Subgroups — #13085501

Order 1440.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4379	C₂	1	1	yes	maximal, characteristic
2	C₅×#3787	C₂	1	1	yes	maximal, characteristic
3	C₃⋊D₃×#168	C₂	1	1	yes	maximal, characteristic
4	C₂×#4379	C₄	1	2	yes	—
5	C₄×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₅×#622	C₄	1	2	yes	complemented
7	C₃⋊(C₃⋊#168)	C₄	1	1	yes	characteristic, complemented
8	C₃²×#168	C₄	1	1	yes	characteristic, complemented
9	#63767	—	5	1	no	maximal, Hall
10	D₃×#168	—	6	2	no	—
11	C₂×C₅×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
12	#4379	—	2	2	no	—
13	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₅×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
15	#3947	—	9	1	no	maximal, Hall
16	C₄×#157	—	5	1	no	—
17	#3787	D₅	1	1	yes	characteristic
18	C₈×C₃⋊D₃	—	5	1	no	—
19	C₄×C₅×D₃	—	6	2	no	—
20	C₃⋊#168	—	6	2	no	—
21	C₃×#168	—	2	2	no	—
22	C₂×C₃²×C₅	C₄²	1	1	yes	characteristic
23	C₅×C₃⋊D₃	M₁₆	1	2	yes	—
24	C₂×C₅×C₈	—	9	1	no	—
25	C₄×Dic₅	—	9	1	no	—
26	C₂×#168	—	9	1	no	—
27	C₂×#157	—	5	2	no	—
28	C₄×C₃⋊D₃	C₂×D₅	1	1	yes	characteristic
29	#622	Dic₅	1	2	yes	complemented
30	C₃²×C₈	—	5	1	no	—
31	C₃⋊(C₃⋊C₈)	—	5	1	no	—
32	C₂×C₅×D₃	—	6	2	no	—
33	C₃×C₄×C₅	—	2	2	no	—
34	C₅×Dic₃	—	6	2	no	—
35	D₃×C₈	—	30	2	no	—
36	C₃²×C₅	#97	1	1	yes	characteristic, complemented, Hall
37	C₂×C₄×C₅	—	9	1	no	—
38	C₂×Dic₅	—	9	2	no	—
39	C₅×C₈	—	9	2	no	—
40	#168	—	9	1	no	—
41	#168	#157	1	1	yes	characteristic, complemented
42	C₂×C₃⋊D₃	C₄×D₅	1	1	yes	characteristic
43	#157	—	10	2	no	—
44	C₃²×C₄	C₄×D₅	1	1	yes	characteristic
45	C₃⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
46	#97	—	45	1	no	Sylow
47	C₂×C₃×C₅	—	2	2	no	—
48	C₅×D₃	—	6	4	no	—
49	C₄×D₃	—	6	2	no	—
50	C₃×C₈	—	10	2	no	—
51	C₃⋊C₈	—	30	2	no	—
52	C₂²×C₅	—	9	1	no	—
53	C₄×C₅	—	9	1	no	—
54	C₄×C₅	C₂×#157	1	1	yes	characteristic
55	Dic₅	—	18	2	no	—
56	C₂×C₃²	C₄×Dic₅	1	1	yes	characteristic
57	C₃⋊D₃	#676	1	2	yes	—
58	C₂×C₈	—	9	1	no	—
59	C₂×C₈	—	45	1	no	—
60	C₄²	—	45	1	no	—
61	C₃×C₅	—	2	2	no	—
62	C₂×D₃	—	6	2	no	—
63	C₃×C₄	—	2	2	no	—
64	Dic₃	—	6	2	no	—
65	C₂×C₅	—	9	2	no	—
66	C₂×C₅	C₄×#157	1	1	yes	characteristic
67	C₃²	#3947	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄	—	9	1	no	—
69	C₂×C₄	—	45	2	no	—
70	C₈	—	5	1	no	—
71	C₈	—	9	2	no	—
72	C₈	—	45	1	no	—
73	C₂×C₃	—	2	2	no	—
74	D₃	—	6	4	no	—
75	C₅	#63767	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	9	1	no	—
77	C₄	—	9	1	no	—
78	C₄	—	90	2	no	—
79	C₄	D₅×#157	1	1	yes	characteristic
80	C₃	—	2	2	no	—
81	C₂	—	9	2	no	—
82	C₂	Dic₅×#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₅×#157
commutator, socle	C₂×C₃²×C₅	C₄²
Fitting	C₃²×C₄×C₅	C₂×C₄
prime radical	C₂×C₅×C₃⋊D₃	C₂×C₄
cosocle	C₄×C₅×C₃⋊D₃	C₂²
hypocenter	C₃²×C₅	#97
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
6. C₅×#622	C₄	semidirect
7. C₃⋊(C₃⋊#168)	C₄	semidirect
8. C₃²×#168	C₄	semidirect
29. #622	Dic₅	semidirect
36. C₃²×C₅	#97	semidirect
41. #168	#157	semidirect
67. C₃²	#3947	semidirect
75. C₅	#63767	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4379	2
C₅×#3787	2
C₃⋊D₃×#168	2
#63767	5
#3947	9