Group #21423963 — subgroups

Subgroups — #21423963

Order 7552.

272 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 157 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	C₂×#14273637	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837156	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
9	C₄⋊(C₅₉⋊C₈)	C₄	1	4	yes	complemented
10	#14273637	—	2	4	no	—
11	C₅₉⋊#95	C₄	1	2	yes	—
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	C₄	1	4	yes	complemented
16	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₉	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₅₉	—	2	1	no	—
21	C₂×C₈×C₅₉	—	2	1	no	—
22	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₅₉	D₄	1	2	yes	—
25	C₂×C₈×C₅₉	Q₈	1	2	yes	—
26	C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	C₄²×C₅₉	C₂×C₄	1	2	yes	—
28	C₄⋊Dic₅₉	—	2	4	no	—
29	C₄×Dic₅₉	—	2	2	no	—
30	C₄×Dic₅₉	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂×C₅₉⋊C₈	—	2	2	no	—
33	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
34	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	—	2	2	no	—
36	C₂×C₄×C₅₉	C₄²	1	2	yes	—
37	C₂×C₄×C₅₉	C₄²	1	2	yes	—
38	C₂×C₄×C₅₉	M₁₆	1	4	yes	—
39	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
44	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
45	C₂×Dic₅₉	—	2	4	no	—
46	C₂×Dic₅₉	—	2	4	no	—
47	C₂×Dic₅₉	—	2	4	no	—
48	C₈×C₅₉	—	2	2	no	—
49	C₈×C₅₉	C₄⋊C₄	1	4	yes	complemented
50	C₅₉⋊C₈	—	2	4	no	—
51	C₂²×C₅₉	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₅₉	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₅₉	C₂×M₁₆	1	2	yes	—
55	C₂²×C₅₉	#131	1	2	yes	—
56	C₄×C₅₉	—	2	2	no	—
57	C₄×C₅₉	#97	1	4	yes	complemented
58	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₅₉	#117	1	1	yes	characteristic
60	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
61	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
62	Dic₅₉	—	4	4	no	—
63	#1770	—	59	1	no	maximal, Sylow
64	C₂×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₅₉	C₂×#97	1	1	yes	characteristic
66	C₂×C₅₉	#405	1	1	yes	characteristic
67	C₂×C₅₉	#436	1	2	yes	—
68	C₂×C₅₉	#446	1	2	yes	—
69	C₂×C₄⋊C₈	—	59	1	no	—
70	C₂×#97	—	59	2	no	—
71	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
72	C₄×C₄⋊C₄	—	59	1	no	—
73	#336	—	59	2	no	—
74	C₅₉	#1770	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	59	2	no	—
76	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
78	C₂×C₄⋊C₄	—	59	2	no	—
79	C₂×C₄²	—	59	2	no	—
80	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
81	C₄⋊C₈	—	59	4	no	—
82	#97	—	118	4	no	—
83	C₄×C₈	Dic₅₉	1	4	yes	complemented
84	#95	—	59	2	no	—
85	C₂²×C₄	—	59	4	no	—
86	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	59	4	no	—
92	C₂×C₈	—	118	2	no	—
93	C₂×C₈	—	118	2	no	—
94	C₂×C₈	Dic₁₁₈	1	2	yes	—
95	C₂×C₈	D₂₃₆	1	2	yes	—
96	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
97	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
98	C₄⋊C₄	—	118	4	no	—
99	C₄²	—	118	2	no	—
100	C₄²	—	118	2	no	—
101	C₄²	C₄×D₅₉	1	2	yes	—
102	C₄²	C₂×Dic₅₉	1	2	yes	—
103	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	118	4	no	—
106	C₂×C₄	—	118	4	no	—
107	C₂×C₄	—	118	4	no	—
108	C₂×C₄	#11745659	1	4	yes	—
109	C₂×C₄	C₄×Dic₅₉	1	2	yes	—
110	C₂×C₄	C₄×Dic₅₉	1	2	yes	—
111	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
113	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
114	C₂×C₄	#11745685	1	1	yes	characteristic
115	C₂×C₄	#11745685	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	118	4	no	—
119	C₈	C₄⋊Dic₅₉	1	4	yes	complemented
120	C₂²	C₂×C₄×Dic₅₉	1	1	yes	characteristic
121	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
122	C₂²	C₄×Dic₁₁₈	1	1	yes	characteristic
123	C₂²	C₂×#11745659	1	2	yes	—
124	C₂²	#14273733	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	236	4	no	—
127	C₄	#14273637	1	4	yes	complemented
128	C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
129	C₄	#14273686	1	1	yes	characteristic
130	C₄	C₄×D₂₃₆	1	1	yes	characteristic
131	C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
132	C₂	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
133	C₂	C₂×#14273637	1	1	yes	characteristic
134	C₂	#16837357	1	1	yes	characteristic
135	C₂	#16837481	1	2	yes	—
136	C₂	#16837532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14273637

C₂

yes

maximal

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

#16837156

C₂

yes

maximal

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₄

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₄

yes

complemented

#14273637

—

C₅₉⋊#95

C₄

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

Q₈

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄²

yes

—

C₂×C₄×C₅₉

C₄²

yes

—

C₂×C₄×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

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

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄²

yes

characteristic

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

characteristic

C₂²×C₅₉

C₂×M₁₆

yes

—

C₂²×C₅₉

#131

yes

—

C₄×C₅₉

—

C₄×C₅₉

#97

yes

complemented

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

Dic₅₉

—

#1770

—

maximal, Sylow

C₂×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₉

C₂×#97

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#446

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

D₅₉

yes

characteristic

C₄×C₄⋊C₄

—

#336

—

C₅₉

#1770

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

#97

—

118

—

C₄×C₈

Dic₅₉

yes

complemented

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

100

C₄²

—

118

—

101

C₄²

C₄×D₅₉

yes

—

102

C₄²

C₂×Dic₅₉

yes

—

103

C₂³

C₂×C₄×D₅₉

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

118

—

106

C₂×C₄

—

118

—

107

C₂×C₄

—

118

—

108

C₂×C₄

#11745659

yes

—

109

C₂×C₄

C₄×Dic₅₉

yes

—

110

C₂×C₄

C₄×Dic₅₉

yes

—

111

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

112

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

113

C₂×C₄

C₂×D₂₃₆

yes

characteristic

114

C₂×C₄

#11745685

yes

characteristic

115

C₂×C₄

#11745685

yes

characteristic

116

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

117

C₈

—

118

C₈

—

118

—

119

C₈

C₄⋊Dic₅₉

yes

complemented

120

C₂²

C₂×C₄×Dic₅₉

yes

characteristic

121

C₂²

C₄×D₂₃₆

yes

characteristic

122

C₂²

C₄×Dic₁₁₈

yes

characteristic

123

C₂²

C₂×#11745659

yes

—

124

C₂²

#14273733

yes

—

125

C₄

—

126

C₄

—

236

—

127

C₄

#14273637

yes

complemented

128

C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

129

C₄

#14273686

yes

characteristic

130

C₄

C₄×D₂₃₆

yes

characteristic

131

C₄

C₄×Dic₁₁₈

yes

characteristic

132

C₂

C₄×C₄⋊Dic₅₉

yes

characteristic

133

C₂

C₂×#14273637

yes

characteristic

134

C₂

#16837357

yes

characteristic

135

C₂

#16837481

yes

—

136

C₂

#16837532

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂×C₄²
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₄
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#1770
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₉

commutator

C₂²×C₅₉

C₂×C₄²

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂²×C₄

hypercenter

C₂×C₄×C₈

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₄⋊(C₅₉⋊C₈)	C₄	semidirect
15. C₄×C₈×C₅₉	C₄	semidirect
49. C₈×C₅₉	C₄⋊C₄	semidirect
57. C₄×C₅₉	#97	semidirect
74. C₅₉	#1770	semidirect
83. C₄×C₈	Dic₅₉	semidirect
119. C₈	C₄⋊Dic₅₉	semidirect
127. C₄	#14273637	semidirect

H ≅ G/N

type

9. C₄⋊(C₅₉⋊C₈)

C₄

semidirect

15. C₄×C₈×C₅₉

C₄

semidirect

49. C₈×C₅₉

C₄⋊C₄

semidirect

57. C₄×C₅₉

#97

semidirect

74. C₅₉

#1770

semidirect

83. C₄×C₈

Dic₅₉

semidirect

119. C₈

C₄⋊Dic₅₉

semidirect

127. C₄

#14273637

semidirect

Subgroups — #21423963

Conjugacy classes of subgroups

Complements