Group #23763937 — subgroups

Subgroups — #23763937

Order 10112.

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

normal subgroups only

This group has 314 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	#19099137	C₂	1	1	yes	maximal, characteristic
2	#19098581	C₂	1	2	yes	maximal
3	C₂×#14579080	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098319	C₂	1	2	yes	maximal, complemented
5	#519×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11929624	C₂²	1	1	yes	characteristic
7	C₂×#11929608	C₂²	1	2	yes	—
8	#14579159	C₄	1	2	yes	complemented
9	#14579150	C₄	1	2	yes	complemented
10	#14579144	—	2	2	no	—
11	#14579137	—	2	2	no	—
12	#14579120	C₂²	1	1	yes	characteristic
13	#14579089	—	2	4	no	—
14	#14579080	—	2	2	no	—
15	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₇₉	—	2	4	no	—
17	#117×C₇₉	—	2	2	no	—
18	#117×C₇₉	C₂²	1	2	yes	—
19	#11929624	—	2	2	no	—
20	#11929624	—	2	2	no	—
21	C₂×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
22	C₂×D₃₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₇₉	—	2	1	no	—
24	C₂×C₄×D₇₉	—	2	1	no	—
25	#11929608	—	2	2	no	—
26	#11929608	—	2	2	no	—
27	C₂×C₇₉⋊C₈	—	2	2	no	—
28	C₄⋊Dic₇₉	—	2	1	no	—
29	Dic₇₉⋊C₄	—	2	4	no	—
30	C₂²⋊Dic₇₉	—	2	1	no	—
31	(C₂×D₇₉)⋊C₄	—	2	4	no	—
32	C₄×Dic₇₉	C₂×C₄	1	2	yes	—
33	C₂×Q₈×C₇₉	—	2	2	no	—
34	C₂×Q₈×C₇₉	—	2	2	no	—
35	C₂×Q₈×C₇₉	C₂×C₄	1	2	yes	—
36	C₂×Q₈×C₇₉	D₄	1	2	yes	—
37	C₂²×C₄×C₇₉	D₄	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₉	—	2	4	no	—
40	C₄⋊C₄×C₇₉	—	2	2	no	—
41	C₂²⋊C₄×C₇₉	—	2	2	no	—
42	C₄²×C₇₉	—	2	4	no	—
43	C₄²×C₇₉	D₄	1	4	yes	—
44	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₇₉	—	2	1	no	—
46	Q₈×C₇₉	—	2	4	no	—
47	Q₈×C₇₉	—	2	4	no	—
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	4	no	—
51	C₂×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#41	1	2	yes	—
56	C₂²⋊D₇₉	—	4	1	no	—
57	C₂²⋊D₇₉	—	4	1	no	—
58	C₂×Dic₇₉	—	2	2	no	—
59	C₂×Dic₇₉	—	2	1	no	—
60	D₃₁₆	—	2	2	no	—
61	C₄×D₇₉	—	4	2	no	—
62	Dic₁₅₈	—	2	2	no	—
63	C₇₉⋊C₈	—	4	2	no	—
64	C₂²×C₇₉	—	4	1	no	—
65	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₇₉	#120	1	1	yes	characteristic
67	C₂²×C₇₉	#123	1	1	yes	characteristic
68	C₂×D₇₉	—	4	2	no	—
69	C₂×D₇₉	—	4	1	no	—
70	C₄×C₇₉	—	2	4	no	—
71	C₄×C₇₉	—	4	4	no	—
72	C₄×C₇₉	C₄×D₄	1	2	yes	—
73	C₄×C₇₉	#121	1	2	yes	—
74	Dic₇₉	—	4	2	no	—
75	Dic₇₉	—	4	1	no	—
76	C₂×C₇₉	—	2	1	no	—
77	C₂×C₇₉	—	2	1	no	—
78	C₂×C₇₉	—	2	1	no	—
79	C₂×C₇₉	#386	1	1	yes	characteristic
80	D₇₉	—	8	1	no	—
81	#1912	—	79	1	no	maximal, Sylow
82	C₇₉	#1912	1	1	yes	characteristic, complemented, Sylow
83	#536	—	79	1	no	—
84	#519	D₇₉	1	1	yes	characteristic, complemented
85	#421	—	79	2	no	—
86	C₂×#101	—	79	1	no	—
87	H(3,ℤ/4)	—	79	2	no	—
88	C₂×#41	—	79	1	no	—
89	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
90	C₂×M₁₆	—	79	2	no	—
91	C₄⋊Q₈	—	79	2	no	—
92	#124	—	79	2	no	—
93	#123	—	158	2	no	—
94	C₂²⋊Q₈	—	158	2	no	—
95	C₄×Q₈	—	2	4	no	—
96	#117	—	2	2	no	—
97	#117	—	79	1	no	—
98	#117	C₂×D₇₉	1	2	yes	—
99	#104	—	158	4	no	—
100	#101	—	158	2	no	—
101	#41	—	158	2	no	—
102	#41	—	158	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	79	1	no	—
106	C₂×Q₈	C₄×D₇₉	1	2	yes	—
107	C₂×Q₈	D₃₁₆	1	2	yes	—
108	C₂×D₄	—	79	1	no	—
109	C₂×D₄	—	158	1	no	—
110	C₂²×C₄	—	158	1	no	—
111	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
113	M₁₆	—	158	2	no	—
114	M₁₆	—	158	2	no	—
115	C₂×C₈	—	158	2	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	158	4	no	—
119	C₄⋊C₄	—	158	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	158	1	no	—
122	C₂²⋊C₄	—	158	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	—	79	2	no	—
125	C₄²	C₂²⋊D₇₉	1	4	yes	—
126	C₂³	—	158	1	no	—
127	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	158	2	no	—
131	D₄	—	316	1	no	—
132	D₄	—	316	1	no	—
133	D₄	—	158	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	316	2	no	—
138	C₂×C₄	—	158	2	no	—
139	C₂×C₄	—	158	1	no	—
140	C₂×C₄	(C₂×D₇₉)⋊C₄	1	4	yes	—
141	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
143	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
144	C₂×C₄	#11929624	1	2	yes	—
145	C₈	—	316	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	316	2	no	—
148	C₂²	—	316	1	no	—
149	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
150	C₂²	#14579130	1	1	yes	characteristic
151	C₂²	#14579144	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	4	4	no	—
154	C₄	—	316	2	no	—
155	C₄	—	316	1	no	—
156	C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
157	C₄	#14579135	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	632	1	no	—
162	C₂	#19098442	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099137

C₂

yes

maximal, characteristic

#19098581

C₂

yes

maximal

C₂×#14579080

C₂

yes

maximal, characteristic, complemented

#19098319

C₂

yes

maximal, complemented

#519×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

—

#14579159

C₄

yes

complemented

#14579150

C₄

yes

—

—

C₂²

yes

characteristic

#14579089

—

#14579080

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

—

#117×C₇₉

—

#117×C₇₉

C₂²

yes

—

#11929624

—

#11929624

—

C₂×Dic₁₅₈

C₂×C₄

yes

characteristic

C₂×D₃₁₆

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

C₂×C₄

yes

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂×C₄

yes

—

C₂×Q₈×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

—

C₄×C₇₉

#121

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#386

yes

characteristic

D₇₉

—

#1912

—

maximal, Sylow

C₇₉

#1912

yes

characteristic, complemented, Sylow

#536

—

#519

D₇₉

yes

characteristic, complemented

#421

—

C₂×#101

—

H(3,ℤ/4)

—

C₂×#41

—

C₂²×Q₈

C₂×D₇₉

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

#124

—

#123

—

158

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

#117

—

#117

—

#117

C₂×D₇₉

yes

—

#104

—

158

—

100

#101

—

158

—

101

#41

—

158

—

102

#41

—

158

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₄×D₇₉

yes

—

107

C₂×Q₈

D₃₁₆

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

158

—

110

C₂²×C₄

—

158

—

111

C₂²×C₄

C₂²⋊D₇₉

yes

—

112

C₂²×C₄

C₂²×D₇₉

yes

characteristic

113

M₁₆

—

158

—

114

M₁₆

—

158

—

115

C₂×C₈

—

158

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

158

—

119

C₄⋊C₄

—

158

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

158

—

122

C₂²⋊C₄

—

158

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂²⋊D₇₉

yes

—

126

C₂³

—

158

—

127

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

158

—

131

D₄

—

316

—

132

D₄

—

316

—

133

D₄

—

158

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

316

—

138

C₂×C₄

—

158

—

139

C₂×C₄

—

158

—

140

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

141

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

143

C₂×C₄

C₂×D₃₁₆

yes

characteristic

144

C₂×C₄

#11929624

yes

—

145

C₈

—

316

—

146

C₂²

—

147

C₂²

—

316

—

148

C₂²

—

316

—

149

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

150

C₂²

#14579130

yes

characteristic

151

C₂²

#14579144

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

316

—

155

C₄

—

316

—

156

C₄

C₄×C₂²⋊D₇₉

yes

—

157

C₄

#14579135

yes

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

632

—

162

C₂

#19098442

yes

characteristic

	subgroup	G/H
center	C₂	#19098442
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#519×C₇₉	C₂
socle	C₂×C₇₉	#386
prime radical	C₂×#11929624	C₂²
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#519	D₇₉
hypocenter	C₇₉	#1912
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #519

subgroup

G/H

center

C₂

#19098442

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#519×C₇₉

C₂

C₂×C₇₉

C₂²

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #519

N	H ≅ G/N	type
3. C₂×#14579080	C₂	semidirect
4. #19098319	C₂	semidirect
5. #519×C₇₉	C₂	semidirect
8. #14579159	C₄	semidirect
9. #14579150	C₄	semidirect
82. C₇₉	#1912	semidirect
84. #519	D₇₉	semidirect

H ≅ G/N

type

3. C₂×#14579080

C₂

semidirect

4. #19098319

C₂

semidirect

5. #519×C₇₉

C₂

semidirect

8. #14579159

C₄

semidirect

9. #14579150

C₄

semidirect

82. C₇₉

#1912

semidirect

84. #519

D₇₉

semidirect

Subgroups — #23763937

Conjugacy classes of subgroups

Complements