Subgroups — C₂²×C₄²×D₃₁ #16858865

Order 3968.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₃₁	C₂	1	3	yes	maximal
2	C₂²×C₄×Dic₃₁	C₂	1	3	yes	maximal, complemented
3	C₂×C₄²×D₃₁	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₃₁	C₂²	1	1	yes	characteristic
6	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
7	C₂³×Dic₃₁	C₂²	1	3	yes	—
8	C₂²×C₄×D₃₁	C₄	1	48	yes	direct product factor
9	C₂²×C₄×D₃₁	C₂²	1	36	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	12	yes	complemented
11	C₂×C₄×Dic₃₁	C₂²	1	36	yes	complemented
12	C₄²×D₃₁	C₂²	1	64	yes	direct product factor
13	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
14	C₂³×D₃₁	C₂×C₄	1	24	yes	—
15	C₂³×D₃₁	C₂³	1	6	yes	—
16	C₂²×C₄×C₃₁	C₂×C₄	1	24	yes	complemented
17	C₂²×C₄×C₃₁	C₂³	1	18	yes	—
18	C₂²×Dic₃₁	C₂×C₄	1	24	yes	complemented
19	C₂²×Dic₃₁	C₂³	1	18	yes	—
20	C₂×C₄×D₃₁	C₂×C₄	1	288	yes	direct product factor
21	C₂×C₄×D₃₁	C₂³	1	48	yes	—
22	C₄²×C₃₁	C₂³	1	16	yes	complemented
23	C₄×Dic₃₁	C₂³	1	48	yes	complemented
24	C₂³×C₃₁	C₂²×C₄	1	12	yes	—
25	C₂³×C₃₁	C₂⁴	1	3	yes	—
26	C₂²×D₃₁	C₄²	1	64	yes	direct product factor
27	C₂²×D₃₁	C₂²×C₄	1	72	yes	—
28	C₂²×D₃₁	C₂⁴	1	4	yes	—
29	C₂×C₄×C₃₁	C₂²×C₄	1	72	yes	complemented
30	C₂×C₄×C₃₁	C₂⁴	1	12	yes	—
31	C₂×Dic₃₁	C₂²×C₄	1	72	yes	complemented
32	C₂×Dic₃₁	C₂⁴	1	12	yes	—
33	C₄×D₃₁	C₂²×C₄	1	192	yes	direct product factor
34	C₂³×C₄²	—	31	1	no	maximal, Sylow
35	C₂²×C₃₁	C₂×C₄²	1	16	yes	complemented
36	C₂²×C₃₁	C₂³×C₄	1	18	yes	—
37	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
38	C₂×D₃₁	C₂×C₄²	1	96	yes	direct product factor
39	C₂×D₃₁	C₂³×C₄	1	24	yes	—
40	C₄×C₃₁	C₂³×C₄	1	24	yes	complemented
41	Dic₃₁	C₂³×C₄	1	24	yes	complemented
42	C₂⁴×C₄	—	31	3	no	—
43	C₂²×C₄²	—	31	3	no	—
44	C₂²×C₄²	—	31	24	no	—
45	C₂²×C₄²	D₃₁	1	1	yes	characteristic, direct product factor
46	C₂×C₃₁	C₂²×C₄²	1	12	yes	complemented
47	C₂×C₃₁	C₂⁴×C₄	1	3	yes	—
48	D₃₁	C₂²×C₄²	1	16	yes	direct product factor
49	C₂⁵	—	31	1	no	—
50	C₂³×C₄	—	31	3	no	—
51	C₂³×C₄	—	31	48	no	—
52	C₂³×C₄	—	31	36	no	—
53	C₂³×C₄	C₂×D₃₁	1	3	yes	—
54	C₂×C₄²	—	31	36	no	—
55	C₂×C₄²	—	31	64	no	—
56	C₂×C₄²	C₂×D₃₁	1	12	yes	direct product factor
57	C₃₁	C₂³×C₄²	1	1	yes	characteristic, complemented, Sylow
58	C₂⁴	—	31	24	no	—
59	C₂⁴	—	31	6	no	—
60	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
61	C₂²×C₄	—	31	24	no	—
62	C₂²×C₄	—	31	18	no	—
63	C₂²×C₄	—	31	288	no	—
64	C₂²×C₄	—	31	48	no	—
65	C₂²×C₄	C₄×D₃₁	1	24	yes	direct product factor
66	C₂²×C₄	C₂²×D₃₁	1	18	yes	—
67	C₄²	—	31	48	no	—
68	C₄²	C₂²×D₃₁	1	16	yes	direct product factor
69	C₂³	—	31	64	no	—
70	C₂³	—	31	72	no	—
71	C₂³	—	31	4	no	—
72	C₂³	C₂×C₄×D₃₁	1	12	yes	—
73	C₂³	C₂³×D₃₁	1	3	yes	—
74	C₂×C₄	—	31	72	no	—
75	C₂×C₄	—	31	12	no	—
76	C₂×C₄	—	31	192	no	—
77	C₂×C₄	C₂×C₄×D₃₁	1	72	yes	direct product factor
78	C₂×C₄	C₂³×D₃₁	1	12	yes	—
79	C₂²	—	31	24	no	—
80	C₂²	—	31	96	no	—
81	C₂²	C₄²×D₃₁	1	16	yes	direct product factor
82	C₂²	C₂²×C₄×D₃₁	1	18	yes	—
83	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
84	C₄	—	31	24	no	—
85	C₄	C₂²×C₄×D₃₁	1	24	yes	direct product factor
86	C₂	—	31	16	no	—
87	C₂	C₂×C₄²×D₃₁	1	12	yes	direct product factor
88	C₂	C₂³×C₄×D₃₁	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₃₁
commutator, hypocenter	C₃₁	C₂³×C₄²
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂²×C₄²×C₃₁	C₂
socle	C₂⁴×C₃₁	C₂³
prime radical	C₂⁴×D₃₁	C₂²
cosocle	C₂²×C₃₁	C₂⁵
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₃₁	C₂	semidirect
3. C₂×C₄²×D₃₁	C₂	direct (N × H)
4. C₂²×C₄²×C₃₁	C₂	semidirect
8. C₂²×C₄×D₃₁	C₄	direct (N × H)
10. C₂×C₄²×C₃₁	C₂²	semidirect
11. C₂×C₄×Dic₃₁	C₂²	semidirect
12. C₄²×D₃₁	C₂²	direct (N × H)
16. C₂²×C₄×C₃₁	C₂×C₄	semidirect
18. C₂²×Dic₃₁	C₂×C₄	semidirect
20. C₂×C₄×D₃₁	C₂×C₄	direct (N × H)
22. C₄²×C₃₁	C₂³	semidirect
23. C₄×Dic₃₁	C₂³	semidirect
26. C₂²×D₃₁	C₄²	direct (N × H)
29. C₂×C₄×C₃₁	C₂²×C₄	semidirect
31. C₂×Dic₃₁	C₂²×C₄	semidirect
33. C₄×D₃₁	C₂²×C₄	direct (N × H)
35. C₂²×C₃₁	C₂×C₄²	semidirect
38. C₂×D₃₁	C₂×C₄²	direct (N × H)
40. C₄×C₃₁	C₂³×C₄	semidirect
41. Dic₃₁	C₂³×C₄	semidirect
45. C₂²×C₄²	D₃₁	direct (N × H)
46. C₂×C₃₁	C₂²×C₄²	semidirect
48. D₃₁	C₂²×C₄²	direct (N × H)
56. C₂×C₄²	C₂×D₃₁	direct (N × H)
57. C₃₁	C₂³×C₄²	semidirect
65. C₂²×C₄	C₄×D₃₁	direct (N × H)
68. C₄²	C₂²×D₃₁	direct (N × H)
77. C₂×C₄	C₂×C₄×D₃₁	direct (N × H)
81. C₂²	C₄²×D₃₁	direct (N × H)
85. C₄	C₂²×C₄×D₃₁	direct (N × H)
87. C₂	C₂×C₄²×D₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₃₁	2
C₂²×C₄×Dic₃₁	2
C₂×C₄²×D₃₁	2
C₂²×C₄²×C₃₁	2
C₂³×C₄²	31

Subgroups — C22×C42×D31 #16858865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×D₃₁ #16858865