Group #21426157 — subgroups

Subgroups — #21426157

Order 7552.

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

normal subgroups only

This group has 158 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	C₂×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273671	C₂	1	1	yes	maximal, characteristic
3	#16837785	C₂	1	1	yes	maximal, characteristic
4	#16837496	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837205	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	C₂×#11745670	—	2	1	no	—
11	#14273714	—	2	1	no	—
12	#14273692	—	2	1	no	—
13	Dic₄₇₂	—	2	2	no	—
14	#14273671	—	2	2	no	—
15	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14273650	—	2	1	no	—
17	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	#107×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₅₉	—	2	1	no	—
22	C₂×Dic₁₁₈	—	2	1	no	—
23	C₂×Dic₁₁₈	—	2	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	C₂²⋊Dic₅₉	—	4	1	no	—
29	C₂²⋊Dic₅₉	—	4	1	no	—
30	#11745670	—	4	2	no	—
31	Dic₅₉⋊C₄	—	4	1	no	—
32	C₄×Dic₅₉	—	4	1	no	—
33	C₄×Dic₅₉	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	Dic₂₃₆	—	2	1	no	—
36	Dic₂₃₆	—	2	1	no	—
37	Dic₂₃₆	—	2	1	no	—
38	Dic₂₃₆	D₄	1	2	yes	—
39	C₁₆×C₅₉	—	2	1	no	—
40	C₅₉⋊C₁₆	—	2	1	no	—
41	C₂³×C₅₉	—	4	1	no	—
42	D₄×C₅₉	—	4	1	no	—
43	D₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₅₉	—	4	1	no	—
47	C₂×Dic₅₉	—	4	1	no	—
48	C₂×Dic₅₉	D₈	1	1	yes	characteristic
49	C₂×Dic₅₉	D₈	1	1	yes	characteristic
50	Dic₁₁₈	—	2	2	no	—
51	Dic₁₁₈	—	4	1	no	—
52	Dic₁₁₈	—	4	1	no	—
53	Dic₁₁₈	—	4	1	no	—
54	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₈×C₅₉	#41	1	1	yes	characteristic
56	C₅₉⋊C₈	—	2	1	no	—
57	C₂²×C₅₉	—	8	1	no	—
58	C₂²×C₅₉	—	8	1	no	—
59	C₂²×C₅₉	—	8	1	no	—
60	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
61	C₄×C₅₉	—	8	1	no	—
62	C₄×C₅₉	#121	1	1	yes	characteristic
63	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	Dic₅₉	—	2	1	no	—
65	Dic₅₉	—	2	1	no	—
66	Dic₅₉	—	4	2	no	—
67	Dic₅₉	—	8	1	no	—
68	#2211	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	—	16	1	no	—
70	C₂×C₅₉	#459	1	1	yes	characteristic
71	C₂×C₅₉	#508	1	1	yes	characteristic
72	C₂×C₅₉	#510	1	1	yes	characteristic
73	C₂×Q₃₂	—	59	1	no	—
74	C₂×SD₃₂	—	59	1	no	—
75	#495	—	59	1	no	—
76	C₄×Q₁₆	—	59	1	no	—
77	C₄⋊C₁₆	—	59	1	no	—
78	#358	—	59	1	no	—
79	#357	D₅₉	1	1	yes	characteristic
80	C₅₉	#2211	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	59	1	no	—
82	C₂×Q₁₆	—	59	1	no	—
83	C₂×SD₁₆	—	118	1	no	—
84	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
85	#124	—	118	1	no	—
86	C₄×Q₈	—	118	1	no	—
87	Q₃₂	—	118	2	no	—
88	SD₃₂	—	118	2	no	—
89	C₂×C₁₆	—	59	1	no	—
90	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
91	#107	C₂×D₅₉	1	1	yes	characteristic
92	#103	—	118	1	no	—
93	C₄×C₈	—	59	1	no	—
94	C₂×Q₈	—	118	1	no	—
95	C₂×Q₈	—	118	1	no	—
96	C₂×D₄	—	2	1	no	—
97	Q₁₆	—	59	2	no	—
98	Q₁₆	—	118	1	no	—
99	Q₁₆	—	118	1	no	—
100	Q₁₆	—	118	1	no	—
101	SD₁₆	—	236	2	no	—
102	D₈	—	2	1	no	—
103	D₈	—	2	1	no	—
104	C₂×C₈	—	59	1	no	—
105	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	236	1	no	—
108	C₂²⋊C₄	—	236	1	no	—
109	C₂²⋊C₄	—	236	1	no	—
110	C₄²	—	59	1	no	—
111	C₄²	—	236	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	118	1	no	—
114	C₂³	—	4	1	no	—
115	Q₈	—	118	2	no	—
116	Q₈	—	236	1	no	—
117	Q₈	—	236	1	no	—
118	Q₈	—	236	1	no	—
119	D₄	—	4	1	no	—
120	D₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	59	1	no	—
123	C₂×C₄	—	59	1	no	—
124	C₂×C₄	—	236	1	no	—
125	C₂×C₄	—	236	1	no	—
126	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
127	C₈	—	118	1	no	—
128	C₈	#11745685	1	1	yes	characteristic
129	C₈	D₄×D₅₉	1	1	yes	characteristic
130	C₂²	—	8	1	no	—
131	C₂²	—	8	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	D₈×D₅₉	1	1	yes	characteristic
134	C₄	—	8	1	no	—
135	C₄	—	118	1	no	—
136	C₄	—	118	1	no	—
137	C₄	—	472	1	no	—
138	C₄	—	236	2	no	—
139	C₄	#14273700	1	1	yes	characteristic
140	C₄	#14273754	1	1	yes	characteristic
141	C₂	—	16	1	no	—
142	C₂	#16837596	1	1	yes	characteristic
143	C₂	#16837831	1	1	yes	characteristic
144	C₂	#16837841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₇₂

C₂

yes

maximal, characteristic, complemented

C₂×#14273671

C₂

yes

maximal, characteristic

#16837785

C₂

yes

maximal, characteristic

#16837496

C₂

yes

maximal, characteristic, complemented

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#16837205

C₂

yes

maximal, characteristic, complemented

C₅₉×#357

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₂²

yes

—

—

—

Dic₄₇₂

—

#14273671

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273650

—

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×Dic₁₁₈

—

C₂×Dic₁₁₈

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

#11745670

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

D₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2211

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×Q₃₂

—

C₂×SD₃₂

—

#495

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#357

D₅₉

yes

characteristic

C₅₉

#2211

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

118

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#124

—

118

—

C₄×Q₈

—

118

—

Q₃₂

—

118

—

SD₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

C₂×D₅₉

yes

characteristic

#103

—

118

—

C₄×C₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

118

—

Q₁₆

—

118

—

100

Q₁₆

—

118

—

101

SD₁₆

—

236

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×D₅₉

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

236

—

108

C₂²⋊C₄

—

236

—

109

C₂²⋊C₄

—

236

—

110

C₄²

—

111

C₄²

—

236

—

112

C₁₆

—

113

C₁₆

—

118

—

114

C₂³

—

115

Q₈

—

118

—

116

Q₈

—

236

—

117

Q₈

—

236

—

118

Q₈

—

236

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

236

—

125

C₂×C₄

—

236

—

126

C₂×C₄

D₄×D₅₉

yes

characteristic

127

C₈

—

118

—

128

C₈

#11745685

yes

characteristic

129

C₈

D₄×D₅₉

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

D₈×D₅₉

yes

characteristic

134

C₄

—

135

C₄

—

118

—

136

C₄

—

118

—

137

C₄

—

472

—

138

C₄

—

236

—

139

C₄

#14273700

yes

characteristic

140

C₄

#14273754

yes

characteristic

141

C₂

—

142

C₂

#16837596

yes

characteristic

143

C₂

#16837831

yes

characteristic

144

C₂

#16837841

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	#2211
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. C₂×Dic₄₇₂	C₂	semidirect
4. #16837496	C₂	semidirect
5. Dic₅₉⋊C₁₆	C₂	semidirect
6. #16837205	C₂	semidirect
80. C₅₉	#2211	semidirect

H ≅ G/N

type

1. C₂×Dic₄₇₂

C₂

semidirect

4. #16837496

C₂

semidirect

5. Dic₅₉⋊C₁₆

C₂

semidirect

6. #16837205

C₂

semidirect

80. C₅₉

#2211

semidirect

Subgroups — #21426157

Conjugacy classes of subgroups

Complements