Group #28503361 — subgroups

Subgroups — #28503361

Order 15104.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21426378	C₂	1	1	yes	maximal, characteristic
2	#21426323	C₂	1	2	yes	maximal
3	#21423690	C₂	1	2	yes	maximal
4	#21422012	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#2247	C₂	1	1	yes	maximal, characteristic
6	#16837807	C₂²	1	2	yes	—
7	#16837798	—	2	2	no	—
8	#16837754	—	2	2	no	—
9	#16837230	—	2	2	no	—
10	#16837206	—	2	4	no	—
11	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₅₉	—	2	2	no	—
14	C₅₉×#494	C₂²	1	2	yes	—
15	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	#14273727	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
18	#14273661	—	2	2	no	—
19	#14273661	—	2	2	no	—
20	#14273658	—	4	4	no	—
21	#14273650	—	4	4	no	—
22	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₅₉	—	2	2	no	—
24	C₂×Q₁₆×C₅₉	—	2	2	no	—
25	C₄⋊Q₈×C₅₉	—	2	2	no	—
26	Q₃₂×C₅₉	—	4	4	no	—
27	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
28	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₅₉	—	4	2	no	—
30	C₂×Dic₁₁₈	—	4	2	no	—
31	Q₁₆×C₅₉	—	4	4	no	—
32	Q₁₆×C₅₉	—	4	4	no	—
33	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	—	4	2	no	—
36	C₄⋊C₄×C₅₉	—	4	2	no	—
37	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₅₉	—	4	2	no	—
39	C₄⋊Dic₅₉	—	4	2	no	—
40	C₂×C₅₉⋊C₈	D₈	1	4	yes	—
41	C₁₆×C₅₉	SD₁₆	1	4	yes	—
42	C₅₉⋊C₁₆	—	2	2	no	—
43	Q₈×C₅₉	—	8	2	no	—
44	Q₈×C₅₉	—	8	2	no	—
45	C₂×C₄×C₅₉	—	8	2	no	—
46	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
48	C₂×Dic₅₉	—	8	2	no	—
49	Dic₁₁₈	—	8	4	no	—
50	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
51	C₈×C₅₉	#135	1	2	yes	—
52	C₅₉⋊C₈	—	2	4	no	—
53	#12507	—	59	1	no	maximal, Sylow
54	C₂²×C₅₉	#493	1	1	yes	characteristic
55	C₄×C₅₉	—	16	2	no	—
56	C₄×C₅₉	#492	1	1	yes	characteristic
57	C₄×C₅₉	#495	1	1	yes	characteristic
58	C₄×C₅₉	#510	1	4	yes	—
59	Dic₅₉	—	16	2	no	—
60	#2254	—	59	1	no	—
61	#2247	D₅₉	1	1	yes	characteristic
62	#2241	—	59	2	no	—
63	#1707	—	59	2	no	—
64	#1311	—	59	1	no	—
65	C₂×C₅₉	#1706	1	1	yes	characteristic
66	C₂×C₅₉	#2251	1	2	yes	—
67	C₂×Q₃₂	—	2	2	no	—
68	#500	—	59	2	no	—
69	#498	—	118	2	no	—
70	#494	C₂×D₅₉	1	2	yes	—
71	#487	—	118	2	no	—
72	#367	—	118	2	no	—
73	#358	—	118	4	no	—
74	C₄×C₁₆	—	59	1	no	—
75	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
76	C₈²	—	59	1	no	—
77	C₅₉	#12507	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	118	2	no	—
82	Q₃₂	—	4	4	no	—
83	C₂×C₁₆	—	59	2	no	—
84	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
85	#107	—	118	2	no	—
86	#107	—	118	2	no	—
87	#106	—	236	4	no	—
88	#103	—	236	4	no	—
89	C₄×C₈	—	59	2	no	—
90	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	236	2	no	—
93	Q₁₆	—	4	4	no	—
94	Q₁₆	—	4	4	no	—
95	C₂×C₈	—	59	4	no	—
96	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
97	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	236	2	no	—
101	C₄⋊C₄	—	236	2	no	—
102	C₄²	D₄×D₅₉	1	1	yes	characteristic
103	C₁₆	—	118	2	no	—
104	C₁₆	#11745670	1	4	yes	—
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	472	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	472	2	no	—
110	C₂×C₄	#14273723	1	1	yes	characteristic
111	C₂×C₄	D₈×D₅₉	1	2	yes	—
112	C₈	—	118	4	no	—
113	C₈	C₂×#11745670	1	2	yes	—
114	C₈	#14273746	1	2	yes	—
115	C₂²	#16837777	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	944	2	no	—
118	C₄	#16837774	1	1	yes	characteristic
119	C₄	#16837785	1	1	yes	characteristic
120	C₄	#16837844	1	4	yes	—
121	C₂	#21423685	1	1	yes	characteristic
122	C₂	#21426364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21426378

C₂

yes

maximal, characteristic

#21426323

C₂

yes

maximal

#21423690

C₂

yes

maximal

#21422012

C₂

yes

maximal, characteristic

C₅₉×#2247

C₂

yes

maximal, characteristic

#16837807

C₂²

yes

—

—

—

—

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₅₉

—

C₅₉×#494

C₂²

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

#14273727

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

—

—

—

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

—

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

D₈

yes

—

C₁₆×C₅₉

SD₁₆

yes

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

#135

yes

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

#2254

—

#2247

D₅₉

yes

—

—

—

C₂×C₅₉

#1706

yes

characteristic

C₂×C₅₉

#2251

yes

—

C₂×Q₃₂

—

#500

—

#498

—

118

—

#494

C₂×D₅₉

yes

—

#487

—

118

—

#367

—

118

—

#358

—

118

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₈²

—

C₅₉

#12507

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₉

yes

—

#107

—

118

—

#107

—

118

—

#106

—

236

—

#103

—

236

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

236

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₅₉

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

236

—

101

C₄⋊C₄

—

236

—

102

C₄²

D₄×D₅₉

yes

characteristic

103

C₁₆

—

118

—

104

C₁₆

#11745670

yes

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

472

—

108

C₂×C₄

—

109

C₂×C₄

—

472

—

110

C₂×C₄

#14273723

yes

characteristic

111

C₂×C₄

D₈×D₅₉

yes

—

112

C₈

—

118

—

113

C₈

C₂×#11745670

yes

—

114

C₈

#14273746

yes

—

115

C₂²

#16837777

yes

characteristic

116

C₄

—

117

C₄

—

944

—

118

C₄

#16837774

yes

characteristic

119

C₄

#16837785

yes

characteristic

120

C₄

#16837844

yes

—

121

C₂

#21423685

yes

characteristic

122

C₂

#21426364

yes

—

	subgroup	G/H
center	C₂²	#16837777
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2247	C₂
socle, prime radical	C₂²×C₅₉	#493
hypercenter	#2247	D₅₉
hypocenter	C₅₉	#12507
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16837777

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#2247

C₂

C₂²×C₅₉

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
77. C₅₉	#12507	semidirect

H ≅ G/N

type

77. C₅₉

#12507

semidirect

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

subgroup	index
#21426378	2
#21426323	2
#21423690	2
#21422012	2
C₅₉×#2247	2
#12507	59

subgroup

index

C₅₉×#2247

Subgroups — #28503361

Conjugacy classes of subgroups

Complements

Maximal subgroups