Subgroups — C₃²×C₄⋊#170 #13086255

Order 1440.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#170	C₂	1	2	yes	maximal
2	C₃²×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊#170	C₃	1	4	yes	maximal, direct product factor
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×#170	—	2	2	no	—
8	C₃²×C₄⋊C₈	—	5	1	no	maximal, Hall
9	C₂×C₃×#170	C₂×C₃	1	8	yes	—
10	C₃×C₄×Dic₅	C₂×C₃	1	4	yes	—
11	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
12	C₃²×C₄×C₅	C₈	1	2	yes	complemented
13	C₃²×Dic₅	—	2	1	no	—
14	C₃²×Dic₅	D₄	1	1	yes	characteristic
15	C₃²×Dic₅	Q₈	1	1	yes	characteristic
16	C₄⋊#170	C₃²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃²×C₈	—	5	2	no	—
18	C₃²×C₄²	—	5	1	no	—
19	C₂×C₃×C₄×C₅	C₃×C₄	1	4	yes	—
20	C₂×C₃×Dic₅	C₃×C₄	1	4	yes	—
21	C₂×C₃×Dic₅	C₂²×C₃	1	4	yes	—
22	C₃×#170	—	2	8	no	—
23	C₃×C₄⋊C₈	—	5	4	no	—
24	C₂×C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₃²×C₅	C₂×C₈	1	1	yes	characteristic
26	C₂×C₃²×C₅	M₁₆	1	1	yes	characteristic
27	C₂×#170	C₂×C₃²	1	2	yes	—
28	C₄×Dic₅	C₂×C₃²	1	1	yes	characteristic
29	C₂×C₃²×C₄	—	5	1	no	—
30	C₂×C₃²×C₄	—	5	1	no	—
31	C₂×C₃²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
32	C₃²×C₈	—	10	2	no	—
33	C₂²×C₃×C₅	C₂×C₃×C₄	1	4	yes	—
34	C₃×C₄×C₅	C₃×C₈	1	8	yes	complemented
35	C₃×Dic₅	—	2	4	no	—
36	C₃×Dic₅	C₃×D₄	1	4	yes	—
37	C₃×Dic₅	C₃×Q₈	1	4	yes	—
38	C₂×C₃×C₈	—	5	8	no	—
39	C₃×C₄²	—	5	4	no	—
40	C₃²×C₅	C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
41	C₂×C₄×C₅	C₃²×C₄	1	1	yes	characteristic
42	C₂×Dic₅	C₃²×C₄	1	1	yes	characteristic
43	C₂×Dic₅	C₂²×C₃²	1	1	yes	characteristic
44	#170	—	2	2	no	—
45	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₃²×C₄	—	5	1	no	—
47	C₃²×C₄	—	5	1	no	—
48	C₃²×C₄	—	10	1	no	—
49	C₃²×C₄	#170	1	2	yes	complemented
50	C₄⋊C₈	—	5	1	no	Sylow
51	C₂×C₃×C₅	C₃×C₄⋊C₄	1	4	yes	—
52	C₂×C₃×C₅	C₂×C₃×C₈	1	4	yes	—
53	C₂×C₃×C₅	C₃×M₁₆	1	4	yes	—
54	C₂×C₃×C₄	—	5	4	no	—
55	C₂×C₃×C₄	—	5	4	no	—
56	C₂×C₃×C₄	C₃×AGL(1,𝔽₅)	1	4	yes	—
57	C₃×C₈	—	10	8	no	—
58	C₂²×C₅	C₂×C₃²×C₄	1	1	yes	characteristic
59	C₄×C₅	C₃²×C₈	1	2	yes	complemented
60	Dic₅	—	2	1	no	—
61	Dic₅	C₃²×D₄	1	1	yes	characteristic
62	Dic₅	C₃²×Q₈	1	1	yes	characteristic
63	C₂×C₃²	#700	1	1	yes	characteristic
64	C₂×C₃²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×C₃²	C₂×#170	1	1	yes	characteristic
66	C₂×C₈	—	5	2	no	—
67	C₄²	—	5	1	no	—
68	C₃×C₅	C₃×C₄⋊C₈	1	4	yes	complemented
69	C₂²×C₃	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
70	C₃×C₄	—	5	4	no	—
71	C₃×C₄	—	5	4	no	—
72	C₃×C₄	—	10	4	no	—
73	C₃×C₄	C₃×#170	1	8	yes	complemented
74	C₂×C₅	C₃²×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₅	C₂×C₃²×C₈	1	1	yes	characteristic
76	C₂×C₅	C₃²×M₁₆	1	1	yes	characteristic
77	C₃²	C₄⋊#170	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₄	—	5	1	no	—
79	C₂×C₄	—	5	1	no	—
80	C₂×C₄	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₈	—	10	2	no	—
82	C₂×C₃	C₃×#700	1	4	yes	—
83	C₂×C₃	C₃×C₄⋊AGL(1,𝔽₅)	1	4	yes	—
84	C₂×C₃	C₂×C₃×#170	1	4	yes	—
85	C₅	C₃²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
86	C₂²	C₂×C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₄	—	5	1	no	—
88	C₄	—	5	1	no	—
89	C₄	—	10	1	no	—
90	C₄	C₃²×#170	1	2	yes	complemented
91	C₃	C₃×C₄⋊#170	1	4	yes	direct product factor
92	C₂	C₃²×#700	1	1	yes	characteristic
93	C₂	C₃²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
94	C₂	C₂×C₃²×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃²×C₈
Frattini	C₂²	C₂×C₃²×AGL(1,𝔽₅)
Fitting	C₂×C₃²×C₄×C₅	C₄
socle, prime radical	C₂²×C₃²×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂²×C₃²
hypercenter	C₂×C₃²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃²×C₄⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊#170	C₃	direct (N × H)
12. C₃²×C₄×C₅	C₈	semidirect
16. C₄⋊#170	C₃²	direct (N × H)
34. C₃×C₄×C₅	C₃×C₈	semidirect
40. C₃²×C₅	C₄⋊C₈	semidirect
49. C₃²×C₄	#170	semidirect
59. C₄×C₅	C₃²×C₈	semidirect
68. C₃×C₅	C₃×C₄⋊C₈	semidirect
73. C₃×C₄	C₃×#170	semidirect
77. C₃²	C₄⋊#170	direct (N × H)
85. C₅	C₃²×C₄⋊C₈	semidirect
90. C₄	C₃²×#170	semidirect
91. C₃	C₃×C₄⋊#170	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#170	2
C₃²×C₄×Dic₅	2
C₃×C₄⋊#170	3
C₃²×C₄⋊C₈	5

Subgroups — C32×C4⋊#170 #13086255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄⋊#170 #13086255