Group #23762772 — subgroups

Subgroups — #23762772

Order 10112.

166 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19098850	C₂	1	1	yes	maximal, characteristic
2	#19098765	C₂	1	1	yes	maximal, characteristic
3	C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic
4	#19098314	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098312	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098309	C₂	1	1	yes	maximal, characteristic, complemented
7	#498×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929611	—	2	1	no	—
9	C₂×C₈×D₇₉	—	2	1	no	—
10	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
11	#14579092	—	2	1	no	—
12	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#14579085	—	2	1	no	—
16	#14579081	—	2	1	no	—
17	(C₂×D₇₉)⋊C₈	—	2	1	no	—
18	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
20	#106×C₇₉	—	2	1	no	—
21	#106×C₇₉	—	2	1	no	—
22	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×D₃₁₆	D₄	1	1	yes	characteristic
24	C₂×C₄×D₇₉	—	2	1	no	—
25	#11929611	—	4	2	no	—
26	C₂×C₇₉⋊C₈	—	2	1	no	—
27	C₂×C₇₉⋊C₈	—	2	1	no	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₈×D₇₉	—	4	2	no	—
30	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
31	(C₂×D₇₉)⋊C₄	—	2	1	no	—
32	C₂×Q₈×C₇₉	—	2	1	no	—
33	C₂×Q₈×C₇₉	—	2	1	no	—
34	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₇₉	—	2	1	no	—
37	C₄⋊C₄×C₇₉	—	2	1	no	—
38	C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	C₂²×D₇₉	—	2	1	no	—
42	Q₈×C₇₉	—	4	1	no	—
43	Q₈×C₇₉	—	4	1	no	—
44	Q₈×C₇₉	—	8	1	no	—
45	C₂×C₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	—	4	1	no	—
47	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
50	C₂×Dic₇₉	—	2	1	no	—
51	D₃₁₆	—	2	1	no	—
52	D₃₁₆	SD₁₆	1	2	yes	—
53	C₄×D₇₉	—	4	1	no	—
54	C₄×D₇₉	—	4	1	no	—
55	C₈×C₇₉	—	2	1	no	—
56	C₈×C₇₉	SD₁₆	1	2	yes	complemented
57	C₇₉⋊C₈	—	4	1	no	—
58	C₇₉⋊C₈	—	4	1	no	—
59	C₇₉⋊C₈	—	4	1	no	—
60	C₂²×C₇₉	#121	1	1	yes	characteristic
61	C₂×D₇₉	—	2	2	no	—
62	C₂×D₇₉	—	4	1	no	—
63	C₂×D₇₉	—	4	1	no	—
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	—	8	1	no	—
66	C₄×C₇₉	—	8	1	no	—
67	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₉	#135	1	1	yes	characteristic
70	C₄×C₇₉	#137	1	1	yes	characteristic
71	Dic₇₉	—	4	1	no	—
72	C₂×C₇₉	#461	1	1	yes	characteristic
73	C₂×C₇₉	#465	1	1	yes	characteristic
74	C₂×C₇₉	#471	1	1	yes	characteristic
75	D₇₉	—	4	2	no	—
76	#1671	—	79	1	no	maximal, Sylow
77	C₇₉	#1671	1	1	yes	characteristic, complemented, Sylow
78	#498	D₇₉	1	1	yes	characteristic, complemented
79	#476	—	79	1	no	—
80	#461	—	79	1	no	—
81	C₈×D₄	—	79	1	no	—
82	#334	—	79	1	no	—
83	#333	—	79	1	no	—
84	#332	—	79	1	no	—
85	C₂×SD₁₆	—	158	1	no	—
86	C₂²×C₈	—	158	1	no	—
87	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
88	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
89	C₄×D₄	—	79	1	no	—
90	#106	—	2	1	no	—
91	#106	—	2	1	no	—
92	#106	—	158	1	no	—
93	C₄⋊C₈	—	79	1	no	—
94	C₄⋊C₈	—	79	1	no	—
95	C₄⋊C₈	—	79	1	no	—
96	#103	—	158	1	no	—
97	#102	—	158	1	no	—
98	C₂²⋊C₈	—	158	1	no	—
99	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×D₄	—	79	1	no	—
103	C₂²×C₄	—	158	1	no	—
104	SD₁₆	—	316	2	no	—
105	C₂×C₈	—	316	2	no	—
106	C₂×C₈	—	158	1	no	—
107	C₂×C₈	—	158	1	no	—
108	C₂×C₈	—	158	1	no	—
109	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	79	1	no	—
116	C₂²⋊C₄	—	158	1	no	—
117	C₄²	C₂²×D₇₉	1	1	yes	characteristic
118	C₂³	—	158	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	8	1	no	—
122	D₄	—	79	2	no	—
123	D₄	—	158	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	316	1	no	—
127	C₂×C₄	—	316	1	no	—
128	C₂×C₄	—	158	1	no	—
129	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
131	C₂×C₄	#11929623	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	316	1	no	—
134	C₈	—	316	1	no	—
135	C₈	—	316	1	no	—
136	C₈	#11929611	1	2	yes	complemented
137	C₂²	—	316	1	no	—
138	C₂²	—	316	1	no	—
139	C₂²	—	158	2	no	—
140	C₂²	#14579132	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	316	1	no	—
145	C₄	SD₁₆×D₇₉	1	1	yes	characteristic
146	C₄	C₂×#11929611	1	1	yes	characteristic
147	C₄	#14579182	1	1	yes	characteristic
148	C₄	#14579192	1	1	yes	characteristic
149	C₂	—	316	2	no	—
150	C₂	#19098765	1	1	yes	characteristic
151	C₂	#19098789	1	1	yes	characteristic
152	C₂	#19098823	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098850

C₂

yes

maximal, characteristic

#19098765

C₂

yes

maximal, characteristic

C₈×D₃₁₆

C₂

yes

maximal, characteristic

#19098314

C₂

yes

maximal, characteristic, complemented

#19098312

C₂

yes

maximal, characteristic, complemented

#19098309

C₂

yes

maximal, characteristic, complemented

#498×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929611

—

C₂×C₈×D₇₉

—

C₄×D₃₁₆

C₂²

yes

characteristic

#14579092

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579085

—

#14579081

—

(C₂×D₇₉)⋊C₈

—

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#106×C₇₉

—

#106×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

D₄

yes

characteristic

C₂×C₄×D₇₉

—

#11929611

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₈×D₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

SD₁₆

yes

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

SD₁₆

yes

complemented

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#121

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#1671

—

maximal, Sylow

C₇₉

#1671

yes

characteristic, complemented, Sylow

#498

D₇₉

yes

characteristic, complemented

#476

—

#461

—

C₈×D₄

—

#334

—

#333

—

#332

—

C₂×SD₁₆

—

158

—

C₂²×C₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

C₄×D₄

—

#106

—

#106

—

#106

—

158

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

158

—

#102

—

158

—

C₂²⋊C₈

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

158

—

104

SD₁₆

—

316

—

105

C₂×C₈

—

316

—

106

C₂×C₈

—

158

—

107

C₂×C₈

—

158

—

108

C₂×C₈

—

158

—

109

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

110

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

158

—

117

C₄²

C₂²×D₇₉

yes

characteristic

118

C₂³

—

158

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

158

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

316

—

127

C₂×C₄

—

316

—

128

C₂×C₄

—

158

—

129

C₂×C₄

D₄×D₇₉

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

131

C₂×C₄

#11929623

yes

characteristic

132

C₈

—

133

C₈

—

316

—

134

C₈

—

316

—

135

C₈

—

316

—

136

C₈

#11929611

yes

complemented

137

C₂²

—

316

—

138

C₂²

—

316

—

139

C₂²

—

158

—

140

C₂²

#14579132

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

316

—

145

C₄

SD₁₆×D₇₉

yes

characteristic

146

C₄

C₂×#11929611

yes

characteristic

147

C₄

#14579182

yes

characteristic

148

C₄

#14579192

yes

characteristic

149

C₂

—

316

—

150

C₂

#19098765

yes

characteristic

151

C₂

#19098789

yes

characteristic

152

C₂

#19098823

yes

characteristic

	subgroup	G/H
center	C₂²	#14579132
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#498×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	C₂×D₃₁₆	D₄
hypercenter	#498	D₇₉
hypocenter	C₇₉	#1671
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#14579132

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#498×C₇₉

C₂

socle

C₂²×C₇₉

#121

prime radical

C₂×D₃₁₆

D₄

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
4. #19098314	C₂	semidirect
5. #19098312	C₂	semidirect
6. #19098309	C₂	semidirect
7. #498×C₇₉	C₂	semidirect
56. C₈×C₇₉	SD₁₆	semidirect
77. C₇₉	#1671	semidirect
78. #498	D₇₉	semidirect
136. C₈	#11929611	semidirect

H ≅ G/N

type

4. #19098314

C₂

semidirect

5. #19098312

C₂

semidirect

6. #19098309

C₂

semidirect

7. #498×C₇₉

C₂

semidirect

56. C₈×C₇₉

SD₁₆

semidirect

77. C₇₉

#1671

semidirect

78. #498

D₇₉

semidirect

136. C₈

#11929611

semidirect

Subgroups — #23762772

Conjugacy classes of subgroups

Complements