Subgroups — C₂²×C₁₆×D₃₁ #16858803

Order 3968.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×D₃₁	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×D₃₁	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₃₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×D₃₁	C₄	1	6	yes	—
9	C₂×C₈×D₃₁	C₂²	1	6	yes	—
10	C₂×C₁₆×C₃₁	C₂²	1	6	yes	complemented
11	C₂×C₃₁⋊C₁₆	C₂²	1	6	yes	complemented
12	C₁₆×D₃₁	C₂²	1	16	yes	direct product factor
13	C₂³×D₃₁	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₃₁	C₈	1	1	yes	characteristic
16	C₂×C₄×D₃₁	C₈	1	6	yes	—
17	C₂×C₄×D₃₁	C₂×C₄	1	6	yes	—
18	C₂×C₈×C₃₁	C₂×C₄	1	3	yes	—
19	C₂×C₈×C₃₁	C₂³	1	3	yes	—
20	C₂×C₃₁⋊C₈	C₂×C₄	1	6	yes	—
21	C₈×D₃₁	C₂×C₄	1	12	yes	—
22	C₈×D₃₁	C₂³	1	4	yes	—
23	C₁₆×C₃₁	C₂³	1	4	yes	complemented
24	C₃₁⋊C₁₆	C₂³	1	4	yes	complemented
25	C₂³×C₃₁	C₂×C₈	1	1	yes	characteristic
26	C₂²×D₃₁	C₁₆	1	8	yes	direct product factor
27	C₂²×D₃₁	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₃₁	C₂×C₈	1	3	yes	—
29	C₂×C₄×C₃₁	C₂²×C₄	1	3	yes	—
30	C₂×Dic₃₁	C₂×C₈	1	6	yes	—
31	C₄×D₃₁	C₂×C₈	1	12	yes	—
32	C₄×D₃₁	C₂²×C₄	1	4	yes	—
33	C₈×C₃₁	C₂²×C₄	1	3	yes	—
34	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
35	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
36	C₂³×C₁₆	—	31	1	no	maximal, Sylow
37	C₂²×C₃₁	C₂×C₁₆	1	4	yes	complemented
38	C₂²×C₃₁	C₂²×C₈	1	3	yes	—
39	C₂×D₃₁	C₂×C₁₆	1	24	yes	direct product factor
40	C₂×D₃₁	C₂²×C₈	1	4	yes	—
41	C₄×C₃₁	C₂²×C₈	1	3	yes	—
42	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
43	Dic₃₁	C₂²×C₈	1	4	yes	—
44	C₂³×C₈	—	31	1	no	—
45	C₂²×C₁₆	—	31	1	no	—
46	C₂²×C₁₆	—	31	12	no	—
47	C₂²×C₁₆	D₃₁	1	1	yes	characteristic, direct product factor
48	C₂×C₃₁	C₂²×C₁₆	1	6	yes	complemented
49	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
50	D₃₁	C₂²×C₁₆	1	8	yes	direct product factor
51	C₂³×C₄	—	31	1	no	—
52	C₂²×C₈	—	31	1	no	—
53	C₂²×C₈	—	31	6	no	—
54	C₂²×C₈	—	31	6	no	—
55	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
56	C₂×C₁₆	—	31	6	no	—
57	C₂×C₁₆	—	31	16	no	—
58	C₂×C₁₆	C₂×D₃₁	1	6	yes	direct product factor
59	C₃₁	C₂³×C₁₆	1	1	yes	characteristic, complemented, Sylow
60	C₂⁴	—	31	1	no	—
61	C₂²×C₄	—	31	1	no	—
62	C₂²×C₄	—	31	6	no	—
63	C₂²×C₄	—	31	6	no	—
64	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
65	C₂×C₈	—	31	6	no	—
66	C₂×C₈	—	31	12	no	—
67	C₂×C₈	—	31	4	no	—
68	C₂×C₈	C₄×D₃₁	1	3	yes	—
69	C₂×C₈	C₂²×D₃₁	1	3	yes	—
70	C₁₆	—	31	4	no	—
71	C₁₆	C₂²×D₃₁	1	4	yes	direct product factor
72	C₂³	—	31	8	no	—
73	C₂³	—	31	6	no	—
74	C₂³	C₈×D₃₁	1	1	yes	characteristic
75	C₂×C₄	—	31	6	no	—
76	C₂×C₄	—	31	12	no	—
77	C₂×C₄	—	31	4	no	—
78	C₂×C₄	C₈×D₃₁	1	3	yes	—
79	C₂×C₄	C₂×C₄×D₃₁	1	3	yes	—
80	C₈	—	31	4	no	—
81	C₈	C₂×C₄×D₃₁	1	3	yes	—
82	C₈	C₂³×D₃₁	1	1	yes	characteristic
83	C₂²	—	31	24	no	—
84	C₂²	—	31	4	no	—
85	C₂²	C₁₆×D₃₁	1	4	yes	direct product factor
86	C₂²	C₂×C₈×D₃₁	1	3	yes	—
87	C₄	—	31	4	no	—
88	C₄	C₂×C₈×D₃₁	1	3	yes	—
89	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
90	C₂	—	31	8	no	—
91	C₂	C₂×C₁₆×D₃₁	1	6	yes	direct product factor
92	C₂	C₂²×C₈×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	D₃₁
commutator, hypocenter	C₃₁	C₂³×C₁₆
Frattini	C₈	C₂³×D₃₁
Fitting	C₂²×C₁₆×C₃₁	C₂
socle	C₂³×C₃₁	C₂×C₈
prime radical	C₂³×D₃₁	C₈
cosocle	C₈×C₃₁	C₂⁴
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁⋊C₁₆	C₂	semidirect
3. C₂×C₁₆×D₃₁	C₂	direct (N × H)
4. C₂²×C₁₆×C₃₁	C₂	semidirect
10. C₂×C₁₆×C₃₁	C₂²	semidirect
11. C₂×C₃₁⋊C₁₆	C₂²	semidirect
12. C₁₆×D₃₁	C₂²	direct (N × H)
23. C₁₆×C₃₁	C₂³	semidirect
24. C₃₁⋊C₁₆	C₂³	semidirect
26. C₂²×D₃₁	C₁₆	direct (N × H)
37. C₂²×C₃₁	C₂×C₁₆	semidirect
39. C₂×D₃₁	C₂×C₁₆	direct (N × H)
47. C₂²×C₁₆	D₃₁	direct (N × H)
48. C₂×C₃₁	C₂²×C₁₆	semidirect
50. D₃₁	C₂²×C₁₆	direct (N × H)
58. C₂×C₁₆	C₂×D₃₁	direct (N × H)
59. C₃₁	C₂³×C₁₆	semidirect
71. C₁₆	C₂²×D₃₁	direct (N × H)
85. C₂²	C₁₆×D₃₁	direct (N × H)
91. C₂	C₂×C₁₆×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
C₂²×C₈×D₃₁	2
C₂²×C₃₁⋊C₁₆	2
C₂×C₁₆×D₃₁	2
C₂²×C₁₆×C₃₁	2
C₂³×C₁₆	31

Subgroups — C22×C16×D31 #16858803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₆×D₃₁ #16858803