Group #21422985 — subgroups

Subgroups — #21422985

Order 7552.

166 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16837529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11745660	C₂	1	1	yes	maximal, characteristic
3	#16837479	C₂	1	1	yes	maximal, characteristic
4	#16837132	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837129	C₂	1	1	yes	maximal, characteristic
6	C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745660	C₄	1	1	yes	characteristic
9	C₂×#11745659	—	2	1	no	—
10	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
11	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
12	#14273659	C₄	1	1	yes	characteristic
13	Dic₅₉⋊C₈	—	2	1	no	—
14	#14273652	C₄	1	1	yes	characteristic
15	#14273649	C₄	1	1	yes	characteristic
16	(C₂×D₅₉)⋊C₈	—	2	1	no	—
17	#14273637	—	2	1	no	—
18	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
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×D₂₃₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₉	—	2	1	no	—
25	C₂×Dic₁₁₈	C₂×C₄	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	(C₂×D₅₉)⋊C₄	—	2	1	no	—
32	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
34	Dic₅₉⋊C₄	—	2	1	no	—
35	C₄×Dic₅₉	—	2	1	no	—
36	C₂×C₅₉⋊C₈	—	2	1	no	—
37	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
39	#11745660	—	2	2	no	—
40	#11745659	—	4	2	no	—
41	C₂²×D₅₉	—	2	1	no	—
42	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
45	C₂×Dic₅₉	—	2	1	no	—
46	C₂×Dic₅₉	—	2	1	no	—
47	C₂×Dic₅₉	—	2	1	no	—
48	D₂₃₆	—	2	1	no	—
49	D₂₃₆	—	2	1	no	—
50	C₄×D₅₉	—	4	1	no	—
51	C₄×D₅₉	—	4	1	no	—
52	Dic₁₁₈	—	2	1	no	—
53	Dic₁₁₈	—	2	1	no	—
54	C₈×C₅₉	—	2	1	no	—
55	C₈×C₅₉	—	4	1	no	—
56	C₈×C₅₉	—	4	1	no	—
57	C₈×C₅₉	M₁₆	1	2	yes	complemented
58	C₅₉⋊C₈	—	2	2	no	—
59	C₅₉⋊C₈	—	2	1	no	—
60	C₅₉⋊C₈	SD₁₆	1	2	yes	complemented
61	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
62	C₂×D₅₉	—	4	1	no	—
63	C₂×D₅₉	—	4	1	no	—
64	C₂×D₅₉	—	4	1	no	—
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₅₉	#131	1	1	yes	characteristic
68	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₅₉	#135	1	1	yes	characteristic
70	Dic₅₉	—	4	1	no	—
71	Dic₅₉	—	4	1	no	—
72	Dic₅₉	—	4	1	no	—
73	#1590	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	#436	1	1	yes	characteristic
75	C₂×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₅₉	#443	1	1	yes	characteristic
77	D₅₉	—	8	1	no	—
78	#446	—	59	1	no	—
79	C₄×SD₁₆	—	59	1	no	—
80	#436	—	59	1	no	—
81	#334	D₅₉	1	1	yes	characteristic, complemented
82	Q₈⋊C₈	—	59	1	no	—
83	D₄⋊C₈	—	59	1	no	—
84	C₈²	—	59	1	no	—
85	C₅₉	#1590	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	59	1	no	—
87	C₂×M₁₆	—	118	1	no	—
88	C₄×Q₈	—	59	1	no	—
89	C₄×D₄	—	59	1	no	—
90	#106	—	59	1	no	—
91	C₄⋊C₈	—	118	1	no	—
92	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
94	#103	—	59	1	no	—
95	#102	—	59	1	no	—
96	C₂²⋊C₈	—	118	1	no	—
97	#97	—	118	1	no	—
98	C₄×C₈	—	59	1	no	—
99	C₄×C₈	—	59	1	no	—
100	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	C₂×Q₈	—	59	1	no	—
102	C₂×D₄	—	59	1	no	—
103	C₂²×C₄	—	118	1	no	—
104	SD₁₆	—	118	2	no	—
105	M₁₆	—	236	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	59	1	no	—
109	C₂×C₈	—	59	1	no	—
110	C₂×C₈	—	118	1	no	—
111	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
112	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
113	C₄⋊C₄	—	59	1	no	—
114	C₄⋊C₄	—	59	1	no	—
115	C₄⋊C₄	—	118	1	no	—
116	C₂²⋊C₄	—	118	1	no	—
117	C₄²	—	118	1	no	—
118	C₄²	C₂²×D₅₉	1	1	yes	characteristic
119	C₂³	—	118	1	no	—
120	Q₈	—	118	1	no	—
121	Q₈	—	118	1	no	—
122	D₄	—	118	1	no	—
123	D₄	—	118	1	no	—
124	C₂×C₄	—	118	1	no	—
125	C₂×C₄	—	118	1	no	—
126	C₂×C₄	—	118	1	no	—
127	C₂×C₄	—	236	1	no	—
128	C₂×C₄	—	236	1	no	—
129	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
130	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
131	C₂×C₄	#11745689	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	59	2	no	—
136	C₈	—	118	2	no	—
137	C₈	—	118	1	no	—
138	C₈	#11745659	1	2	yes	complemented
139	C₂²	—	236	1	no	—
140	C₂²	—	236	1	no	—
141	C₂²	—	236	1	no	—
142	C₂²	#14273689	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	236	1	no	—
145	C₄	—	236	1	no	—
146	C₄	—	236	1	no	—
147	C₄	C₂×#11745659	1	1	yes	characteristic
148	C₄	#14273735	1	1	yes	characteristic
149	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
150	C₄	#14273748	1	1	yes	characteristic
151	C₂	—	472	1	no	—
152	C₂	#16837479	1	1	yes	characteristic
153	C₂	#16837489	1	1	yes	characteristic
154	C₂	#16837518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837529

C₂

yes

maximal, characteristic, complemented

C₄×#11745660

C₂

yes

maximal, characteristic

#16837479

C₂

yes

maximal, characteristic

#16837132

C₂

yes

maximal, characteristic, complemented

#16837129

C₂

yes

maximal, characteristic

C₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₅₉×#334

C₂

yes

maximal, characteristic, complemented

C₂×#11745660

C₄

yes

characteristic

C₂×#11745659

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

C₄×D₂₃₆

C₂²

yes

characteristic

#14273659

C₄

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

C₄

yes

characteristic

#14273649

C₄

yes

characteristic

(C₂×D₅₉)⋊C₈

—

#14273637

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

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×D₂₃₆

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

C₂×C₄

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

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

#11745660

—

#11745659

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

M₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

#135

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×SD₁₆

yes

characteristic

C₂×C₅₉

#443

yes

characteristic

D₅₉

—

#446

—

C₄×SD₁₆

—

#436

—

#334

D₅₉

yes

characteristic, complemented

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

—

C₅₉

#1590

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

118

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

118

—

#97

—

118

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₅₉

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

118

—

104

SD₁₆

—

118

—

105

M₁₆

—

236

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

118

—

111

C₂×C₈

C₄×D₅₉

yes

characteristic

112

C₂×C₈

C₄×D₅₉

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

118

—

116

C₂²⋊C₄

—

118

—

117

C₄²

—

118

—

118

C₄²

C₂²×D₅₉

yes

characteristic

119

C₂³

—

118

—

120

Q₈

—

118

—

121

Q₈

—

118

—

122

D₄

—

118

—

123

D₄

—

118

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

118

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

236

—

128

C₂×C₄

—

236

—

129

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

130

C₂×C₄

D₄×D₅₉

yes

characteristic

131

C₂×C₄

#11745689

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

118

—

137

C₈

—

118

—

138

C₈

#11745659

yes

complemented

139

C₂²

—

236

—

140

C₂²

—

236

—

141

C₂²

—

236

—

142

C₂²

#14273689

yes

characteristic

143

C₄

—

144

C₄

—

236

—

145

C₄

—

236

—

146

C₄

—

236

—

147

C₄

C₂×#11745659

yes

characteristic

148

C₄

#14273735

yes

characteristic

149

C₄

SD₁₆×D₅₉

yes

characteristic

150

C₄

#14273748

yes

characteristic

151

C₂

—

472

—

152

C₂

#16837479

yes

characteristic

153

C₂

#16837489

yes

characteristic

154

C₂

#16837518

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	C₄×D₄
hypercenter	#334	D₅₉
hypocenter	C₅₉	#1590
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

C₄×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #16837529	C₂	semidirect
4. #16837132	C₂	semidirect
6. C₈×C₅₉⋊C₈	C₂	semidirect
7. C₅₉×#334	C₂	semidirect
57. C₈×C₅₉	M₁₆	semidirect
60. C₅₉⋊C₈	SD₁₆	semidirect
81. #334	D₅₉	semidirect
85. C₅₉	#1590	semidirect
138. C₈	#11745659	semidirect

H ≅ G/N

type

1. #16837529

C₂

semidirect

4. #16837132

C₂

semidirect

6. C₈×C₅₉⋊C₈

C₂

semidirect

7. C₅₉×#334

C₂

semidirect

57. C₈×C₅₉

M₁₆

semidirect

60. C₅₉⋊C₈

SD₁₆

semidirect

81. #334

D₅₉

semidirect

85. C₅₉

#1590

semidirect

138. C₈

#11745659

semidirect

Subgroups — #21422985

Conjugacy classes of subgroups

Complements