Group #28503382 — subgroups

Subgroups — #28503382

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	#21426323	C₂	1	2	yes	maximal, complemented
2	#21426319	C₂	1	2	yes	maximal
3	#21423706	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1709	C₂	1	1	yes	maximal, characteristic
6	#16837807	C₂²	1	2	yes	—
7	#16837753	—	2	2	no	—
8	#16837206	—	2	4	no	—
9	#16837203	—	2	4	no	—
10	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
11	C₅₉×#494	C₂²	1	1	yes	characteristic
12	C₅₉×#493	C₂²	1	1	yes	characteristic
13	C₅₉×#486	—	2	2	no	—
14	C₈²×C₅₉	C₂²	1	1	yes	characteristic
15	#14273727	—	2	2	no	—
16	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	—
17	#14273661	—	2	2	no	—
18	#14273661	—	2	2	no	—
19	#14273652	—	4	4	no	—
20	C₂×Q₁₆×C₅₉	—	2	2	no	—
21	C₂×D₈×C₅₉	—	2	2	no	—
22	C₄⋊Q₈×C₅₉	—	2	1	no	—
23	#127×C₅₉	—	2	1	no	—
24	#102×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	1	no	—
28	C₂×D₄×C₅₉	—	4	1	no	—
29	C₂×D₄×C₅₉	—	4	2	no	—
30	C₂×Dic₁₁₈	—	4	2	no	—
31	Q₁₆×C₅₉	—	4	4	no	—
32	D₈×C₅₉	—	4	4	no	—
33	C₂×C₈×C₅₉	D₈	1	4	yes	—
34	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
35	C₄⋊C₄×C₅₉	—	4	2	no	—
36	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅₉	—	4	2	no	—
38	C₄⋊Dic₅₉	—	4	2	no	—
39	C₅₉⋊C₁₆	—	2	4	no	—
40	C₂³×C₅₉	—	8	1	no	—
41	Q₈×C₅₉	—	8	2	no	—
42	D₄×C₅₉	—	8	2	no	—
43	D₄×C₅₉	—	8	4	no	—
44	C₂×C₄×C₅₉	—	8	1	no	—
45	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
47	C₂×Dic₅₉	—	8	2	no	—
48	Dic₁₁₈	—	8	4	no	—
49	C₈×C₅₉	—	2	4	no	—
50	C₈×C₅₉	#135	1	4	yes	—
51	#12511	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	—	16	1	no	—
53	C₂²×C₅₉	—	16	2	no	—
54	C₂²×C₅₉	#493	1	1	yes	characteristic
55	C₄×C₅₉	—	16	1	no	—
56	C₄×C₅₉	#495	1	2	yes	—
57	C₄×C₅₉	#508	1	4	yes	—
58	Dic₅₉	—	16	2	no	—
59	#2241	—	59	2	no	—
60	#2240	—	59	2	no	—
61	#1710	—	59	1	no	—
62	#1709	D₅₉	1	1	yes	characteristic
63	C₈×C₁₆	—	59	1	no	—
64	C₂×C₅₉	—	32	1	no	—
65	C₂×C₅₉	#1709	1	1	yes	characteristic
66	C₂×C₅₉	#2249	1	2	yes	—
67	#500	—	59	2	no	—
68	#494	C₂×D₅₉	1	1	yes	characteristic
69	#493	C₂×D₅₉	1	1	yes	characteristic
70	#487	—	118	2	no	—
71	#486	—	2	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	—	2	2	no	—
78	C₂×D₈	—	2	2	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	—	118	2	no	—
81	#127	—	2	1	no	—
82	C₂×C₁₆	—	59	4	no	—
83	#107	—	118	2	no	—
84	#107	—	118	2	no	—
85	#103	—	236	4	no	—
86	#102	—	4	4	no	—
87	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
88	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	236	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	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	—	236	2	no	—
99	C₄⋊C₄	—	236	2	no	—
100	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
101	C₁₆	—	118	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	472	4	no	—
105	D₄	—	8	2	no	—
106	D₄	—	8	4	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	472	2	no	—
109	C₂×C₄	#14273723	1	1	yes	characteristic
110	C₂×C₄	C₂×#11745669	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14273746	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	#16837778	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	944	2	no	—
118	C₄	#16837785	1	2	yes	—
119	C₄	#16837832	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	#21423701	1	1	yes	characteristic
122	C₂	#21426355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21426323

C₂

yes

maximal, complemented

#21426319

C₂

yes

maximal

#21423706

C₂

yes

maximal, characteristic, complemented

C₈×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₅₉×#1709

C₂

yes

maximal, characteristic

#16837807

C₂²

yes

—

#16837753

—

#16837206

—

#16837203

—

C₄×C₅₉⋊C₁₆

C₂²

yes

—

C₅₉×#494

C₂²

yes

characteristic

C₅₉×#493

C₂²

yes

characteristic

C₅₉×#486

—

C₈²×C₅₉

C₂²

yes

characteristic

#14273727

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

#14273661

—

#14273661

—

#14273652

—

C₂×Q₁₆×C₅₉

—

C₂×D₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

#127×C₅₉

—

#102×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#135

yes

—

#12511

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#493

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#495

yes

—

C₄×C₅₉

#508

yes

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

C₈×C₁₆

—

C₂×C₅₉

—

C₂×C₅₉

#1709

yes

characteristic

C₂×C₅₉

#2249

yes

—

#500

—

#494

C₂×D₅₉

yes

characteristic

#493

C₂×D₅₉

yes

characteristic

#487

—

118

—

#486

—

#358

—

118

—

#357

—

118

—

C₄×C₁₆

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12511

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

#127

—

C₂×C₁₆

—

#107

—

118

—

#107

—

118

—

#103

—

236

—

#102

—

C₄×C₈

C₂²⋊D₅₉

yes

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

236

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#11745669

yes

—

C₂×C₈

D₄×D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

100

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

101

C₁₆

—

118

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

472

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

472

—

109

C₂×C₄

#14273723

yes

characteristic

110

C₂×C₄

C₂×#11745669

yes

—

111

C₈

—

112

C₈

#14273746

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

#16837778

yes

characteristic

116

C₄

—

117

C₄

—

944

—

118

C₄

#16837785

yes

—

119

C₄

#16837832

yes

—

120

C₂

—

121

C₂

#21423701

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×#1709	C₂
socle	C₂²×C₅₉	#493
prime radical	C₅₉×#493	C₂²
hypercenter	#1709	D₅₉
hypocenter	C₅₉	#12511
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16837778

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1709

C₂

socle

C₂²×C₅₉

#493

prime radical

C₅₉×#493

C₂²

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #21426323	C₂	semidirect
3. #21423706	C₂	semidirect
4. C₈×C₅₉⋊C₁₆	C₂	semidirect
76. C₅₉	#12511	semidirect

H ≅ G/N

type

1. #21426323

C₂

semidirect

3. #21423706

C₂

semidirect

4. C₈×C₅₉⋊C₁₆

C₂

semidirect

76. C₅₉

#12511

semidirect

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

subgroup	index
#21426323	2
#21426319	2
#21423706	2
C₈×C₅₉⋊C₁₆	2
C₅₉×#1709	2
#12511	59

subgroup

index

#21426323

#21426319

#21423706

C₈×C₅₉⋊C₁₆

C₅₉×#1709

#12511

Subgroups — #28503382

Conjugacy classes of subgroups

Complements

Maximal subgroups