Group #28482445 — subgroups

Subgroups — #28482445

Order 15104.

402 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 211 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	#21424248	C₂	1	2	yes	maximal
2	C₂×#16837155	C₂	1	4	yes	maximal
3	C₅₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
5	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
6	#16837155	—	2	8	no	—
7	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
8	C₄³×C₅₉	C₂²	1	1	yes	characteristic
9	C₅₉×#336	C₄	1	4	yes	—
10	C₂²×C₅₉⋊C₈	—	2	4	no	—
11	C₄⋊(C₅₉⋊C₈)	—	2	8	no	—
12	C₄⋊(C₅₉⋊C₈)	—	4	8	no	—
13	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₉	D₄	1	2	yes	—
16	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
17	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₅₉	D₄	1	4	yes	—
19	C₄×C₈×C₅₉	Q₈	1	4	yes	—
20	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
21	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₈×C₅₉	—	2	8	no	—
25	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
26	C₄²×C₅₉	—	2	4	no	—
27	C₄²×C₅₉	—	4	2	no	—
28	C₄²×C₅₉	D₈	1	4	yes	—
29	C₄²×C₅₉	Q₁₆	1	4	yes	—
30	C₄²×C₅₉	C₂×D₄	1	2	yes	—
31	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
32	C₂×C₅₉⋊C₈	—	4	4	no	—
33	C₂×C₅₉⋊C₈	—	4	8	no	—
34	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	—	2	4	no	—
36	C₂×C₄×C₅₉	—	2	4	no	—
37	C₂×C₄×C₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	#107	1	8	yes	—
39	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
42	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#131	1	4	yes	—
44	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
46	C₈×C₅₉	—	2	8	no	—
47	C₅₉⋊C₈	—	8	4	no	—
48	#8810	—	59	1	no	maximal, Sylow
49	C₂²×C₅₉	#384	1	1	yes	characteristic
50	C₂²×C₅₉	#424	1	2	yes	—
51	C₂²×C₅₉	C₂×#107	1	2	yes	—
52	C₂²×C₅₉	#430	1	2	yes	—
53	C₄×C₅₉	—	2	8	no	—
54	C₄×C₅₉	—	2	2	no	—
55	C₄×C₅₉	#424	1	4	yes	—
56	C₄×C₅₉	#493	1	1	yes	characteristic
57	C₄×C₅₉	#494	1	1	yes	characteristic
58	C₄×C₅₉	#500	1	2	yes	—
59	#1842	—	59	2	no	—
60	#1744	D₅₉	1	1	yes	characteristic
61	C₂×#335	—	59	4	no	—
62	C₂×C₅₉	#1571	1	4	yes	—
63	C₂×C₅₉	#1843	1	1	yes	characteristic
64	C₂×C₅₉	#1845	1	1	yes	characteristic
65	C₂×C₅₉	#1849	1	1	yes	characteristic
66	C₂×C₄⋊C₈	—	59	4	no	—
67	C₂×C₄⋊C₈	—	118	4	no	—
68	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
69	C₄³	C₂×D₅₉	1	1	yes	characteristic
70	#336	Dic₅₉	1	4	yes	—
71	#335	—	118	8	no	—
72	C₅₉	#8810	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	118	4	no	—
74	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
75	C₂×C₄²	Dic₁₁₈	1	2	yes	—
76	C₂×C₄²	D₂₃₆	1	2	yes	—
77	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
78	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
79	C₄⋊C₈	—	118	8	no	—
80	C₄⋊C₈	—	236	8	no	—
81	C₄×C₈	Dic₁₁₈	1	4	yes	—
82	C₄×C₈	D₂₃₆	1	4	yes	—
83	C₂²×C₄	C₄⋊Dic₅₉	1	4	yes	—
84	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
86	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	236	4	no	—
89	C₂×C₈	—	236	8	no	—
90	C₂×C₈	C₄⋊Dic₅₉	1	8	yes	—
91	C₄²	—	2	4	no	—
92	C₄²	—	4	2	no	—
93	C₄²	D₄₇₂	1	4	yes	—
94	C₄²	Dic₂₃₆	1	4	yes	—
95	C₄²	C₂×Dic₁₁₈	1	2	yes	—
96	C₄²	C₂×D₂₃₆	1	2	yes	—
97	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	#14273661	1	8	yes	—
102	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
103	C₂×C₄	#14273724	1	1	yes	characteristic
104	C₂×C₄	#14273727	1	2	yes	—
105	C₂×C₄	#14273727	1	1	yes	characteristic
106	C₂×C₄	#14273734	1	4	yes	—
107	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
109	C₈	—	2	8	no	—
110	C₈	—	472	4	no	—
111	C₂²	#16837272	1	1	yes	characteristic
112	C₂²	#16837429	1	2	yes	—
113	C₂²	C₂×#14273661	1	2	yes	—
114	C₂²	#16837451	1	2	yes	—
115	C₄	—	2	8	no	—
116	C₄	—	2	2	no	—
117	C₄	#16837429	1	4	yes	—
118	C₄	#16837779	1	1	yes	characteristic
119	C₄	#16837782	1	1	yes	characteristic
120	C₄	#16837807	1	2	yes	—
121	C₂	#21422871	1	4	yes	—
122	C₂	#21424250	1	1	yes	characteristic
123	C₂	#21424256	1	1	yes	characteristic
124	C₂	#21424271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424248

C₂

yes

maximal

C₂×#16837155

C₂

yes

maximal

C₅₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#16837155

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

C₄

yes

—

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

Q₈

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

D₈

yes

—

C₄²×C₅₉

Q₁₆

yes

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×Q₈

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#107

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

C₂×#107

yes

—

C₂²×C₅₉

#430

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#424

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#1842

—

#1744

D₅₉

yes

characteristic

C₂×#335

—

C₂×C₅₉

#1571

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₄³

C₂×D₅₉

yes

characteristic

#336

Dic₅₉

yes

—

#335

—

118

—

C₅₉

#8810

yes

characteristic, complemented, Sylow

C₂²×C₈

—

118

—

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂×C₄²

Dic₁₁₈

yes

—

C₂×C₄²

D₂₃₆

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

C₄⋊C₈

—

236

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

D₂₃₆

yes

—

C₂²×C₄

C₄⋊Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

—

236

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₄²

—

C₄²

—

C₄²

D₄₇₂

yes

—

C₄²

Dic₂₃₆

yes

—

C₄²

C₂×Dic₁₁₈

yes

—

C₄²

C₂×D₂₃₆

yes

—

C₂³

C₂×C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#14273661

yes

—

102

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

—

103

C₂×C₄

#14273724

yes

characteristic

104

C₂×C₄

#14273727

yes

—

105

C₂×C₄

#14273727

yes

characteristic

106

C₂×C₄

#14273734

yes

—

107

C₂×C₄

C₂×D₄₇₂

yes

—

108

C₂×C₄

C₂×Dic₂₃₆

yes

—

109

C₈

—

110

C₈

—

472

—

111

C₂²

#16837272

yes

characteristic

112

C₂²

#16837429

yes

—

113

C₂²

C₂×#14273661

yes

—

114

C₂²

#16837451

yes

—

115

C₄

—

116

C₄

—

117

C₄

#16837429

yes

—

118

C₄

#16837779

yes

characteristic

119

C₄

#16837782

yes

characteristic

120

C₄

#16837807

yes

—

121

C₂

#21422871

yes

—

122

C₂

#21424250

yes

characteristic

123

C₂

#21424256

yes

characteristic

124

C₂

#21424271

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1744	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1744	D₅₉
hypocenter	C₅₉	#8810
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₃₆

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1744

C₂

socle, prime radical

C₂³×C₅₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
72. C₅₉	#8810	semidirect

H ≅ G/N

type

72. C₅₉

#8810

semidirect

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

subgroup	index
#21424248	2
C₂×#16837155	2
C₅₉×#1744	2
#8810	59

subgroup

index

#21424248

C₂×#16837155

C₅₉×#1744

#8810

Subgroups — #28482445

Conjugacy classes of subgroups

Complements

Maximal subgroups