Group #21425074 — subgroups

Subgroups — #21425074

Order 7552.

284 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 105 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	#16837426	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837161	C₂	1	4	yes	maximal
4	C₅₉×#533	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	C₂²	1	2	yes	—
6	C₂×C₄×Dic₅₉	—	2	2	no	—
7	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
8	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
9	C₄⋊Q₈×C₅₉	C₄	1	2	yes	complemented
10	#125×C₅₉	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₅₉	—	2	2	no	—
12	C₄×Q₈×C₅₉	—	2	2	no	—
13	#117×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	D₄	1	2	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	—	2	2	no	—
19	C₄⋊C₄×C₅₉	—	2	4	no	—
20	C₄⋊C₄×C₅₉	—	2	2	no	—
21	C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
22	C₂²⋊C₄×C₅₉	—	2	2	no	—
23	C₄²×C₅₉	—	2	2	no	—
24	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₅₉	D₄	1	2	yes	—
27	C₄×Dic₅₉	—	2	8	no	—
28	C₄×Dic₅₉	—	2	2	no	—
29	C₄×Dic₅₉	Q₈	1	4	yes	complemented
30	#11745664	—	2	4	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₅₉	—	4	4	no	—
34	C₂×C₄×C₅₉	—	2	4	no	—
35	C₂×C₄×C₅₉	—	2	4	no	—
36	C₂×C₄×C₅₉	C₂²⋊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	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₅₉	#41	1	2	yes	—
42	C₂×Dic₅₉	—	2	8	no	—
43	C₂×Dic₅₉	—	2	4	no	—
44	C₅₉⋊C₈	—	4	2	no	—
45	C₂²×C₅₉	—	2	2	no	—
46	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₅₉	—	2	2	no	—
49	C₄×C₅₉	—	4	4	no	—
50	C₄×C₅₉	#104	1	4	yes	—
51	C₄×C₅₉	C₄×Q₈	1	2	yes	—
52	C₄×C₅₉	#124	1	1	yes	characteristic
53	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₅₉	—	2	8	no	—
55	#1994	—	59	1	no	maximal, Sylow
56	C₂×C₅₉	—	2	2	no	—
57	C₂×C₅₉	#391	1	1	yes	characteristic
58	C₂×C₅₉	C₂×#104	1	2	yes	—
59	#533	D₅₉	1	1	yes	characteristic
60	#423	—	59	1	no	—
61	C₄³	—	59	1	no	—
62	#339	—	59	4	no	—
63	C₅₉	#1994	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₁₆	—	59	2	no	—
65	C₄⋊Q₈	Dic₅₉	1	2	yes	complemented
66	#125	Dic₅₉	1	2	yes	complemented
67	C₂²⋊Q₈	—	2	2	no	—
68	C₄×Q₈	—	2	2	no	—
69	#117	C₂×D₅₉	1	2	yes	—
70	C₂×C₄²	—	59	2	no	—
71	C₂×C₄²	—	118	2	no	—
72	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
73	C₄⋊C₈	—	118	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂²×C₄	—	118	2	no	—
76	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
77	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
78	M₁₆	—	118	4	no	—
79	C₂×C₈	—	118	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
83	C₂²⋊C₄	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	59	4	no	—
86	C₄²	—	118	8	no	—
87	C₄²	—	118	2	no	—
88	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
89	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
90	C₄²	C₂²⋊D₅₉	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	118	8	no	—
96	C₂×C₄	—	118	4	no	—
97	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
99	C₂×C₄	Q₈×D₅₉	1	2	yes	—
100	C₂×C₄	#11745689	1	2	yes	—
101	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
103	C₈	—	236	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
106	C₂²	#14273703	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	118	8	no	—
110	C₄	#14273654	1	4	yes	—
111	C₄	Q₈×Dic₅₉	1	2	yes	—
112	C₄	#14273716	1	1	yes	characteristic
113	C₄	#14273725	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#16837309	1	1	yes	characteristic
116	C₂	C₂×#14273654	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837426

C₂

yes

maximal, characteristic

C₄²×Dic₅₉

C₂

yes

maximal, characteristic

#16837161

C₂

yes

maximal

C₅₉×#533

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊Q₈×C₅₉

C₄

yes

complemented

#125×C₅₉

C₄

yes

complemented

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

Q₈

yes

complemented

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

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

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#104

yes

—

C₄×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

#1994

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#391

yes

characteristic

C₂×C₅₉

C₂×#104

yes

—

#533

D₅₉

yes

characteristic

#423

—

C₄³

—

#339

—

C₅₉

#1994

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

Dic₅₉

yes

complemented

#125

Dic₅₉

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

C₂×Q₈

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

118

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

118

—

C₄²

—

118

—

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

—

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

C₂²⋊Dic₅₉

yes

—

C₂×C₄

C₂²⋊Dic₅₉

yes

—

C₂×C₄

Q₈×D₅₉

yes

—

100

C₂×C₄

#11745689

yes

—

101

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

103

C₈

—

236

—

104

C₂²

—

105

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

106

C₂²

#14273703

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

118

—

110

C₄

#14273654

yes

—

111

C₄

Q₈×Dic₅₉

yes

—

112

C₄

#14273716

yes

characteristic

113

C₄

#14273725

yes

characteristic

114

C₂

—

115

C₂

#16837309

yes

characteristic

116

C₂

C₂×#14273654

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#533	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#533	D₅₉
hypocenter	C₅₉	#1994
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#533

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₅₉	C₄	semidirect
10. #125×C₅₉	C₄	semidirect
29. C₄×Dic₅₉	Q₈	semidirect
63. C₅₉	#1994	semidirect
65. C₄⋊Q₈	Dic₅₉	semidirect
66. #125	Dic₅₉	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₅₉

C₄

semidirect

10. #125×C₅₉

C₄

semidirect

29. C₄×Dic₅₉

Q₈

semidirect

63. C₅₉

#1994

semidirect

65. C₄⋊Q₈

Dic₅₉

semidirect

66. #125

Dic₅₉

semidirect

Subgroups — #21425074

Conjugacy classes of subgroups

Complements