Subgroups — #13086307

Order 1440.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	#10624097	C₂	1	2	yes	maximal
3	#91234	—	3	4	no	maximal
4	C₂×C₃²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂×C₃²×Dic₅	C₄	1	1	yes	characteristic
6	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
7	C₃²×Dic₁₀	—	2	2	no	—
8	C₃⋊(C₃⋊#170)	—	2	2	no	—
9	#63663	—	5	1	no	maximal, Hall
10	C₂×C₃×Dic₁₀	D₃	1	4	yes	—
11	#6762	—	3	8	no	—
12	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
13	C₃²×C₄×C₅	—	2	1	no	—
14	C₃²×Dic₅	—	2	1	no	—
15	C₃²×Dic₅	D₄	1	2	yes	—
16	#4005	—	9	1	no	Hall
17	C₂×C₃²×Q₈	—	5	1	no	—
18	#3748	—	5	2	no	—
19	C₂×C₃×C₄×C₅	Dic₃	1	4	yes	—
20	C₂×C₃×Dic₅	Dic₃	1	4	yes	—
21	C₂×C₃×Dic₅	C₂×D₃	1	4	yes	—
22	C₃×Dic₁₀	—	2	8	no	—
23	C₃⋊#170	—	6	8	no	—
24	#836	—	15	4	no	—
25	C₂×C₃²×C₅	—	2	1	no	—
26	C₂×C₃²×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₁₀	C₃⋊D₃	1	1	yes	characteristic
28	#704	—	9	2	no	—
29	C₃²×Q₈	—	10	2	no	—
30	C₂×C₃²×C₄	—	5	1	no	—
31	C₂×C₃²×C₄	—	5	1	no	—
32	C₂×C₃²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
33	C₃⋊(C₃⋊C₈)	—	10	2	no	—
34	C₂²×C₃×C₅	C₂×Dic₃	1	4	yes	—
35	C₃×C₄×C₅	—	2	4	no	—
36	C₃×Dic₅	—	2	4	no	—
37	C₃×Dic₅	#67	1	8	yes	—
38	C₂×C₃×Q₈	—	5	4	no	—
39	#208	—	15	8	no	—
40	C₃²×C₅	#101	1	1	yes	characteristic, complemented, Hall
41	C₂×C₄×C₅	C₃⋊Dic₃	1	1	yes	characteristic
42	C₂×Dic₅	C₃⋊Dic₃	1	1	yes	characteristic
43	C₂×Dic₅	C₂×C₃⋊D₃	1	1	yes	characteristic
44	Dic₁₀	—	2	2	no	—
45	#170	—	18	2	no	—
46	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₃²×C₄	—	2	1	no	—
48	C₃²×C₄	—	5	2	no	—
49	C₃²×C₄	—	10	1	no	—
50	#101	—	45	1	no	Sylow
51	C₂×C₃×C₅	—	2	4	no	—
52	C₂×C₃×C₅	#217	1	4	yes	—
53	C₃×Q₈	—	10	8	no	—
54	C₂×C₃×C₄	—	5	4	no	—
55	C₂×C₃×C₄	—	5	4	no	—
56	C₂×C₃×C₄	ΓL(1,𝔽₁₆)	1	4	yes	—
57	C₃⋊C₈	—	30	8	no	—
58	C₂²×C₅	C₂×C₃⋊Dic₃	1	1	yes	characteristic
59	C₄×C₅	—	2	1	no	—
60	Dic₅	—	2	1	no	—
61	Dic₅	#638	1	2	yes	—
62	C₂×C₃²	—	2	1	no	—
63	C₂×C₃²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₂×Q₈	—	5	1	no	—
65	M₁₆	—	45	2	no	—
66	C₃×C₅	#836	1	4	yes	complemented
67	C₂²×C₃	C₂×ΓL(1,𝔽₁₆)	1	4	yes	—
68	C₃×C₄	—	2	4	no	—
69	C₃×C₄	—	5	8	no	—
70	C₃×C₄	—	10	4	no	—
71	C₂×C₅	—	2	1	no	—
72	C₂×C₅	#3757	1	1	yes	characteristic
73	C₃²	#4005	1	1	yes	characteristic, complemented, Sylow
74	Q₈	—	10	2	no	—
75	C₂×C₄	—	5	1	no	—
76	C₂×C₄	—	5	1	no	—
77	C₂×C₄	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
78	C₈	—	90	2	no	—
79	C₂×C₃	—	2	4	no	—
80	C₂×C₃	#6763	1	4	yes	—
81	C₅	#63663	1	1	yes	characteristic, complemented, Sylow
82	C₂²	C₂×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	5	2	no	—
85	C₄	—	10	1	no	—
86	C₃	#91234	1	4	yes	complemented
87	C₂	—	2	1	no	—
88	C₂	#10624098	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10624098
commutator, prime radical	C₂²×C₃²×C₅	C₂×C₄
Frattini	C₂²	C₂×C₃⋊ΓL(1,𝔽₁₆)
Fitting	C₂×C₃²×C₄×C₅	C₄
socle	C₂×C₃²×C₅	C₂²⋊C₄
cosocle	C₂×C₃²×Dic₅	C₂²
hypercenter	C₂×C₄	C₃⋊ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	#101
LCS	G ▷ C₂²×C₃²×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
40. C₃²×C₅	#101	semidirect
66. C₃×C₅	#836	semidirect
73. C₃²	#4005	semidirect
81. C₅	#63663	semidirect
86. C₃	#91234	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Dic₁₀	2
#10624097	2
#91234	3
#63663	5