Group #20274834 — subgroups

Subgroups — #20274834

Order 6784.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 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	1	yes	maximal, characteristic
2	#16810469	C₂	1	1	yes	maximal, characteristic
3	#16810377	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133124	C₄	1	2	yes	—
9	#13133124	C₄	1	2	yes	—
10	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	—	2	2	no	—
13	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	4	no	—
19	C₄²×C₅₃	D₄	1	2	yes	—
20	C₄²×C₅₃	Q₈	1	2	yes	—
21	C₄×Dic₅₃	—	2	2	no	—
22	C₄×Dic₅₃	—	2	2	no	—
23	C₂×#88322	—	2	2	no	—
24	C₂×#88322	—	2	4	no	—
25	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	—	2	2	no	—
28	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	#41	1	2	yes	—
33	C₂×C₄×C₅₃	#41	1	2	yes	—
34	C₂×Dic₅₃	—	2	4	no	—
35	C₂×Dic₅₃	—	2	4	no	—
36	C₂×Dic₅₃	—	2	4	no	—
37	C₈×C₅₃	—	4	2	no	—
38	#88322	—	4	2	no	—
39	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₅₃	#117	1	2	yes	—
41	C₂²×C₅₃	#131	1	2	yes	—
42	C₂²×C₅₃	#131	1	2	yes	—
43	C₄×C₅₃	—	2	2	no	—
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	#124	1	1	yes	characteristic
46	C₄×C₅₃	#124	1	1	yes	characteristic
47	C₄×C₅₃	#125	1	2	yes	—
48	Dic₅₃	—	4	4	no	—
49	#1832	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	#382	1	1	yes	characteristic
51	C₂×C₅₃	#424	1	1	yes	characteristic
52	C₂×C₅₃	#424	1	1	yes	characteristic
53	C₂×C₅₃	#433	1	4	yes	—
54	C₂×C₄⋊C₈	—	53	1	no	—
55	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
56	#376	—	53	1	no	—
57	#336	—	53	4	no	—
58	C₅₃	#1832	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	53	2	no	—
60	C₂²×C₈	C₂×D₅₃	1	2	yes	—
61	C₂×C₄²	—	53	2	no	—
62	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	106	2	no	—
65	#95	—	53	2	no	—
66	#95	—	53	2	no	—
67	C₂²×C₄	—	53	4	no	—
68	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	106	4	no	—
74	C₂×C₈	—	106	2	no	—
75	C₄²	—	106	2	no	—
76	C₄²	—	106	2	no	—
77	C₄²	Dic₁₀₆	1	2	yes	—
78	C₄²	C₂²⋊D₅₃	1	2	yes	—
79	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	106	4	no	—
83	C₂×C₄	—	106	4	no	—
84	C₂×C₄	—	106	4	no	—
85	C₂×C₄	#11719919	1	2	yes	—
86	C₂×C₄	#11719919	1	2	yes	—
87	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
88	C₂×C₄	#11719947	1	2	yes	—
89	C₂×C₄	#11719949	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	212	2	no	—
93	C₂²	C₂×#11719919	1	1	yes	characteristic
94	C₂²	#13133175	1	2	yes	—
95	C₂²	#13133223	1	2	yes	—
96	C₂²	#13133225	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	212	4	no	—
100	C₄	#13133204	1	1	yes	characteristic
101	C₄	#13133206	1	1	yes	characteristic
102	C₄	#13133208	1	2	yes	—
103	C₂	#16810488	1	1	yes	characteristic
104	C₂	#16810649	1	1	yes	characteristic
105	C₂	#16810651	1	1	yes	characteristic
106	C₂	#16810683	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

#16810469

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄⋊#88322

—

#13133124

C₄

yes

—

#13133124

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

#131

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic

#376

—

#336

—

C₅₃

#1832

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

#11719919

yes

—

C₂×C₄

#11719919

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

#11719947

yes

—

C₂×C₄

#11719949

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₂²

C₂×#11719919

yes

characteristic

C₂²

#13133175

yes

—

C₂²

#13133223

yes

—

C₂²

#13133225

yes

—

C₄

—

C₄

—

C₄

—

212

—

100

C₄

#13133204

yes

characteristic

101

C₄

#13133206

yes

characteristic

102

C₄

#13133208

yes

—

103

C₂

#16810488

yes

characteristic

104

C₂

#16810649

yes

characteristic

105

C₂

#16810651

yes

characteristic

106

C₂

#16810683

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1832
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

hypercenter

C₂×C₄⋊C₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
58. C₅₃	#1832	semidirect

H ≅ G/N

type

58. C₅₃

#1832

semidirect

Subgroups — #20274834

Conjugacy classes of subgroups

Complements