Group #21423813 — subgroups

Subgroups — #21423813

Order 7552.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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×#14273686	C₂	1	2	yes	maximal
2	#16837165	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	#14273686	—	2	4	no	—
5	#14273686	C₄	1	8	yes	complemented
6	C₂×C₄⋊Dic₅₉	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₅₉	—	2	2	no	—
8	C₂×C₄×Dic₅₉	—	2	2	no	—
9	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₅₉	—	2	4	no	—
11	C₂²×C₈×C₅₉	C₄	1	4	yes	complemented
12	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
13	C₂⁴×C₅₉	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₅₉	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	D₄	1	4	yes	—
18	C₂²×C₄×C₅₉	Q₈	1	2	yes	—
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₉	—	2	4	no	—
21	C₂×C₈×C₅₉	—	2	8	no	—
22	C₂×C₈×C₅₉	—	2	4	no	—
23	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	—
24	C₂²⋊Dic₅₉	—	2	8	no	—
25	C₄⋊Dic₅₉	—	2	4	no	—
26	C₄⋊Dic₅₉	C₂×C₄	1	8	yes	—
27	C₄×Dic₅₉	—	2	8	no	—
28	C₂³×C₅₉	—	2	4	no	—
29	C₂³×C₅₉	C₂²⋊C₄	1	4	yes	—
30	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₄²	1	4	yes	—
33	C₂×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₅₉	C₄⋊C₄	1	8	yes	—
36	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₅₉	—	2	8	no	—
42	C₂×Dic₅₉	—	4	8	no	—
43	C₈×C₅₉	—	2	4	no	—
44	C₂²×C₅₉	—	2	8	no	—
45	C₂²×C₅₉	—	2	4	no	—
46	C₂²×C₅₉	#95	1	4	yes	—
47	C₂²×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
48	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₅₉	#135	1	4	yes	—
50	C₄×C₅₉	#95	1	4	yes	—
51	C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
54	Dic₅₉	—	4	8	no	—
55	#1735	—	59	1	no	maximal, Sylow
56	C₂×C₅₉	—	2	4	no	—
57	C₂×C₅₉	C₂×#95	1	1	yes	characteristic
58	C₂×C₅₉	#416	1	4	yes	—
59	C₂×C₅₉	#427	1	2	yes	—
60	C₂³×C₈	D₅₉	1	1	yes	characteristic
61	C₂×#117	—	59	2	no	—
62	#340	—	59	4	no	—
63	C₅₉	#1735	1	1	yes	characteristic, complemented, Sylow
64	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
65	C₂²×C₈	—	2	4	no	—
66	C₂²×C₈	Dic₅₉	1	4	yes	complemented
67	C₂²×C₈	C₂×D₅₉	1	2	yes	—
68	#117	—	59	8	no	—
69	#117	—	118	4	no	—
70	C₂×C₄⋊C₄	—	59	4	no	—
71	C₂×C₂²⋊C₄	—	118	2	no	—
72	C₂×C₄²	—	118	2	no	—
73	C₂⁴	D₂₃₆	1	1	yes	characteristic
74	C₂²×C₄	—	118	4	no	—
75	C₂²×C₄	Dic₁₁₈	1	2	yes	—
76	C₂²×C₄	C₄×D₅₉	1	4	yes	—
77	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
79	C₂²×C₄	C₂²⋊D₅₉	1	4	yes	—
80	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×Dic₅₉	1	4	yes	—
84	C₄⋊C₄	—	59	8	no	—
85	C₄⋊C₄	—	118	4	no	—
86	C₂²⋊C₄	—	118	8	no	—
87	C₄²	—	118	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₄⋊Dic₅₉	1	2	yes	—
90	C₂³	(C₂×D₅₉)⋊C₄	1	4	yes	—
91	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
92	C₂×C₄	—	118	8	no	—
93	C₂×C₄	—	236	8	no	—
94	C₂×C₄	C₄×Dic₅₉	1	4	yes	—
95	C₂×C₄	Dic₅₉⋊C₄	1	8	yes	—
96	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
97	C₂×C₄	(C₂×D₅₉)⋊C₄	1	4	yes	—
98	C₂×C₄	C₂²⋊Dic₅₉	1	4	yes	—
99	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
101	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	C₅₉⋊#95	1	4	yes	—
107	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
109	C₂²	#14273749	1	4	yes	—
110	C₄	—	236	8	no	—
111	C₄	C₅₉⋊#95	1	4	yes	—
112	C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
113	C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
114	C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	C₂×C₅₉⋊#95	1	1	yes	characteristic
117	C₂	#16837399	1	4	yes	—
118	C₂	#16837440	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273686

C₂

yes

maximal

#16837165

C₂

yes

maximal, complemented

C₂³×C₈×C₅₉

C₂

yes

maximal, characteristic

#14273686

—

#14273686

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₂²⋊Dic₅₉

—

C₂×C₄×Dic₅₉

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

C₄

yes

complemented

C₂²×C₈×C₅₉

C₂²

yes

—

C₂⁴×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²⋊C₄

yes

—

C₂³×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₄²

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#95

yes

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#135

yes

—

C₄×C₅₉

#95

yes

—

C₄×C₅₉

C₂×C₄²

yes

characteristic

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₄

yes

—

Dic₅₉

—

#1735

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×#95

yes

characteristic

C₂×C₅₉

#416

yes

—

C₂×C₅₉

#427

yes

—

C₂³×C₈

D₅₉

yes

characteristic

C₂×#117

—

#340

—

C₅₉

#1735

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₅₉

yes

complemented

C₂²×C₈

C₂×D₅₉

yes

—

#117

—

#117

—

118

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

118

—

C₂×C₄²

—

118

—

C₂⁴

D₂₃₆

yes

characteristic

C₂²×C₄

—

118

—

C₂²×C₄

Dic₁₁₈

yes

—

C₂²×C₄

C₄×D₅₉

yes

—

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₄²

—

118

—

C₂³

—

C₂³

C₄⋊Dic₅₉

yes

—

C₂³

(C₂×D₅₉)⋊C₄

yes

—

C₂³

C₂×D₂₃₆

yes

characteristic

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

C₄×Dic₅₉

yes

—

C₂×C₄

Dic₅₉⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₅₉

yes

—

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

100

C₂×C₄

C₂×C₄×D₅₉

yes

—

101

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₅₉⋊#95

yes

—

107

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

—

108

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

109

C₂²

#14273749

yes

—

110

C₄

—

236

—

111

C₄

C₅₉⋊#95

yes

—

112

C₄

C₂×C₄×Dic₅₉

yes

characteristic

113

C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

114

C₄

C₂×Dic₅₉⋊C₄

yes

—

115

C₂

—

116

C₂

C₂×C₅₉⋊#95

yes

characteristic

117

C₂

#16837399

yes

—

118

C₂

#16837440

yes

—

	subgroup	G/H
center	C₂²×C₄	D₂₃₆
commutator	C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂³×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂³×C₈	D₅₉
hypocenter	C₅₉	#1735
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₂₃₆

commutator

C₄×C₅₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂³×C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂³×C₈

D₅₉

hypocenter

C₅₉

#1735

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #16837165	C₂	semidirect
5. #14273686	C₄	semidirect
11. C₂²×C₈×C₅₉	C₄	semidirect
63. C₅₉	#1735	semidirect
66. C₂²×C₈	Dic₅₉	semidirect

H ≅ G/N

type

2. #16837165

C₂

semidirect

5. #14273686

C₄

semidirect

11. C₂²×C₈×C₅₉

C₄

semidirect

63. C₅₉

#1735

semidirect

66. C₂²×C₈

Dic₅₉

semidirect

Subgroups — #21423813

Conjugacy classes of subgroups

Complements