Group #20274572 — subgroups

Subgroups — #20274572

Order 6784.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	#16810378	C₂	1	2	yes	maximal
5	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	2	yes	—
8	C₄⋊#88322	C₄	1	8	yes	complemented
9	#13133128	—	2	2	no	—
10	C₄×#88322	—	2	2	no	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#95×C₅₃	C₄	1	4	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₄²×C₅₃	—	2	2	no	—
20	C₄²×C₅₃	C₂×C₄	1	4	yes	—
21	C₄²×C₅₃	D₄	1	2	yes	—
22	C₄²×C₅₃	Q₈	1	2	yes	—
23	C₂×#88322	—	2	2	no	—
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	2	2	no	—
27	C₂×#88322	C₂×C₄	1	4	yes	—
28	C₂×#88322	C₂×C₄	1	4	yes	—
29	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	4	no	—
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	C₄²	1	4	yes	—
35	C₂×C₄×C₅₃	C₄⋊C₄	1	4	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×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#41	1	2	yes	—
40	#88322	—	2	4	no	—
41	#88322	—	2	4	no	—
42	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₅₃	#117	1	1	yes	characteristic
45	C₂²×C₅₃	#131	1	4	yes	—
46	C₄×C₅₃	—	2	4	no	—
47	C₄×C₅₃	—	2	4	no	—
48	C₄×C₅₃	—	4	2	no	—
49	C₄×C₅₃	C₄×D₄	1	2	yes	—
50	C₄×C₅₃	C₄×Q₈	1	2	yes	—
51	#1768	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₅₃	#405	1	2	yes	—
54	C₂×C₅₃	#424	1	2	yes	—
55	C₂×C₅₃	#433	1	2	yes	—
56	C₂×C₄⋊C₈	—	53	2	no	—
57	C₂×#97	—	53	1	no	—
58	C₂×C₄×C₈	—	53	1	no	—
59	#376	D₅₃	1	1	yes	characteristic
60	#336	—	53	2	no	—
61	C₅₃	#1768	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₈	—	53	2	no	—
63	C₂²×C₈	—	53	2	no	—
64	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
67	C₄⋊C₈	—	53	8	no	—
68	#97	—	106	2	no	—
69	C₄×C₈	—	106	2	no	—
70	#95	Dic₅₃	1	4	yes	—
71	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
72	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
73	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
75	C₂×C₈	—	53	4	no	—
76	C₂×C₈	—	53	4	no	—
77	C₂×C₈	—	106	2	no	—
78	C₂×C₈	—	106	2	no	—
79	C₂×C₈	—	106	2	no	—
80	C₂×C₈	—	106	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	Dic₁₀₆	1	2	yes	—
83	C₄²	C₄×D₅₃	1	4	yes	—
84	C₄²	D₂₁₂	1	2	yes	—
85	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
92	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
94	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
95	C₂×C₄	#11719945	1	2	yes	—
96	C₈	—	106	4	no	—
97	C₈	—	106	4	no	—
98	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
100	C₂²	#13133176	1	1	yes	characteristic
101	C₂²	#13133224	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₂₁₂	1	2	yes	—
106	C₄	C₄×Dic₁₀₆	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
108	C₂	#16810579	1	2	yes	—
109	C₂	#16810650	1	2	yes	—
110	C₂	#16810684	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal

C₅₃×#376

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

C₄

yes

complemented

#13133128

—

C₄×#88322

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

#1768

—

maximal, Sylow

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#405

yes

—

C₂×C₅₃

#424

yes

—

C₂×C₅₃

#433

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

D₅₃

yes

characteristic

#336

—

C₅₃

#1768

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#97

—

106

—

C₄×C₈

—

106

—

#95

Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₄×D₅₃

yes

—

C₄²

D₂₁₂

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₅₃

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719945

yes

—

C₈

—

106

—

C₈

—

106

—

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

100

C₂²

#13133176

yes

characteristic

101

C₂²

#13133224

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₂₁₂

yes

—

106

C₄

C₄×Dic₁₀₆

yes

—

107

C₂

C₄×C₄⋊Dic₅₃

yes

characteristic

108

C₂

#16810579

yes

—

109

C₂

#16810650

yes

—

110

C₂

#16810684

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₅₃×#376	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	#376	D₅₃
hypocenter	C₅₃	#1768
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Fitting

C₅₃×#376

C₂

socle

C₂³×C₅₃

C₂²×C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
8. C₄⋊#88322	C₄	semidirect
61. C₅₃	#1768	semidirect

H ≅ G/N

type

8. C₄⋊#88322

C₄

semidirect

61. C₅₃

#1768

semidirect

Subgroups — #20274572

Conjugacy classes of subgroups

Complements