Group #16811187 — subgroups

Subgroups — #16811187

Order 3392.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂²×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133213	C₂	1	2	yes	maximal, complemented
3	#13133206	C₂	1	2	yes	maximal, complemented
4	#13133191	C₂	1	4	yes	maximal, complemented
5	#13133185	C₂	1	4	yes	maximal, complemented
6	#13133176	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₅₃	C₂²	1	2	yes	—
9	#41×C₅₃	—	2	4	no	—
10	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄×C₅₃	C₂²	1	2	yes	—
12	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂²	1	2	yes	complemented
15	C₂×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
16	C₂×D₂₁₂	—	2	4	no	—
17	C₂×D₂₁₂	C₂²	1	2	yes	—
18	C₂×D₂₁₂	C₂²	1	2	yes	—
19	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
20	#11719921	C₂²	1	8	yes	complemented
21	C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
22	C₄×Dic₅₃	C₂²	1	2	yes	complemented
23	C₂³×C₅₃	C₂³	1	2	yes	—
24	C₂³×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×D₅₃	—	2	4	no	—
26	C₂²×D₅₃	—	4	4	no	—
27	C₂²×D₅₃	C₂³	1	4	yes	—
28	Q₈×C₅₃	—	2	2	no	—
29	D₄×C₅₃	—	2	4	no	—
30	D₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	D₄	1	4	yes	complemented
33	C₂×C₄×C₅₃	C₂³	1	2	yes	—
34	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²⋊D₅₃	—	4	8	no	—
37	C₂×Dic₅₃	C₂³	1	4	yes	—
38	D₂₁₂	—	2	4	no	—
39	D₂₁₂	—	2	4	no	—
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	—	2	2	no	—
42	C₂²×C₅₃	—	2	4	no	—
43	C₂²×C₅₃	C₂×D₄	1	2	yes	—
44	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₅₃	—	2	4	no	—
46	C₂×D₅₃	—	2	8	no	—
47	C₂×D₅₃	—	4	8	no	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₄×C₅₃	C₂×D₄	1	2	yes	—
51	Dic₅₃	—	2	4	no	—
52	C₂×C₅₃	—	2	2	no	—
53	C₂×C₅₃	—	4	2	no	—
54	C₂×C₅₃	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₅₃	H(5,𝔽₂)	1	2	yes	—
56	D₅₃	—	4	4	no	—
57	#535	—	53	1	no	maximal, Sylow
58	C₅₃	#535	1	1	yes	characteristic, complemented, Sylow
59	C₂×#41	D₅₃	1	1	yes	characteristic, complemented
60	C₂²×D₄	—	53	1	no	—
61	#127	—	53	2	no	—
62	#124	—	53	2	no	—
63	#121	—	53	4	no	—
64	#120	—	53	4	no	—
65	#117	—	53	1	no	—
66	C₂⁴	—	53	2	no	—
67	#41	—	2	4	no	—
68	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
69	C₂×D₄	—	53	8	no	—
70	C₂×D₄	—	53	2	no	—
71	C₂×D₄	—	53	2	no	—
72	C₂×D₄	—	106	4	no	—
73	C₂×D₄	C₂×D₅₃	1	2	yes	—
74	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
75	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₅₃	1	2	yes	complemented
77	C₄⋊C₄	—	53	2	no	—
78	C₂²⋊C₄	—	53	2	no	—
79	C₂²⋊C₄	—	53	8	no	—
80	C₄²	—	53	2	no	—
81	C₂³	—	53	4	no	—
82	C₂³	—	106	4	no	—
83	C₂³	—	212	4	no	—
84	C₂³	C₂²×D₅₃	1	2	yes	—
85	C₂³	C₂²×D₅₃	1	1	yes	characteristic
86	Q₈	—	2	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	2	no	—
89	D₄	—	106	4	no	—
90	D₄	—	106	4	no	—
91	D₄	—	212	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	53	4	no	—
94	C₂×C₄	C₂²⋊D₅₃	1	4	yes	complemented
95	C₂×C₄	C₂²×D₅₃	1	2	yes	—
96	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	4	no	—
101	C₂²	—	106	4	no	—
102	C₂²	—	106	8	no	—
103	C₂²	—	212	8	no	—
104	C₂²	C₂×C₂²⋊D₅₃	1	2	yes	—
105	C₂²	C₂³×D₅₃	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	106	4	no	—
108	C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
109	C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	2	no	—
112	C₂	—	212	4	no	—
113	C₂	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂	#13133263	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₂₁₂

C₂

yes

maximal, characteristic, complemented

#13133213

C₂

yes

maximal, complemented

#13133206

C₂

yes

maximal, complemented

#13133191

C₂

yes

maximal, complemented

#13133185

C₂

yes

maximal, complemented

#13133176

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₅₃

C₂²

yes

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₅₃

C₂²

yes

—

C₂×D₄×C₅₃

C₂²

yes

characteristic, complemented

C₂²×C₄×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂²

yes

complemented

C₂×C₂²⋊D₅₃

C₂²

yes

complemented

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂²

yes

—

C₂×D₂₁₂

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

#11719921

C₂²

yes

complemented

C₄⋊Dic₅₃

C₂²

yes

complemented

C₄×Dic₅₃

C₂²

yes

complemented

C₂³×C₅₃

C₂³

yes

—

C₂³×C₅₃

C₂³

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂³

yes

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₄

yes

complemented

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂³

yes

—

D₂₁₂

—

D₂₁₂

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₄

yes

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₄

yes

—

C₄×C₅₃

C₂×D₄

yes

—

Dic₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×D₄

yes

characteristic

C₂×C₅₃

H(5,𝔽₂)

yes

—

D₅₃

—

#535

—

maximal, Sylow

C₅₃

#535

yes

characteristic, complemented, Sylow

C₂×#41

D₅₃

yes

characteristic, complemented

C₂²×D₄

—

—

—

—

—

—

C₂⁴

—

#41

—

C₂×Q₈

C₂×D₅₃

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

C₂×D₅₃

yes

—

C₂×D₄

C₂×D₅₃

yes

characteristic, complemented

C₂²×C₄

C₂×D₅₃

yes

characteristic

C₂²×C₄

C₂×D₅₃

yes

complemented

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

106

—

C₂³

—

212

—

C₂³

C₂²×D₅₃

yes

—

C₂³

C₂²×D₅₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

106

—

D₄

—

106

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₅₃

yes

complemented

C₂×C₄

C₂²×D₅₃

yes

—

C₂×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₄

C₂²×D₅₃

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

106

—

102

C₂²

—

106

—

103

C₂²

—

212

—

104

C₂²

C₂×C₂²⋊D₅₃

yes

—

105

C₂²

C₂³×D₅₃

yes

characteristic

106

C₄

—

107

C₄

—

106

—

108

C₄

C₂×C₂²⋊D₅₃

yes

—

109

C₄

C₂×C₂²⋊D₅₃

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

212

—

113

C₂

C₂²×C₂²⋊D₅₃

yes

characteristic

114

C₂

#13133263

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₂×#41×C₅₃	C₂
hypercenter	C₂×#41	D₅₃
hypocenter	C₅₃	#535
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

C₂×#41×C₅₃

C₂

hypercenter

C₂×#41

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×D₂₁₂	C₂	semidirect
2. #13133213	C₂	semidirect
3. #13133206	C₂	semidirect
4. #13133191	C₂	semidirect
5. #13133185	C₂	semidirect
6. #13133176	C₂	semidirect
7. C₂×#41×C₅₃	C₂	semidirect
10. C₂×Q₈×C₅₃	C₂²	semidirect
12. C₂×D₄×C₅₃	C₂²	semidirect
14. C₂²×C₄×C₅₃	C₂²	semidirect
15. C₂×C₂²⋊D₅₃	C₂²	semidirect
20. #11719921	C₂²	semidirect
21. C₄⋊Dic₅₃	C₂²	semidirect
22. C₄×Dic₅₃	C₂²	semidirect
32. C₂×C₄×C₅₃	D₄	semidirect
58. C₅₃	#535	semidirect
59. C₂×#41	D₅₃	semidirect
68. C₂×Q₈	C₂×D₅₃	semidirect
74. C₂×D₄	C₂×D₅₃	semidirect
76. C₂²×C₄	C₂×D₅₃	semidirect
94. C₂×C₄	C₂²⋊D₅₃	semidirect

H ≅ G/N

type

1. C₂²×D₂₁₂

C₂

semidirect

2. #13133213

C₂

semidirect

3. #13133206

C₂

semidirect

4. #13133191

C₂

semidirect

5. #13133185

C₂

semidirect

6. #13133176

C₂

semidirect

7. C₂×#41×C₅₃

C₂

semidirect

10. C₂×Q₈×C₅₃

C₂²

semidirect

12. C₂×D₄×C₅₃

C₂²

semidirect

14. C₂²×C₄×C₅₃

C₂²

semidirect

15. C₂×C₂²⋊D₅₃

C₂²

semidirect

20. #11719921

C₂²

semidirect

21. C₄⋊Dic₅₃

C₂²

semidirect

22. C₄×Dic₅₃

C₂²

semidirect

32. C₂×C₄×C₅₃

D₄

semidirect

58. C₅₃

#535

semidirect

59. C₂×#41

D₅₃

semidirect

68. C₂×Q₈

C₂×D₅₃

semidirect

74. C₂×D₄

C₂×D₅₃

semidirect

76. C₂²×C₄

C₂×D₅₃

semidirect

94. C₂×C₄

C₂²⋊D₅₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₂₁₂	2
#13133213	2
#13133206	2
#13133191	2
#13133185	2
#13133176	2
C₂×#41×C₅₃	2
#535	53

subgroup

index

C₂²×D₂₁₂

C₂×#41×C₅₃

Subgroups — #16811187

Conjugacy classes of subgroups

Complements

Maximal subgroups