Group #21422005 — subgroups

Subgroups — #21422005

Order 7552.

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

normal subgroups only

This group has 192 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	#16837366	C₂	1	1	yes	maximal, characteristic
2	#16837265	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#407	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745664	—	2	1	no	—
5	C₂×C₂²⋊Dic₅₉	C₄	1	2	yes	—
6	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	complemented
8	C₅₉⋊#95	—	2	1	no	—
9	C₅₉⋊#95	—	2	1	no	—
10	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₅₉	—	2	1	no	—
12	C₂²⋊C₈×C₅₉	—	2	1	no	—
13	C₂²⋊C₈×C₅₉	C₄	1	2	yes	complemented
14	C₂⁴×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	—	4	2	no	—
16	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
22	C₂²×Dic₅₉	—	2	2	no	—
23	M₁₆×C₅₉	—	4	2	no	—
24	C₂×C₈×C₅₉	—	2	2	no	—
25	C₂²⋊Dic₅₉	—	4	2	no	—
26	#11745664	—	4	2	no	—
27	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂³×C₅₉	—	2	2	no	—
29	C₂³×C₅₉	—	4	1	no	—
30	C₂³×C₅₉	—	4	1	no	—
31	C₂³×C₅₉	C₄²	1	1	yes	characteristic
32	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	—	2	2	no	—
35	C₂×C₄×C₅₉	—	2	2	no	—
36	C₂×C₄×C₅₉	—	4	1	no	—
37	C₂×C₄×C₅₉	—	4	1	no	—
38	C₂×C₄×C₅₉	—	4	1	no	—
39	C₂×C₄×C₅₉	—	4	1	no	—
40	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₅₉	—	4	2	no	—
45	C₂×Dic₅₉	—	4	2	no	—
46	C₂×Dic₅₉	—	4	2	no	—
47	C₈×C₅₉	—	4	2	no	—
48	C₅₉⋊C₈	—	4	2	no	—
49	C₂²×C₅₉	—	2	1	no	—
50	C₂²×C₅₉	—	2	1	no	—
51	C₂²×C₅₉	—	2	1	no	—
52	C₂²×C₅₉	—	2	1	no	—
53	C₂²×C₅₉	—	2	1	no	—
54	C₂²×C₅₉	—	2	1	no	—
55	C₂²×C₅₉	—	2	1	no	—
56	C₂²×C₅₉	—	2	1	no	—
57	C₂²×C₅₉	—	4	4	no	—
58	C₂²×C₅₉	#95	1	1	yes	characteristic
59	C₂²×C₅₉	#99	1	2	yes	—
60	C₄×C₅₉	—	2	1	no	—
61	C₄×C₅₉	—	2	1	no	—
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	—	2	1	no	—
64	Dic₅₉	—	8	2	no	—
65	#1306	—	59	1	no	maximal, Sylow
66	C₂×C₅₉	—	2	2	no	—
67	C₂×C₅₉	—	4	1	no	—
68	C₂×C₅₉	—	4	1	no	—
69	C₂×C₅₉	#342	1	1	yes	characteristic
70	C₂×C₅₉	#344	1	1	yes	characteristic
71	C₂×C₅₉	#344	1	1	yes	characteristic
72	#407	—	59	1	no	—
73	#407	D₅₉	1	1	yes	characteristic
74	#381	—	59	1	no	—
75	C₅₉	#1306	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
77	C₂×M₁₆	—	2	1	no	—
78	C₂×M₁₆	—	118	1	no	—
79	C₂×C₂²⋊C₄	—	59	2	no	—
80	C₂²⋊C₈	—	2	1	no	—
81	C₂²⋊C₈	—	59	2	no	—
82	C₂²⋊C₈	—	118	1	no	—
83	C₂²⋊C₈	Dic₅₉	1	2	yes	complemented
84	#95	—	118	1	no	—
85	#95	—	118	1	no	—
86	C₂⁴	C₄×D₅₉	1	1	yes	characteristic
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	118	2	no	—
89	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
90	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
91	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
95	M₁₆	—	4	2	no	—
96	M₁₆	—	236	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	118	2	no	—
99	C₂²⋊C₄	—	236	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	—	4	1	no	—
102	C₂³	—	4	1	no	—
103	C₂³	C₄×Dic₅₉	1	1	yes	characteristic
104	C₂³	Dic₅₉⋊C₄	1	1	yes	characteristic
105	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	236	2	no	—
113	C₂×C₄	—	236	2	no	—
114	C₂×C₄	—	236	2	no	—
115	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
117	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	236	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	4	4	no	—
130	C₂²	C₅₉⋊#95	1	1	yes	characteristic
131	C₂²	#14273641	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	472	2	no	—
137	C₂	—	2	2	no	—
138	C₂	—	4	1	no	—
139	C₂	—	4	1	no	—
140	C₂	#16837168	1	1	yes	characteristic
141	C₂	#16837173	1	1	yes	characteristic
142	C₂	#16837174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837366

C₂

yes

maximal, characteristic

#16837265

C₂

yes

maximal, characteristic

C₅₉×#407

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₂×C₂²⋊Dic₅₉

C₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

C₄

yes

complemented

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

C₄

yes

complemented

C₂⁴×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂²⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₄²

yes

characteristic

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#95

yes

characteristic

C₂²×C₅₉

#99

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

#1306

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

#381

—

C₅₉

#1306

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

118

—

C₂²⋊C₈

Dic₅₉

yes

complemented

#95

—

118

—

#95

—

118

—

C₂⁴

C₄×D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂²⋊C₄

—

236

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

C₄×Dic₅₉

yes

characteristic

104

C₂³

Dic₅₉⋊C₄

yes

characteristic

105

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

236

—

113

C₂×C₄

—

236

—

114

C₂×C₄

—

236

—

115

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

116

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

117

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

118

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

119

C₈

—

120

C₈

—

236

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₅₉⋊#95

yes

characteristic

131

C₂²

#14273641

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

472

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

#16837168

yes

characteristic

141

C₂

#16837173

yes

characteristic

142

C₂

#16837174

yes

characteristic

	subgroup	G/H
center	C₂²	C₅₉⋊#95
commutator	C₂³×C₅₉	C₄²
Frattini	C₂³×C₄	C₂×D₅₉
Fitting	C₅₉×#407	C₂
socle	C₂²×C₅₉	#95
hypercenter	#407	D₅₉
hypocenter	C₅₉	#1306
LCS	G ▷ C₂³×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₅₉⋊#95

commutator

C₂³×C₅₉

C₄²

Frattini

C₂³×C₄

C₂×D₅₉

Fitting

C₅₉×#407

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
7. C₂²⋊(C₅₉⋊C₈)	C₄	semidirect
13. C₂²⋊C₈×C₅₉	C₄	semidirect
75. C₅₉	#1306	semidirect
83. C₂²⋊C₈	Dic₅₉	semidirect

H ≅ G/N

type

7. C₂²⋊(C₅₉⋊C₈)

C₄

semidirect

13. C₂²⋊C₈×C₅₉

C₄

semidirect

75. C₅₉

#1306

semidirect

83. C₂²⋊C₈

Dic₅₉

semidirect

Subgroups — #21422005

Conjugacy classes of subgroups

Complements