Group #20274832 — subgroups

Subgroups — #20274832

Order 6784.

260 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×C₄⋊#88322	C₂	1	2	yes	maximal
2	#16810378	C₂	1	4	yes	maximal
3	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₂²	1	4	yes	—
5	C₄⋊#88322	—	2	4	no	—
6	C₂×C₄²×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₄²×C₅₃	—	2	4	no	—
17	C₄²×C₅₃	—	2	2	no	—
18	C₄²×C₅₃	—	2	4	no	—
19	C₄²×C₅₃	C₈	1	4	yes	complemented
20	C₄²×C₅₃	D₄	1	2	yes	—
21	C₄²×C₅₃	Q₈	1	2	yes	—
22	C₂×#88322	—	2	4	no	—
23	C₂×#88322	—	2	8	no	—
24	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	—	2	2	no	—
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	—	2	2	no	—
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
33	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
35	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#41	1	4	yes	—
39	#88322	—	4	4	no	—
40	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₅₃	#117	1	2	yes	—
42	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
43	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
45	C₄×C₅₃	—	2	2	no	—
46	C₄×C₅₃	—	2	4	no	—
47	C₄×C₅₃	—	2	4	no	—
48	C₄×C₅₃	C₄⋊C₈	1	4	yes	—
49	C₄×C₅₃	#124	1	2	yes	—
50	C₄×C₅₃	#125	1	2	yes	—
51	#1831	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	#382	1	1	yes	characteristic
53	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₅₃	#423	1	1	yes	characteristic
55	C₂×C₅₃	#431	1	2	yes	—
56	C₂×C₅₃	#432	1	2	yes	—
57	C₂×C₄⋊C₈	—	53	2	no	—
58	C₄³	D₅₃	1	1	yes	characteristic
59	#336	—	53	4	no	—
60	C₅₃	#1831	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₈	—	53	4	no	—
62	C₂×C₄²	Dic₅₃	1	2	yes	—
63	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
64	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₅₃	1	2	yes	—
66	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
67	C₄⋊C₈	—	106	4	no	—
68	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
69	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
70	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
71	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
73	C₂×C₈	—	106	4	no	—
74	C₂×C₈	—	106	8	no	—
75	C₄²	—	2	4	no	—
76	C₄²	—	2	4	no	—
77	C₄²	—	2	2	no	—
78	C₄²	#88322	1	4	yes	complemented
79	C₄²	Dic₁₀₆	1	2	yes	—
80	C₄²	D₂₁₂	1	2	yes	—
81	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₂×#88322	1	4	yes	—
88	C₂×C₄	C₂×#88322	1	2	yes	—
89	C₂×C₄	#11719917	1	4	yes	—
90	C₂×C₄	#11719917	1	2	yes	—
91	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
92	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
93	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
94	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
95	C₂×C₄	#11719945	1	4	yes	—
96	C₈	—	212	4	no	—
97	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
98	C₂²	#13133176	1	2	yes	—
99	C₂²	C₂²×#88322	1	1	yes	characteristic
100	C₂²	C₂×#11719917	1	1	yes	characteristic
101	C₂²	C₂×#11719917	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	4	no	—
104	C₄	—	2	4	no	—
105	C₄	C₄⋊#88322	1	4	yes	—
106	C₄	#13133205	1	2	yes	—
107	C₄	#13133209	1	2	yes	—
108	C₂	#16810489	1	1	yes	characteristic
109	C₂	C₂×C₄⋊#88322	1	1	yes	characteristic
110	C₂	#16810647	1	1	yes	characteristic
111	C₂	#16810676	1	2	yes	—
112	C₂	#16810680	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal

#16810378

C₂

yes

maximal

C₄³×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₈

yes

complemented

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

#125

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#432

yes

—

C₂×C₄⋊C₈

—

C₄³

D₅₃

yes

characteristic

#336

—

C₅₃

#1831

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

Dic₅₃

yes

characteristic

C₂×C₄²

Dic₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#88322

yes

complemented

C₄²

Dic₁₀₆

yes

—

C₄²

D₂₁₂

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#88322

yes

—

C₂×C₄

C₂×#88322

yes

—

C₂×C₄

#11719917

yes

—

C₂×C₄

#11719917

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719945

yes

—

C₈

—

212

—

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂²

#13133176

yes

—

C₂²

C₂²×#88322

yes

characteristic

100

C₂²

C₂×#11719917

yes

characteristic

101

C₂²

C₂×#11719917

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄⋊#88322

yes

—

106

C₄

#13133205

yes

—

107

C₄

#13133209

yes

—

108

C₂

#16810489

yes

characteristic

109

C₂

C₂×C₄⋊#88322

yes

characteristic

110

C₂

#16810647

yes

characteristic

111

C₂

#16810676

yes

—

112

C₂

#16810680

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#1831
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₄³×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₄³

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
19. C₄²×C₅₃	C₈	semidirect
60. C₅₃	#1831	semidirect
78. C₄²	#88322	semidirect

H ≅ G/N

type

19. C₄²×C₅₃

C₈

semidirect

60. C₅₃

#1831

semidirect

78. C₄²

#88322

semidirect

Subgroups — #20274832

Conjugacy classes of subgroups

Complements