Group #21421993 — subgroups

Subgroups — #21421993

Order 7552.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16837455	C₂	1	1	yes	maximal, characteristic
2	#16837426	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745664	—	2	1	no	—
5	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
6	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
9	C₄×C₅₉⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₅₉	—	2	1	no	—
11	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₉	—	2	1	no	—
13	C₄⋊C₈×C₅₉	C₄	1	2	yes	—
14	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	—	4	1	no	—
19	C₂×C₈×C₅₉	—	4	1	no	—
20	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₉	D₄	1	1	yes	characteristic
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	#11745664	—	4	1	no	—
25	#11745664	—	4	1	no	—
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	—	2	2	no	—
30	C₂×C₄×C₅₉	—	4	1	no	—
31	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
38	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
41	C₈×C₅₉	—	4	2	no	—
42	C₅₉⋊C₈	—	4	2	no	—
43	C₅₉⋊C₈	—	4	2	no	—
44	C₂²×C₅₉	—	2	1	no	—
45	C₂²×C₅₉	—	2	1	no	—
46	C₂²×C₅₉	#95	1	1	yes	characteristic
47	C₂²×C₅₉	#106	1	1	yes	characteristic
48	C₂²×C₅₉	#107	1	1	yes	characteristic
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	—	2	1	no	—
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	#102	1	1	yes	characteristic
53	C₄×C₅₉	#102	1	1	yes	characteristic
54	C₄×C₅₉	#103	1	1	yes	characteristic
55	C₄×C₅₉	#103	1	1	yes	characteristic
56	C₄×C₅₉	#108	1	2	yes	—
57	#1301	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	—	2	1	no	—
59	C₂×C₅₉	—	2	1	no	—
60	C₂×C₅₉	#340	1	1	yes	characteristic
61	C₂×C₅₉	#341	1	1	yes	characteristic
62	C₂×C₅₉	#344	1	1	yes	characteristic
63	#431	—	59	1	no	—
64	#423	—	59	1	no	—
65	C₂×C₄⋊C₈	D₅₉	1	1	yes	characteristic
66	C₅₉	#1301	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₁₆	—	118	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	59	2	no	—
72	C₄⋊C₈	—	118	1	no	—
73	C₄⋊C₈	—	118	1	no	—
74	C₄⋊C₈	Dic₅₉	1	2	yes	—
75	C₂²⋊C₈	—	59	2	no	—
76	C₄×C₈	—	118	1	no	—
77	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
80	M₁₆	—	236	1	no	—
81	M₁₆	—	236	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	118	2	no	—
86	C₂×C₈	—	118	2	no	—
87	C₄²	C₄×D₅₉	1	1	yes	characteristic
88	C₄²	D₂₃₆	1	1	yes	characteristic
89	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
90	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
91	C₂³	Dic₅₉⋊C₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
95	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
97	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
98	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
99	C₂×C₄	#11745669	1	1	yes	characteristic
100	C₂×C₄	#11745670	1	1	yes	characteristic
101	C₂×C₄	#11745671	1	1	yes	characteristic
102	C₂×C₄	#11745672	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	236	2	no	—
106	C₈	—	236	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₅₉⋊#95	1	1	yes	characteristic
110	C₂²	#14273658	1	1	yes	characteristic
111	C₂²	#14273660	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#14273647	1	1	yes	characteristic
116	C₄	#14273648	1	1	yes	characteristic
117	C₄	#14273650	1	1	yes	characteristic
118	C₄	#14273651	1	1	yes	characteristic
119	C₄	#14273663	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#16837164	1	1	yes	characteristic
123	C₂	#16837167	1	1	yes	characteristic
124	C₂	#16837174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837455

C₂

yes

maximal, characteristic

#16837426

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₄

yes

—

C₂²⋊(C₅₉⋊C₈)

C₄

yes

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

Q₁₆

yes

characteristic

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#1301

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

C₂×C₄⋊C₈

D₅₉

yes

characteristic

C₅₉

#1301

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

C₄⋊C₈

Dic₅₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

118

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

M₁₆

—

236

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄²

C₄×D₅₉

yes

characteristic

C₄²

D₂₃₆

yes

characteristic

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

characteristic

C₂³

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₅₉

yes

characteristic

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

C₂×C₄

#11745669

yes

characteristic

100

C₂×C₄

#11745670

yes

characteristic

101

C₂×C₄

#11745671

yes

characteristic

102

C₂×C₄

#11745672

yes

characteristic

103

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

104

C₈

—

105

C₈

—

236

—

106

C₈

—

236

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₅₉⋊#95

yes

characteristic

110

C₂²

#14273658

yes

characteristic

111

C₂²

#14273660

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#14273647

yes

characteristic

116

C₄

#14273648

yes

characteristic

117

C₄

#14273650

yes

characteristic

118

C₄

#14273651

yes

characteristic

119

C₄

#14273663

yes

—

120

C₂

—

121

C₂

—

122

C₂

#16837164

yes

characteristic

123

C₂

#16837167

yes

characteristic

124

C₂

#16837174

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₅₉)⋊C₄
commutator	C₂×C₄×C₅₉	C₄²
Frattini	C₂×C₄²	C₂×D₅₉
Fitting	C₂×C₄⋊C₈×C₅₉	C₂
socle	C₂²×C₅₉	#95
hypercenter	C₂×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#1301
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

(C₂×D₅₉)⋊C₄

commutator

C₂×C₄×C₅₉

C₄²

Frattini

C₂×C₄²

C₂×D₅₉

Fitting

C₂×C₄⋊C₈×C₅₉

C₂

socle

C₂²×C₅₉

#95

hypercenter

C₂×C₄⋊C₈

D₅₉

hypocenter

C₅₉

#1301

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
66. C₅₉	#1301	semidirect

H ≅ G/N

type

66. C₅₉

#1301

semidirect

Subgroups — #21421993

Conjugacy classes of subgroups

Complements