Group #28493371 — subgroups

Subgroups — #28493371

Order 15104.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 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	#21422887	C₂	1	3	yes	maximal, complemented
2	#21422765	C₂	1	3	yes	maximal
3	C₅₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14273724	C₂²	1	1	yes	characteristic
5	#16837455	C₂²	1	3	yes	—
6	#16837154	—	2	3	no	—
7	#16837152	—	2	3	no	—
8	#16837145	—	2	3	no	—
9	#16837145	—	2	3	no	—
10	C₅₉×#423	C₂²	1	3	yes	—
11	C₂²×D₂₃₆	—	2	3	no	—
12	#14273724	—	2	3	no	—
13	#14273724	Q₈	1	2	yes	—
14	C₄⋊(C₅₉⋊C₈)	D₄	1	3	yes	—
15	C₄⋊(C₅₉⋊C₈)	D₄	1	3	yes	—
16	C₂²⋊(C₅₉⋊C₈)	—	2	3	no	—
17	C₄×C₅₉⋊C₈	—	2	3	no	—
18	C₂×M₁₆×C₅₉	—	2	3	no	—
19	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₅₉	—	2	3	no	—
21	C₄⋊C₈×C₅₉	—	2	3	no	—
22	C₂³×D₅₉	—	4	1	no	—
23	C₂²×C₄×C₅₉	#41	1	3	yes	—
24	C₂×D₂₃₆	—	2	6	no	—
25	C₂×D₂₃₆	—	4	3	no	—
26	C₂×D₂₃₆	—	8	3	no	—
27	C₂×D₂₃₆	—	8	3	no	—
28	M₁₆×C₅₉	—	8	3	no	—
29	C₂×C₈×C₅₉	—	4	3	no	—
30	C₄²×C₅₉	C₂×D₄	1	3	yes	—
31	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₅₉⋊C₈	—	2	3	no	—
33	C₂×C₅₉⋊C₈	—	2	3	no	—
34	C₂³×C₅₉	#126	1	1	yes	characteristic
35	C₂²×D₅₉	—	4	2	no	—
36	C₂²×D₅₉	—	8	6	no	—
37	C₂×C₄×C₅₉	—	2	3	no	—
38	C₂×C₄×C₅₉	—	2	3	no	—
39	C₂×C₄×C₅₉	#121	1	3	yes	—
40	C₂×C₄×C₅₉	C₂²⋊Q₈	1	3	yes	—
41	D₂₃₆	—	8	6	no	—
42	D₂₃₆	—	8	3	no	—
43	D₂₃₆	—	8	3	no	—
44	C₈×C₅₉	—	8	3	no	—
45	C₅₉⋊C₈	—	4	3	no	—
46	C₅₉⋊C₈	—	4	3	no	—
47	#10611	—	59	1	no	maximal, Sylow
48	C₂²×C₅₉	—	2	3	no	—
49	C₂²×C₅₉	#396	1	1	yes	characteristic
50	C₂×D₅₉	—	8	6	no	—
51	C₂×D₅₉	—	16	4	no	—
52	C₄×C₅₉	—	2	3	no	—
53	C₄×C₅₉	#471	1	3	yes	—
54	C₄×C₅₉	#472	1	3	yes	—
55	#1574	—	59	3	no	—
56	#1545	—	59	3	no	—
57	#1309	D₅₉	1	1	yes	characteristic, complemented
58	C₂×C₅₉	—	4	1	no	—
59	C₂×C₅₉	#2063	1	3	yes	—
60	D₅₉	—	16	2	no	—
61	C₂×#127	—	59	1	no	—
62	#431	—	59	3	no	—
63	#423	C₂×D₅₉	1	3	yes	—
64	#335	—	118	3	no	—
65	#334	—	118	3	no	—
66	#331	—	118	3	no	—
67	#331	—	118	3	no	—
68	C₅₉	#10611	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	118	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	59	2	no	—
72	#127	—	118	3	no	—
73	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	59	3	no	—
77	C₄⋊C₈	—	59	3	no	—
78	C₂²⋊C₈	—	118	3	no	—
79	C₄×C₈	—	118	3	no	—
80	C₂⁴	—	236	1	no	—
81	C₂×D₄	—	118	6	no	—
82	C₂×D₄	—	472	3	no	—
83	C₂×D₄	—	472	3	no	—
84	C₂×D₄	—	236	3	no	—
85	C₂²×C₄	#11745685	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	118	3	no	—
89	C₂×C₈	—	118	3	no	—
90	C₄²	D₄×D₅₉	1	3	yes	—
91	C₄²	Q₈×D₅₉	1	1	yes	characteristic
92	C₂³	—	472	6	no	—
93	C₂³	—	236	2	no	—
94	C₂³	#14273721	1	1	yes	characteristic
95	D₄	—	472	6	no	—
96	D₄	—	472	3	no	—
97	D₄	—	472	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#14273700	1	3	yes	—
101	C₂×C₄	#14273705	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	236	3	no	—
104	C₈	—	236	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	944	4	no	—
107	C₂²	—	472	6	no	—
108	C₂²	#16837326	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#16837668	1	3	yes	—
111	C₄	#16837671	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	944	2	no	—
114	C₂	#21425457	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21422887

C₂

yes

maximal, complemented

#21422765

C₂

yes

maximal

C₅₉×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#14273724

C₂²

yes

characteristic

#16837455

C₂²

yes

—

—

—

—

—

C₅₉×#423

C₂²

yes

—

C₂²×D₂₃₆

—

#14273724

—

#14273724

Q₈

yes

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

#41

yes

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#126

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10611

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#396

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#471

yes

—

C₄×C₅₉

#472

yes

—

#1574

—

#1545

—

#1309

D₅₉

yes

characteristic, complemented

C₂×C₅₉

—

C₂×C₅₉

#2063

yes

—

D₅₉

—

C₂×#127

—

#431

—

#423

C₂×D₅₉

yes

—

#335

—

118

—

#334

—

118

—

#331

—

118

—

#331

—

118

—

C₅₉

#10611

yes

characteristic, complemented, Sylow

C₂²×D₄

—

118

—

C₂×M₁₆

—

#127

—

#127

—

118

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

118

—

C₄×C₈

—

118

—

C₂⁴

—

236

—

C₂×D₄

—

118

—

C₂×D₄

—

472

—

C₂×D₄

—

472

—

C₂×D₄

—

236

—

C₂²×C₄

#11745685

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄²

D₄×D₅₉

yes

—

C₄²

Q₈×D₅₉

yes

characteristic

C₂³

—

472

—

C₂³

—

236

—

C₂³

#14273721

yes

characteristic

D₄

—

472

—

D₄

—

472

—

D₄

—

472

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#14273700

yes

—

101

C₂×C₄

#14273705

yes

—

102

C₈

—

103

C₈

—

236

—

104

C₈

—

236

—

105

C₂²

—

106

C₂²

—

944

—

107

C₂²

—

472

—

108

C₂²

#16837326

yes

characteristic

109

C₄

—

110

C₄

#16837668

yes

—

111

C₄

#16837671

yes

—

112

C₂

—

113

C₂

—

944

—

114

C₂

#21425457

yes

—

	subgroup	G/H
center	C₂²	#16837326
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1309	C₂
socle	C₂²×C₅₉	#396
prime radical	C₂×#14273724	C₂²
hypercenter	#1309	D₅₉
hypocenter	C₅₉	#10611
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#16837326

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1309

C₂

C₂²×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #21422887	C₂	semidirect
3. C₅₉×#1309	C₂	semidirect
57. #1309	D₅₉	semidirect
68. C₅₉	#10611	semidirect

H ≅ G/N

type

1. #21422887

C₂

semidirect

3. C₅₉×#1309

C₂

semidirect

57. #1309

D₅₉

semidirect

68. C₅₉

#10611

semidirect

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

subgroup	index
#21422887	2
#21422765	2
C₅₉×#1309	2
#10611	59

subgroup

index

#21422887

#21422765

C₅₉×#1309

#10611

Subgroups — #28493371

Conjugacy classes of subgroups

Complements

Maximal subgroups