Subgroups — C₂²×C₅×#226 #11756878

Order 960.

306 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#226	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×Q₁₆	—	3	1	no	maximal, Hall
4	C₂²×C₅×Dic₃	—	4	1	no	maximal
5	C₂×C₅×SL(2,𝔽₃)	C₂²	1	3	yes	—
6	C₅×#226	C₂²	1	4	yes	direct product factor
7	C₂²×#226	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₅×Q₈	—	3	1	no	—
9	C₂²×C₅×Q₈	D₃	1	1	yes	characteristic
10	C₂×C₅×Q₁₆	—	3	6	no	—
11	C₂×C₅×Q₁₆	—	3	6	no	—
12	C₂²×C₅×C₈	—	3	1	no	—
13	C₂³×C₃×C₅	—	4	1	no	—
14	C₂×C₅×Dic₃	—	4	6	no	—
15	C₅×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
16	C₂²×SL(2,𝔽₃)	C₂×C₅	1	1	yes	characteristic
17	C₂×#226	C₂×C₅	1	6	yes	direct product factor
18	C₂×C₅×Q₈	—	3	3	no	—
19	C₂×C₅×Q₈	—	3	6	no	—
20	C₂×C₅×Q₈	—	6	3	no	—
21	C₂×C₅×Q₈	C₂×D₃	1	3	yes	—
22	C₂²×C₄×C₅	—	3	1	no	—
23	C₂²×C₄×C₅	—	6	1	no	—
24	C₅×Q₁₆	—	3	12	no	—
25	C₅×Q₁₆	—	3	4	no	—
26	C₂×C₅×C₈	—	3	6	no	—
27	C₂²×Q₁₆	—	3	1	no	Sylow
28	C₂²×C₃×C₅	—	4	4	no	—
29	C₂²×C₃×C₅	—	4	3	no	—
30	C₅×Dic₃	—	4	4	no	—
31	C₂²×Dic₃	—	4	1	no	—
32	C₂×SL(2,𝔽₃)	C₂²×C₅	1	3	yes	—
33	#226	C₂²×C₅	1	4	yes	direct product factor
34	C₂³×C₅	S₄	1	1	yes	characteristic
35	C₅×Q₈	—	3	3	no	—
36	C₅×Q₈	—	3	4	no	—
37	C₅×Q₈	—	6	1	no	—
38	C₅×Q₈	—	6	6	no	—
39	C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
40	C₂×C₄×C₅	—	3	3	no	—
41	C₂×C₄×C₅	—	3	3	no	—
42	C₂×C₄×C₅	—	6	6	no	—
43	C₅×C₈	—	3	4	no	—
44	C₂²×Q₈	—	3	1	no	—
45	C₂²×Q₈	C₅×D₃	1	1	yes	characteristic
46	C₂×Q₁₆	—	3	6	no	—
47	C₂×Q₁₆	—	3	6	no	—
48	C₂²×C₈	—	3	1	no	—
49	C₂×C₃×C₅	—	4	6	no	—
50	C₂×C₃×C₅	—	4	1	no	—
51	C₂³×C₃	—	4	1	no	—
52	C₂×Dic₃	—	4	6	no	—
53	SL(2,𝔽₃)	C₂³×C₅	1	1	yes	characteristic
54	C₂²×C₅	#226	1	4	yes	direct product factor
55	C₂²×C₅	C₂×S₄	1	3	yes	—
56	C₄×C₅	—	3	1	no	—
57	C₄×C₅	—	3	3	no	—
58	C₄×C₅	—	6	4	no	—
59	C₂×Q₈	—	3	3	no	—
60	C₂×Q₈	—	3	6	no	—
61	C₂×Q₈	—	6	3	no	—
62	C₂×Q₈	C₂×C₅×D₃	1	3	yes	—
63	C₂²×C₄	—	3	1	no	—
64	C₂²×C₄	—	6	1	no	—
65	Q₁₆	—	3	12	no	—
66	Q₁₆	—	3	4	no	—
67	C₂×C₈	—	3	6	no	—
68	C₃×C₅	—	4	1	no	Hall
69	C₂²×C₃	—	4	4	no	—
70	C₂²×C₃	—	4	3	no	—
71	Dic₃	—	4	4	no	—
72	C₂×C₅	C₂×#226	1	6	yes	direct product factor
73	C₂×C₅	C₂²×S₄	1	1	yes	characteristic
74	C₂³	C₅×S₄	1	1	yes	characteristic
75	Q₈	—	3	3	no	—
76	Q₈	—	3	4	no	—
77	Q₈	—	6	1	no	—
78	Q₈	—	6	6	no	—
79	Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
80	C₂×C₄	—	3	3	no	—
81	C₂×C₄	—	3	3	no	—
82	C₂×C₄	—	6	6	no	—
83	C₈	—	3	4	no	—
84	C₂×C₃	—	4	6	no	—
85	C₂×C₃	—	4	1	no	—
86	C₅	C₂²×#226	1	1	yes	characteristic, direct product factor, Sylow
87	C₂²	C₅×#226	1	4	yes	direct product factor
88	C₂²	C₂×C₅×S₄	1	3	yes	—
89	C₄	—	3	1	no	—
90	C₄	—	3	3	no	—
91	C₄	—	6	4	no	—
92	C₃	—	4	1	no	Sylow
93	C₂	C₂×C₅×#226	1	6	yes	direct product factor
94	C₂	C₂²×C₅×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂³×C₅	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂	C₂²×C₅×S₄
Fitting	C₂²×C₅×Q₈	D₃
prime radical	C₂²×C₅×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂³×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#226	C₂	direct (N × H)
6. C₅×#226	C₂²	direct (N × H)
7. C₂²×#226	C₅	direct (N × H)
17. C₂×#226	C₂×C₅	direct (N × H)
33. #226	C₂²×C₅	direct (N × H)
54. C₂²×C₅	#226	direct (N × H)
72. C₂×C₅	C₂×#226	direct (N × H)
86. C₅	C₂²×#226	direct (N × H)
87. C₂²	C₅×#226	direct (N × H)
93. C₂	C₂×C₅×#226	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×C₅×SL(2,𝔽₃)	2
C₂×C₅×#226	2
C₂²×C₅×Q₁₆	3
C₂²×C₅×Dic₃	4
C₂²×#226	5

Subgroups — C22×C5×#226 #11756878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#226 #11756878