Group #21423669 — subgroups

Subgroups — #21423669

Order 7552.

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

normal subgroups only

This group has 154 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	#16837692	C₂	1	1	yes	maximal, characteristic
3	#16837480	C₂	1	1	yes	maximal, characteristic
4	#16837152	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837150	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×#11745664	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273658	—	2	1	no	—
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	#14273648	—	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	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	#11745664	—	4	2	no	—
36	C₂×C₅₉⋊C₈	—	2	1	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	C₂³×C₅₉	—	2	1	no	—
41	Q₈×C₅₉	—	4	1	no	—
42	Q₈×C₅₉	—	4	1	no	—
43	D₄×C₅₉	—	2	1	no	—
44	D₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	2	1	no	—
46	C₂×C₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	—	4	1	no	—
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
52	C₂×Dic₅₉	—	4	1	no	—
53	Dic₁₁₈	—	8	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	—	4	1	no	—
60	C₂²×C₅₉	—	4	1	no	—
61	C₂²×C₅₉	—	4	1	no	—
62	C₂²×C₅₉	#121	1	1	yes	characteristic
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	—	4	1	no	—
65	C₄×C₅₉	—	8	1	no	—
66	C₄×C₅₉	#135	1	1	yes	characteristic
67	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₄×C₅₉	#137	1	1	yes	characteristic
69	C₄×C₅₉	#137	1	1	yes	characteristic
70	Dic₅₉	—	8	1	no	—
71	#1703	—	59	1	no	maximal, Sylow
72	C₂×C₅₉	—	8	1	no	—
73	C₂×C₅₉	#463	1	1	yes	characteristic
74	C₂×C₅₉	#470	1	1	yes	characteristic
75	C₂×C₅₉	#471	1	1	yes	characteristic
76	#487	—	59	1	no	—
77	#476	—	59	1	no	—
78	#474	D₅₉	1	1	yes	characteristic
79	#436	—	59	1	no	—
80	#334	—	59	1	no	—
81	#333	—	59	1	no	—
82	#332	—	59	1	no	—
83	C₅₉	#1703	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₁₆	—	118	1	no	—
85	C₄⋊Q₈	—	59	1	no	—
86	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
87	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
88	#107	—	2	1	no	—
89	#106	—	118	1	no	—
90	C₄⋊C₈	—	59	1	no	—
91	C₄⋊C₈	—	59	1	no	—
92	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
93	#103	—	118	1	no	—
94	#103	—	118	1	no	—
95	#102	—	2	1	no	—
96	#102	—	118	1	no	—
97	C₂²⋊C₈	—	118	1	no	—
98	C₄×C₈	—	59	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	118	1	no	—
101	C₂×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂²×C₄	—	2	1	no	—
103	M₁₆	—	236	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	59	1	no	—
106	C₂×C₈	—	59	1	no	—
107	C₂×C₈	—	118	1	no	—
108	C₂×C₈	—	118	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	118	1	no	—
112	C₄⋊C₄	—	118	1	no	—
113	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
114	C₂²⋊C₄	—	2	1	no	—
115	C₄²	C₂²×D₅₉	1	1	yes	characteristic
116	C₂³	—	2	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	472	1	no	—
120	D₄	—	2	1	no	—
121	D₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	236	1	no	—
127	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
128	C₂×C₄	#11745687	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
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	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	#14273699	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	472	1	no	—
143	C₄	#14273746	1	1	yes	characteristic
144	C₄	#14273751	1	1	yes	characteristic
145	C₄	#14273757	1	1	yes	characteristic
146	C₄	#14273758	1	1	yes	characteristic
147	C₂	—	8	1	no	—
148	C₂	#16837615	1	1	yes	characteristic
149	C₂	#16837658	1	1	yes	characteristic
150	C₂	#16837667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837752

C₂

yes

maximal, characteristic, complemented

#16837692

C₂

yes

maximal, characteristic

#16837480

C₂

yes

maximal, characteristic

#16837152

C₂

yes

maximal, characteristic, complemented

#16837150

C₂

yes

maximal, characteristic, complemented

#16837148

C₂

yes

maximal, characteristic, complemented

C₅₉×#474

C₂

yes

maximal, characteristic

C₂×#11745664

—

#14273727

C₂²

yes

characteristic

#14273658

—

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

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

—

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

—

#1703

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

D₅₉

yes

—

—

—

—

C₅₉

#1703

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

C₂×D₅₉

yes

characteristic

#107

—

#106

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

#103

—

118

—

#102

—

#102

—

118

—

C₂²⋊C₈

—

118

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

118

—

101

C₂×D₄

C₂²⋊D₅₉

yes

characteristic

102

C₂²×C₄

—

103

M₁₆

—

236

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

118

—

108

C₂×C₈

—

118

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

118

—

112

C₄⋊C₄

—

118

—

113

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

114

C₂²⋊C₄

—

115

C₄²

C₂²×D₅₉

yes

characteristic

116

C₂³

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

472

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

236

—

127

C₂×C₄

D₄×D₅₉

yes

characteristic

128

C₂×C₄

#11745687

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

130

C₈

—

131

C₈

—

118

—

132

C₈

—

118

—

133

C₈

—

236

—

134

C₈

—

236

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14273699

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

472

—

143

C₄

#14273746

yes

characteristic

144

C₄

#14273751

yes

characteristic

145

C₄

#14273757

yes

characteristic

146

C₄

#14273758

yes

characteristic

147

C₂

—

148

C₂

#16837615

yes

characteristic

149

C₂

#16837658

yes

characteristic

150

C₂

#16837667

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	#1703
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. #16837152	C₂	semidirect
5. #16837150	C₂	semidirect
6. #16837148	C₂	semidirect
83. C₅₉	#1703	semidirect

H ≅ G/N

type

1. #16837752

C₂

semidirect

4. #16837152

C₂

semidirect

5. #16837150

C₂

semidirect

6. #16837148

C₂

semidirect

83. C₅₉

#1703

semidirect

Subgroups — #21423669

Conjugacy classes of subgroups

Complements