Group #13183446 — subgroups

Subgroups — #13183446

Order 1760.

332 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 119 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×C₅×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₅₅	C₂	1	4	yes	maximal, complemented
3	#11725033	C₂	1	4	yes	maximal, complemented
4	D₄×D₅₅	C₂	1	4	yes	maximal, complemented
5	#11725031	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₁₁	C₂²	1	2	yes	complemented
7	C₂²×D₅₅	C₂²	1	4	yes	complemented
8	C₅×D₄×C₁₁	C₂²	1	4	yes	complemented
9	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic, complemented
10	C₂²⋊D₅₅	C₂²	1	4	yes	complemented
11	C₂²⋊D₅₅	C₂²	1	8	yes	complemented
12	C₂×Dic₅₅	C₂²	1	4	yes	complemented
13	D₂₂₀	C₂²	1	2	yes	complemented
14	C₄×D₅₅	C₂²	1	4	yes	complemented
15	Dic₁₁₀	C₂²	1	2	yes	complemented
16	#67037	—	5	1	no	maximal, Hall
17	C₂²×C₅×C₁₁	—	4	2	no	—
18	C₂²×C₅×C₁₁	C₂³	1	4	yes	—
19	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
20	C₂×D₅₅	—	4	4	no	—
21	C₂×D₅₅	C₂³	1	4	yes	—
22	C₄×C₅×C₁₁	C₂³	1	2	yes	—
23	Dic₅₅	C₂³	1	4	yes	—
24	C₂×D₄×C₁₁	D₅	1	1	yes	characteristic, complemented
25	C₂×C₂²⋊D₁₁	—	5	4	no	—
26	#4341	—	5	4	no	—
27	D₄×D₁₁	—	5	4	no	—
28	#4339	—	5	2	no	—
29	#4144	—	11	1	no	maximal, Hall
30	C₂×C₅×C₁₁	—	2	4	no	—
31	C₂×C₅×C₁₁	—	2	1	no	—
32	C₂×C₅×C₁₁	C₂⁴	1	1	yes	characteristic
33	D₅₅	—	2	4	no	—
34	C₂³×C₁₁	C₂×D₅	1	2	yes	complemented
35	C₂²×D₁₁	—	5	4	no	—
36	D₄×C₁₁	C₂×D₅	1	4	yes	complemented
37	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic, complemented
38	C₂²⋊D₁₁	—	5	4	no	—
39	C₂²⋊D₁₁	—	5	8	no	—
40	C₂×Dic₁₁	—	5	4	no	—
41	D₄₄	—	5	2	no	—
42	C₄×D₁₁	—	5	4	no	—
43	Dic₂₂	—	5	2	no	—
44	C₂×C₅×D₄	D₁₁	1	1	yes	characteristic, complemented
45	C₂×C₂²⋊D₅	—	11	4	no	—
46	#711	—	11	4	no	—
47	D₄×D₅	—	11	4	no	—
48	#709	—	11	2	no	—
49	C₅×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₁₁	—	4	2	no	—
51	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
52	C₂²×C₁₁	C₂²×D₅	1	4	yes	—
53	C₂×D₁₁	—	5	4	no	—
54	C₂×D₁₁	—	20	4	no	—
55	C₄×C₁₁	C₂²×D₅	1	2	yes	—
56	Dic₁₁	—	5	4	no	—
57	C₂³×C₅	C₂×D₁₁	1	2	yes	complemented
58	C₂²×D₅	—	11	4	no	—
59	C₅×D₄	C₂×D₁₁	1	4	yes	complemented
60	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic, complemented
61	C₂²⋊D₅	—	11	4	no	—
62	C₂²⋊D₅	—	11	8	no	—
63	C₂×Dic₅	—	11	4	no	—
64	D₂₀	—	11	2	no	—
65	C₄×D₅	—	11	4	no	—
66	Dic₁₀	—	11	2	no	—
67	H(5,𝔽₂)	—	55	1	no	Sylow
68	C₂×C₁₁	—	2	4	no	—
69	C₂×C₁₁	—	2	1	no	—
70	C₂×C₁₁	C₂³×D₅	1	1	yes	characteristic
71	D₁₁	—	10	4	no	—
72	C₂²×C₅	—	4	2	no	—
73	C₂²×C₅	C₂²×D₁₁	1	4	yes	—
74	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
75	C₂×D₅	—	11	4	no	—
76	C₂×D₅	—	44	4	no	—
77	C₄×C₅	C₂²×D₁₁	1	2	yes	—
78	Dic₅	—	11	4	no	—
79	#41	—	55	4	no	—
80	#41	—	55	2	no	—
81	C₂×D₄	—	55	4	no	—
82	C₂×D₄	—	55	4	no	—
83	C₂×D₄	D₅₅	1	1	yes	characteristic, complemented
84	C₁₁	#4144	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₅	—	2	4	no	—
86	C₂×C₅	—	2	1	no	—
87	C₂×C₅	C₂³×D₁₁	1	1	yes	characteristic
88	D₅	—	22	4	no	—
89	C₂³	—	55	4	no	—
90	C₂³	C₂×D₅₅	1	2	yes	complemented
91	Q₈	—	55	2	no	—
92	D₄	—	55	8	no	—
93	D₄	—	55	4	no	—
94	D₄	—	55	2	no	—
95	D₄	C₂×D₅₅	1	4	yes	complemented
96	C₂×C₄	—	55	4	no	—
97	C₂×C₄	—	55	4	no	—
98	C₂×C₄	C₂×D₅₅	1	1	yes	characteristic, complemented
99	C₅	#67037	1	1	yes	characteristic, complemented, Sylow
100	C₂²	—	4	2	no	—
101	C₂²	—	55	4	no	—
102	C₂²	—	220	4	no	—
103	C₂²	C₂²×D₅₅	1	4	yes	—
104	C₂²	C₂²×D₅₅	1	1	yes	characteristic
105	C₄	—	55	4	no	—
106	C₄	C₂²×D₅₅	1	2	yes	—
107	C₂	—	2	1	no	—
108	C₂	—	2	4	no	—
109	C₂	—	110	4	no	—
110	C₂	C₂³×D₅₅	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×D₄×C₁₁

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₅₅

C₂

yes

maximal, complemented

#11725033

C₂

yes

maximal, complemented

D₄×D₅₅

C₂

yes

maximal, complemented

#11725031

C₂

yes

maximal, complemented

C₂³×C₅×C₁₁

C₂²

yes

complemented

C₂²×D₅₅

C₂²

yes

complemented

C₅×D₄×C₁₁

C₂²

yes

complemented

C₂×C₄×C₅×C₁₁

C₂²

yes

characteristic, complemented

C₂²⋊D₅₅

C₂²

yes

complemented

C₂²⋊D₅₅

C₂²

yes

complemented

C₂×Dic₅₅

C₂²

yes

complemented

D₂₂₀

C₂²

yes

complemented

C₄×D₅₅

C₂²

yes

complemented

Dic₁₁₀

C₂²

yes

complemented

#67037

—

maximal, Hall

C₂²×C₅×C₁₁

—

C₂²×C₅×C₁₁

C₂³

yes

—

C₂²×C₅×C₁₁

C₂³

yes

characteristic

C₂×D₅₅

—

C₂×D₅₅

C₂³

yes

—

C₄×C₅×C₁₁

C₂³

yes

—

Dic₅₅

C₂³

yes

—

C₂×D₄×C₁₁

D₅

yes

characteristic, complemented

C₂×C₂²⋊D₁₁

—

#4341

—

D₄×D₁₁

—

#4339

—

#4144

—

maximal, Hall

C₂×C₅×C₁₁

—

C₂×C₅×C₁₁

—

C₂×C₅×C₁₁

C₂⁴

yes

characteristic

D₅₅

—

C₂³×C₁₁

C₂×D₅

yes

complemented

C₂²×D₁₁

—

D₄×C₁₁

C₂×D₅

yes

complemented

C₂×C₄×C₁₁

C₂×D₅

yes

characteristic, complemented

C₂²⋊D₁₁

—

C₂²⋊D₁₁

—

C₂×Dic₁₁

—

D₄₄

—

C₄×D₁₁

—

Dic₂₂

—

C₂×C₅×D₄

D₁₁

yes

characteristic, complemented

C₂×C₂²⋊D₅

—

#711

—

D₄×D₅

—

#709

—

C₅×C₁₁

H(5,𝔽₂)

yes

characteristic, complemented, Hall

C₂²×C₁₁

—

C₂²×C₁₁

C₂²×D₅

yes

characteristic

C₂²×C₁₁

C₂²×D₅

yes

—

C₂×D₁₁

—

C₂×D₁₁

—

C₄×C₁₁

C₂²×D₅

yes

—

Dic₁₁

—

C₂³×C₅

C₂×D₁₁

yes

complemented

C₂²×D₅

—

C₅×D₄

C₂×D₁₁

yes

complemented

C₂×C₄×C₅

C₂×D₁₁

yes

characteristic, complemented

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

H(5,𝔽₂)

—

Sylow

C₂×C₁₁

—

C₂×C₁₁

—

C₂×C₁₁

C₂³×D₅

yes

characteristic

D₁₁

—

C₂²×C₅

—

C₂²×C₅

C₂²×D₁₁

yes

—

C₂²×C₅

C₂²×D₁₁

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

C₂²×D₁₁

yes

—

Dic₅

—

#41

—

#41

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₅₅

yes

characteristic, complemented

C₁₁

#4144

yes

characteristic, complemented, Sylow

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂³×D₁₁

yes

characteristic

D₅

—

C₂³

—

C₂³

C₂×D₅₅

yes

complemented

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

C₂×D₅₅

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₅₅

yes

characteristic, complemented

C₅

#67037

yes

characteristic, complemented, Sylow

100

C₂²

—

101

C₂²

—

102

C₂²

—

220

—

103

C₂²

C₂²×D₅₅

yes

—

104

C₂²

C₂²×D₅₅

yes

characteristic

105

C₄

—

106

C₄

C₂²×D₅₅

yes

—

107

C₂

—

108

C₂

—

109

C₂

—

110

—

110

C₂

C₂³×D₅₅

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₅₅
commutator, socle, cosocle	C₂×C₅×C₁₁	C₂⁴
Fitting	C₂×C₅×D₄×C₁₁	C₂
hypercenter	C₂×D₄	D₅₅
hypocenter	C₅×C₁₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄

subgroup

G/H

center, Frattini

C₂

C₂³×D₅₅

commutator, socle, cosocle

C₂×C₅×C₁₁

C₂⁴

Fitting

C₂×C₅×D₄×C₁₁

C₂

hypercenter

C₂×D₄

D₅₅

hypocenter

C₅×C₁₁

H(5,𝔽₂)

LCS

G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁

UCS

C₁ ◁ C₂ ◁ C₂×D₄

N	H ≅ G/N	type
1. C₂×C₅×D₄×C₁₁	C₂	semidirect
2. C₂×C₂²⋊D₅₅	C₂	semidirect
3. #11725033	C₂	semidirect
4. D₄×D₅₅	C₂	semidirect
5. #11725031	C₂	semidirect
6. C₂³×C₅×C₁₁	C₂²	semidirect
7. C₂²×D₅₅	C₂²	semidirect
8. C₅×D₄×C₁₁	C₂²	semidirect
9. C₂×C₄×C₅×C₁₁	C₂²	semidirect
10. C₂²⋊D₅₅	C₂²	semidirect
11. C₂²⋊D₅₅	C₂²	semidirect
12. C₂×Dic₅₅	C₂²	semidirect
13. D₂₂₀	C₂²	semidirect
14. C₄×D₅₅	C₂²	semidirect
15. Dic₁₁₀	C₂²	semidirect
24. C₂×D₄×C₁₁	D₅	semidirect
34. C₂³×C₁₁	C₂×D₅	semidirect
36. D₄×C₁₁	C₂×D₅	semidirect
37. C₂×C₄×C₁₁	C₂×D₅	semidirect
44. C₂×C₅×D₄	D₁₁	semidirect
49. C₅×C₁₁	H(5,𝔽₂)	semidirect
57. C₂³×C₅	C₂×D₁₁	semidirect
59. C₅×D₄	C₂×D₁₁	semidirect
60. C₂×C₄×C₅	C₂×D₁₁	semidirect
83. C₂×D₄	D₅₅	semidirect
84. C₁₁	#4144	semidirect
90. C₂³	C₂×D₅₅	semidirect
95. D₄	C₂×D₅₅	semidirect
98. C₂×C₄	C₂×D₅₅	semidirect
99. C₅	#67037	semidirect

H ≅ G/N

type

1. C₂×C₅×D₄×C₁₁

C₂

semidirect

2. C₂×C₂²⋊D₅₅

C₂

semidirect

3. #11725033

C₂

semidirect

4. D₄×D₅₅

C₂

semidirect

5. #11725031

C₂

semidirect

6. C₂³×C₅×C₁₁

C₂²

semidirect

7. C₂²×D₅₅

C₂²

semidirect

8. C₅×D₄×C₁₁

C₂²

semidirect

9. C₂×C₄×C₅×C₁₁

C₂²

semidirect

10. C₂²⋊D₅₅

C₂²

semidirect

11. C₂²⋊D₅₅

C₂²

semidirect

12. C₂×Dic₅₅

C₂²

semidirect

13. D₂₂₀

C₂²

semidirect

14. C₄×D₅₅

C₂²

semidirect

15. Dic₁₁₀

C₂²

semidirect

24. C₂×D₄×C₁₁

D₅

semidirect

34. C₂³×C₁₁

C₂×D₅

semidirect

36. D₄×C₁₁

C₂×D₅

semidirect

37. C₂×C₄×C₁₁

C₂×D₅

semidirect

44. C₂×C₅×D₄

D₁₁

semidirect

49. C₅×C₁₁

H(5,𝔽₂)

semidirect

57. C₂³×C₅

C₂×D₁₁

semidirect

59. C₅×D₄

C₂×D₁₁

semidirect

60. C₂×C₄×C₅

C₂×D₁₁

semidirect

83. C₂×D₄

D₅₅

semidirect

84. C₁₁

#4144

semidirect

90. C₂³

C₂×D₅₅

semidirect

95. D₄

C₂×D₅₅

semidirect

98. C₂×C₄

C₂×D₅₅

semidirect

99. C₅

#67037

semidirect

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

subgroup	index
C₂×C₅×D₄×C₁₁	2
C₂×C₂²⋊D₅₅	2
#11725033	2
D₄×D₅₅	2
#11725031	2
#67037	5
#4144	11

subgroup

index

C₂×C₅×D₄×C₁₁

C₂×C₂²⋊D₅₅

D₄×D₅₅

Subgroups — #13183446

Conjugacy classes of subgroups

Complements

Maximal subgroups