Subgroups — #13183225

Order 1760.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊Dic₅₅	C₂	1	2	yes	maximal
3	C₄⋊Dic₅₅	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₅₅	C₂	1	2	yes	maximal, complemented
5	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₅×C₁₁	C₄	1	4	yes	complemented
7	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₅	C₂²	1	4	yes	—
10	#66979	—	5	1	no	maximal, Hall
11	C₂²×C₅×C₁₁	—	2	2	no	—
12	C₂²×C₅×C₁₁	C₂×C₄	1	2	yes	—
13	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
14	C₄×C₅×C₁₁	C₂×C₄	1	2	yes	—
15	C₄×C₅×C₁₁	C₂×C₄	1	2	yes	—
16	Dic₅₅	—	2	4	no	—
17	C₂²×C₄×C₁₁	D₅	1	1	yes	characteristic
18	C₂²⋊Dic₁₁	—	5	2	no	—
19	C₄⋊Dic₁₁	—	5	2	no	—
20	C₄×Dic₁₁	—	5	2	no	—
21	#4072	—	11	1	no	maximal, Hall
22	C₂×C₅×C₁₁	—	2	2	no	—
23	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₅×C₁₁	#41	1	2	yes	—
25	C₂³×C₁₁	C₂×D₅	1	1	yes	characteristic
26	C₂×C₄×C₁₁	Dic₅	1	4	yes	complemented
27	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
28	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
29	C₂×Dic₁₁	—	5	4	no	—
30	C₂²×C₄×C₅	D₁₁	1	1	yes	characteristic
31	C₂²⋊Dic₅	—	11	2	no	—
32	C₄⋊Dic₅	—	11	2	no	—
33	C₄×Dic₅	—	11	2	no	—
34	C₅×C₁₁	#117	1	1	yes	characteristic, complemented, Hall
35	C₂²×C₁₁	—	2	2	no	—
36	C₂²×C₁₁	C₂×Dic₅	1	2	yes	—
37	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
38	C₄×C₁₁	C₂×Dic₅	1	2	yes	—
39	C₄×C₁₁	C₂×Dic₅	1	2	yes	—
40	Dic₁₁	—	10	4	no	—
41	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
42	C₂×C₄×C₅	Dic₁₁	1	4	yes	complemented
43	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
44	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
45	C₂×Dic₅	—	11	4	no	—
46	#117	—	55	1	no	Sylow
47	C₂×C₁₁	—	2	2	no	—
48	C₂×C₁₁	#709	1	2	yes	—
49	C₂×C₁₁	C₂²×Dic₅	1	1	yes	characteristic
50	C₂²×C₅	—	2	2	no	—
51	C₂²×C₅	C₂×Dic₁₁	1	2	yes	—
52	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
53	C₄×C₅	C₂×Dic₁₁	1	2	yes	—
54	C₄×C₅	C₂×Dic₁₁	1	2	yes	—
55	Dic₅	—	22	4	no	—
56	C₂²×C₄	D₅₅	1	1	yes	characteristic
57	C₄⋊C₄	—	55	2	no	—
58	C₂²⋊C₄	—	55	2	no	—
59	C₄²	—	55	2	no	—
60	C₁₁	#4072	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₅	—	2	2	no	—
62	C₂×C₅	#4339	1	2	yes	—
63	C₂×C₅	C₂²×Dic₁₁	1	1	yes	characteristic
64	C₂³	C₂×D₅₅	1	1	yes	characteristic
65	C₂×C₄	—	55	4	no	—
66	C₂×C₄	Dic₅₅	1	4	yes	complemented
67	C₂×C₄	C₂×D₅₅	1	1	yes	characteristic
68	C₂×C₄	C₂×D₅₅	1	1	yes	characteristic
69	C₅	#66979	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	2	2	no	—
71	C₂²	C₂×Dic₅₅	1	2	yes	—
72	C₂²	C₂²×D₅₅	1	1	yes	characteristic
73	C₄	—	110	4	no	—
74	C₄	C₂×Dic₅₅	1	2	yes	—
75	C₄	C₂×Dic₅₅	1	2	yes	—
76	C₂	—	2	2	no	—
77	C₂	#11725031	1	2	yes	—
78	C₂	C₂²×Dic₅₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₅
commutator	C₂×C₅×C₁₁	C₂²×C₄
Frattini	C₂²	C₂²×D₅₅
Fitting	C₂²×C₄×C₅×C₁₁	C₂
socle, cosocle	C₂²×C₅×C₁₁	C₂³
prime radical	C₂³×C₅×C₁₁	C₂²
hypercenter	C₂²×C₄	D₅₅
hypocenter	C₅×C₁₁	#117
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄⋊Dic₅₅	C₂	semidirect
4. C₄×Dic₅₅	C₂	semidirect
6. C₂×C₄×C₅×C₁₁	C₄	semidirect
26. C₂×C₄×C₁₁	Dic₅	semidirect
34. C₅×C₁₁	#117	semidirect
42. C₂×C₄×C₅	Dic₁₁	semidirect
60. C₁₁	#4072	semidirect
66. C₂×C₄	Dic₅₅	semidirect
69. C₅	#66979	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₁₁	2
C₂²⋊Dic₅₅	2
C₄⋊Dic₅₅	2
C₄×Dic₅₅	2
#66979	5
#4072	11