Group #28493739 — subgroups

Subgroups — #28493739

Order 15104.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#21424902	C₂	1	3	yes	maximal
2	#21424256	C₂	1	3	yes	maximal, complemented
3	C₅₉×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14273724	C₂²	1	1	yes	characteristic
5	C₂×#14273661	—	2	6	no	—
6	C₂×#14273649	—	2	6	no	—
7	#16837272	C₂²	1	3	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
9	C₂²×D₂₃₆	—	2	3	no	—
10	#14273724	—	2	3	no	—
11	#14273724	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₅₉	—	2	3	no	—
13	C₂×C₄⋊Dic₅₉	—	2	6	no	—
14	#14273661	—	4	12	no	—
15	#14273649	—	4	12	no	—
16	C₂²×C₈×C₅₉	D₄	1	6	yes	—
17	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₅₉	—	2	6	no	—
19	C₂³×D₅₉	—	4	1	no	—
20	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
21	C₂²×Dic₅₉	—	4	3	no	—
22	C₂×D₂₃₆	—	2	6	no	—
23	C₂×D₂₃₆	—	4	6	no	—
24	C₂×D₂₃₆	—	4	6	no	—
25	C₂×D₂₃₆	—	4	3	no	—
26	C₂×C₈×C₅₉	—	2	6	no	—
27	C₂×C₈×C₅₉	—	2	6	no	—
28	C₂×C₈×C₅₉	D₈	1	12	yes	—
29	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₅₉	#41	1	3	yes	—
31	C₄⋊Dic₅₉	—	4	6	no	—
32	C₄⋊Dic₅₉	—	4	12	no	—
33	C₂³×C₅₉	#127	1	1	yes	characteristic
34	C₂²×D₅₉	—	4	2	no	—
35	C₂²×D₅₉	—	8	3	no	—
36	C₂²×D₅₉	—	8	3	no	—
37	C₂×C₄×C₅₉	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₅₉	#123	1	3	yes	—
39	C₂×C₄×C₅₉	C₂×D₈	1	6	yes	—
40	C₂×C₄×C₅₉	#137	1	6	yes	—
41	C₂×Dic₅₉	—	8	3	no	—
42	C₂×Dic₅₉	—	8	6	no	—
43	D₂₃₆	—	4	12	no	—
44	D₂₃₆	—	8	6	no	—
45	C₈×C₅₉	—	2	12	no	—
46	#10673	—	59	1	no	maximal, Sylow
47	C₂²×C₅₉	#399	1	1	yes	characteristic
48	C₂²×C₅₉	#493	1	3	yes	—
49	C₂²×C₅₉	#497	1	3	yes	—
50	C₂×D₅₉	—	8	6	no	—
51	C₂×D₅₉	—	16	1	no	—
52	C₂×D₅₉	—	16	3	no	—
53	C₄×C₅₉	#474	1	6	yes	—
54	C₄×C₅₉	#480	1	6	yes	—
55	Dic₅₉	—	16	3	no	—
56	#1957	—	59	3	no	—
57	#1845	—	59	3	no	—
58	#1273	D₅₉	1	1	yes	characteristic, complemented
59	C₂×C₅₉	#1712	1	1	yes	characteristic
60	C₂×C₅₉	#1722	1	3	yes	—
61	C₂×C₅₉	#2093	1	3	yes	—
62	D₅₉	—	16	2	no	—
63	C₂×#127	—	59	1	no	—
64	C₂×#107	—	118	6	no	—
65	C₂×#102	—	118	6	no	—
66	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
67	#384	—	59	3	no	—
68	C₅₉	#10673	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	118	3	no	—
70	C₂²×C₈	D₂₃₆	1	6	yes	—
71	#127	—	59	2	no	—
72	#127	—	118	3	no	—
73	C₂×C₄⋊C₄	—	118	3	no	—
74	C₂×C₄⋊C₄	—	118	6	no	—
75	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
76	#107	—	236	12	no	—
77	#102	—	236	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	236	1	no	—
80	C₂×D₄	—	118	6	no	—
81	C₂×D₄	—	236	3	no	—
82	C₂×D₄	—	236	6	no	—
83	C₂×D₄	—	236	6	no	—
84	C₂²×C₄	—	236	3	no	—
85	C₂²×C₄	C₂×D₂₃₆	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	D₄₇₂	1	12	yes	—
89	C₄⋊C₄	—	236	6	no	—
90	C₄⋊C₄	—	236	12	no	—
91	C₄²	#11745687	1	3	yes	—
92	C₄²	Q₈×D₅₉	1	1	yes	characteristic
93	C₂³	—	472	3	no	—
94	C₂³	—	472	3	no	—
95	C₂³	—	236	2	no	—
96	C₂³	#14273724	1	1	yes	characteristic
97	D₄	—	472	6	no	—
98	D₄	—	236	12	no	—
99	C₂×C₄	—	472	3	no	—
100	C₂×C₄	—	472	6	no	—
101	C₂×C₄	#14273707	1	3	yes	—
102	C₂×C₄	#14273711	1	3	yes	—
103	C₂×C₄	C₂×D₄₇₂	1	6	yes	—
104	C₂×C₄	#14273760	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	944	1	no	—
107	C₂²	—	944	3	no	—
108	C₂²	—	472	6	no	—
109	C₂²	#16837339	1	1	yes	characteristic
110	C₂²	#16837779	1	3	yes	—
111	C₂²	#16837796	1	3	yes	—
112	C₄	—	944	3	no	—
113	C₄	#16837683	1	6	yes	—
114	C₄	#16837716	1	6	yes	—
115	C₂	—	944	2	no	—
116	C₂	#21423711	1	1	yes	characteristic
117	C₂	#21423767	1	3	yes	—
118	C₂	#21425623	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424902

C₂

yes

maximal

#21424256

C₂

yes

maximal, complemented

C₅₉×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14273724

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂²×D₂₃₆

—

#14273724

—

#14273724

Q₈

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273649

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

#127

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#497

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#474

yes

—

C₄×C₅₉

#480

yes

—

Dic₅₉

—

#1957

—

#1845

—

#1273

D₅₉

yes

characteristic, complemented

C₂×C₅₉

#1712

yes

characteristic

C₂×C₅₉

#1722

yes

—

C₂×C₅₉

#2093

yes

—

D₅₉

—

C₂×#127

—

C₂×#107

—

118

—

C₂×#102

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

#384

—

C₅₉

#10673

yes

characteristic, complemented, Sylow

C₂²×D₄

—

118

—

C₂²×C₈

D₂₃₆

yes

—

#127

—

#127

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#107

—

236

—

#102

—

236

—

C₄×C₈

—

C₂⁴

—

236

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

C₂²×C₄

—

236

—

C₂²×C₄

C₂×D₂₃₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄₇₂

yes

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

#11745687

yes

—

C₄²

Q₈×D₅₉

yes

characteristic

C₂³

—

472

—

C₂³

—

472

—

C₂³

—

236

—

C₂³

#14273724

yes

characteristic

D₄

—

472

—

D₄

—

236

—

C₂×C₄

—

472

—

100

C₂×C₄

—

472

—

101

C₂×C₄

#14273707

yes

—

102

C₂×C₄

#14273711

yes

—

103

C₂×C₄

C₂×D₄₇₂

yes

—

104

C₂×C₄

#14273760

yes

—

105

C₈

—

106

C₂²

—

944

—

107

C₂²

—

944

—

108

C₂²

—

472

—

109

C₂²

#16837339

yes

characteristic

110

C₂²

#16837779

yes

—

111

C₂²

#16837796

yes

—

112

C₄

—

944

—

113

C₄

#16837683

yes

—

114

C₄

#16837716

yes

—

115

C₂

—

944

—

116

C₂

#21423711

yes

characteristic

117

C₂

#21423767

yes

—

118

C₂

#21425623

yes

—

	subgroup	G/H
center	C₂³	#14273724
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1273	C₂
socle	C₂³×C₅₉	#127
prime radical	C₂×#14273724	C₂²
hypercenter	#1273	D₅₉
hypocenter	C₅₉	#10673
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14273724

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1273

C₂

C₂³×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #21424256	C₂	semidirect
3. C₅₉×#1273	C₂	semidirect
58. #1273	D₅₉	semidirect
68. C₅₉	#10673	semidirect

H ≅ G/N

type

2. #21424256

C₂

semidirect

3. C₅₉×#1273

C₂

semidirect

58. #1273

D₅₉

semidirect

68. C₅₉

#10673

semidirect

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

subgroup	index
#21424902	2
#21424256	2
C₅₉×#1273	2
#10673	59

subgroup

index

#21424902

#21424256

C₅₉×#1273

#10673

Subgroups — #28493739

Conjugacy classes of subgroups

Complements

Maximal subgroups