Group #21422878 — subgroups

Subgroups — #21422878

Order 7552.

160 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16837458	C₂	1	1	yes	maximal, characteristic
2	#16837453	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837155	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837154	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837152	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₅₉	—	2	1	no	—
9	C₄²×D₅₉	C₂²	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₅₉⋊C₈	—	2	1	no	—
13	Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
14	Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
15	(C₂×D₅₉)⋊C₈	—	2	1	no	—
16	(C₂×D₅₉)⋊C₈	C₄	1	1	yes	characteristic
17	(C₂×D₅₉)⋊C₈	C₄	1	1	yes	characteristic
18	#14273637	—	2	1	no	—
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₉	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	—	2	1	no	—
28	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₅₉	D₄	1	1	yes	characteristic
33	C₄×Dic₅₉	Q₈	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	—	2	1	no	—
35	C₂×C₅₉⋊C₈	—	2	1	no	—
36	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₈×D₅₉	—	4	2	no	—
39	C₂²×D₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄×D₅₉	—	2	2	no	—
47	C₄×D₅₉	—	2	1	no	—
48	C₄×D₅₉	—	2	1	no	—
49	C₄×D₅₉	—	2	1	no	—
50	C₄×D₅₉	—	2	1	no	—
51	C₈×C₅₉	—	2	1	no	—
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	—	4	1	no	—
54	C₈×C₅₉	—	4	1	no	—
55	C₅₉⋊C₈	—	2	1	no	—
56	C₅₉⋊C₈	—	2	1	no	—
57	C₅₉⋊C₈	—	4	1	no	—
58	C₅₉⋊C₈	—	4	1	no	—
59	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×D₅₉	—	2	1	no	—
61	C₂×D₅₉	—	2	1	no	—
62	C₂×D₅₉	#108	1	2	yes	—
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	#131	1	1	yes	characteristic
65	C₄×C₅₉	#131	1	1	yes	characteristic
66	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₄×C₅₉	#137	1	1	yes	characteristic
68	Dic₅₉	—	2	1	no	—
69	Dic₅₉	—	2	1	no	—
70	Dic₅₉	—	2	1	no	—
71	#1573	—	59	1	no	maximal, Sylow
72	C₂×C₅₉	#424	1	1	yes	characteristic
73	C₂×C₅₉	#428	1	1	yes	characteristic
74	C₂×C₅₉	C₂×#108	1	1	yes	characteristic
75	D₅₉	—	2	2	no	—
76	#432	—	59	1	no	—
77	#431	—	59	1	no	—
78	C₂×C₄⋊C₈	—	59	1	no	—
79	#335	—	59	1	no	—
80	#335	—	59	1	no	—
81	#334	—	59	1	no	—
82	#334	D₅₉	1	1	yes	characteristic, complemented
83	C₅₉	#1573	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	118	1	no	—
85	C₂×C₄²	—	59	1	no	—
86	C₄⋊C₈	—	59	1	no	—
87	C₄⋊C₈	—	59	1	no	—
88	C₄⋊C₈	—	59	1	no	—
89	C₄⋊C₈	—	59	1	no	—
90	C₄⋊C₈	—	118	1	no	—
91	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
92	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
93	C₂²⋊C₈	—	59	1	no	—
94	C₂²⋊C₈	—	59	1	no	—
95	C₂²⋊C₈	—	118	1	no	—
96	#97	—	118	1	no	—
97	C₄×C₈	—	59	1	no	—
98	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
99	C₂²×C₄	—	59	1	no	—
100	C₂²×C₄	—	59	1	no	—
101	C₂²×C₄	—	59	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	59	1	no	—
105	C₂×C₈	—	59	1	no	—
106	C₂×C₈	—	118	1	no	—
107	C₂×C₈	—	118	1	no	—
108	C₂×C₈	—	236	2	no	—
109	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
110	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
111	C₄²	—	59	1	no	—
112	C₄²	—	59	1	no	—
113	C₄²	—	59	1	no	—
114	C₄²	C₂²×D₅₉	1	1	yes	characteristic
115	C₂³	—	59	1	no	—
116	C₂×C₄	—	59	1	no	—
117	C₂×C₄	—	59	1	no	—
118	C₂×C₄	—	59	1	no	—
119	C₂×C₄	—	118	2	no	—
120	C₂×C₄	—	118	1	no	—
121	C₂×C₄	—	118	1	no	—
122	C₂×C₄	—	118	1	no	—
123	C₂×C₄	—	118	1	no	—
124	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
125	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	4	1	no	—
131	C₈	—	118	1	no	—
132	C₈	—	118	1	no	—
133	C₈	—	236	1	no	—
134	C₈	—	236	1	no	—
135	C₂²	—	59	2	no	—
136	C₂²	—	118	1	no	—
137	C₂²	—	118	1	no	—
138	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	118	1	no	—
141	C₄	—	118	1	no	—
142	C₄	—	118	1	no	—
143	C₄	#14273733	1	1	yes	characteristic
144	C₄	#14273735	1	1	yes	characteristic
145	C₄	#14273754	1	1	yes	characteristic
146	C₄	#14273756	1	1	yes	characteristic
147	C₂	—	118	2	no	—
148	C₂	#16837427	1	1	yes	characteristic
149	C₂	#16837441	1	1	yes	characteristic
150	C₂	#108×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837458

C₂

yes

maximal, characteristic

#16837453

C₂

yes

maximal, characteristic

C₄⋊C₈×D₅₉

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, characteristic, complemented

#16837154

C₂

yes

maximal, characteristic, complemented

#16837152

C₂

yes

maximal, characteristic, complemented

C₅₉×#334

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₉

—

C₄²×D₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

Dic₅₉⋊C₈

C₄

yes

characteristic

Dic₅₉⋊C₈

C₄

yes

characteristic

(C₂×D₅₉)⋊C₈

—

(C₂×D₅₉)⋊C₈

C₄

yes

characteristic

(C₂×D₅₉)⋊C₈

C₄

yes

characteristic

#14273637

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

Q₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₈×D₅₉

—

C₂²×D₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×Dic₅₉

C₄⋊C₄

yes

characteristic

C₂×Dic₅₉

C₄⋊C₄

yes

characteristic

C₂×Dic₅₉

C₄⋊C₄

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

#108

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#432

—

#431

—

C₂×C₄⋊C₈

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#1573

yes

characteristic, complemented, Sylow

C₂²×C₈

—

118

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

118

—

#97

—

118

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

118

—

107

C₂×C₈

—

118

—

108

C₂×C₈

—

236

—

109

C₂×C₈

C₄×D₅₉

yes

characteristic

110

C₂×C₈

C₄×D₅₉

yes

characteristic

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂²×D₅₉

yes

characteristic

115

C₂³

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

118

—

120

C₂×C₄

—

118

—

121

C₂×C₄

—

118

—

122

C₂×C₄

—

118

—

123

C₂×C₄

—

118

—

124

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

125

C₂×C₄

D₄×D₅₉

yes

characteristic

126

C₂×C₄

Q₈×D₅₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

118

—

132

C₈

—

118

—

133

C₈

—

236

—

134

C₈

—

236

—

135

C₂²

—

136

C₂²

—

118

—

137

C₂²

—

118

—

138

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

139

C₄

—

140

C₄

—

118

—

141

C₄

—

118

—

142

C₄

—

118

—

143

C₄

#14273733

yes

characteristic

144

C₄

#14273735

yes

characteristic

145

C₄

#14273754

yes

characteristic

146

C₄

#14273756

yes

characteristic

147

C₂

—

118

—

148

C₂

#16837427

yes

characteristic

149

C₂

#16837441

yes

characteristic

150

C₂

#108×D₅₉

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#334	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#334	D₅₉
hypocenter	C₅₉	#1573
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#334

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
4. #16837155	C₂	semidirect
5. #16837154	C₂	semidirect
6. #16837152	C₂	semidirect
7. C₅₉×#334	C₂	semidirect
82. #334	D₅₉	semidirect
83. C₅₉	#1573	semidirect

H ≅ G/N

type

4. #16837155

C₂

semidirect

5. #16837154

C₂

semidirect

6. #16837152

C₂

semidirect

7. C₅₉×#334

C₂

semidirect

82. #334

D₅₉

semidirect

83. C₅₉

#1573

semidirect

Subgroups — #21422878

Conjugacy classes of subgroups

Complements