Group #20288342 — subgroups

Subgroups — #20288342

Order 6784.

1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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	#16811339	C₂	1	9	yes	maximal, complemented
2	#16811295	C₂	1	6	yes	maximal, complemented
3	#16811174	C₂	1	9	yes	maximal, complemented
4	C₄×#11719949	C₂	1	6	yes	maximal, complemented
5	Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719949	C₂²	1	6	yes	—
7	C₂×#11719949	C₂²	1	6	yes	—
8	#13133214	C₂²	1	9	yes	—
9	#13133206	C₂²	1	18	yes	—
10	#13133192	C₂²	1	36	yes	—
11	Q₈×Dic₅₃	C₂²	1	2	yes	—
12	C₄×D₂₁₂	C₂²	1	18	yes	—
13	#13133179	C₂²	1	18	yes	—
14	#13133175	C₂²	1	18	yes	—
15	C₄²×D₅₃	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₅₃	C₂²	1	9	yes	—
17	C₄×Q₈×C₅₃	C₂²	1	6	yes	—
18	C₂×Q₈×C₅₃	C₂³	1	6	yes	—
19	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
20	#11719949	—	2	24	no	—
21	#11719949	—	4	12	no	—
22	C₂×D₂₁₂	C₂³	1	18	yes	—
23	C₂×D₂₁₂	C₂³	1	18	yes	—
24	C₂×C₄×D₅₃	C₂³	1	18	yes	—
25	C₂×C₄×D₅₃	C₂³	1	9	yes	—
26	C₄⋊C₄×C₅₃	C₂³	1	18	yes	—
27	C₄²×C₅₃	C₂³	1	9	yes	—
28	#11719921	C₂³	1	36	yes	—
29	C₄⋊Dic₅₃	C₂³	1	6	yes	—
30	C₄×Dic₅₃	C₂³	1	6	yes	—
31	C₄×Dic₅₃	C₂³	1	9	yes	—
32	C₂²×D₅₃	C₂⁴	1	9	yes	—
33	Q₈×C₅₃	—	4	6	no	—
34	Q₈×C₅₃	#41	1	8	yes	complemented
35	C₂×C₄×C₅₃	C₂⁴	1	9	yes	—
36	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
37	C₂×Dic₅₃	C₂⁴	1	6	yes	—
38	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
39	D₂₁₂	—	2	36	no	—
40	D₂₁₂	—	4	18	no	—
41	C₄×D₅₃	—	2	36	no	—
42	C₄×D₅₃	—	2	18	no	—
43	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
44	C₂×D₅₃	—	2	18	no	—
45	C₂×D₅₃	—	2	9	no	—
46	C₄×C₅₃	—	2	9	no	—
47	C₄×C₅₃	C₂×#41	1	12	yes	—
48	Dic₅₃	—	2	6	no	—
49	Dic₅₃	H(5,𝔽₂)	1	2	yes	—
50	#3483	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	C₂²×#41	1	2	yes	—
52	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
53	D₅₃	—	4	9	no	—
54	Q₈²	D₅₃	1	1	yes	characteristic, complemented
55	#555	—	53	9	no	—
56	#550	—	53	6	no	—
57	#532	—	53	9	no	—
58	C₄×#41	—	53	6	no	—
59	C₅₃	#3483	1	1	yes	characteristic, complemented, Sylow
60	C₂×#41	—	53	6	no	—
61	C₂×#41	—	53	6	no	—
62	C₄⋊Q₈	C₂×D₅₃	1	9	yes	—
63	#127	—	53	9	no	—
64	#124	—	53	18	no	—
65	#121	—	53	36	no	—
66	C₄×Q₈	—	53	2	no	—
67	C₄×Q₈	C₂×D₅₃	1	6	yes	—
68	C₄×D₄	—	53	18	no	—
69	C₄×D₄	—	53	18	no	—
70	#117	—	53	18	no	—
71	C₂×C₄²	—	53	9	no	—
72	#41	—	106	24	no	—
73	#41	—	212	12	no	—
74	C₂×Q₈	C₂²×D₅₃	1	6	yes	—
75	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
76	C₂×D₄	—	53	18	no	—
77	C₂×D₄	—	53	18	no	—
78	C₂²×C₄	—	53	18	no	—
79	C₂²×C₄	—	53	9	no	—
80	C₄⋊C₄	—	53	6	no	—
81	C₄⋊C₄	C₂²×D₅₃	1	18	yes	—
82	C₂²⋊C₄	—	53	36	no	—
83	C₄²	—	53	6	no	—
84	C₄²	—	53	9	no	—
85	C₄²	C₂²×D₅₃	1	9	yes	—
86	C₂³	—	53	9	no	—
87	Q₈	—	4	6	no	—
88	Q₈	#11719949	1	8	yes	complemented
89	D₄	—	106	36	no	—
90	D₄	—	212	18	no	—
91	C₂×C₄	—	53	1	no	—
92	C₂×C₄	—	53	6	no	—
93	C₂×C₄	—	106	18	no	—
94	C₂×C₄	—	106	36	no	—
95	C₂×C₄	C₂³×D₅₃	1	9	yes	—
96	C₂×C₄	C₂³×D₅₃	1	6	yes	—
97	C₂²	—	106	18	no	—
98	C₂²	—	106	9	no	—
99	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	53	2	no	—
102	C₄	—	106	6	no	—
103	C₄	C₂×#11719949	1	12	yes	—
104	C₂	—	212	9	no	—
105	C₂	C₂²×#11719949	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811339

C₂

yes

maximal, complemented

#16811295

C₂

yes

maximal, complemented

#16811174

C₂

yes

maximal, complemented

C₄×#11719949

C₂

yes

maximal, complemented

Q₈²×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719949

C₂²

yes

—

C₂×#11719949

C₂²

yes

—

#13133214

C₂²

yes

—

#13133206

C₂²

yes

—

#13133192

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133179

C₂²

yes

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

C₂³

yes

—

#11719949

—

#11719949

—

C₂×D₂₁₂

C₂³

yes

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂²×D₅₃

C₂⁴

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

#41

yes

complemented

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

H(5,𝔽₂)

yes

—

#3483

—

maximal, Sylow

C₂×C₅₃

C₂²×#41

yes

—

C₂×C₅₃

C₂×H(5,𝔽₂)

yes

characteristic

D₅₃

—

Q₈²

D₅₃

yes

characteristic, complemented

#555

—

#550

—

#532

—

C₄×#41

—

C₅₃

#3483

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#127

—

#124

—

#121

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#41

—

106

—

#41

—

212

—

C₂×Q₈

C₂²×D₅₃

yes

—

C₂×Q₈

C₂²×D₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₃

yes

—

C₂³

—

Q₈

—

Q₈

#11719949

yes

complemented

D₄

—

106

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂³×D₅₃

yes

—

C₂×C₄

C₂³×D₅₃

yes

—

C₂²

—

106

—

C₂²

—

106

—

C₂²

C₂⁴×D₅₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

106

—

103

C₄

C₂×#11719949

yes

—

104

C₂

—

212

—

105

C₂

C₂²×#11719949

yes

—

106

C₂

H(5,𝔽₂)×D₅₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	Q₈²×C₅₃	C₂
hypercenter	Q₈²	D₅₃
hypocenter	C₅₃	#3483
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₅₃

commutator, socle

C₂²×C₅₃

C₂⁵

Fitting

Q₈²×C₅₃

C₂

hypercenter

Q₈²

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. #16811339	C₂	semidirect
2. #16811295	C₂	semidirect
3. #16811174	C₂	semidirect
4. C₄×#11719949	C₂	semidirect
5. Q₈²×C₅₃	C₂	semidirect
34. Q₈×C₅₃	#41	semidirect
54. Q₈²	D₅₃	semidirect
59. C₅₃	#3483	semidirect
88. Q₈	#11719949	semidirect

H ≅ G/N

type

1. #16811339

C₂

semidirect

2. #16811295

C₂

semidirect

3. #16811174

C₂

semidirect

4. C₄×#11719949

C₂

semidirect

5. Q₈²×C₅₃

C₂

semidirect

34. Q₈×C₅₃

#41

semidirect

54. Q₈²

D₅₃

semidirect

59. C₅₃

#3483

semidirect

88. Q₈

#11719949

semidirect

Subgroups — #20288342

Conjugacy classes of subgroups

Complements