Group #21422947 — subgroups

Subgroups — #21422947

Order 7552.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16837533	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#16837155	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#16837124	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	#14273638	—	2	1	no	—
13	C₄×C₅₉⋊C₈	—	2	1	no	—
14	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₅₉	C₄	1	1	yes	characteristic
17	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	#107×C₅₉	C₄	1	1	yes	characteristic
20	#103×C₅₉	C₄	1	1	yes	characteristic
21	#103×C₅₉	C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	Q₁₆×C₅₉	—	2	2	no	—
26	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₅₉	—	2	1	no	—
33	C₄²×C₅₉	—	2	1	no	—
34	C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₅₉⋊C₈	—	2	1	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	Q₈×C₅₉	—	2	1	no	—
41	Q₈×C₅₉	—	2	1	no	—
42	Q₈×C₅₉	—	2	1	no	—
43	Q₈×C₅₉	M₁₆	1	2	yes	—
44	C₂×C₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	2	1	no	—
46	C₂×C₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
51	C₈×C₅₉	—	2	1	no	—
52	C₈×C₅₉	—	2	1	no	—
53	C₅₉⋊C₈	—	2	1	no	—
54	C₅₉⋊C₈	—	4	1	no	—
55	C₅₉⋊C₈	—	4	1	no	—
56	C₅₉⋊C₈	—	4	1	no	—
57	C₅₉⋊C₈	Q₁₆	1	2	yes	complemented
58	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₅₉	—	2	2	no	—
60	C₄×C₅₉	—	2	1	no	—
61	C₄×C₅₉	—	4	1	no	—
62	C₄×C₅₉	—	4	1	no	—
63	C₄×C₅₉	—	4	1	no	—
64	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
65	C₄×C₅₉	#131	1	1	yes	characteristic
66	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₄×C₅₉	#135	1	1	yes	characteristic
68	#1584	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	#435	1	1	yes	characteristic
70	C₂×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
71	C₂×C₅₉	#444	1	1	yes	characteristic
72	#446	—	59	1	no	—
73	C₈×Q₈	—	59	1	no	—
74	C₄×Q₁₆	D₅₉	1	1	yes	characteristic
75	#335	—	59	1	no	—
76	Q₈⋊C₈	—	59	1	no	—
77	Q₈⋊C₈	—	59	1	no	—
78	#322	—	59	1	no	—
79	C₅₉	#1584	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	Dic₅₉	1	1	yes	characteristic
81	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
83	#107	Dic₅₉	1	1	yes	characteristic
84	C₄⋊C₈	—	59	1	no	—
85	C₄⋊C₈	—	59	1	no	—
86	C₄⋊C₈	—	118	1	no	—
87	C₄⋊C₈	—	118	1	no	—
88	#103	Dic₅₉	1	1	yes	characteristic
89	#103	Dic₅₉	1	1	yes	characteristic
90	#97	—	118	1	no	—
91	C₄×C₈	—	59	1	no	—
92	C₄×C₈	—	59	1	no	—
93	C₄×C₈	—	118	1	no	—
94	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
95	C₂×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
96	C₂×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
97	Q₁₆	—	2	2	no	—
98	C₂×C₈	—	59	1	no	—
99	C₂×C₈	—	59	1	no	—
100	C₂×C₈	—	118	1	no	—
101	C₂×C₈	—	118	1	no	—
102	C₂×C₈	—	118	1	no	—
103	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	C₂×Dic₅₉	1	1	yes	characteristic
108	C₄⋊C₄	C₂×Dic₅₉	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	C₂²×D₅₉	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	#11745664	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
121	C₂×C₄	#11745687	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	59	2	no	—
126	C₈	—	118	1	no	—
127	C₈	—	236	1	no	—
128	C₈	—	236	1	no	—
129	C₈	—	236	1	no	—
130	C₂²	D₄×Dic₅₉	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	4	1	no	—
136	C₄	C₂×#11745664	1	1	yes	characteristic
137	C₄	#14273734	1	1	yes	characteristic
138	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
139	C₄	#14273750	1	1	yes	characteristic
140	C₂	#16837474	1	1	yes	characteristic
141	C₂	Q₁₆×Dic₅₉	1	1	yes	characteristic
142	C₂	#16837524	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837533

C₂

yes

maximal, characteristic

Q₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, characteristic

Q₈⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

Q₈⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837124

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₅₉

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273638

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₄

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₄

yes

characteristic

#103×C₅₉

C₄

yes

characteristic

#103×C₅₉

C₄

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

Q₁₆

yes

complemented

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×Q₁₆

yes

C₂×C₅₉

yes

—

C₈×Q₈

—

C₄×Q₁₆

D₅₉

yes

characteristic

#335

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₅₉

#1584

yes

characteristic, complemented, Sylow

C₂×Q₁₆

Dic₅₉

yes

characteristic

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

C₂×D₅₉

yes

characteristic

#107

Dic₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

#103

Dic₅₉

yes

characteristic

#103

Dic₅₉

yes

characteristic

#97

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

C₂×Dic₅₉

yes

characteristic

C₂×Q₈

C₂×Dic₅₉

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

118

—

101

C₂×C₈

—

118

—

102

C₂×C₈

—

118

—

103

C₂×C₈

C₂×Dic₅₉

yes

characteristic

104

C₂×C₈

C₂×Dic₅₉

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×Dic₅₉

yes

characteristic

108

C₄⋊C₄

C₂×Dic₅₉

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₅₉

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

#11745664

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×D₅₉

yes

characteristic

121

C₂×C₄

#11745687

yes

characteristic

122

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

118

—

127

C₈

—

236

—

128

C₈

—

236

—

129

C₈

—

236

—

130

C₂²

D₄×Dic₅₉

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×#11745664

yes

characteristic

137

C₄

#14273734

yes

characteristic

138

C₄

Q₁₆×D₅₉

yes

characteristic

139

C₄

#14273750

yes

characteristic

140

C₂

#16837474

yes

characteristic

141

C₂

Q₁₆×Dic₅₉

yes

characteristic

142

C₂

#16837524

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₄×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	C₄×Q₁₆	D₅₉
hypocenter	C₅₉	#1584
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₄×Q₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₄×D₄

hypercenter

C₄×Q₁₆

D₅₉

hypocenter

C₅₉

#1584

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
57. C₅₉⋊C₈	Q₁₆	semidirect
79. C₅₉	#1584	semidirect

H ≅ G/N

type

57. C₅₉⋊C₈

Q₁₆

semidirect

79. C₅₉

#1584

semidirect

Subgroups — #21422947

Conjugacy classes of subgroups

Complements