Group #20275709 — subgroups

Subgroups — #20275709

Order 6784.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16810647	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#16810392	C₂	1	4	yes	maximal
4	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133128	—	2	1	no	—
9	C₄×#88322	—	2	1	no	—
10	#125×C₅₃	—	2	1	no	—
11	#125×C₅₃	—	2	1	no	—
12	#125×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₅₃	C₄	1	4	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	—	2	4	no	—
20	C₄⋊C₄×C₅₃	—	2	4	no	—
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₄²×C₅₃	D₄	1	2	yes	—
26	#11719917	—	2	2	no	—
27	#11719917	—	2	4	no	—
28	#11719917	Q₈	1	4	yes	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	4	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	—	4	4	no	—
35	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₅₃	#41	1	2	yes	—
41	#88322	—	2	4	no	—
42	#88322	—	4	2	no	—
43	C₂²×C₅₃	—	2	2	no	—
44	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
46	C₄×C₅₃	—	2	2	no	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	4	4	no	—
49	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₅₃	#124	1	1	yes	characteristic
51	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
52	#1998	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	2	no	—
54	C₂×C₅₃	#391	1	1	yes	characteristic
55	C₂×C₅₃	#413	1	2	yes	—
56	C₂×#125	D₅₃	1	1	yes	characteristic
57	#423	—	53	1	no	—
58	C₄×M₁₆	—	53	1	no	—
59	#343	—	53	4	no	—
60	C₅₃	#1998	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	53	2	no	—
62	C₂×M₁₆	—	53	2	no	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	Dic₅₃	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
68	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
69	C₄⋊C₈	—	106	2	no	—
70	#97	—	106	1	no	—
71	C₄×C₈	—	106	1	no	—
72	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
73	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
75	M₁₆	—	53	4	no	—
76	M₁₆	—	106	2	no	—
77	M₁₆	—	106	4	no	—
78	C₂×C₈	—	106	2	no	—
79	C₂×C₈	—	106	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	4	no	—
83	C₄⋊C₄	—	2	4	no	—
84	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
85	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
86	C₄²	C₂²⋊D₅₃	1	2	yes	—
87	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
93	C₂×C₄	Q₈×D₅₃	1	2	yes	—
94	C₂×C₄	#11719949	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
97	C₈	—	106	4	no	—
98	C₈	—	212	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
101	C₂²	Q₈×Dic₅₃	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#13133193	1	2	yes	—
106	C₄	#13133206	1	1	yes	characteristic
107	C₄	#13133215	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#16810530	1	1	yes	characteristic
110	C₂	#16810607	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

#16810392

C₂

yes

maximal

C₂×#125×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

C₄⋊#88322

—

#13133128

—

C₄×#88322

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

#11719917

—

#11719917

—

#11719917

Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

#1998

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#391

yes

characteristic

C₂×C₅₃

#413

yes

—

C₂×#125

D₅₃

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₅₃

#1998

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

Q₈×D₅₃

yes

—

C₂×C₄

#11719949

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

106

—

C₈

—

212

—

C₂²

—

100

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

101

C₂²

Q₈×Dic₅₃

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#13133193

yes

—

106

C₄

#13133206

yes

characteristic

107

C₄

#13133215

yes

characteristic

108

C₂

—

109

C₂

#16810530

yes

characteristic

110

C₂

#16810607

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	#1998
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#125×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
60. C₅₃	#1998	semidirect

H ≅ G/N

type

60. C₅₃

#1998

semidirect

Subgroups — #20275709

Conjugacy classes of subgroups

Complements