Group #21425812 — subgroups

Subgroups — #21425812

Order 7552.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14273667	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837440	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837203	C₂	1	4	yes	maximal, complemented
4	C₂²×D₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#14273686	—	2	1	no	—
6	#14273667	—	2	2	no	—
7	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	complemented
8	#14273661	C₂²	1	2	yes	complemented
9	#14273659	—	2	1	no	—
10	C₂²×D₄×C₅₉	—	2	1	no	—
11	C₂×D₈×C₅₉	—	2	1	no	—
12	C₂×D₈×C₅₉	—	2	2	no	—
13	C₂×D₈×C₅₉	C₄	1	4	yes	complemented
14	C₂×D₈×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂⁴×C₅₉	—	4	1	no	—
17	C₂×D₄×C₅₉	—	2	4	no	—
18	C₂×D₄×C₅₉	—	2	2	no	—
19	C₂×D₄×C₅₉	—	4	1	no	—
20	C₂×D₄×C₅₉	—	4	2	no	—
21	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
22	D₈×C₅₉	—	2	4	no	—
23	D₈×C₅₉	—	2	2	no	—
24	D₈×C₅₉	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	D₄	1	2	yes	—
28	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂²⋊Dic₅₉	—	4	1	no	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄×Dic₅₉	—	4	1	no	—
32	C₅₉⋊C₁₆	—	2	2	no	—
33	C₂³×C₅₉	—	4	4	no	—
34	C₂³×C₅₉	—	4	2	no	—
35	C₂³×C₅₉	—	8	4	no	—
36	C₂³×C₅₉	D₈	1	1	yes	characteristic
37	D₄×C₅₉	—	2	4	no	—
38	D₄×C₅₉	—	4	4	no	—
39	D₄×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₅₉	—	4	2	no	—
45	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₅₉	—	2	2	no	—
49	C₂²×C₅₉	—	4	4	no	—
50	C₂²×C₅₉	—	8	8	no	—
51	C₂²×C₅₉	—	8	4	no	—
52	C₂²×C₅₉	#102	1	2	yes	—
53	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
54	C₄×C₅₉	#102	1	2	yes	—
55	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
57	Dic₅₉	—	8	2	no	—
58	#2140	—	59	1	no	maximal, Sylow
59	C₂×C₅₉	—	2	2	no	—
60	C₂×C₅₉	—	8	4	no	—
61	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
62	C₂×C₅₉	#509	1	2	yes	—
63	C₂²×D₈	D₅₉	1	1	yes	characteristic
64	C₂×M₃₂	—	59	1	no	—
65	#427	—	59	1	no	—
66	#357	—	59	4	no	—
67	C₅₉	#2140	1	1	yes	characteristic, complemented, Sylow
68	C₂²×D₄	—	2	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	—	2	2	no	—
71	C₂×D₈	Dic₅₉	1	4	yes	complemented
72	C₂×D₈	C₂×D₅₉	1	2	yes	—
73	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
74	#117	—	118	1	no	—
75	M₃₂	—	118	2	no	—
76	C₂×C₁₆	—	59	2	no	—
77	#107	—	59	2	no	—
78	#106	—	118	1	no	—
79	C₂⁴	—	4	1	no	—
80	C₂×D₄	—	2	4	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	4	1	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
85	D₈	—	2	4	no	—
86	D₈	—	2	2	no	—
87	D₈	C₂×Dic₅₉	1	4	yes	—
88	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
89	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
91	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
92	C₄⋊C₄	—	118	2	no	—
93	C₂²⋊C₄	—	236	1	no	—
94	C₄²	—	236	1	no	—
95	C₁₆	—	118	2	no	—
96	C₂³	—	4	4	no	—
97	C₂³	—	4	2	no	—
98	C₂³	—	8	4	no	—
99	C₂³	#11745669	1	1	yes	characteristic
100	D₄	—	2	4	no	—
101	D₄	—	4	4	no	—
102	D₄	—	4	2	no	—
103	C₂×C₄	—	236	2	no	—
104	C₂×C₄	#11745669	1	1	yes	characteristic
105	C₂×C₄	#11745670	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
108	C₈	C₂²⋊Dic₅₉	1	2	yes	—
109	C₈	C₂²×Dic₅₉	1	1	yes	characteristic
110	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	8	8	no	—
114	C₂²	—	8	4	no	—
115	C₂²	#14273648	1	2	yes	—
116	C₂²	C₂×#11745669	1	1	yes	characteristic
117	C₄	—	472	2	no	—
118	C₄	#14273648	1	2	yes	—
119	C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
120	C₄	C₂×#11745670	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	8	4	no	—
123	C₂	C₂×#14273648	1	1	yes	characteristic
124	C₂	#16837838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273667

C₂

yes

maximal, characteristic, complemented

#16837440

C₂

yes

maximal, characteristic, complemented

#16837203

C₂

yes

maximal, complemented

C₂²×D₈×C₅₉

C₂

yes

maximal, characteristic

#14273686

—

#14273667

—

C₂×C₅₉⋊C₁₆

C₂²

yes

complemented

#14273661

C₂²

yes

complemented

#14273659

—

C₂²×D₄×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

C₄

yes

complemented

C₂×D₈×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

D₈×C₅₉

—

D₈×C₅₉

—

D₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

D₈

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#102

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

#102

yes

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

#2140

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×#102

yes

characteristic

C₂×C₅₉

#509

yes

—

C₂²×D₈

D₅₉

yes

characteristic

C₂×M₃₂

—

#427

—

#357

—

C₅₉

#2140

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

Dic₅₉

yes

complemented

C₂×D₈

C₂×D₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

118

—

M₃₂

—

118

—

C₂×C₁₆

—

#107

—

#106

—

118

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

D₈

—

D₈

—

D₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

236

—

C₄²

—

236

—

C₁₆

—

118

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#11745669

yes

characteristic

100

D₄

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

236

—

104

C₂×C₄

#11745669

yes

characteristic

105

C₂×C₄

#11745670

yes

—

106

C₂×C₄

C₂²⋊Dic₅₉

yes

—

107

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

108

C₈

C₂²⋊Dic₅₉

yes

—

109

C₈

C₂²×Dic₅₉

yes

characteristic

110

C₈

C₂×C₂²⋊D₅₉

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#14273648

yes

—

116

C₂²

C₂×#11745669

yes

characteristic

117

C₄

—

472

—

118

C₄

#14273648

yes

—

119

C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

120

C₄

C₂×#11745670

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₂×#14273648

yes

characteristic

124

C₂

#16837838

yes

—

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂²×D₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂²×D₈	D₅₉
hypocenter	C₅₉	#2140
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂²×D₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂²×D₈

D₅₉

hypocenter

C₅₉

#2140

LCS

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#14273667	C₂	semidirect
2. #16837440	C₂	semidirect
3. #16837203	C₂	semidirect
7. C₂×C₅₉⋊C₁₆	C₂²	semidirect
8. #14273661	C₂²	semidirect
13. C₂×D₈×C₅₉	C₄	semidirect
67. C₅₉	#2140	semidirect
71. C₂×D₈	Dic₅₉	semidirect

H ≅ G/N

type

1. C₂×#14273667

C₂

semidirect

2. #16837440

C₂

semidirect

3. #16837203

C₂

semidirect

7. C₂×C₅₉⋊C₁₆

C₂²

semidirect

8. #14273661

C₂²

semidirect

13. C₂×D₈×C₅₉

C₄

semidirect

67. C₅₉

#2140

semidirect

71. C₂×D₈

Dic₅₉

semidirect

Subgroups — #21425812

Conjugacy classes of subgroups

Complements