Subgroups — C₂²×C₃²×2₋¹⁺⁴ #11770641

Order 1152.

17616 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×2₋¹⁺⁴	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃²×#41	C₂	1	10	yes	maximal, complemented
3	C₂³×C₃²×Q₈	C₂	1	5	yes	maximal, complemented
4	C₂²×C₃×2₋¹⁺⁴	C₃	1	4	yes	maximal, direct product factor
5	C₃²×2₋¹⁺⁴	C₂²	1	256	yes	direct product factor
6	C₂×C₃²×#41	C₂²	1	240	yes	complemented
7	C₂²×C₃²×Q₈	C₂²	1	10	yes	—
8	C₂²×C₃²×Q₈	C₂²	1	120	yes	complemented
9	C₂²×C₃²×D₄	C₂²	1	10	yes	—
10	C₂³×C₃²×C₄	C₂²	1	15	yes	—
11	C₂×C₃×2₋¹⁺⁴	C₂×C₃	1	192	yes	direct product factor
12	C₂²×C₃×#41	C₂×C₃	1	40	yes	complemented
13	C₂³×C₃×Q₈	C₂×C₃	1	20	yes	complemented
14	C₂⁴×C₃²	C₂³	1	5	yes	—
15	C₃²×#41	C₂³	1	640	yes	complemented
16	C₂×C₃²×Q₈	C₂³	1	120	yes	—
17	C₂×C₃²×Q₈	C₂³	1	320	yes	complemented
18	C₂×C₃²×D₄	C₂³	1	120	yes	—
19	C₂²×C₃²×C₄	C₂³	1	180	yes	—
20	C₂²×C₃²×C₄	C₂³	1	10	yes	—
21	C₂²×2₋¹⁺⁴	C₃²	1	1	yes	characteristic, direct product factor, Sylow
22	C₃×2₋¹⁺⁴	C₂²×C₃	1	1024	yes	direct product factor
23	C₂×C₃×#41	C₂²×C₃	1	960	yes	complemented
24	C₂²×C₃×Q₈	C₂²×C₃	1	40	yes	—
25	C₂²×C₃×Q₈	C₂²×C₃	1	480	yes	complemented
26	C₂²×C₃×D₄	C₂²×C₃	1	40	yes	—
27	C₂³×C₃×C₄	C₂²×C₃	1	60	yes	—
28	C₂³×C₃²	—	2	20	no	—
29	C₂³×C₃²	C₂⁴	1	30	yes	—
30	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
31	C₃²×Q₈	C₂⁴	1	160	yes	—
32	C₃²×D₄	C₂⁴	1	160	yes	—
33	C₂×C₃²×C₄	C₂⁴	1	60	yes	—
34	C₂×C₃²×C₄	C₂⁴	1	240	yes	—
35	C₂×2₋¹⁺⁴	C₂×C₃²	1	48	yes	direct product factor
36	C₂²×#41	C₂×C₃²	1	10	yes	complemented
37	C₂³×Q₈	C₂×C₃²	1	5	yes	complemented
38	C₂⁴×C₃	C₂³×C₃	1	20	yes	—
39	C₃×#41	C₂³×C₃	1	2560	yes	complemented
40	C₂×C₃×Q₈	C₂³×C₃	1	480	yes	—
41	C₂×C₃×Q₈	C₂³×C₃	1	1280	yes	complemented
42	C₂×C₃×D₄	C₂³×C₃	1	480	yes	—
43	C₂²×C₃×C₄	C₂³×C₃	1	40	yes	—
44	C₂²×C₃×C₄	C₂³×C₃	1	720	yes	—
45	C₂²×C₃²	—	2	60	no	—
46	C₂²×C₃²	2₋¹⁺⁴	1	4	yes	direct product factor
47	C₂²×C₃²	C₂⁵	1	20	yes	—
48	C₂²×C₃²	C₂⁵	1	3	yes	—
49	C₃²×C₄	C₂⁵	1	40	yes	—
50	2₋¹⁺⁴	C₂²×C₃²	1	256	yes	direct product factor
51	C₂×#41	C₂²×C₃²	1	240	yes	complemented
52	C₂²×Q₈	C₂²×C₃²	1	10	yes	—
53	C₂²×Q₈	C₂²×C₃²	1	120	yes	complemented
54	C₂²×D₄	C₂²×C₃²	1	10	yes	—
55	C₂³×C₄	C₂²×C₃²	1	15	yes	—
56	C₂³×C₃	—	2	80	no	—
57	C₂³×C₃	C₂⁴×C₃	1	120	yes	—
58	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
59	C₃×Q₈	C₂⁴×C₃	1	640	yes	—
60	C₃×D₄	C₂⁴×C₃	1	640	yes	—
61	C₂×C₃×C₄	C₂⁴×C₃	1	240	yes	—
62	C₂×C₃×C₄	C₂⁴×C₃	1	960	yes	—
63	C₂×C₃²	—	2	20	no	—
64	C₂×C₃²	C₂×2₋¹⁺⁴	1	6	yes	direct product factor
65	C₂×C₃²	C₂⁶	1	1	yes	characteristic
66	C₂⁴	C₂³×C₃²	1	5	yes	—
67	#41	C₂³×C₃²	1	640	yes	complemented
68	C₂×Q₈	C₂³×C₃²	1	120	yes	—
69	C₂×Q₈	C₂³×C₃²	1	320	yes	complemented
70	C₂×D₄	C₂³×C₃²	1	120	yes	—
71	C₂²×C₄	C₂³×C₃²	1	10	yes	—
72	C₂²×C₄	C₂³×C₃²	1	180	yes	—
73	C₂²×C₃	—	2	240	no	—
74	C₂²×C₃	C₃×2₋¹⁺⁴	1	16	yes	direct product factor
75	C₂²×C₃	C₂⁵×C₃	1	80	yes	—
76	C₂²×C₃	C₂⁵×C₃	1	12	yes	—
77	C₃×C₄	C₂⁵×C₃	1	160	yes	—
78	C₃²	C₂²×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
79	C₂³	—	2	20	no	—
80	C₂³	C₂⁴×C₃²	1	30	yes	—
81	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
82	Q₈	C₂⁴×C₃²	1	160	yes	—
83	D₄	C₂⁴×C₃²	1	160	yes	—
84	C₂×C₄	C₂⁴×C₃²	1	60	yes	—
85	C₂×C₄	C₂⁴×C₃²	1	240	yes	—
86	C₂×C₃	—	2	80	no	—
87	C₂×C₃	C₂×C₃×2₋¹⁺⁴	1	24	yes	direct product factor
88	C₂×C₃	C₂⁶×C₃	1	4	yes	—
89	C₂²	—	2	60	no	—
90	C₂²	C₃²×2₋¹⁺⁴	1	4	yes	direct product factor
91	C₂²	C₂⁵×C₃²	1	20	yes	—
92	C₂²	C₂⁵×C₃²	1	3	yes	—
93	C₄	C₂⁵×C₃²	1	40	yes	—
94	C₃	C₂²×C₃×2₋¹⁺⁴	1	4	yes	direct product factor
95	C₂	—	2	20	no	—
96	C₂	C₂×C₃²×2₋¹⁺⁴	1	6	yes	direct product factor
97	C₂	C₂⁶×C₃²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×2₋¹⁺⁴	C₂	direct (N × H)
2. C₂²×C₃²×#41	C₂	semidirect
3. C₂³×C₃²×Q₈	C₂	semidirect
4. C₂²×C₃×2₋¹⁺⁴	C₃	direct (N × H)
5. C₃²×2₋¹⁺⁴	C₂²	direct (N × H)
6. C₂×C₃²×#41	C₂²	semidirect
8. C₂²×C₃²×Q₈	C₂²	semidirect
11. C₂×C₃×2₋¹⁺⁴	C₂×C₃	direct (N × H)
12. C₂²×C₃×#41	C₂×C₃	semidirect
13. C₂³×C₃×Q₈	C₂×C₃	semidirect
15. C₃²×#41	C₂³	semidirect
17. C₂×C₃²×Q₈	C₂³	semidirect
21. C₂²×2₋¹⁺⁴	C₃²	direct (N × H)
22. C₃×2₋¹⁺⁴	C₂²×C₃	direct (N × H)
23. C₂×C₃×#41	C₂²×C₃	semidirect
25. C₂²×C₃×Q₈	C₂²×C₃	semidirect
35. C₂×2₋¹⁺⁴	C₂×C₃²	direct (N × H)
36. C₂²×#41	C₂×C₃²	semidirect
37. C₂³×Q₈	C₂×C₃²	semidirect
39. C₃×#41	C₂³×C₃	semidirect
41. C₂×C₃×Q₈	C₂³×C₃	semidirect
46. C₂²×C₃²	2₋¹⁺⁴	direct (N × H)
50. 2₋¹⁺⁴	C₂²×C₃²	direct (N × H)
51. C₂×#41	C₂²×C₃²	semidirect
53. C₂²×Q₈	C₂²×C₃²	semidirect
64. C₂×C₃²	C₂×2₋¹⁺⁴	direct (N × H)
67. #41	C₂³×C₃²	semidirect
69. C₂×Q₈	C₂³×C₃²	semidirect
74. C₂²×C₃	C₃×2₋¹⁺⁴	direct (N × H)
78. C₃²	C₂²×2₋¹⁺⁴	direct (N × H)
87. C₂×C₃	C₂×C₃×2₋¹⁺⁴	direct (N × H)
90. C₂²	C₃²×2₋¹⁺⁴	direct (N × H)
94. C₃	C₂²×C₃×2₋¹⁺⁴	direct (N × H)
96. C₂	C₂×C₃²×2₋¹⁺⁴	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×C₃²×2₋¹⁺⁴	2
C₂²×C₃²×#41	2
C₂³×C₃²×Q₈	2
C₂²×C₃×2₋¹⁺⁴	3

Subgroups — C22×C32×2−1+4 #11770641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×2₋¹⁺⁴ #11770641