Subgroups — #6790

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#67	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#1203	C₂	1	1	yes	maximal, characteristic, complemented
6	#1202	C₂	1	1	yes	maximal, characteristic, complemented
7	#1200	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×D₅	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic, complemented
10	C₂×D₁₅	—	2	2	no	—
11	C₂×D₁₅	C₂²	1	1	yes	characteristic
12	C₂×D₁₅	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₅×D₃	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃×D₅	C₂²	1	1	yes	characteristic, complemented
15	D₃×D₅	—	2	2	no	—
16	C₃×Dic₅	C₂²	1	1	yes	characteristic, complemented
17	C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
18	D₃×D₄	—	5	1	no	maximal, Hall
19	C₂²×D₅	—	3	1	no	—
20	C₂²×D₅	—	3	1	no	—
21	C₅×D₄	—	3	1	no	—
22	C₂²⋊D₅	—	3	1	no	—
23	C₂²⋊D₅	D₃	1	1	yes	characteristic, complemented
24	D₂₀	—	3	1	no	—
25	C₄×D₅	—	3	1	no	—
26	C₂×C₃×C₅	—	2	1	no	—
27	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
28	D₁₅	—	2	1	no	—
29	D₁₅	D₄	1	2	yes	complemented
30	C₃×D₅	—	2	1	no	—
31	C₅×D₃	—	2	1	no	—
32	C₂²×D₃	—	5	1	no	—
33	C₂²×D₃	—	5	1	no	—
34	C₃×D₄	—	5	1	no	—
35	#67	—	5	1	no	—
36	#67	D₅	1	1	yes	characteristic, complemented
37	D₁₂	—	5	1	no	—
38	C₄×D₃	—	5	1	no	—
39	C₂²×C₅	—	3	1	no	—
40	C₂²×C₅	C₂×D₃	1	1	yes	characteristic, complemented
41	C₂×D₅	—	3	1	no	—
42	C₂×D₅	—	3	1	no	—
43	C₂×D₅	—	6	2	no	—
44	C₂×D₅	—	6	2	no	—
45	C₂×D₅	C₂×D₃	1	1	yes	characteristic, complemented
46	C₄×C₅	—	3	1	no	—
47	Dic₅	C₂×D₃	1	1	yes	characteristic, complemented
48	C₂×D₄	—	15	1	no	Sylow
49	C₃×C₅	C₂×D₄	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₃	—	5	1	no	—
51	C₂²×C₃	C₂×D₅	1	1	yes	characteristic, complemented
52	C₂×D₃	—	5	1	no	—
53	C₂×D₃	—	5	1	no	—
54	C₂×D₃	—	10	2	no	—
55	C₂×D₃	—	10	2	no	—
56	C₂×D₃	C₂×D₅	1	1	yes	characteristic, complemented
57	C₃×C₄	—	5	1	no	—
58	Dic₃	C₂×D₅	1	1	yes	characteristic, complemented
59	C₂×C₅	—	2	1	no	—
60	C₂×C₅	—	6	1	no	—
61	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
62	D₅	—	2	1	no	—
63	D₅	—	3	2	no	—
64	D₅	—	6	1	no	—
65	C₂³	—	15	1	no	—
66	C₂³	—	15	1	no	—
67	D₄	—	3	1	no	—
68	D₄	—	5	1	no	—
69	D₄	—	15	1	no	—
70	D₄	—	15	1	no	—
71	C₂×C₄	—	15	1	no	—
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	10	1	no	—
74	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
75	D₃	—	2	1	no	—
76	D₃	—	5	2	no	—
77	D₃	—	10	1	no	—
78	C₅	D₃×D₄	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	3	1	no	—
80	C₂²	—	5	1	no	—
81	C₂²	—	15	1	no	—
82	C₂²	—	15	1	no	—
83	C₂²	—	30	2	no	—
84	C₂²	—	30	2	no	—
85	C₂²	D₃×D₅	1	1	yes	characteristic, complemented
86	C₄	—	3	1	no	—
87	C₄	—	5	1	no	—
88	C₃	D₄×D₅	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	2	1	no	—
90	C₂	—	6	1	no	—
91	C₂	—	10	1	no	—
92	C₂	—	15	2	no	—
93	C₂	—	30	1	no	—
94	C₂	C₂×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₂²×C₃×C₅	C₂²
hypercenter	C₂²	D₃×D₅
hypocenter	C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₅	C₂	semidirect
2. C₂×D₃×D₅	C₂	semidirect
3. C₅×#67	C₂	semidirect
4. C₃×C₂²⋊D₅	C₂	semidirect
5. #1203	C₂	semidirect
6. #1202	C₂	semidirect
7. #1200	C₂	semidirect
9. C₂²×C₃×C₅	C₂²	semidirect
12. C₂×D₁₅	C₂²	semidirect
13. C₂×C₅×D₃	C₂²	semidirect
14. C₂×C₃×D₅	C₂²	semidirect
16. C₃×Dic₅	C₂²	semidirect
17. C₅×Dic₃	C₂²	semidirect
23. C₂²⋊D₅	D₃	semidirect
29. D₁₅	D₄	semidirect
36. #67	D₅	semidirect
40. C₂²×C₅	C₂×D₃	semidirect
45. C₂×D₅	C₂×D₃	semidirect
47. Dic₅	C₂×D₃	semidirect
49. C₃×C₅	C₂×D₄	semidirect
51. C₂²×C₃	C₂×D₅	semidirect
56. C₂×D₃	C₂×D₅	semidirect
58. Dic₃	C₂×D₅	semidirect
78. C₅	D₃×D₄	semidirect
85. C₂²	D₃×D₅	semidirect
88. C₃	D₄×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₅	2
C₂×D₃×D₅	2
C₅×#67	2
C₃×C₂²⋊D₅	2
#1203	2
#1202	2
#1200	2
D₄×D₅	3
D₃×D₄	5