Group #21425049 — subgroups

Subgroups — #21425049

Order 7552.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837164	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
7	C₂×C₄×Dic₅₉	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
9	C₅₉⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₅₉	—	2	1	no	—
11	C₄⋊Q₈×C₅₉	—	2	2	no	—
12	C₄⋊Q₈×C₅₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₉	—	2	4	no	—
17	C₂×Q₈×C₅₉	—	4	2	no	—
18	C₂×Q₈×C₅₉	—	4	2	no	—
19	C₂²×C₄×C₅₉	—	2	2	no	—
20	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₉	—	2	2	no	—
24	C₄⋊C₄×C₅₉	—	2	4	no	—
25	C₄⋊C₄×C₅₉	—	2	2	no	—
26	C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
27	C₄²×C₅₉	C₂×C₄	1	2	yes	—
28	C₄²×C₅₉	D₄	1	2	yes	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	Q₈	1	4	yes	complemented
31	C₄×Dic₅₉	—	4	2	no	—
32	C₂×C₅₉⋊C₈	—	2	2	no	—
33	C₂×C₅₉⋊C₈	—	2	4	no	—
34	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
35	Q₈×C₅₉	—	4	8	no	—
36	C₂×C₄×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	—	2	4	no	—
38	C₂×C₄×C₅₉	—	4	2	no	—
39	C₂×C₄×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₉	D₈	1	2	yes	—
43	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
44	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₅₉	#41	1	2	yes	—
48	C₂×Dic₅₉	—	2	4	no	—
49	C₂×Dic₅₉	—	4	2	no	—
50	C₂×Dic₅₉	—	4	2	no	—
51	C₅₉⋊C₈	—	4	2	no	—
52	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₅₉	#137	1	2	yes	—
57	C₄×C₅₉	—	2	2	no	—
58	C₄×C₅₉	—	4	4	no	—
59	C₄×C₅₉	#102	1	4	yes	—
60	C₄×C₅₉	C₄×Q₈	1	2	yes	—
61	C₄×C₅₉	#124	1	1	yes	characteristic
62	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₅₉	—	4	4	no	—
64	#1988	—	59	1	no	maximal, Sylow
65	C₂×C₅₉	#391	1	1	yes	characteristic
66	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
67	C₂×C₅₉	#419	1	1	yes	characteristic
68	C₂×C₅₉	#474	1	2	yes	—
69	C₂×C₅₉	#475	1	2	yes	—
70	C₂×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
71	C₂×C₄⋊C₈	—	59	1	no	—
72	C₄×C₄⋊C₄	—	59	1	no	—
73	#340	—	59	4	no	—
74	C₅₉	#1988	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	59	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	Dic₅₉	1	4	yes	complemented
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	59	2	no	—
81	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
82	C₂×C₄²	—	118	1	no	—
83	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
84	C₄⋊C₈	—	118	2	no	—
85	#95	—	118	1	no	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	—	4	2	no	—
88	C₂×Q₈	—	4	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	118	2	no	—
91	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
94	C₂×C₈	—	118	4	no	—
95	C₂×C₈	—	118	2	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	59	4	no	—
99	C₄⋊C₄	—	118	2	no	—
100	C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
101	C₄²	—	236	2	no	—
102	C₄²	C₂×Dic₅₉	1	2	yes	—
103	C₄²	C₂²⋊D₅₉	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
105	Q₈	—	4	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	118	4	no	—
111	C₂×C₄	—	236	2	no	—
112	C₂×C₄	—	236	2	no	—
113	C₂×C₄	#11745669	1	2	yes	—
114	C₂×C₄	#11745670	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
116	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
117	C₂×C₄	Q₈×D₅₉	1	2	yes	—
118	C₂×C₄	#11745689	1	2	yes	—
119	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
121	C₈	—	236	2	no	—
122	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
123	C₂²	#14273703	1	2	yes	—
124	C₂²	C₂×#11745669	1	1	yes	characteristic
125	C₂²	C₂×#11745670	1	1	yes	characteristic
126	C₂²	#14273758	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	236	4	no	—
130	C₄	#14273648	1	4	yes	—
131	C₄	Q₈×Dic₅₉	1	2	yes	—
132	C₄	#14273716	1	1	yes	characteristic
133	C₄	#14273725	1	1	yes	characteristic
134	C₂	#16837309	1	1	yes	characteristic
135	C₂	C₂×#14273648	1	1	yes	characteristic
136	C₂	#16837411	1	1	yes	characteristic
137	C₂	#16837679	1	2	yes	—
138	C₂	#16837684	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

Q₈

yes

complemented

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂²×C₅₉

C₂×SD₁₆

yes

characteristic

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#102

yes

—

C₄×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#475

yes

—

C₂×C₄⋊Q₈

D₅₉

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₅₉

#1988

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

#95

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

100

C₄⋊C₄

C₂×Dic₅₉

yes

—

101

C₄²

—

236

—

102

C₄²

C₂×Dic₅₉

yes

—

103

C₄²

C₂²⋊D₅₉

yes

—

104

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

118

—

111

C₂×C₄

—

236

—

112

C₂×C₄

—

236

—

113

C₂×C₄

#11745669

yes

—

114

C₂×C₄

#11745670

yes

—

115

C₂×C₄

C₂²⋊Dic₅₉

yes

—

116

C₂×C₄

C₂²⋊Dic₅₉

yes

—

117

C₂×C₄

Q₈×D₅₉

yes

—

118

C₂×C₄

#11745689

yes

—

119

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

121

C₈

—

236

—

122

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

123

C₂²

#14273703

yes

—

124

C₂²

C₂×#11745669

yes

characteristic

125

C₂²

C₂×#11745670

yes

characteristic

126

C₂²

#14273758

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

236

—

130

C₄

#14273648

yes

—

131

C₄

Q₈×Dic₅₉

yes

—

132

C₄

#14273716

yes

characteristic

133

C₄

#14273725

yes

characteristic

134

C₂

#16837309

yes

characteristic

135

C₂

C₂×#14273648

yes

characteristic

136

C₂

#16837411

yes

characteristic

137

C₂

#16837679

yes

—

138

C₂

#16837684

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×C₄⋊Q₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#1988
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×C₄⋊Q₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₉

hypocenter

C₅₉

#1988

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₅₉	C₄	semidirect
30. C₄⋊Dic₅₉	Q₈	semidirect
74. C₅₉	#1988	semidirect
78. C₄⋊Q₈	Dic₅₉	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₅₉

C₄

semidirect

30. C₄⋊Dic₅₉

Q₈

semidirect

74. C₅₉

#1988

semidirect

78. C₄⋊Q₈

Dic₅₉

semidirect

Subgroups — #21425049

Conjugacy classes of subgroups

Complements