Subgroups — C₂×#11724888 #13182956

Order 1760.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	#11724888	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×Dic₅₅	C₄	1	4	yes	complemented
7	C₂×Dic₅₅	C₂²	1	2	yes	—
8	C₂×C₁₁×Dic₅	—	2	2	no	—
9	C₂×C₁₁×Dic₅	C₂²	1	2	yes	—
10	C₂×C₅×Dic₁₁	—	2	2	no	—
11	C₂×C₅×Dic₁₁	C₂²	1	2	yes	—
12	C₂×Dic₁₁⋊C₄	—	5	1	no	maximal, Hall
13	C₂²×C₅×C₁₁	C₂×C₄	1	2	yes	—
14	C₂²×C₅×C₁₁	D₄	1	2	yes	—
15	C₂²×C₅×C₁₁	Q₈	1	2	yes	—
16	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
17	Dic₅₅	C₂×C₄	1	4	yes	complemented
18	C₅×Dic₁₁	—	2	2	no	—
19	C₁₁×Dic₅	—	2	2	no	—
20	C₂²×C₄×C₁₁	—	5	1	no	—
21	C₂²×Dic₁₁	—	5	1	no	—
22	C₂²×Dic₁₁	D₅	1	1	yes	characteristic
23	Dic₁₁⋊C₄	—	5	4	no	—
24	C₂×#683	—	11	1	no	maximal, Hall
25	C₂×C₅×C₁₁	C₄⋊C₄	1	4	yes	complemented
26	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₅×C₁₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₅×C₁₁	C₂×Q₈	1	1	yes	characteristic
29	C₂³×C₁₁	C₂×D₅	1	1	yes	characteristic
30	C₂×C₄×C₁₁	—	5	2	no	—
31	C₂×C₄×C₁₁	—	10	2	no	—
32	C₂×Dic₁₁	—	2	2	no	—
33	C₂×Dic₁₁	—	5	4	no	—
34	C₂×Dic₁₁	—	5	2	no	—
35	C₂×Dic₁₁	C₂×D₅	1	2	yes	—
36	C₂²×C₄×C₅	—	11	1	no	—
37	C₂²×Dic₅	—	11	1	no	—
38	C₂²×Dic₅	D₁₁	1	1	yes	characteristic
39	#683	—	11	4	no	—
40	C₅×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
41	C₂²×C₁₁	Dic₁₀	1	2	yes	—
42	C₂²×C₁₁	C₄×D₅	1	2	yes	—
43	C₂²×C₁₁	C₂²⋊D₅	1	2	yes	—
44	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
45	C₄×C₁₁	—	10	2	no	—
46	Dic₁₁	—	2	2	no	—
47	Dic₁₁	—	5	4	no	—
48	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
49	C₂×C₄×C₅	—	11	2	no	—
50	C₂×C₄×C₅	—	22	2	no	—
51	C₂×Dic₅	—	2	2	no	—
52	C₂×Dic₅	—	11	2	no	—
53	C₂×Dic₅	—	11	4	no	—
54	C₂×Dic₅	C₂×D₁₁	1	2	yes	—
55	C₂×C₄⋊C₄	—	55	1	no	Sylow
56	C₂×C₁₁	#683	1	4	yes	complemented
57	C₂×C₁₁	C₂×Dic₁₀	1	1	yes	characteristic
58	C₂×C₁₁	C₂×C₄×D₅	1	1	yes	characteristic
59	C₂×C₁₁	C₂×C₂²⋊D₅	1	1	yes	characteristic
60	C₂²×C₅	Dic₂₂	1	2	yes	—
61	C₂²×C₅	C₄×D₁₁	1	2	yes	—
62	C₂²×C₅	C₂²⋊D₁₁	1	2	yes	—
63	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
64	C₄×C₅	—	22	2	no	—
65	Dic₅	—	2	2	no	—
66	Dic₅	—	11	4	no	—
67	C₂²×C₄	—	5	1	no	—
68	C₂²×C₄	—	11	1	no	—
69	C₂²×C₄	—	55	1	no	—
70	C₄⋊C₄	—	55	4	no	—
71	C₁₁	C₂×#683	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₅	Dic₁₁⋊C₄	1	4	yes	complemented
73	C₂×C₅	C₂×Dic₂₂	1	1	yes	characteristic
74	C₂×C₅	C₂×C₄×D₁₁	1	1	yes	characteristic
75	C₂×C₅	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
76	C₂³	D₅×D₁₁	1	1	yes	characteristic
77	C₂×C₄	—	5	2	no	—
78	C₂×C₄	—	10	2	no	—
79	C₂×C₄	—	11	2	no	—
80	C₂×C₄	—	22	2	no	—
81	C₂×C₄	—	55	4	no	—
82	C₂×C₄	—	55	2	no	—
83	C₅	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic, complemented, Sylow
84	C₂²	#89166	1	2	yes	—
85	C₂²	#89167	1	2	yes	—
86	C₂²	#89170	1	2	yes	—
87	C₂²	C₂×D₅×D₁₁	1	1	yes	characteristic
88	C₄	—	10	2	no	—
89	C₄	—	22	2	no	—
90	C₄	—	55	4	no	—
91	C₂	#11724888	1	4	yes	direct product factor
92	C₂	C₂×#89166	1	1	yes	characteristic
93	C₂	C₂×#89167	1	1	yes	characteristic
94	C₂	C₂×#89170	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅×D₁₁
commutator	C₂×C₅×C₁₁	C₂²×C₄
Frattini	C₂²	C₂×D₅×D₁₁
Fitting, socle, prime radical	C₂³×C₅×C₁₁	C₂²
cosocle	C₂²×C₅×C₁₁	C₂³
hypocenter	C₅×C₁₁	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
4. #11724888	C₂	direct (N × H)
6. C₂×Dic₅₅	C₄	semidirect
17. Dic₅₅	C₂×C₄	semidirect
25. C₂×C₅×C₁₁	C₄⋊C₄	semidirect
40. C₅×C₁₁	C₂×C₄⋊C₄	semidirect
56. C₂×C₁₁	#683	semidirect
71. C₁₁	C₂×#683	semidirect
72. C₂×C₅	Dic₁₁⋊C₄	semidirect
83. C₅	C₂×Dic₁₁⋊C₄	semidirect
91. C₂	#11724888	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₅	2
C₂²×C₁₁×Dic₅	2
C₂²×C₅×Dic₁₁	2
#11724888	2
C₂×Dic₁₁⋊C₄	5
C₂×#683	11

Subgroups — C2×#11724888 #13182956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11724888 #13182956