Subgroups — C₂²×C₃×#672 #11749993

Order 960.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#672	C₂	1	6	yes	maximal, direct product factor
3	C₂²×#672	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂×C₃×C₅×C₈	C₄	1	3	yes	—
6	C₂×C₃×C₅×C₈	C₂²	1	3	yes	—
7	C₃×#672	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₁₆	—	5	1	no	maximal, Hall
9	C₂²×C₅×C₈	C₂×C₃	1	1	yes	characteristic
10	C₂×#672	C₂×C₃	1	6	yes	direct product factor
11	C₂³×C₃×C₅	C₈	1	1	yes	characteristic
12	C₂×C₃×C₄×C₅	C₈	1	3	yes	—
13	C₂×C₃×C₄×C₅	C₂×C₄	1	3	yes	—
14	C₃×C₅×C₈	C₂×C₄	1	3	yes	—
15	C₃×C₅×C₈	C₂³	1	1	yes	characteristic
16	C₂²×C₃×C₈	D₅	1	1	yes	characteristic
17	C₂×C₃×C₁₆	—	5	6	no	—
18	C₂²×C₄×C₅	C₃×C₄	1	1	yes	characteristic
19	C₂×C₅×C₈	C₃×C₄	1	3	yes	—
20	C₂×C₅×C₈	C₂²×C₃	1	3	yes	—
21	#672	C₂²×C₃	1	4	yes	direct product factor
22	C₂²×C₁₆	—	5	1	no	Sylow
23	C₂²×C₃×C₅	C₁₆	1	4	yes	complemented
24	C₂²×C₃×C₅	C₂×C₈	1	3	yes	—
25	C₃×C₄×C₅	C₂×C₈	1	3	yes	—
26	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄	Dic₅	1	1	yes	characteristic
28	C₂×C₃×C₈	Dic₅	1	3	yes	—
29	C₂×C₃×C₈	C₂×D₅	1	3	yes	—
30	C₃×C₁₆	—	5	4	no	—
31	C₂³×C₅	C₃×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅	C₃×C₈	1	3	yes	—
33	C₂×C₄×C₅	C₂×C₃×C₄	1	3	yes	—
34	C₅×C₈	C₂×C₃×C₄	1	3	yes	—
35	C₅×C₈	C₂³×C₃	1	1	yes	characteristic
36	C₂²×C₈	C₃×D₅	1	1	yes	characteristic
37	C₂×C₁₆	—	5	6	no	—
38	C₂×C₃×C₅	C₂×C₁₆	1	6	yes	complemented
39	C₂×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
40	C₂³×C₃	#168	1	1	yes	characteristic
41	C₂×C₃×C₄	#168	1	3	yes	—
42	C₂×C₃×C₄	C₂×Dic₅	1	3	yes	—
43	C₃×C₈	C₂×Dic₅	1	3	yes	—
44	C₃×C₈	C₂²×D₅	1	1	yes	characteristic
45	C₂²×C₅	C₃×C₁₆	1	4	yes	complemented
46	C₂²×C₅	C₂×C₃×C₈	1	3	yes	—
47	C₄×C₅	C₂×C₃×C₈	1	3	yes	—
48	C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
49	C₂²×C₄	C₃×Dic₅	1	1	yes	characteristic
50	C₂×C₈	C₃×Dic₅	1	3	yes	—
51	C₂×C₈	C₂×C₃×D₅	1	3	yes	—
52	C₁₆	—	5	4	no	—
53	C₃×C₅	C₂²×C₁₆	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₃	#672	1	4	yes	direct product factor
55	C₂²×C₃	C₂×#168	1	3	yes	—
56	C₃×C₄	C₂×#168	1	3	yes	—
57	C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
58	C₂×C₅	C₂×C₃×C₁₆	1	6	yes	complemented
59	C₂×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
60	C₂³	C₃×#168	1	1	yes	characteristic
61	C₂×C₄	C₃×#168	1	3	yes	—
62	C₂×C₄	C₂×C₃×Dic₅	1	3	yes	—
63	C₈	C₂×C₃×Dic₅	1	3	yes	—
64	C₈	C₂²×C₃×D₅	1	1	yes	characteristic
65	C₂×C₃	C₂×#672	1	6	yes	direct product factor
66	C₂×C₃	C₂²×#168	1	1	yes	characteristic
67	C₅	C₂²×C₃×C₁₆	1	1	yes	characteristic, complemented, Sylow
68	C₂²	C₃×#672	1	4	yes	direct product factor
69	C₂²	C₂×C₃×#168	1	3	yes	—
70	C₄	C₂×C₃×#168	1	3	yes	—
71	C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
72	C₃	C₂²×#672	1	1	yes	characteristic, direct product factor, Sylow
73	C₂	C₂×C₃×#672	1	6	yes	direct product factor
74	C₂	C₂²×C₃×#168	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₈	D₅
commutator, hypocenter	C₅	C₂²×C₃×C₁₆
Frattini	C₈	C₂²×C₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₈
cosocle	C₅×C₈	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#672	C₂	direct (N × H)
3. C₂²×#672	C₃	direct (N × H)
7. C₃×#672	C₂²	direct (N × H)
10. C₂×#672	C₂×C₃	direct (N × H)
21. #672	C₂²×C₃	direct (N × H)
23. C₂²×C₃×C₅	C₁₆	semidirect
38. C₂×C₃×C₅	C₂×C₁₆	semidirect
45. C₂²×C₅	C₃×C₁₆	semidirect
53. C₃×C₅	C₂²×C₁₆	semidirect
54. C₂²×C₃	#672	direct (N × H)
58. C₂×C₅	C₂×C₃×C₁₆	semidirect
65. C₂×C₃	C₂×#672	direct (N × H)
67. C₅	C₂²×C₃×C₁₆	semidirect
68. C₂²	C₃×#672	direct (N × H)
72. C₃	C₂²×#672	direct (N × H)
73. C₂	C₂×C₃×#672	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×C₃×C₅×C₈	2
C₂×C₃×#672	2
C₂²×#672	3
C₂²×C₃×C₁₆	5

Subgroups — C22×C3×#672 #11749993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#672 #11749993