Group #23763548 — subgroups

Subgroups — #23763548

Order 10112.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic
2	#19098334	C₂	1	1	yes	maximal, characteristic
3	#19098324	C₂	1	1	yes	maximal, characteristic
4	#19098322	C₂	1	1	yes	maximal, characteristic
5	C₇₉⋊#338	C₂	1	1	yes	maximal, characteristic
6	#19098319	C₂	1	1	yes	maximal, characteristic
7	#344×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929608	C₂²	1	1	yes	characteristic
9	C₂×#11929608	C₂²	1	1	yes	characteristic
10	#14579120	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
12	#14579071	C₄	1	2	yes	complemented
13	C₈×Dic₇₉	C₄	1	2	yes	complemented
14	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	#117×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₇₉	—	2	1	no	—
18	#11929608	—	2	1	no	—
19	#11929608	—	2	1	no	—
20	#11929608	—	2	1	no	—
21	#11929608	—	2	1	no	—
22	C₂×C₇₉⋊C₈	—	2	1	no	—
23	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄⋊Dic₇₉	—	2	1	no	—
25	C₂²⋊Dic₇₉	—	2	1	no	—
26	C₄×Dic₇₉	—	2	1	no	—
27	C₄×Dic₇₉	—	2	1	no	—
28	C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₇₉	—	2	1	no	—
32	M₁₆×C₇₉	—	2	1	no	—
33	M₁₆×C₇₉	—	2	1	no	—
34	M₁₆×C₇₉	—	2	1	no	—
35	C₂×C₈×C₇₉	—	2	1	no	—
36	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₇₉	—	2	1	no	—
38	C₂²⋊C₄×C₇₉	—	2	1	no	—
39	C₄²×C₇₉	—	2	1	no	—
40	C₂³×C₇₉	#41	1	1	yes	characteristic
41	C₂×C₄×C₇₉	—	4	1	no	—
42	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
44	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
47	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
48	C₂×Dic₇₉	—	2	1	no	—
49	C₂×Dic₇₉	—	2	1	no	—
50	C₂×Dic₇₉	—	4	2	no	—
51	C₂×Dic₇₉	—	4	1	no	—
52	C₈×C₇₉	—	2	2	no	—
53	C₈×C₇₉	—	4	1	no	—
54	C₇₉⋊C₈	—	2	2	no	—
55	C₇₉⋊C₈	—	4	1	no	—
56	C₂²×C₇₉	—	4	1	no	—
57	C₂²×C₇₉	#117	1	1	yes	characteristic
58	C₂²×C₇₉	#126	1	1	yes	characteristic
59	C₂²×C₇₉	#126	1	1	yes	characteristic
60	C₄×C₇₉	—	8	1	no	—
61	C₄×C₇₉	#117	1	1	yes	characteristic
62	C₄×C₇₉	#117	1	1	yes	characteristic
63	C₄×C₇₉	#126	1	1	yes	characteristic
64	C₄×C₇₉	#126	1	1	yes	characteristic
65	Dic₇₉	—	2	2	no	—
66	Dic₇₉	—	4	1	no	—
67	Dic₇₉	—	8	1	no	—
68	C₂×C₇₉	—	2	1	no	—
69	C₂×C₇₉	—	2	1	no	—
70	C₂×C₇₉	—	2	1	no	—
71	C₂×C₇₉	#383	1	1	yes	characteristic
72	#1841	—	79	1	no	maximal, Sylow
73	C₇₉	#1841	1	1	yes	characteristic, complemented, Sylow
74	C₄×M₁₆	—	79	1	no	—
75	#344	—	79	1	no	—
76	#344	D₇₉	1	1	yes	characteristic
77	#339	—	79	1	no	—
78	#339	—	79	1	no	—
79	#338	—	79	1	no	—
80	H(3,ℤ/4)	—	79	1	no	—
81	C₂×M₁₆	—	79	1	no	—
82	C₂×M₁₆	—	79	1	no	—
83	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
84	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
85	#117	—	79	1	no	—
86	#117	C₂×D₇₉	1	1	yes	characteristic
87	C₂×C₄²	—	79	1	no	—
88	#97	—	79	2	no	—
89	C₄×C₈	—	79	2	no	—
90	C₂²×C₄	—	158	1	no	—
91	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	158	1	no	—
97	M₁₆	—	158	1	no	—
98	M₁₆	—	158	1	no	—
99	M₁₆	—	158	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	79	2	no	—
102	C₂×C₈	—	158	1	no	—
103	C₂×C₈	C₄×D₇₉	1	2	yes	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	158	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	158	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	79	1	no	—
110	C₄²	—	79	1	no	—
111	C₄²	—	158	1	no	—
112	C₄²	—	158	1	no	—
113	C₂³	#11929624	1	1	yes	characteristic
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	316	2	no	—
116	C₂×C₄	—	316	1	no	—
117	C₂×C₄	—	158	1	no	—
118	C₂×C₄	—	158	1	no	—
119	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
120	C₂×C₄	#11929623	1	1	yes	characteristic
121	C₂×C₄	#11929623	1	1	yes	characteristic
122	C₂×C₄	#11929623	1	1	yes	characteristic
123	C₂×C₄	#11929624	1	1	yes	characteristic
124	C₂×C₄	#11929625	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	4	1	no	—
127	C₈	—	316	1	no	—
128	C₈	—	158	2	no	—
129	C₂²	—	4	1	no	—
130	C₂²	#14579118	1	1	yes	characteristic
131	C₂²	#14579154	1	1	yes	characteristic
132	C₂²	#14579156	1	1	yes	characteristic
133	C₄	—	8	1	no	—
134	C₄	—	316	1	no	—
135	C₄	—	632	1	no	—
136	C₄	—	158	2	no	—
137	C₄	#14579117	1	1	yes	characteristic
138	C₄	#14579119	1	1	yes	characteristic
139	C₄	#14579154	1	1	yes	characteristic
140	C₄	#14579156	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#19098430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₇₉

C₂

yes

maximal, characteristic

#19098334

C₂

yes

maximal, characteristic

#19098324

C₂

yes

maximal, characteristic

#19098322

C₂

yes

maximal, characteristic

C₇₉⋊#338

C₂

yes

maximal, characteristic

#19098319

C₂

yes

maximal, characteristic

#344×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

#14579120

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579071

C₄

yes

complemented

C₈×Dic₇₉

C₄

yes

complemented

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

#41

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

#344

—

#344

D₇₉

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

characteristic

C₂×M₁₆

C₂×D₇₉

yes

characteristic

#117

—

#117

C₂×D₇₉

yes

characteristic

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

158

—

M₁₆

—

158

—

M₁₆

—

158

—

M₁₆

—

158

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

158

—

103

C₂×C₈

C₄×D₇₉

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

158

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

158

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

158

—

112

C₄²

—

158

—

113

C₂³

#11929624

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

316

—

116

C₂×C₄

—

316

—

117

C₂×C₄

—

158

—

118

C₂×C₄

—

158

—

119

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

120

C₂×C₄

#11929623

yes

characteristic

121

C₂×C₄

#11929623

yes

characteristic

122

C₂×C₄

#11929623

yes

characteristic

123

C₂×C₄

#11929624

yes

characteristic

124

C₂×C₄

#11929625

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

316

—

128

C₈

—

158

—

129

C₂²

—

130

C₂²

#14579118

yes

characteristic

131

C₂²

#14579154

yes

characteristic

132

C₂²

#14579156

yes

characteristic

133

C₄

—

134

C₄

—

316

—

135

C₄

—

632

—

136

C₄

—

158

—

137

C₄

#14579117

yes

characteristic

138

C₄

#14579119

yes

characteristic

139

C₄

#14579154

yes

characteristic

140

C₄

#14579156

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#19098430

yes

characteristic

	subgroup	G/H
center	C₄	#14579117
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#344×C₇₉	C₂
socle	C₂×C₇₉	#383
prime radical	C₂³×C₇₉	#41
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#344	D₇₉
hypocenter	C₇₉	#1841
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#14579117

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#344×C₇₉

C₂

C₂×C₇₉

C₂³×C₇₉

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
12. #14579071	C₄	semidirect
13. C₈×Dic₇₉	C₄	semidirect
73. C₇₉	#1841	semidirect

H ≅ G/N

type

12. #14579071

C₄

semidirect

13. C₈×Dic₇₉

C₄

semidirect

73. C₇₉

#1841

semidirect

Subgroups — #23763548

Conjugacy classes of subgroups

Complements