Group #21424883 — subgroups

Subgroups — #21424883

Order 7552.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#16837450	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273662	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273662	C₂	1	1	yes	maximal, characteristic
4	#16837360	C₂	1	1	yes	maximal, characteristic
5	#16837174	C₂	1	1	yes	maximal, characteristic
6	#16837173	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#430	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745664	C₂²	1	1	yes	characteristic
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	#14273686	C₂²	1	1	yes	characteristic
12	#14273662	—	2	2	no	—
13	#14273662	—	2	1	no	—
14	#14273662	—	2	1	no	—
15	#14273662	C₄	1	4	yes	complemented
16	#14273637	—	2	1	no	—
17	C₈×Dic₅₉	—	2	1	no	—
18	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	#108×C₅₉	—	2	2	no	—
22	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₅₉	—	2	1	no	—
24	M₁₆×C₅₉	—	2	1	no	—
25	M₁₆×C₅₉	—	2	1	no	—
26	M₁₆×C₅₉	—	2	1	no	—
27	M₁₆×C₅₉	C₂×C₄	1	2	yes	—
28	M₁₆×C₅₉	D₄	1	2	yes	—
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	—	2	1	no	—
32	C₂²⋊Dic₅₉	Q₈	1	2	yes	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄×Dic₅₉	—	2	1	no	—
35	#11745664	—	2	1	no	—
36	#11745664	—	2	1	no	—
37	#11745664	—	2	1	no	—
38	#11745664	C₂×C₄	1	2	yes	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₅₉⋊C₈	—	2	1	no	—
41	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
45	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
52	C₂×Dic₅₉	—	2	2	no	—
53	C₈×C₅₉	—	2	2	no	—
54	C₈×C₅₉	—	2	2	no	—
55	C₈×C₅₉	—	4	1	no	—
56	C₅₉⋊C₈	—	2	2	no	—
57	C₅₉⋊C₈	—	4	1	no	—
58	C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₅₉	—	4	1	no	—
61	C₂²×C₅₉	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
66	C₄×C₅₉	#121	1	1	yes	characteristic
67	C₄×C₅₉	#125	1	1	yes	characteristic
68	Dic₅₉	—	4	2	no	—
69	#1953	—	59	1	no	maximal, Sylow
70	C₂×C₅₉	—	2	1	no	—
71	C₂×C₅₉	—	2	1	no	—
72	C₂×C₅₉	—	2	1	no	—
73	C₂×C₅₉	#389	1	1	yes	characteristic
74	#430	—	59	1	no	—
75	#430	D₅₉	1	1	yes	characteristic
76	C₂×#108	—	59	1	no	—
77	C₂×#108	—	59	1	no	—
78	#405	—	59	1	no	—
79	#344	—	59	1	no	—
80	#344	—	59	1	no	—
81	C₅₉	#1953	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₁₆	—	59	1	no	—
83	C₂×M₁₆	—	59	1	no	—
84	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
87	C₂²×C₈	—	59	1	no	—
88	#117	—	59	1	no	—
89	#108	—	2	2	no	—
90	#108	—	59	4	no	—
91	#108	—	118	1	no	—
92	#108	—	118	1	no	—
93	#108	—	118	2	no	—
94	#97	—	118	1	no	—
95	C₄×C₈	—	118	1	no	—
96	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	59	2	no	—
102	M₁₆	—	118	1	no	—
103	M₁₆	—	118	1	no	—
104	M₁₆	—	118	1	no	—
105	M₁₆	C₄×D₅₉	1	2	yes	—
106	M₁₆	C₂²⋊D₅₉	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	59	2	no	—
111	C₂×C₈	—	59	1	no	—
112	C₂×C₈	—	59	1	no	—
113	C₂×C₈	—	59	2	no	—
114	C₂×C₈	—	118	1	no	—
115	C₂×C₈	—	118	1	no	—
116	C₄⋊C₄	—	118	1	no	—
117	C₂²⋊C₄	—	59	2	no	—
118	C₄²	—	118	1	no	—
119	C₂³	Q₈×D₅₉	1	1	yes	characteristic
120	C₂×C₄	—	118	2	no	—
121	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
122	C₂×C₄	#11745685	1	1	yes	characteristic
123	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
124	C₂×C₄	#11745687	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	2	2	no	—
129	C₈	—	4	1	no	—
130	C₈	—	59	2	no	—
131	C₈	—	59	2	no	—
132	C₈	—	118	2	no	—
133	C₈	—	236	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	#14273692	1	1	yes	characteristic
136	C₂²	#14273705	1	1	yes	characteristic
137	C₂²	#14273725	1	1	yes	characteristic
138	C₄	—	236	2	no	—
139	C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
140	C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
141	C₄	#14273699	1	1	yes	characteristic
142	C₄	#14273717	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#16837297	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837450

C₂

yes

maximal, characteristic

C₂×#14273662

C₂

yes

maximal, characteristic

C₂×#14273662

C₂

yes

maximal, characteristic

#16837360

C₂

yes

maximal, characteristic

#16837174

C₂

yes

maximal, characteristic

#16837173

C₂

yes

maximal, characteristic

C₅₉×#430

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273686

C₂²

yes

—

—

—

C₄

yes

complemented

#14273637

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#108×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

C₂×C₄

yes

—

M₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂²⋊Dic₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

—

—

—

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₄⋊C₄

yes

—

C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×Q₈

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1953

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

—

—

C₅₉

#1953

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

—

—

—

—

118

—

#108

—

118

—

#108

—

118

—

#97

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

118

—

103

M₁₆

—

118

—

104

M₁₆

—

118

—

105

M₁₆

C₄×D₅₉

yes

—

106

M₁₆

C₂²⋊D₅₉

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

118

—

115

C₂×C₈

—

118

—

116

C₄⋊C₄

—

118

—

117

C₂²⋊C₄

—

118

C₄²

—

118

—

119

C₂³

Q₈×D₅₉

yes

characteristic

120

C₂×C₄

—

118

—

121

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

122

C₂×C₄

#11745685

yes

characteristic

123

C₂×C₄

D₄×D₅₉

yes

characteristic

124

C₂×C₄

#11745687

yes

characteristic

125

C₂×C₄

Q₈×D₅₉

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

118

—

133

C₈

—

236

—

134

C₂²

—

135

C₂²

#14273692

yes

characteristic

136

C₂²

#14273705

yes

characteristic

137

C₂²

#14273725

yes

characteristic

138

C₄

—

236

—

139

C₄

C₄⋊C₄×D₅₉

yes

characteristic

140

C₄

C₄×C₂²⋊D₅₉

yes

characteristic

141

C₄

#14273699

yes

characteristic

142

C₄

#14273717

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#16837297

yes

characteristic

	subgroup	G/H
center	C₄	#14273699
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#430	C₂
socle	C₂×C₅₉	#389
hypercenter	#430	D₅₉
hypocenter	C₅₉	#1953
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#14273699

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#430

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
15. #14273662	C₄	semidirect
81. C₅₉	#1953	semidirect

H ≅ G/N

type

15. #14273662

C₄

semidirect

81. C₅₉

#1953

semidirect

Subgroups — #21424883

Conjugacy classes of subgroups

Complements