Group #28482374 — subgroups

Subgroups — #28482374

Order 15104.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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	#21424234	C₂	1	1	yes	maximal, characteristic
2	#21424227	C₂	1	1	yes	maximal, characteristic
3	#21421955	C₂	1	4	yes	maximal
4	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic
5	#16837426	C₂²	1	2	yes	—
6	#16837268	C₂²	1	2	yes	—
7	#16837167	—	2	2	no	—
8	#16837165	—	2	2	no	—
9	C₅₉×#431	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#322	C₄	1	2	yes	complemented
12	C₈²×C₅₉	C₄	1	2	yes	complemented
13	C₂×#11745664	—	2	2	no	—
14	C₂×C₄⋊Dic₅₉	—	2	2	no	—
15	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
16	C₅₉⋊#95	—	2	4	no	—
17	C₂²×C₈×C₅₉	D₄	1	2	yes	—
18	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₅₉	—	2	2	no	—
20	C₂²⋊C₈×C₅₉	—	2	2	no	—
21	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₅₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	#41	1	2	yes	—
27	C₂²×Dic₅₉	—	4	2	no	—
28	C₂×C₈×C₅₉	—	2	4	no	—
29	C₂×C₈×C₅₉	—	2	4	no	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
32	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₅₉	#41	1	2	yes	—
35	C₄⋊Dic₅₉	—	4	4	no	—
36	#11745664	—	4	4	no	—
37	C₂×C₅₉⋊C₈	—	4	2	no	—
38	C₂³×C₅₉	#124	1	1	yes	characteristic
39	C₂×C₄×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	#117	1	2	yes	—
42	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#125	1	2	yes	—
44	C₂×C₄×C₅₉	#135	1	4	yes	—
45	C₂×Dic₅₉	—	8	4	no	—
46	C₂×Dic₅₉	—	8	2	no	—
47	C₈×C₅₉	—	2	2	no	—
48	C₈×C₅₉	—	2	2	no	—
49	C₈×C₅₉	—	4	4	no	—
50	C₅₉⋊C₈	—	8	2	no	—
51	#8791	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	#382	1	1	yes	characteristic
54	C₂²×C₅₉	#488	1	2	yes	—
55	C₄×C₅₉	—	2	2	no	—
56	C₄×C₅₉	—	2	2	no	—
57	C₄×C₅₉	#427	1	2	yes	—
58	C₄×C₅₉	#488	1	2	yes	—
59	Dic₅₉	—	16	2	no	—
60	#1837	—	59	1	no	—
61	#1835	—	59	1	no	—
62	#1450	D₅₉	1	1	yes	characteristic
63	#1286	—	59	4	no	—
64	C₂×C₅₉	—	2	2	no	—
65	C₂×C₅₉	#1833	1	2	yes	—
66	C₂×C₅₉	#1835	1	1	yes	characteristic
67	#431	C₂×D₅₉	1	2	yes	—
68	#423	—	59	2	no	—
69	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
70	#382	—	59	2	no	—
71	#341	—	118	2	no	—
72	#340	—	118	2	no	—
73	#322	Dic₅₉	1	2	yes	complemented
74	C₈²	Dic₅₉	1	2	yes	complemented
75	C₅₉	#8791	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	118	2	no	—
77	C₂²×C₈	D₂₃₆	1	2	yes	—
78	C₂×C₄⋊C₄	—	118	2	no	—
79	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	118	4	no	—
82	C₂²⋊C₈	—	2	2	no	—
83	C₄×C₈	Dic₁₁₈	1	2	yes	—
84	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
85	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
87	#95	—	118	4	no	—
88	C₂²×C₄	—	236	2	no	—
89	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
90	C₂²×C₄	#11745685	1	2	yes	—
91	M₁₆	—	236	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	236	2	no	—
95	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
97	C₄⋊C₄	—	236	4	no	—
98	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
99	C₄²	#11745685	1	2	yes	—
100	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
101	C₂³	#14273715	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	472	4	no	—
104	C₂×C₄	—	472	2	no	—
105	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
106	C₂×C₄	#14273686	1	2	yes	—
107	C₂×C₄	#14273715	1	1	yes	characteristic
108	C₂×C₄	#14273719	1	2	yes	—
109	C₂×C₄	#14273749	1	4	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	4	4	no	—
113	C₈	—	472	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#16837268	1	1	yes	characteristic
116	C₂²	#16837757	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	944	2	no	—
120	C₄	#16837440	1	2	yes	—
121	C₄	#16837757	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#21424220	1	2	yes	—
124	C₂	#21424227	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424234

C₂

yes

maximal, characteristic

#21424227

C₂

yes

maximal, characteristic

#21421955

C₂

yes

maximal

C₅₉×#1450

C₂

yes

maximal, characteristic

#16837426

C₂²

yes

—

#16837268

C₂²

yes

—

#16837167

—

#16837165

—

C₅₉×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

C₄

yes

complemented

C₈²×C₅₉

C₄

yes

complemented

C₂×#11745664

—

C₂×C₄⋊Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

#125

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

#8791

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#382

yes

characteristic

C₂²×C₅₉

#488

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#427

yes

—

C₄×C₅₉

#488

yes

—

Dic₅₉

—

#1837

—

#1835

—

#1450

D₅₉

yes

characteristic

#1286

—

C₂×C₅₉

—

C₂×C₅₉

#1833

yes

—

C₂×C₅₉

#1835

yes

characteristic

#431

C₂×D₅₉

yes

—

#423

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#382

—

#341

—

118

—

#340

—

118

—

#322

Dic₅₉

yes

complemented

C₈²

Dic₅₉

yes

complemented

C₅₉

#8791

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

D₂₃₆

yes

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₄×C₈

C₂×Dic₅₉

yes

characteristic

C₄×C₈

C₂×Dic₅₉

yes

characteristic

#95

—

118

—

C₂²×C₄

—

236

—

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

C₂²×C₄

#11745685

yes

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₄⋊C₄

—

236

—

C₄²

C₂×Dic₁₁₈

yes

characteristic

C₄²

#11745685

yes

—

100

C₄²

C₂²×Dic₅₉

yes

characteristic

101

C₂³

#14273715

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

472

—

104

C₂×C₄

—

472

—

105

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

106

C₂×C₄

#14273686

yes

—

107

C₂×C₄

#14273715

yes

characteristic

108

C₂×C₄

#14273719

yes

—

109

C₂×C₄

#14273749

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

472

—

114

C₂²

—

115

C₂²

#16837268

yes

characteristic

116

C₂²

#16837757

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

944

—

120

C₄

#16837440

yes

—

121

C₄

#16837757

yes

—

122

C₂

—

123

C₂

#21424220

yes

—

124

C₂

#21424227

yes

characteristic

	subgroup	G/H
center	C₂²	#16837268
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂²×C₅₉	#382
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#8791
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#16837268

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1450

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
11. C₅₉×#322	C₄	semidirect
12. C₈²×C₅₉	C₄	semidirect
73. #322	Dic₅₉	semidirect
74. C₈²	Dic₅₉	semidirect
75. C₅₉	#8791	semidirect

H ≅ G/N

type

11. C₅₉×#322

C₄

semidirect

12. C₈²×C₅₉

C₄

semidirect

73. #322

Dic₅₉

semidirect

74. C₈²

Dic₅₉

semidirect

75. C₅₉

#8791

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21424234	2
#21424227	2
#21421955	2
C₅₉×#1450	2
#8791	59

subgroup

index

C₅₉×#1450

Subgroups — #28482374

Conjugacy classes of subgroups

Complements

Maximal subgroups