Group #21423641 — subgroups

Subgroups — #21423641

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	#16837752	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837607	C₂	1	1	yes	maximal, characteristic
3	#16837480	C₂	1	1	yes	maximal, characteristic
4	#16837154	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837149	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837148	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#474	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745670	—	2	1	no	—
9	C₂×#11745664	—	2	1	no	—
10	#14273727	C₂²	1	1	yes	characteristic
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	#14273651	—	2	1	no	—
14	#14273651	—	2	1	no	—
15	#14273650	—	2	1	no	—
16	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
20	#107×C₅₉	—	2	1	no	—
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#102×C₅₉	—	2	1	no	—
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	—	2	1	no	—
26	C₂×Dic₁₁₈	—	2	1	no	—
27	C₂×C₈×C₅₉	—	2	1	no	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₂²⋊C₄×C₅₉	—	2	1	no	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	#11745670	—	4	2	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	#11745664	—	4	2	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂³×C₅₉	—	2	1	no	—
42	Q₈×C₅₉	—	4	1	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	D₄×C₅₉	—	2	1	no	—
45	D₄×C₅₉	—	2	1	no	—
46	C₂×C₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	—	4	1	no	—
49	C₂×C₄×C₅₉	—	4	1	no	—
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
53	C₂×Dic₅₉	—	4	1	no	—
54	Dic₁₁₈	—	4	1	no	—
55	Dic₁₁₈	—	4	1	no	—
56	C₈×C₅₉	—	4	1	no	—
57	C₅₉⋊C₈	—	2	1	no	—
58	C₅₉⋊C₈	—	2	1	no	—
59	C₅₉⋊C₈	—	4	1	no	—
60	C₅₉⋊C₈	—	4	1	no	—
61	C₂²×C₅₉	—	4	1	no	—
62	C₂²×C₅₉	—	4	1	no	—
63	C₂²×C₅₉	—	4	1	no	—
64	C₂²×C₅₉	#121	1	1	yes	characteristic
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	—	4	1	no	—
67	C₄×C₅₉	—	8	1	no	—
68	C₄×C₅₉	#135	1	1	yes	characteristic
69	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₅₉	#137	1	1	yes	characteristic
72	Dic₅₉	—	8	1	no	—
73	#1699	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	—	8	1	no	—
75	C₂×C₅₉	#463	1	1	yes	characteristic
76	C₂×C₅₉	#468	1	1	yes	characteristic
77	C₂×C₅₉	#473	1	1	yes	characteristic
78	#487	—	59	1	no	—
79	#474	D₅₉	1	1	yes	characteristic
80	#461	—	59	1	no	—
81	#436	—	59	1	no	—
82	#335	—	59	1	no	—
83	#332	—	59	1	no	—
84	#332	—	59	1	no	—
85	C₅₉	#1699	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	118	1	no	—
87	C₂×M₁₆	—	118	1	no	—
88	C₄⋊Q₈	—	59	1	no	—
89	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
90	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
91	#107	—	2	1	no	—
92	C₄⋊C₈	—	59	1	no	—
93	C₄⋊C₈	—	59	1	no	—
94	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
95	#103	—	118	1	no	—
96	#103	—	118	1	no	—
97	#103	—	118	1	no	—
98	#102	—	2	1	no	—
99	C₂²⋊C₈	—	118	1	no	—
100	C₄×C₈	—	59	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	118	1	no	—
103	C₂×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	SD₁₆	—	236	2	no	—
106	M₁₆	—	236	2	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	—	118	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	118	1	no	—
115	C₄⋊C₄	—	118	1	no	—
116	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	1	no	—
118	C₄²	C₂²×D₅₉	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	236	1	no	—
123	Q₈	—	236	1	no	—
124	D₄	—	2	1	no	—
125	D₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	236	1	no	—
131	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
132	C₂×C₄	#11745687	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
134	C₈	—	4	1	no	—
135	C₈	—	118	1	no	—
136	C₈	—	118	1	no	—
137	C₈	—	236	1	no	—
138	C₈	—	236	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	#14273699	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	—	472	1	no	—
147	C₄	#14273746	1	1	yes	characteristic
148	C₄	#14273751	1	1	yes	characteristic
149	C₄	#14273752	1	1	yes	characteristic
150	C₄	#14273758	1	1	yes	characteristic
151	C₂	—	8	1	no	—
152	C₂	#16837615	1	1	yes	characteristic
153	C₂	#16837650	1	1	yes	characteristic
154	C₂	#16837675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837752

C₂

yes

maximal, characteristic, complemented

#16837607

C₂

yes

maximal, characteristic

#16837480

C₂

yes

maximal, characteristic

#16837154

C₂

yes

maximal, characteristic, complemented

#16837149

C₂

yes

maximal, characteristic, complemented

#16837148

C₂

yes

maximal, characteristic, complemented

C₅₉×#474

C₂

yes

maximal, characteristic

C₂×#11745670

—

C₂×#11745664

—

#14273727

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

—

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄×D₄×C₅₉

C₂²

yes

characteristic

#107×C₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

#102×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745670

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1699

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

—

—

C₅₉

#1699

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

C₂×D₅₉

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

#103

—

118

—

#103

—

118

—

#102

—

C₂²⋊C₈

—

118

—

100

C₄×C₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

118

—

103

C₂×D₄

C₂²⋊D₅₉

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

236

—

106

M₁₆

—

236

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

118

—

111

C₂×C₈

—

118

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

118

—

115

C₄⋊C₄

—

118

—

116

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

C₂²×D₅₉

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

236

—

123

Q₈

—

236

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

236

—

131

C₂×C₄

D₄×D₅₉

yes

characteristic

132

C₂×C₄

#11745687

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

134

C₈

—

135

C₈

—

118

—

136

C₈

—

118

—

137

C₈

—

236

—

138

C₈

—

236

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#14273699

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

472

—

147

C₄

#14273746

yes

characteristic

148

C₄

#14273751

yes

characteristic

149

C₄

#14273752

yes

characteristic

150

C₄

#14273758

yes

characteristic

151

C₂

—

152

C₂

#16837615

yes

characteristic

153

C₂

#16837650

yes

characteristic

154

C₂

#16837675

yes

characteristic

	subgroup	G/H
center	C₂²	#14273699
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#474	C₂
socle	C₂²×C₅₉	#121
hypercenter	#474	D₅₉
hypocenter	C₅₉	#1699
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

subgroup

G/H

center

C₂²

#14273699

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#474

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #474

N	H ≅ G/N	type
1. #16837752	C₂	semidirect
4. #16837154	C₂	semidirect
5. #16837149	C₂	semidirect
6. #16837148	C₂	semidirect
85. C₅₉	#1699	semidirect

H ≅ G/N

type

1. #16837752

C₂

semidirect

4. #16837154

C₂

semidirect

5. #16837149

C₂

semidirect

6. #16837148

C₂

semidirect

85. C₅₉

#1699

semidirect

Subgroups — #21423641

Conjugacy classes of subgroups

Complements