Group #28298949 — subgroups

Subgroups — #28298949

Order 13568.

136 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 38 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	#20289661	C₂	1	2	yes	maximal
2	#20274853	C₂	1	1	yes	maximal, characteristic
3	#16811519	—	2	2	no	—
4	#16810647	C₄	1	1	yes	characteristic
5	#16810647	C₂²	1	1	yes	characteristic
6	#16810388	—	2	2	no	—
7	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
8	#13133291	—	4	2	no	—
9	C₂×#11719917	—	2	1	no	—
10	C₂×#11719917	D₄	1	2	yes	—
11	C₄⋊#88322	—	4	1	no	—
12	C₄⋊#88322	—	4	1	no	—
13	C₂²×C₈×C₅₃	—	2	1	no	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₈	1	2	yes	—
17	C₂²×C₄×C₅₃	—	2	1	no	—
18	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	4	1	no	—
21	C₂×C₈×C₅₃	—	4	1	no	—
22	C₄²×C₅₃	—	2	1	no	—
23	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
25	#11719917	—	4	2	no	—
26	#11719917	—	4	2	no	—
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	—	4	1	no	—
29	#11719910	—	8	2	no	—
30	C₂³×C₅₃	#101	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
38	C₈×C₅₃	—	4	2	no	—
39	#88322	—	4	2	no	—
40	#88322	—	8	1	no	—
41	#7368	—	53	1	no	maximal, Sylow
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	4	1	no	—
44	C₂²×C₅₃	#324	1	1	yes	characteristic
45	C₄×C₅₃	—	2	2	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	4	1	no	—
48	C₄×C₅₃	#354	1	1	yes	characteristic
49	C₄×C₅₃	#355	1	1	yes	characteristic
50	#1837	—	53	1	no	—
51	#1327	—	53	2	no	—
52	C₂×C₅₃	—	2	1	no	—
53	C₂×C₅₃	—	4	1	no	—
54	C₂×C₅₃	#1324	1	1	yes	characteristic
55	#423	—	53	1	no	—
56	#423	—	53	1	no	—
57	C₂×C₄×C₈	#6157	1	1	yes	characteristic
58	#349	—	106	2	no	—
59	#341	—	106	2	no	—
60	C₅₃	#7368	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	53	2	no	—
62	C₂×M₁₆	—	106	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
65	M₃₂	—	212	2	no	—
66	C₄⋊C₈	—	212	1	no	—
67	C₄⋊C₈	—	212	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	#88324	1	2	yes	—
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
72	M₁₆	—	212	2	no	—
73	M₁₆	—	212	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	106	2	no	—
78	C₂×C₈	—	212	1	no	—
79	C₄²	—	2	1	no	—
80	C₄²	C₂×#88324	1	1	yes	characteristic
81	C₄²	#11719940	1	1	yes	characteristic
82	C₁₆	—	424	2	no	—
83	C₂³	#13133278	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
90	C₂×C₄	#13133276	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	424	1	no	—
93	C₈	—	212	2	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	#16811468	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	2	1	no	—
99	C₄	—	4	1	no	—
100	C₄	#16811530	1	1	yes	characteristic
101	C₄	#16811532	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	4	1	no	—
104	C₂	#20289655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20289661

C₂

yes

maximal

#20274853

C₂

yes

maximal, characteristic

#16811519

—

#16810647

C₄

yes

characteristic

#16810647

C₂²

yes

characteristic

#16810388

—

C₂×C₄×C₈×C₅₃

C₄

yes

—

—

D₄

yes

—

C₄⋊#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

—

—

—

—

—

C₂³×C₅₃

#101

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

#99

yes

characteristic

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#324

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

C₂×C₄×C₈

#6157

yes

characteristic

#349

—

106

—

#341

—

106

—

C₅₃

#7368

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄²

C₂×#6157

yes

characteristic

M₃₂

—

212

—

C₄⋊C₈

—

212

—

C₄⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

#88324

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊#6157

yes

characteristic

M₁₆

—

212

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₄²

—

C₄²

yes

C₄²

yes

C₁₆

—

424

—

C₂³

#13133278

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

424

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

#16811468

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

#16811530

yes

characteristic

101

C₄

#16811532

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#20289655

yes

characteristic

	subgroup	G/H
center	C₄	#16811530
commutator	C₄²×C₅₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂×C₅₃	#1324
prime radical	C₂³×C₅₃	#101
cosocle	#16810647	C₂²
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#7368
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄

#16811530

commutator

C₄²×C₅₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

C₂×C₅₃

C₂³×C₅₃

C₂²

C₂×C₄×C₈

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
60. C₅₃	#7368	semidirect

H ≅ G/N

type

60. C₅₃

#7368

semidirect

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

subgroup	index
#20289661	2
#20274853	2
#7368	53

subgroup

index

#20289661

#20274853

#7368

Subgroups — #28298949

Conjugacy classes of subgroups

Complements

Maximal subgroups