Group #23762912 — subgroups

Subgroups — #23762912

Order 10112.

166 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 51 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	#19098966	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098769	C₂	1	1	yes	maximal, characteristic
3	#19098765	C₂	1	1	yes	maximal, characteristic
4	#19098640	C₂	1	1	yes	maximal, characteristic
5	#19098313	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098308	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929613	—	2	1	no	—
9	C₂×#11929611	—	2	1	no	—
10	C₂×#11929607	—	2	1	no	—
11	#14579161	C₂²	1	1	yes	characteristic
12	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
13	#14579094	—	2	1	no	—
14	#14579094	—	2	1	no	—
15	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	#14579086	—	2	1	no	—
17	#14579085	—	2	1	no	—
18	(C₂×D₇₉)⋊C₈	—	2	1	no	—
19	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₈	—	2	1	no	—
24	C₂×D₃₁₆	D₄	1	1	yes	characteristic
25	C₂×C₄×D₇₉	—	2	1	no	—
26	#11929613	—	4	2	no	—
27	#11929611	—	4	2	no	—
28	#11929607	—	4	2	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
32	C₄⋊Dic₇₉	—	2	1	no	—
33	C₄⋊Dic₇₉	—	2	1	no	—
34	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
35	(C₂×D₇₉)⋊C₄	—	2	1	no	—
36	C₂×Q₈×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₄⋊C₄×C₇₉	—	2	1	no	—
41	C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	C₂²×D₇₉	—	2	1	no	—
43	Q₈×C₇₉	—	4	1	no	—
44	Q₈×C₇₉	—	4	1	no	—
45	C₂×C₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
49	C₂×Dic₇₉	—	2	1	no	—
50	C₂×Dic₇₉	—	4	1	no	—
51	D₃₁₆	—	2	1	no	—
52	D₃₁₆	—	2	1	no	—
53	C₄×D₇₉	—	4	1	no	—
54	C₄×D₇₉	—	4	1	no	—
55	Dic₁₅₈	—	4	1	no	—
56	Dic₁₅₈	—	4	1	no	—
57	C₈×C₇₉	—	4	1	no	—
58	C₈×C₇₉	—	4	1	no	—
59	C₇₉⋊C₈	—	2	1	no	—
60	C₇₉⋊C₈	—	4	1	no	—
61	C₇₉⋊C₈	SD₁₆	1	2	yes	complemented
62	C₂²×C₇₉	#121	1	1	yes	characteristic
63	C₂×D₇₉	—	4	1	no	—
64	C₂×D₇₉	—	4	1	no	—
65	C₂×D₇₉	—	4	1	no	—
66	C₄×C₇₉	—	2	1	no	—
67	C₄×C₇₉	—	8	1	no	—
68	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₇₉	#137	1	1	yes	characteristic
72	Dic₇₉	—	4	1	no	—
73	Dic₇₉	—	8	1	no	—
74	C₂×C₇₉	#461	1	1	yes	characteristic
75	C₂×C₇₉	#469	1	1	yes	characteristic
76	C₂×C₇₉	#473	1	1	yes	characteristic
77	D₇₉	—	8	1	no	—
78	#1693	—	79	1	no	maximal, Sylow
79	C₇₉	#1693	1	1	yes	characteristic, complemented, Sylow
80	#500	—	79	1	no	—
81	#461	—	79	1	no	—
82	#461	—	79	1	no	—
83	#436	—	79	1	no	—
84	#334	—	79	1	no	—
85	#332	—	79	1	no	—
86	#332	D₇₉	1	1	yes	characteristic, complemented
87	C₂×SD₁₆	—	158	1	no	—
88	C₂×SD₁₆	—	158	1	no	—
89	C₂×M₁₆	—	158	1	no	—
90	C₄⋊Q₈	—	79	1	no	—
91	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
92	C₄×D₄	—	79	1	no	—
93	#107	—	158	1	no	—
94	#107	—	158	1	no	—
95	C₄⋊C₈	—	79	1	no	—
96	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
97	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
98	#103	—	158	1	no	—
99	#103	—	158	1	no	—
100	C₂²⋊C₈	—	158	1	no	—
101	C₄×C₈	—	79	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	158	1	no	—
104	C₂×D₄	—	79	1	no	—
105	C₂²×C₄	—	158	1	no	—
106	SD₁₆	—	316	2	no	—
107	SD₁₆	—	316	2	no	—
108	M₁₆	—	316	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	79	1	no	—
112	C₂×C₈	—	79	1	no	—
113	C₂×C₈	—	158	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	79	1	no	—
117	C₄⋊C₄	—	158	1	no	—
118	C₄⋊C₄	—	158	1	no	—
119	C₂²⋊C₄	—	158	1	no	—
120	C₄²	C₂²×D₇₉	1	1	yes	characteristic
121	C₂³	—	158	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	316	1	no	—
125	Q₈	—	316	1	no	—
126	D₄	—	158	1	no	—
127	D₄	—	158	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	316	1	no	—
130	C₂×C₄	—	316	1	no	—
131	C₂×C₄	—	316	1	no	—
132	C₂×C₄	—	158	1	no	—
133	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
134	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
135	C₂×C₄	#11929624	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	316	1	no	—
139	C₈	—	79	2	no	—
140	C₈	—	158	1	no	—
141	C₂²	—	316	1	no	—
142	C₂²	—	316	1	no	—
143	C₂²	—	316	1	no	—
144	C₂²	#14579134	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	1	no	—
147	C₄	—	316	1	no	—
148	C₄	—	632	1	no	—
149	C₄	SD₁₆×D₇₉	1	1	yes	characteristic
150	C₄	#14579185	1	1	yes	characteristic
151	C₄	#14579188	1	1	yes	characteristic
152	C₄	#14579193	1	1	yes	characteristic
153	C₂	—	632	1	no	—
154	C₂	#19098767	1	1	yes	characteristic
155	C₂	#19098815	1	1	yes	characteristic
156	C₂	#19098837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098966

C₂

yes

maximal, characteristic, complemented

#19098769

C₂

yes

maximal, characteristic

#19098765

C₂

yes

maximal, characteristic

#19098640

C₂

yes

maximal, characteristic

#19098313

C₂

yes

maximal, characteristic, complemented

#19098308

C₂

yes

maximal, characteristic, complemented

#332×C₇₉

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄×D₃₁₆

C₂²

yes

characteristic

#14579094

—

#14579094

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579086

—

#14579085

—

(C₂×D₇₉)⋊C₈

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

D₄

yes

characteristic

C₂×C₄×D₇₉

—

#11929613

—

#11929611

—

#11929607

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×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

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₇₉

#121

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#1693

—

maximal, Sylow

C₇₉

#1693

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₂×SD₁₆

—

158

—

C₂×SD₁₆

—

158

—

C₂×M₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

C₄×D₄

—

#107

—

158

—

#107

—

158

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₈

C₂×D₇₉

yes

characteristic

#103

—

158

—

#103

—

158

—

100

C₂²⋊C₈

—

158

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

158

—

104

C₂×D₄

—

105

C₂²×C₄

—

158

—

106

SD₁₆

—

316

—

107

SD₁₆

—

316

—

108

M₁₆

—

316

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

158

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

158

—

118

C₄⋊C₄

—

158

—

119

C₂²⋊C₄

—

158

—

120

C₄²

C₂²×D₇₉

yes

characteristic

121

C₂³

—

158

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

316

—

125

Q₈

—

316

—

126

D₄

—

158

—

127

D₄

—

158

—

128

C₂×C₄

—

129

C₂×C₄

—

316

—

130

C₂×C₄

—

316

—

131

C₂×C₄

—

316

—

132

C₂×C₄

—

158

—

133

C₂×C₄

D₄×D₇₉

yes

characteristic

134

C₂×C₄

D₄×D₇₉

yes

characteristic

135

C₂×C₄

#11929624

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

316

—

139

C₈

—

140

C₈

—

158

—

141

C₂²

—

316

—

142

C₂²

—

316

—

143

C₂²

—

316

—

144

C₂²

#14579134

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

316

—

148

C₄

—

632

—

149

C₄

SD₁₆×D₇₉

yes

characteristic

150

C₄

#14579185

yes

characteristic

151

C₄

#14579188

yes

characteristic

152

C₄

#14579193

yes

characteristic

153

C₂

—

632

—

154

C₂

#19098767

yes

characteristic

155

C₂

#19098815

yes

characteristic

156

C₂

#19098837

yes

characteristic

	subgroup	G/H
center	C₂²	#14579134
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#332×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	C₂×D₃₁₆	D₄
hypercenter	#332	D₇₉
hypocenter	C₇₉	#1693
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14579134

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#332×C₇₉

C₂

socle

C₂²×C₇₉

#121

prime radical

C₂×D₃₁₆

D₄

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #19098966	C₂	semidirect
5. #19098313	C₂	semidirect
6. #19098308	C₂	semidirect
7. #332×C₇₉	C₂	semidirect
61. C₇₉⋊C₈	SD₁₆	semidirect
79. C₇₉	#1693	semidirect
86. #332	D₇₉	semidirect

H ≅ G/N

type

1. #19098966

C₂

semidirect

5. #19098313

C₂

semidirect

6. #19098308

C₂

semidirect

7. #332×C₇₉

C₂

semidirect

61. C₇₉⋊C₈

SD₁₆

semidirect

79. C₇₉

#1693

semidirect

86. #332

D₇₉

semidirect

Subgroups — #23762912

Conjugacy classes of subgroups

Complements