes

characteristic

136

C₂²

C₂×D₆₃₂

yes

characteristic

137

C₂²

C₂×Dic₃₁₆

yes

characteristic

138

C₂²

#14579183

yes

characteristic

139

C₂²

#14579183

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

316

—

143

C₄

—

632

—

144

C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

145

C₄

C₄×Dic₁₅₈

yes

characteristic

146

C₄

#14579140

yes

characteristic

147

C₄

#14579153

yes

characteristic

148

C₂

#19098461

yes

characteristic

149

C₂

C₄×D₆₃₂

yes

characteristic

150

C₂

C₄×Dic₃₁₆

yes

characteristic

151

C₂

#19098799

yes

characteristic

152

C₂

#19098804

yes

characteristic

153

C₂

#19098965

yes

characteristic

154

C₂

#19098968

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×C₄×C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×C₄×C₈	D₇₉
hypocenter	C₇₉	#1942
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×C₄×C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₂×C₄×C₈

D₇₉

hypocenter

C₇₉

#1942

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
10. #14579095	C₄	semidirect
54. C₈×C₇₉	C₄⋊C₄	semidirect
78. C₇₉	#1942	semidirect
132. C₈	Dic₇₉⋊C₄	semidirect

H ≅ G/N

type

10. #14579095

C₄

semidirect

54. C₈×C₇₉

C₄⋊C₄

semidirect

78. C₇₉

#1942

semidirect

132. C₈

Dic₇₉⋊C₄

semidirect

Subgroups — #23764133

Conjugacy classes of subgroups

Complements