Group #28503385 — subgroups

Subgroups — #28503385

Order 15104.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21426325	C₂	1	2	yes	maximal, complemented
2	#21426321	C₂	1	2	yes	maximal
3	#21423702	C₂	1	1	yes	maximal, characteristic
4	C₈×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837782	C₂²	1	1	yes	characteristic
7	#16837779	C₂²	1	1	yes	characteristic
8	#16837749	—	2	2	no	—
9	#16837205	—	2	4	no	—
10	#16837202	—	2	4	no	—
11	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
12	C₅₉×#500	C₂²	1	2	yes	—
13	C₅₉×#487	—	2	2	no	—
14	C₈²×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₃₆	—	2	2	no	—
16	C₂×D₄₇₂	—	2	2	no	—
17	#14273727	—	2	1	no	—
18	#14273724	—	2	1	no	—
19	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	—
20	#14273649	—	4	4	no	—
21	C₄⋊Q₈×C₅₉	—	2	2	no	—
22	#107×C₅₉	—	2	2	no	—
23	#107×C₅₉	—	2	2	no	—
24	#103×C₅₉	—	4	4	no	—
25	C₄×C₈×C₅₉	D₄	1	2	yes	—
26	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₅₉	—	4	2	no	—
28	C₂×D₂₃₆	—	4	1	no	—
29	C₂×D₂₃₆	—	4	2	no	—
30	C₂×Dic₁₁₈	—	4	1	no	—
31	C₂×C₈×C₅₉	D₈	1	4	yes	—
32	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
33	C₄⋊C₄×C₅₉	—	4	2	no	—
34	C₄⋊C₄×C₅₉	—	4	2	no	—
35	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₅₉	—	4	2	no	—
37	Dic₂₃₆	—	4	4	no	—
38	D₄₇₂	—	4	4	no	—
39	C₅₉⋊C₁₆	—	2	4	no	—
40	C₂²×D₅₉	—	8	1	no	—
41	Q₈×C₅₉	—	8	4	no	—
42	C₂×C₄×C₅₉	—	8	2	no	—
43	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
45	C₂×Dic₅₉	—	8	1	no	—
46	D₂₃₆	—	8	2	no	—
47	D₂₃₆	—	8	4	no	—
48	Dic₁₁₈	—	8	2	no	—
49	C₈×C₅₉	—	2	4	no	—
50	C₈×C₅₉	#135	1	4	yes	—
51	#12511	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	#493	1	1	yes	characteristic
53	C₂×D₅₉	—	16	1	no	—
54	C₂×D₅₉	—	16	2	no	—
55	C₄×C₅₉	—	16	2	no	—
56	C₄×C₅₉	#495	1	2	yes	—
57	C₄×C₅₉	#508	1	4	yes	—
58	Dic₅₉	—	16	1	no	—
59	#2241	—	59	2	no	—
60	#2240	—	59	2	no	—
61	#1710	D₅₉	1	1	yes	characteristic, complemented
62	#1709	—	59	1	no	—
63	C₈×C₁₆	—	59	1	no	—
64	C₂×C₅₉	#1709	1	1	yes	characteristic
65	C₂×C₅₉	#2249	1	2	yes	—
66	D₅₉	—	32	1	no	—
67	#500	C₂×D₅₉	1	2	yes	—
68	#494	—	59	1	no	—
69	#493	—	59	1	no	—
70	#487	—	2	2	no	—
71	#486	—	118	2	no	—
72	#358	—	118	4	no	—
73	#357	—	118	4	no	—
74	C₄×C₁₆	—	59	2	no	—
75	C₈²	C₂×D₅₉	1	1	yes	characteristic
76	C₅₉	#12511	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	118	2	no	—
78	C₂×D₈	—	118	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	118	1	no	—
81	#127	—	118	1	no	—
82	C₂×C₁₆	—	59	4	no	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#103	—	4	4	no	—
86	#102	—	236	4	no	—
87	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
88	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	236	1	no	—
91	C₂×D₄	—	236	1	no	—
92	C₂×D₄	—	236	2	no	—
93	Q₁₆	—	236	4	no	—
94	D₈	—	236	4	no	—
95	C₂×C₈	#11745669	1	4	yes	—
96	C₂×C₈	D₄×D₅₉	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	236	2	no	—
100	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
101	C₁₆	—	118	4	no	—
102	C₂³	—	472	1	no	—
103	Q₈	—	8	4	no	—
104	Q₈	—	472	2	no	—
105	D₄	—	472	2	no	—
106	D₄	—	472	4	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	472	1	no	—
109	C₂×C₄	#14273723	1	1	yes	characteristic
110	C₂×C₄	C₂×#11745669	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14273750	1	4	yes	—
113	C₂²	—	944	1	no	—
114	C₂²	—	944	2	no	—
115	C₂²	#16837778	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	944	1	no	—
118	C₄	#16837787	1	2	yes	—
119	C₄	#16837832	1	4	yes	—
120	C₂	—	1888	1	no	—
121	C₂	#21423703	1	1	yes	characteristic
122	C₂	#21426355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21426325

C₂

yes

maximal, complemented

#21426321

C₂

yes

maximal

#21423702

C₂

yes

maximal, characteristic

C₈×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₅₉×#1710

C₂

yes

maximal, characteristic, complemented

#16837782

C₂²

yes

characteristic

#16837779

C₂²

yes

—

—

—

C₄×C₅₉⋊C₁₆

C₂²

yes

—

C₅₉×#500

C₂²

yes

—

C₅₉×#487

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

C₂×D₄₇₂

—

#14273727

—

#14273724

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

#14273649

—

C₄⋊Q₈×C₅₉

—

#107×C₅₉

—

#107×C₅₉

—

#103×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₂₃₆

—

D₄₇₂

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#135

yes

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#495

yes

—

C₄×C₅₉

#508

yes

—

Dic₅₉

—

#2241

—

#2240

—

#1710

D₅₉

yes

characteristic, complemented

#1709

—

C₈×C₁₆

—

C₂×C₅₉

#1709

yes

characteristic

C₂×C₅₉

#2249

yes

—

D₅₉

—

#500

C₂×D₅₉

yes

—

—

—

—

—

118

—

#358

—

118

—

#357

—

118

—

C₄×C₁₆

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12511

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×D₈

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

#127

—

118

—

C₂×C₁₆

—

—

—

—

—

236

—

C₄×C₈

C₂²⋊D₅₉

yes

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

236

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

Q₁₆

—

236

—

D₈

—

236

—

C₂×C₈

#11745669

yes

—

C₂×C₈

D₄×D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

236

—

100

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

101

C₁₆

—

118

—

102

C₂³

—

472

—

103

Q₈

—

104

Q₈

—

472

—

105

D₄

—

472

—

106

D₄

—

472

—

107

C₂×C₄

—

108

C₂×C₄

—

472

—

109

C₂×C₄

#14273723

yes

characteristic

110

C₂×C₄

C₂×#11745669

yes

—

111

C₈

—

112

C₈

#14273750

yes

—

113

C₂²

—

944

—

114

C₂²

—

944

—

115

C₂²

#16837778

yes

characteristic

116

C₄

—

117

C₄

—

944

—

118

C₄

#16837787

yes

—

119

C₄

#16837832

yes

—

120

C₂

—

1888

—

121

C₂

#21423703

yes

characteristic

122

C₂

#21426355

yes

—

	subgroup	G/H
center	C₂²	#16837778
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1710	C₂
socle	C₂²×C₅₉	#493
prime radical	#16837779	C₂²
hypercenter	#1710	D₅₉
hypocenter	C₅₉	#12511
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#16837778

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1710

C₂

C₂²×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #21426325	C₂	semidirect
4. C₈×C₅₉⋊C₁₆	C₂	semidirect
5. C₅₉×#1710	C₂	semidirect
61. #1710	D₅₉	semidirect
76. C₅₉	#12511	semidirect

H ≅ G/N

type

1. #21426325

C₂

semidirect

4. C₈×C₅₉⋊C₁₆

C₂

semidirect

5. C₅₉×#1710

C₂

semidirect

61. #1710

D₅₉

semidirect

76. C₅₉

#12511

semidirect

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

subgroup	index
#21426325	2
#21426321	2
#21423702	2
C₈×C₅₉⋊C₁₆	2
C₅₉×#1710	2
#12511	59

subgroup

index

#21426325

#21426321

#21423702

C₈×C₅₉⋊C₁₆

C₅₉×#1710

#12511

Subgroups — #28503385

Conjugacy classes of subgroups

Complements

Maximal subgroups