Group #21426318 — subgroups

Subgroups — #21426318

Order 7552.

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

normal subgroups only

This group has 160 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	#16837806	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	C₁₆×Dic₅₉	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	#14273726	—	2	1	no	—
11	#14273723	—	2	1	no	—
12	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14273661	C₂²	1	1	yes	characteristic
14	#14273660	—	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	C₂×Dic₁₁₈	—	4	1	no	—
24	D₈×C₅₉	—	2	1	no	—
25	D₈×C₅₉	—	2	1	no	—
26	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₅₉	—	2	1	no	—
28	#11745669	—	4	2	no	—
29	C₄⋊Dic₅₉	—	2	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	Dic₁₁₈	—	8	2	no	—
51	C₈×C₅₉	#41	1	1	yes	characteristic
52	C₈×C₅₉	#41	1	1	yes	characteristic
53	C₅₉⋊C₈	SD₁₆	1	2	yes	—
54	C₂²×C₅₉	—	8	1	no	—
55	C₂²×C₅₉	—	8	1	no	—
56	C₂²×C₅₉	—	8	1	no	—
57	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₂×D₅₉	—	8	1	no	—
59	C₂×D₅₉	—	8	1	no	—
60	C₂×D₅₉	—	8	1	no	—
61	C₄×C₅₉	—	8	1	no	—
62	C₄×C₅₉	#124	1	1	yes	characteristic
63	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
64	Dic₅₉	—	8	1	no	—
65	Dic₅₉	D₁₆	1	2	yes	—
66	Dic₅₉	SD₃₂	1	2	yes	—
67	#2240	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	—	16	1	no	—
69	C₂×C₅₉	#486	1	1	yes	characteristic
70	C₂×C₅₉	C₂×D₁₆	1	1	yes	characteristic
71	C₂×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
72	D₅₉	—	16	1	no	—
73	#500	—	59	1	no	—
74	#493	—	59	1	no	—
75	#357	—	59	1	no	—
76	#357	—	59	1	no	—
77	#357	—	59	1	no	—
78	#357	D₅₉	1	1	yes	characteristic, complemented
79	C₄×C₁₆	—	59	1	no	—
80	C₅₉	#2240	1	1	yes	characteristic, complemented, Sylow
81	C₂×D₈	—	59	1	no	—
82	C₂×D₈	—	118	1	no	—
83	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
84	C₄⋊Q₈	—	118	1	no	—
85	#127	—	118	1	no	—
86	C₂×C₁₆	—	59	1	no	—
87	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
88	#107	—	59	1	no	—
89	#107	—	118	1	no	—
90	#107	C₂×D₅₉	1	1	yes	characteristic
91	C₄×C₈	—	59	1	no	—
92	C₂×Q₈	—	236	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	118	1	no	—
95	C₂×D₄	—	236	1	no	—
96	C₂×D₄	—	236	1	no	—
97	D₈	—	2	1	no	—
98	D₈	—	2	1	no	—
99	D₈	—	118	1	no	—
100	D₈	—	118	1	no	—
101	D₈	—	236	2	no	—
102	C₂×C₈	—	59	1	no	—
103	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	118	1	no	—
106	C₄⋊C₄	—	236	1	no	—
107	C₄²	—	59	1	no	—
108	C₁₆	—	2	1	no	—
109	C₁₆	—	118	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	236	1	no	—
112	Q₈	—	472	2	no	—
113	D₄	—	4	1	no	—
114	D₄	—	4	1	no	—
115	D₄	—	472	2	no	—
116	D₄	—	472	2	no	—
117	D₄	—	236	1	no	—
118	D₄	—	236	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	59	1	no	—
121	C₂×C₄	—	59	1	no	—
122	C₂×C₄	—	236	1	no	—
123	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
124	C₈	—	59	2	no	—
125	C₈	#11745685	1	1	yes	characteristic
126	C₈	#11745687	1	1	yes	characteristic
127	C₂²	—	8	1	no	—
128	C₂²	—	8	1	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	472	1	no	—
131	C₂²	—	472	1	no	—
132	C₂²	—	472	1	no	—
133	C₂²	D₈×D₅₉	1	1	yes	characteristic
134	C₄	—	8	1	no	—
135	C₄	—	59	2	no	—
136	C₄	—	59	2	no	—
137	C₄	—	472	1	no	—
138	C₄	#14273713	1	1	yes	characteristic
139	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
140	C₂	—	16	1	no	—
141	C₂	—	944	1	no	—
142	C₂	#16837747	1	1	yes	characteristic
143	C₂	D₁₆×D₅₉	1	1	yes	characteristic
144	C₂	SD₃₂×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837806

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

C₁₆×Dic₅₉

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

#14273660

—

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

—

C₂×Dic₁₁₈

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745669

—

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

—

Dic₁₁₈

—

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

D₁₆

yes

—

Dic₅₉

SD₃₂

yes

—

#2240

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#486

yes

characteristic

C₂×C₅₉

C₂×D₁₆

yes

characteristic

C₂×C₅₉

C₂×SD₃₂

yes

characteristic

D₅₉

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₄×C₁₆

—

C₅₉

#2240

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

118

—

C₂×D₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

118

—

#127

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#107

—

118

—

#107

C₂×D₅₉

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

236

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

D₈

—

D₈

—

D₈

—

118

—

100

D₈

—

118

—

101

D₈

—

236

—

102

C₂×C₈

—

103

C₂×C₈

C₂²×D₅₉

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

118

—

106

C₄⋊C₄

—

236

—

107

C₄²

—

108

C₁₆

—

109

C₁₆

—

118

—

110

C₂³

—

111

C₂³

—

236

—

112

Q₈

—

472

—

113

D₄

—

114

D₄

—

115

D₄

—

472

—

116

D₄

—

472

—

117

D₄

—

236

—

118

D₄

—

236

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

236

—

123

C₂×C₄

D₄×D₅₉

yes

characteristic

124

C₈

—

125

C₈

#11745685

yes

characteristic

126

C₈

#11745687

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

472

—

131

C₂²

—

472

—

132

C₂²

—

472

—

133

C₂²

D₈×D₅₉

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

472

—

138

C₄

#14273713

yes

characteristic

139

C₄

SD₁₆×D₅₉

yes

characteristic

140

C₂

—

141

C₂

—

944

—

142

C₂

#16837747

yes

characteristic

143

C₂

D₁₆×D₅₉

yes

characteristic

144

C₂

SD₃₂×D₅₉

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	#2240
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. #16837806	C₂	semidirect
3. #16837204	C₂	semidirect
4. #16837203	C₂	semidirect
5. #16837202	C₂	semidirect
6. C₁₆×Dic₅₉	C₂	semidirect
7. C₅₉×#357	C₂	semidirect
78. #357	D₅₉	semidirect
80. C₅₉	#2240	semidirect

H ≅ G/N

type

1. #16837806

C₂

semidirect

3. #16837204

C₂

semidirect

4. #16837203

C₂

semidirect

5. #16837202

C₂

semidirect

6. C₁₆×Dic₅₉

C₂

semidirect

7. C₅₉×#357

C₂

semidirect

78. #357

D₅₉

semidirect

80. C₅₉

#2240

semidirect

Subgroups — #21426318

Conjugacy classes of subgroups

Complements