Group #23765514 — subgroups

Subgroups — #23765514

Order 10112.

160 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 49 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×#14579106	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579100	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098805	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098800	C₂	1	1	yes	maximal, characteristic
5	#19098366	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098363	C₂	1	1	yes	maximal, characteristic, complemented
7	#366×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
9	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₇₉	C₂²	1	1	yes	characteristic
11	#14579141	—	2	1	no	—
12	#14579136	—	2	1	no	—
13	#14579106	—	2	2	no	—
14	#14579100	—	2	2	no	—
15	C₂×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14579084	—	2	1	no	—
17	#14579081	—	2	1	no	—
18	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	#107×C₇₉	C₂²	1	1	yes	characteristic
20	#107×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₅₈	—	2	1	no	—
22	C₂×D₃₁₆	—	2	1	no	—
23	C₂×D₃₁₆	—	4	1	no	—
24	C₂×C₄×D₇₉	D₄	1	1	yes	characteristic
25	Dic₃₁₆	—	2	1	no	—
26	Dic₃₁₆	—	2	1	no	—
27	D₆₃₂	—	2	1	no	—
28	D₆₃₂	—	2	1	no	—
29	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
30	C₈×D₇₉	—	2	1	no	—
31	C₈×D₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	—	4	1	no	—
33	(C₂×D₇₉)⋊C₄	—	4	1	no	—
34	(C₂×D₇₉)⋊C₄	—	4	1	no	—
35	C₇₉⋊C₁₆	—	2	1	no	—
36	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₇₉	—	2	1	no	—
38	C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₁₆×C₇₉	D₄	1	2	yes	complemented
40	C₂²×D₇₉	—	4	1	no	—
41	C₂²×D₇₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₇₉	—	4	1	no	—
43	C₂×C₄×C₇₉	—	4	1	no	—
44	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₇₉	—	4	1	no	—
46	C₂×Dic₇₉	D₈	1	1	yes	characteristic
47	D₃₁₆	—	4	1	no	—
48	D₃₁₆	—	4	1	no	—
49	D₃₁₆	—	8	2	no	—
50	C₄×D₇₉	—	2	1	no	—
51	C₄×D₇₉	—	2	1	no	—
52	Dic₁₅₈	—	4	1	no	—
53	Dic₁₅₈	—	4	1	no	—
54	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₈×C₇₉	#41	1	1	yes	characteristic
56	C₇₉⋊C₈	—	2	1	no	—
57	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
58	C₂×D₇₉	—	2	1	no	—
59	C₂×D₇₉	—	2	1	no	—
60	C₂×D₇₉	—	2	1	no	—
61	C₂×D₇₉	—	8	1	no	—
62	C₂×D₇₉	—	8	1	no	—
63	C₂×D₇₉	—	8	1	no	—
64	C₄×C₇₉	—	8	1	no	—
65	C₄×C₇₉	—	8	1	no	—
66	C₄×C₇₉	#121	1	1	yes	characteristic
67	C₄×C₇₉	#135	1	1	yes	characteristic
68	Dic₇₉	—	2	1	no	—
69	Dic₇₉	—	8	1	no	—
70	C₂×C₇₉	#466	1	1	yes	characteristic
71	C₂×C₇₉	#509	1	1	yes	characteristic
72	C₂×C₇₉	#510	1	1	yes	characteristic
73	D₇₉	—	4	1	no	—
74	D₇₉	—	16	1	no	—
75	#2220	—	79	1	no	maximal, Sylow
76	C₇₉	#2220	1	1	yes	characteristic, complemented, Sylow
77	C₂×SD₃₂	—	79	1	no	—
78	C₂×M₃₂	—	79	1	no	—
79	#467	—	79	1	no	—
80	#466	—	79	1	no	—
81	#366	D₇₉	1	1	yes	characteristic, complemented
82	#358	—	79	1	no	—
83	#357	—	79	1	no	—
84	C₂×Q₁₆	—	79	1	no	—
85	C₂×D₈	—	79	1	no	—
86	C₂²×C₈	—	79	1	no	—
87	C₂²⋊Q₈	—	158	1	no	—
88	#121	—	158	1	no	—
89	SD₃₂	—	158	2	no	—
90	M₃₂	—	158	2	no	—
91	C₂×C₁₆	—	79	1	no	—
92	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
93	#107	C₂×D₇₉	1	1	yes	characteristic
94	#107	C₂×D₇₉	1	1	yes	characteristic
95	#103	—	158	1	no	—
96	#102	—	158	1	no	—
97	C₂×Q₈	—	158	1	no	—
98	C₂×D₄	—	316	1	no	—
99	C₂×D₄	—	158	1	no	—
100	C₂²×C₄	—	79	1	no	—
101	Q₁₆	—	158	1	no	—
102	Q₁₆	—	158	1	no	—
103	D₈	—	158	1	no	—
104	D₈	—	158	1	no	—
105	C₂×C₈	—	79	1	no	—
106	C₂×C₈	—	158	1	no	—
107	C₂×C₈	—	158	1	no	—
108	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	316	1	no	—
112	C₂²⋊C₄	—	316	1	no	—
113	C₂²⋊C₄	—	316	1	no	—
114	C₁₆	—	158	1	no	—
115	C₁₆	D₃₁₆	1	2	yes	complemented
116	C₂³	—	316	1	no	—
117	C₂³	—	79	1	no	—
118	Q₈	—	316	1	no	—
119	Q₈	—	316	1	no	—
120	D₄	—	316	1	no	—
121	D₄	—	316	1	no	—
122	D₄	—	632	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	316	1	no	—
126	C₂×C₄	—	79	1	no	—
127	C₂×C₄	—	158	1	no	—
128	C₂×C₄	—	158	1	no	—
129	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
130	C₈	—	158	1	no	—
131	C₈	C₂×D₃₁₆	1	1	yes	characteristic
132	C₈	#11929625	1	1	yes	characteristic
133	C₂²	—	632	1	no	—
134	C₂²	—	632	1	no	—
135	C₂²	—	632	1	no	—
136	C₂²	—	158	1	no	—
137	C₂²	—	158	1	no	—
138	C₂²	—	158	1	no	—
139	C₂²	D₈×D₇₉	1	1	yes	characteristic
140	C₄	—	8	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	632	1	no	—
143	C₄	—	158	1	no	—
144	C₄	#14579136	1	1	yes	characteristic
145	C₄	#14579184	1	1	yes	characteristic
146	C₂	—	316	1	no	—
147	C₂	—	1264	1	no	—
148	C₂	#19098800	1	1	yes	characteristic
149	C₂	#19098997	1	1	yes	characteristic
150	C₂	#19099005	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579106

C₂

yes

maximal, characteristic, complemented

C₂×#14579100

C₂

yes

maximal, characteristic, complemented

#19098805

C₂

yes

maximal, characteristic, complemented

#19098800

C₂

yes

maximal, characteristic

#19098366

C₂

yes

maximal, characteristic, complemented

#19098363

C₂

yes

maximal, characteristic, complemented

#366×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₂²

yes

—

—

—

—

C₂×C₇₉⋊C₁₆

C₂²

yes

characteristic

#14579084

—

#14579081

—

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₄×D₇₉

D₄

yes

characteristic

Dic₃₁₆

—

Dic₃₁₆

—

D₆₃₂

—

D₆₃₂

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₈×D₇₉

—

C₈×D₇₉

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

D₄

yes

complemented

C₂²×D₇₉

—

C₂²×D₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

D₈

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

D₇₉

—

#2220

—

maximal, Sylow

C₇₉

#2220

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×M₃₂

—

#467

—

#466

—

#366

D₇₉

yes

characteristic, complemented

#358

—

#357

—

C₂×Q₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₂²⋊Q₈

—

158

—

#121

—

158

—

SD₃₂

—

158

—

M₃₂

—

158

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₉

yes

characteristic

#107

C₂×D₇₉

yes

characteristic

#107

C₂×D₇₉

yes

characteristic

#103

—

158

—

#102

—

158

—

C₂×Q₈

—

158

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

100

C₂²×C₄

—

101

Q₁₆

—

158

—

102

Q₁₆

—

158

—

103

D₈

—

158

—

104

D₈

—

158

—

105

C₂×C₈

—

106

C₂×C₈

—

158

—

107

C₂×C₈

—

158

—

108

C₂×C₈

C₂²×D₇₉

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

316

—

112

C₂²⋊C₄

—

316

—

113

C₂²⋊C₄

—

316

—

114

C₁₆

—

158

—

115

C₁₆

D₃₁₆

yes

complemented

116

C₂³

—

316

—

117

C₂³

—

118

Q₈

—

316

—

119

Q₈

—

316

—

120

D₄

—

316

—

121

D₄

—

316

—

122

D₄

—

632

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

316

—

126

C₂×C₄

—

127

C₂×C₄

—

158

—

128

C₂×C₄

—

158

—

129

C₂×C₄

D₄×D₇₉

yes

characteristic

130

C₈

—

158

—

131

C₈

C₂×D₃₁₆

yes

characteristic

132

C₈

#11929625

yes

characteristic

133

C₂²

—

632

—

134

C₂²

—

632

—

135

C₂²

—

632

—

136

C₂²

—

158

—

137

C₂²

—

158

—

138

C₂²

—

158

—

139

C₂²

D₈×D₇₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

632

—

143

C₄

—

158

—

144

C₄

#14579136

yes

characteristic

145

C₄

#14579184

yes

characteristic

146

C₂

—

316

—

147

C₂

—

1264

—

148

C₂

#19098800

yes

characteristic

149

C₂

#19098997

yes

characteristic

150

C₂

#19099005

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#366×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	#19098800	C₂
hypercenter	#366	D₇₉
hypocenter	C₇₉	#2220
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₇₉

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#366×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

C₂

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
1. C₂×#14579106	C₂	semidirect
2. C₂×#14579100	C₂	semidirect
3. #19098805	C₂	semidirect
5. #19098366	C₂	semidirect
6. #19098363	C₂	semidirect
7. #366×C₇₉	C₂	semidirect
39. C₁₆×C₇₉	D₄	semidirect
76. C₇₉	#2220	semidirect
81. #366	D₇₉	semidirect
115. C₁₆	D₃₁₆	semidirect

H ≅ G/N

type

1. C₂×#14579106

C₂

semidirect

2. C₂×#14579100

C₂

semidirect

3. #19098805

C₂

semidirect

5. #19098366

C₂

semidirect

6. #19098363

C₂

semidirect

7. #366×C₇₉

C₂

semidirect

39. C₁₆×C₇₉

D₄

semidirect

76. C₇₉

#2220

semidirect

81. #366

D₇₉

semidirect

115. C₁₆

D₃₁₆

semidirect

Subgroups — #23765514

Conjugacy classes of subgroups

Complements