Group #21426336 — subgroups

Subgroups — #21426336

Order 7552.

152 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16837803	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837777	C₂	1	1	yes	maximal, characteristic
3	#16837204	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837203	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837202	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×#11745669	—	2	1	no	—
10	#14273723	—	2	1	no	—
11	#14273718	—	2	1	no	—
12	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14273661	C₂²	1	1	yes	characteristic
14	#14273658	—	2	1	no	—
15	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	#107×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₅₉	—	2	1	no	—
20	C₂×C₂²⋊D₅₉	—	4	1	no	—
21	C₂×C₂²⋊D₅₉	—	4	1	no	—
22	C₂×D₂₃₆	—	2	1	no	—
23	D₈×C₅₉	—	2	1	no	—
24	D₈×C₅₉	—	2	1	no	—
25	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	1	no	—
27	#11745669	—	4	2	no	—
28	C₄⋊Dic₅₉	—	2	1	no	—
29	C₄⋊Dic₅₉	—	4	1	no	—
30	Dic₅₉⋊C₄	—	4	1	no	—
31	C₄×Dic₅₉	D₄	1	1	yes	characteristic
32	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	D₄₇₂	—	2	1	no	—
34	D₄₇₂	—	2	1	no	—
35	C₁₆×C₅₉	—	2	1	no	—
36	C₅₉⋊C₁₆	—	2	1	no	—
37	C₂³×C₅₉	—	4	1	no	—
38	C₂²×D₅₉	—	4	1	no	—
39	D₄×C₅₉	—	4	1	no	—
40	D₄×C₅₉	—	4	1	no	—
41	C₂×C₄×C₅₉	—	4	1	no	—
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂²⋊D₅₉	—	8	2	no	—
44	C₂²⋊D₅₉	—	8	2	no	—
45	C₂×Dic₅₉	—	4	1	no	—
46	C₂×Dic₅₉	D₈	1	1	yes	characteristic
47	C₂×Dic₅₉	D₈	1	1	yes	characteristic
48	D₂₃₆	—	4	1	no	—
49	D₂₃₆	—	4	1	no	—
50	C₈×C₅₉	#41	1	1	yes	characteristic
51	C₈×C₅₉	#41	1	1	yes	characteristic
52	C₅₉⋊C₈	SD₁₆	1	2	yes	—
53	C₂²×C₅₉	—	8	1	no	—
54	C₂²×C₅₉	—	8	1	no	—
55	C₂²×C₅₉	—	8	1	no	—
56	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
57	C₂×D₅₉	—	8	1	no	—
58	C₂×D₅₉	—	8	1	no	—
59	C₂×D₅₉	—	8	1	no	—
60	C₄×C₅₉	—	8	1	no	—
61	C₄×C₅₉	#124	1	1	yes	characteristic
62	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
63	Dic₅₉	—	2	1	no	—
64	Dic₅₉	—	2	1	no	—
65	Dic₅₉	—	8	1	no	—
66	#2244	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	—	16	1	no	—
68	C₂×C₅₉	#486	1	1	yes	characteristic
69	C₂×C₅₉	#509	1	1	yes	characteristic
70	C₂×C₅₉	#509	1	1	yes	characteristic
71	D₅₉	—	16	1	no	—
72	#499	—	59	1	no	—
73	#493	—	59	1	no	—
74	#357	—	59	1	no	—
75	#357	—	59	1	no	—
76	#357	—	59	1	no	—
77	#357	D₅₉	1	1	yes	characteristic, complemented
78	#346	—	59	1	no	—
79	C₅₉	#2244	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₈	—	59	1	no	—
81	C₂×D₈	—	118	1	no	—
82	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
83	#127	—	118	1	no	—
84	#125	—	118	1	no	—
85	C₂×C₁₆	—	59	1	no	—
86	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
87	#107	—	59	1	no	—
88	#107	C₂×D₅₉	1	1	yes	characteristic
89	#106	—	118	1	no	—
90	C₄×C₈	—	59	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	118	1	no	—
93	C₂×D₄	—	236	1	no	—
94	C₂×D₄	—	236	1	no	—
95	D₈	—	2	1	no	—
96	D₈	—	2	1	no	—
97	D₈	—	118	1	no	—
98	D₈	—	118	1	no	—
99	D₈	—	236	2	no	—
100	C₂×C₈	—	59	1	no	—
101	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	118	1	no	—
104	C₄⋊C₄	—	236	1	no	—
105	C₄⋊C₄	—	236	1	no	—
106	C₄²	—	59	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	118	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	236	1	no	—
111	D₄	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	472	2	no	—
114	D₄	—	472	2	no	—
115	D₄	—	236	1	no	—
116	D₄	—	236	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	59	1	no	—
119	C₂×C₄	—	59	1	no	—
120	C₂×C₄	—	236	1	no	—
121	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
122	C₈	—	59	2	no	—
123	C₈	#11745685	1	1	yes	characteristic
124	C₈	#11745687	1	1	yes	characteristic
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	472	1	no	—
129	C₂²	—	472	1	no	—
130	C₂²	—	472	1	no	—
131	C₂²	D₈×D₅₉	1	1	yes	characteristic
132	C₄	—	8	1	no	—
133	C₄	—	118	1	no	—
134	C₄	—	118	1	no	—
135	C₄	—	472	1	no	—
136	C₄	#14273713	1	1	yes	characteristic
137	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
138	C₂	—	16	1	no	—
139	C₂	—	944	1	no	—
140	C₂	#16837747	1	1	yes	characteristic
141	C₂	#16837836	1	1	yes	characteristic
142	C₂	#16837839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837803

C₂

yes

maximal, characteristic, complemented

#16837777

C₂

yes

maximal, characteristic

#16837204

C₂

yes

maximal, characteristic, complemented

#16837203

C₂

yes

maximal, characteristic, complemented

#16837202

C₂

yes

maximal, characteristic, complemented

#16837177

C₂

yes

maximal, characteristic, complemented

C₅₉×#357

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

—

—

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273658

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×C₂²⋊D₅₉

—

C₂×C₂²⋊D₅₉

—

C₂×D₂₃₆

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745669

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

D₄₇₂

—

D₄₇₂

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂²×D₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

SD₁₆

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2244

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

#346

—

C₅₉

#2244

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

118

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#127

—

118

—

#125

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#107

C₂×D₅₉

yes

characteristic

#106

—

118

—

C₄×C₈

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

D₈

—

D₈

—

D₈

—

118

—

D₈

—

118

—

D₈

—

236

—

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₅₉

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

118

—

104

C₄⋊C₄

—

236

—

105

C₄⋊C₄

—

236

—

106

C₄²

—

107

C₁₆

—

108

C₁₆

—

118

—

109

C₂³

—

110

C₂³

—

236

—

111

D₄

—

112

D₄

—

113

D₄

—

472

—

114

D₄

—

472

—

115

D₄

—

236

—

116

D₄

—

236

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

236

—

121

C₂×C₄

D₄×D₅₉

yes

characteristic

122

C₈

—

123

C₈

#11745685

yes

characteristic

124

C₈

#11745687

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

472

—

129

C₂²

—

472

—

130

C₂²

—

472

—

131

C₂²

D₈×D₅₉

yes

characteristic

132

C₄

—

133

C₄

—

118

—

134

C₄

—

118

—

135

C₄

—

472

—

136

C₄

#14273713

yes

characteristic

137

C₄

SD₁₆×D₅₉

yes

characteristic

138

C₂

—

139

C₂

—

944

—

140

C₂

#16837747

yes

characteristic

141

C₂

#16837836

yes

characteristic

142

C₂

#16837839

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#357	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#357	D₅₉
hypocenter	C₅₉	#2244
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#357

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #16837803	C₂	semidirect
3. #16837204	C₂	semidirect
4. #16837203	C₂	semidirect
5. #16837202	C₂	semidirect
6. #16837177	C₂	semidirect
7. C₅₉×#357	C₂	semidirect
77. #357	D₅₉	semidirect
79. C₅₉	#2244	semidirect

H ≅ G/N

type

1. #16837803

C₂

semidirect

3. #16837204

C₂

semidirect

4. #16837203

C₂

semidirect

5. #16837202

C₂

semidirect

6. #16837177

C₂

semidirect

7. C₅₉×#357

C₂

semidirect

77. #357

D₅₉

semidirect

79. C₅₉

#2244

semidirect

Subgroups — #21426336

Conjugacy classes of subgroups

Complements