Group #21424903 — subgroups

Subgroups — #21424903

Order 7552.

436 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₂×#14273648	C₂	1	4	yes	maximal, complemented
2	C₂×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837272	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₉	—	2	2	no	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	#14273648	—	2	8	no	—
9	C₄×C₅₉⋊C₈	—	2	2	no	—
10	C₂²×D₄×C₅₉	—	2	2	no	—
11	C₂²×D₄×C₅₉	C₂²	1	2	yes	—
12	#127×C₅₉	—	2	2	no	—
13	#127×C₅₉	C₄	1	4	yes	complemented
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₂⁴×C₅₉	—	2	2	no	—
16	C₂×D₄×C₅₉	—	2	8	no	—
17	C₂×D₄×C₅₉	—	2	4	no	—
18	C₂×D₄×C₅₉	—	2	2	no	—
19	C₂×D₄×C₅₉	—	2	4	no	—
20	C₂×D₄×C₅₉	—	4	4	no	—
21	C₂×D₄×C₅₉	—	4	4	no	—
22	C₂×D₄×C₅₉	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₅₉	D₄	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₉	—	2	2	no	—
26	C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	C₄²×C₅₉	D₄	1	2	yes	—
28	C₄⋊Dic₅₉	—	2	4	no	—
29	C₄⋊Dic₅₉	—	4	4	no	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂×C₅₉⋊C₈	D₄	1	4	yes	complemented
33	C₂³×C₅₉	—	2	4	no	—
34	C₂³×C₅₉	—	4	2	no	—
35	C₂³×C₅₉	—	4	2	no	—
36	C₂³×C₅₉	—	4	4	no	—
37	C₂³×C₅₉	—	4	4	no	—
38	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
39	D₄×C₅₉	—	2	8	no	—
40	D₄×C₅₉	—	4	16	no	—
41	D₄×C₅₉	—	4	4	no	—
42	C₂×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₅₉	D₈	1	4	yes	—
44	C₂×C₄×C₅₉	SD₁₆	1	4	yes	—
45	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#41	1	2	yes	—
49	C₂×Dic₅₉	—	4	2	no	—
50	C₂×Dic₅₉	—	4	4	no	—
51	C₅₉⋊C₈	—	2	4	no	—
52	C₂²×C₅₉	—	4	4	no	—
53	C₂²×C₅₉	—	4	8	no	—
54	C₂²×C₅₉	—	8	4	no	—
55	C₂²×C₅₉	—	8	2	no	—
56	C₂²×C₅₉	—	8	2	no	—
57	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₅₉	#124	1	1	yes	characteristic
59	C₂²×C₅₉	#127	1	1	yes	characteristic
60	C₂²×C₅₉	C₂×D₈	1	2	yes	—
61	C₂²×C₅₉	C₂×SD₁₆	1	2	yes	—
62	C₄×C₅₉	#102	1	8	yes	complemented
63	C₄×C₅₉	C₄×D₄	1	2	yes	—
64	C₄×C₅₉	#121	1	2	yes	—
65	Dic₅₉	—	8	2	no	—
66	#1957	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	—	8	4	no	—
68	C₂×C₅₉	#390	1	1	yes	characteristic
69	C₂×C₅₉	C₂×#102	1	2	yes	—
70	C₂×C₅₉	#486	1	2	yes	—
71	C₂×C₅₉	#492	1	1	yes	characteristic
72	C₂×C₅₉	#493	1	1	yes	characteristic
73	C₂×#127	D₅₉	1	1	yes	characteristic
74	C₂×#102	—	59	4	no	—
75	C₂×C₄×C₈	—	59	1	no	—
76	#384	—	59	1	no	—
77	C₅₉	#1957	1	1	yes	characteristic, complemented, Sylow
78	C₂²×D₄	—	2	2	no	—
79	C₂²×D₄	C₂×D₅₉	1	2	yes	—
80	C₂²×C₈	—	59	2	no	—
81	#127	—	2	2	no	—
82	#127	Dic₅₉	1	4	yes	complemented
83	C₂×C₄⋊C₄	—	59	2	no	—
84	C₂×C₄⋊C₄	—	118	2	no	—
85	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
86	#102	—	118	8	no	—
87	C₄×C₈	—	118	2	no	—
88	C₂⁴	—	2	2	no	—
89	C₂×D₄	—	2	8	no	—
90	C₂×D₄	—	2	4	no	—
91	C₂×D₄	—	2	4	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	4	4	no	—
94	C₂×D₄	—	4	4	no	—
95	C₂×D₄	C₂×Dic₅₉	1	4	yes	—
96	C₂²×C₄	—	118	2	no	—
97	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
98	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
99	C₂×C₈	—	59	4	no	—
100	C₂×C₈	—	118	2	no	—
101	C₂×C₈	—	118	2	no	—
102	C₄⋊C₄	—	118	4	no	—
103	C₄⋊C₄	—	236	4	no	—
104	C₄²	C₂×Dic₅₉	1	2	yes	—
105	C₄²	C₂²⋊D₅₉	1	2	yes	—
106	C₂³	—	2	4	no	—
107	C₂³	—	4	2	no	—
108	C₂³	—	4	2	no	—
109	C₂³	—	4	4	no	—
110	C₂³	—	4	4	no	—
111	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
112	D₄	—	2	8	no	—
113	D₄	—	4	16	no	—
114	D₄	—	4	4	no	—
115	C₂×C₄	—	236	2	no	—
116	C₂×C₄	—	236	4	no	—
117	C₂×C₄	#11745669	1	4	yes	—
118	C₂×C₄	#11745670	1	4	yes	—
119	C₂×C₄	C₂²⋊Dic₅₉	1	4	yes	—
120	C₂×C₄	D₄×D₅₉	1	2	yes	—
121	C₂×C₄	#11745687	1	2	yes	—
122	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
124	C₈	—	118	4	no	—
125	C₂²	—	4	8	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	8	4	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	2	no	—
130	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
131	C₂²	#14273714	1	1	yes	characteristic
132	C₂²	#14273723	1	1	yes	characteristic
133	C₂²	C₂×#11745669	1	2	yes	—
134	C₂²	C₂×#11745670	1	2	yes	—
135	C₄	—	472	2	no	—
136	C₄	#14273648	1	8	yes	complemented
137	C₄	D₄×Dic₅₉	1	2	yes	—
138	C₄	#14273698	1	2	yes	—
139	C₂	—	8	4	no	—
140	C₂	#16837305	1	1	yes	characteristic
141	C₂	C₂×#14273648	1	2	yes	—
142	C₂	#16837748	1	2	yes	—
143	C₂	#16837774	1	1	yes	characteristic
144	C₂	#16837778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273648

C₂

yes

maximal, complemented

C₂×C₄×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#16837272

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273648

—

C₄×C₅₉⋊C₈

—

C₂²×D₄×C₅₉

—

C₂²×D₄×C₅₉

C₂²

yes

—

#127×C₅₉

—

#127×C₅₉

C₄

yes

complemented

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

complemented

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×D₈

yes

—

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

#102

yes

complemented

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

#121

yes

—

Dic₅₉

—

#1957

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#390

yes

characteristic

C₂×C₅₉

C₂×#102

yes

—

C₂×C₅₉

#486

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

C₂×#102

—

C₂×C₄×C₈

—

#384

—

C₅₉

#1957

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

C₂×D₅₉

yes

—

C₂²×C₈

—

#127

—

#127

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#102

—

118

—

C₄×C₈

—

118

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₉

yes

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

118

—

101

C₂×C₈

—

118

—

102

C₄⋊C₄

—

118

—

103

C₄⋊C₄

—

236

—

104

C₄²

C₂×Dic₅₉

yes

—

105

C₄²

C₂²⋊D₅₉

yes

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

236

—

116

C₂×C₄

—

236

—

117

C₂×C₄

#11745669

yes

—

118

C₂×C₄

#11745670

yes

—

119

C₂×C₄

C₂²⋊Dic₅₉

yes

—

120

C₂×C₄

D₄×D₅₉

yes

—

121

C₂×C₄

#11745687

yes

—

122

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

124

C₈

—

118

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

131

C₂²

#14273714

yes

characteristic

132

C₂²

#14273723

yes

characteristic

133

C₂²

C₂×#11745669

yes

—

134

C₂²

C₂×#11745670

yes

—

135

C₄

—

472

—

136

C₄

#14273648

yes

complemented

137

C₄

D₄×Dic₅₉

yes

—

138

C₄

#14273698

yes

—

139

C₂

—

140

C₂

#16837305

yes

characteristic

141

C₂

C₂×#14273648

yes

—

142

C₂

#16837748

yes

—

143

C₂

#16837774

yes

characteristic

144

C₂

#16837778

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#127×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#127	D₅₉
hypocenter	C₅₉	#1957
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#127×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#14273648	C₂	semidirect
2. C₂×C₄×C₅₉⋊C₈	C₂	semidirect
3. #16837272	C₂	semidirect
13. #127×C₅₉	C₄	semidirect
32. C₂×C₅₉⋊C₈	D₄	semidirect
62. C₄×C₅₉	#102	semidirect
77. C₅₉	#1957	semidirect
82. #127	Dic₅₉	semidirect
136. C₄	#14273648	semidirect

H ≅ G/N

type

1. C₂×#14273648

C₂

semidirect

2. C₂×C₄×C₅₉⋊C₈

C₂

semidirect

3. #16837272

C₂

semidirect

13. #127×C₅₉

C₄

semidirect

32. C₂×C₅₉⋊C₈

D₄

semidirect

62. C₄×C₅₉

#102

semidirect

77. C₅₉

#1957

semidirect

82. #127

Dic₅₉

semidirect

136. C₄

#14273648

semidirect

Subgroups — #21424903

Conjugacy classes of subgroups

Complements