Group #20290249 — subgroups

Subgroups — #20290249

Order 6784.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#16811646	C₂	1	2	yes	maximal
2	#16811483	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133284	C₂²	1	4	yes	—
5	#13133274	—	2	2	no	—
6	#13133272	—	2	2	no	—
7	C₂×Q₈×D₅₃	C₄	1	2	yes	—
8	#13133217	C₄	1	1	yes	characteristic
9	#13133216	—	2	2	no	—
10	#13133215	C₂²	1	2	yes	—
11	C₄⋊C₄×D₅₃	—	2	2	no	—
12	C₄²×D₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
15	Q₈×D₅₃	—	4	4	no	—
16	C₂×C₄×D₅₃	—	2	2	no	—
17	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₅₃	D₄	1	2	yes	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
21	C₂×#88324	—	2	4	no	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄⋊Dic₅₃	—	2	2	no	—
25	#11719919	—	2	4	no	—
26	C₄×Dic₅₃	D₄	1	2	yes	—
27	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
28	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	Q₈×C₅₃	—	4	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	2	no	—
34	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
36	C₄×D₅₃	—	2	2	no	—
37	C₄×D₅₃	—	2	4	no	—
38	C₄×D₅₃	—	4	4	no	—
39	Dic₁₀₆	—	4	4	no	—
40	Dic₁₀₆	—	4	2	no	—
41	#88324	—	4	4	no	—
42	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×D₅₃	—	2	2	no	—
44	C₂×D₅₃	#100	1	2	yes	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	2	2	no	—
47	C₄×C₅₃	—	4	2	no	—
48	Dic₅₃	—	2	1	no	—
49	Dic₅₃	—	4	2	no	—
50	Dic₅₃	#137	1	4	yes	—
51	#1558	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
53	C₂×C₅₃	#419	1	2	yes	—
54	D₅₃	—	2	2	no	—
55	C₂×C₄⋊Q₈	—	53	1	no	—
56	#432	—	53	2	no	—
57	#333	—	53	4	no	—
58	C₅₃	#1558	1	1	yes	characteristic, complemented, Sylow
59	C₂²×Q₈	—	53	2	no	—
60	C₄⋊Q₈	—	53	2	no	—
61	C₄⋊Q₈	—	53	1	no	—
62	C₄⋊Q₈	—	106	2	no	—
63	C₄⋊Q₈	#6157	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	106	2	no	—
65	C₂×C₄²	—	53	1	no	—
66	C₄⋊C₈	—	53	4	no	—
67	C₂²⋊C₈	—	106	2	no	—
68	#97	—	106	2	no	—
69	C₂×Q₈	—	53	2	no	—
70	C₂×Q₈	—	106	2	no	—
71	C₂×Q₈	—	212	4	no	—
72	C₂×Q₈	C₂×#6157	1	2	yes	—
73	C₂²×C₄	—	53	2	no	—
74	C₂²×C₄	—	53	1	no	—
75	C₂²×C₄	—	106	2	no	—
76	C₂×C₈	—	106	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	106	4	no	—
79	C₄⋊C₄	—	106	2	no	—
80	C₄²	—	53	2	no	—
81	C₄²	—	53	1	no	—
82	C₄²	C₂×#6157	1	1	yes	characteristic
83	C₂³	—	53	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	212	2	no	—
86	Q₈	—	212	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	53	2	no	—
89	C₂×C₄	—	53	1	no	—
90	C₂×C₄	—	106	4	no	—
91	C₂×C₄	—	106	2	no	—
92	C₂×C₄	—	106	2	no	—
93	C₂×C₄	—	212	4	no	—
94	C₂×C₄	C₂²⋊#6157	1	2	yes	—
95	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
96	C₈	—	212	4	no	—
97	C₂²	—	53	2	no	—
98	C₂²	—	106	2	no	—
99	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	53	4	no	—
104	C₄	—	106	1	no	—
105	C₄	—	212	2	no	—
106	C₂	—	106	2	no	—
107	C₂	D₅₃⋊#100	1	1	yes	characteristic
108	C₂	#16811616	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811646

C₂

yes

maximal

#16811483

C₂

yes

maximal, complemented

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

#13133284

C₂²

yes

—

#13133274

—

#13133272

—

C₂×Q₈×D₅₃

C₄

yes

—

#13133217

C₄

yes

characteristic

#13133216

—

#13133215

C₂²

yes

—

C₄⋊C₄×D₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×#88324

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#100

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#137

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

C₂×C₄⋊Q₈

—

#432

—

#333

—

C₅₃

#1558

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

#6157

yes

characteristic

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂×Q₈

C₂×#6157

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂²⋊#6157

yes

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

212

—

C₂²

—

C₂²

—

106

—

C₂²

C₂×C₂²⋊#6157

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

106

—

105

C₄

—

212

—

106

C₂

—

106

—

107

C₂

D₅₃⋊#100

yes

characteristic

108

C₂

#16811616

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#1558
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #16811483	C₂	semidirect
58. C₅₃	#1558	semidirect

H ≅ G/N

type

2. #16811483

C₂

semidirect

58. C₅₃

#1558

semidirect

Subgroups — #20290249

Conjugacy classes of subgroups

Complements