Group #28482493 — subgroups

Subgroups — #28482493

Order 15104.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	#21424268	C₂	1	2	yes	maximal
2	#21422855	C₂	1	4	yes	maximal
3	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273663	—	2	4	no	—
5	#16837426	C₂²	1	4	yes	—
6	#16837155	—	2	8	no	—
7	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₈²×C₅₉	C₄	1	1	yes	characteristic
10	C₈²×C₅₉	C₄	1	2	yes	—
11	C₈²×C₅₉	C₄	1	1	yes	characteristic
12	C₂×#11745664	—	2	4	no	—
13	#14273663	—	4	8	no	—
14	C₄⋊(C₅₉⋊C₈)	—	2	8	no	—
15	C₂²×C₈×C₅₉	D₄	1	2	yes	—
16	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
17	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₉	D₄	1	4	yes	—
23	C₄×C₈×C₅₉	Q₈	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
26	C₂×C₈×C₅₉	—	2	8	no	—
27	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₅₉	D₈	1	4	yes	—
32	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
33	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₅₉	C₂×D₄	1	2	yes	—
35	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
36	#11745664	—	4	8	no	—
37	C₂×C₅₉⋊C₈	—	4	4	no	—
38	C₂³×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅₉	—	2	4	no	—
40	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
44	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
46	C₈×C₅₉	—	2	4	no	—
47	C₈×C₅₉	#107	1	4	yes	—
48	C₈×C₅₉	#107	1	4	yes	—
49	C₈×C₅₉	#108	1	8	yes	complemented
50	C₅₉⋊C₈	—	8	4	no	—
51	#8823	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	#384	1	1	yes	characteristic
54	C₂²×C₅₉	#500	1	2	yes	—
55	C₄×C₅₉	—	2	2	no	—
56	C₄×C₅₉	C₂×#107	1	2	yes	—
57	C₄×C₅₉	C₂×#108	1	4	yes	—
58	C₄×C₅₉	#493	1	1	yes	characteristic
59	C₄×C₅₉	#494	1	1	yes	characteristic
60	#1848	—	59	2	no	—
61	#1567	—	59	4	no	—
62	C₂×C₈²	D₅₉	1	1	yes	characteristic
63	C₂×C₅₉	—	2	2	no	—
64	C₂×C₅₉	#1845	1	1	yes	characteristic
65	C₂×C₅₉	#1848	1	2	yes	—
66	C₂×#108	—	118	4	no	—
67	#423	—	59	4	no	—
68	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
70	#335	—	118	8	no	—
71	C₈²	Dic₅₉	1	1	yes	characteristic
72	C₈²	Dic₅₉	1	2	yes	—
73	C₈²	Dic₅₉	1	1	yes	characteristic
74	C₅₉	#8823	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	118	4	no	—
76	C₂²×C₈	Dic₁₁₈	1	4	yes	—
77	C₂²×C₈	D₂₃₆	1	2	yes	—
78	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
79	#108	—	236	8	no	—
80	C₄⋊C₈	—	118	8	no	—
81	C₄×C₈	Dic₁₁₈	1	2	yes	—
82	C₄×C₈	D₂₃₆	1	4	yes	—
83	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
84	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
87	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
88	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
89	M₁₆	—	236	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	236	4	no	—
92	C₂×C₈	D₄₇₂	1	4	yes	—
93	C₂×C₈	Dic₂₃₆	1	4	yes	—
94	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
95	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
97	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
98	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
99	C₄²	C₂×D₂₃₆	1	2	yes	—
100	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
101	C₂³	#14273727	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
105	C₂×C₄	#14273724	1	1	yes	characteristic
106	C₂×C₄	#14273727	1	2	yes	—
107	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	472	4	no	—
111	C₈	#14273661	1	4	yes	—
112	C₈	#14273661	1	4	yes	—
113	C₈	#14273663	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#16837272	1	1	yes	characteristic
116	C₂²	#16837807	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#14273661	1	2	yes	—
119	C₄	C₂×#14273663	1	4	yes	—
120	C₄	#16837779	1	1	yes	characteristic
121	C₄	#16837782	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#21424256	1	1	yes	characteristic
124	C₂	#21424268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal

#21422855

C₂

yes

maximal

C₂×C₈²×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273663

—

#16837426

C₂²

yes

—

#16837155

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₈²×C₅₉

C₄

yes

characteristic

C₈²×C₅₉

C₄

yes

—

C₈²×C₅₉

C₄

yes

characteristic

C₂×#11745664

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

Q₈

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

#107

yes

—

C₈×C₅₉

#107

yes

—

C₈×C₅₉

#108

yes

complemented

C₅₉⋊C₈

—

#8823

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#384

yes

characteristic

C₂²×C₅₉

#500

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#107

yes

—

C₄×C₅₉

C₂×#108

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

—

C₂×C₈²

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#1845

yes

characteristic

C₂×C₅₉

#1848

yes

—

C₂×#108

—

118

—

#423

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

—

#335

—

118

—

C₈²

Dic₅₉

yes

characteristic

C₈²

Dic₅₉

yes

—

C₈²

Dic₅₉

yes

characteristic

C₅₉

#8823

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

Dic₁₁₈

yes

—

C₂²×C₈

D₂₃₆

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#108

—

236

—

C₄⋊C₈

—

118

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

D₂₃₆

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₄×C₈

C₂×Dic₅₉

yes

characteristic

C₄×C₈

C₂×Dic₅₉

yes

characteristic

C₄×C₈

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₁₁₈

yes

—

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

D₄₇₂

yes

—

C₂×C₈

Dic₂₃₆

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₄²

C₂×Dic₁₁₈

yes

characteristic

C₄²

C₂×D₂₃₆

yes

—

100

C₄²

C₂²×Dic₅₉

yes

characteristic

101

C₂³

#14273727

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

104

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

—

105

C₂×C₄

#14273724

yes

characteristic

106

C₂×C₄

#14273727

yes

—

107

C₂×C₄

C₂×D₄₇₂

yes

—

108

C₂×C₄

C₂×Dic₂₃₆

yes

—

109

C₈

—

110

C₈

—

472

—

111

C₈

#14273661

yes

—

112

C₈

#14273661

yes

—

113

C₈

#14273663

yes

complemented

114

C₂²

—

115

C₂²

#16837272

yes

characteristic

116

C₂²

#16837807

yes

—

117

C₄

—

118

C₄

C₂×#14273661

yes

—

119

C₄

C₂×#14273663

yes

—

120

C₄

#16837779

yes

characteristic

121

C₄

#16837782

yes

characteristic

122

C₂

—

123

C₂

#21424256

yes

characteristic

124

C₂

#21424268

yes

—

	subgroup	G/H
center	C₂×C₄	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle	C₂²×C₅₉	#384
prime radical	C₂³×C₅₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	#8823
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#14273724

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₈²×C₅₉

C₂

socle

C₂²×C₅₉

#384

prime radical

C₂³×C₅₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₉

C₂³

hypercenter

C₂×C₈²

D₅₉

hypocenter

C₅₉

#8823

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
49. C₈×C₅₉	#108	semidirect
74. C₅₉	#8823	semidirect
113. C₈	#14273663	semidirect

H ≅ G/N

type

49. C₈×C₅₉

#108

semidirect

74. C₅₉

#8823

semidirect

113. C₈

#14273663

semidirect

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

subgroup	index
#21424268	2
#21422855	2
C₂×C₈²×C₅₉	2
#8823	59

subgroup

index

#21424268

#21422855

C₂×C₈²×C₅₉

#8823

Subgroups — #28482493

Conjugacy classes of subgroups

Complements

Maximal subgroups