Subgroups — C₂×C₃₁×#504 #16843339

Order 3968.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#504	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃₁×M₃₂	C₂	1	3	yes	maximal, complemented
4	C₂²×C₁₆×C₃₁	C₂	1	3	yes	maximal, complemented
5	C₂×#41×C₃₁	C₄	1	1	yes	characteristic
6	C₃₁×#131	C₄	1	4	yes	—
7	C₃₁×#131	C₂²	1	4	yes	—
8	C₂×M₁₆×C₃₁	C₄	1	3	yes	—
9	C₂²×C₈×C₃₁	C₂²	1	3	yes	—
10	C₃₁×M₃₂	C₂²	1	12	yes	complemented
11	C₂×C₁₆×C₃₁	C₂²	1	3	yes	—
12	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₃₁	C₂²	1	12	yes	complemented
14	#41×C₃₁	C₈	1	4	yes	—
15	#41×C₃₁	C₂×C₄	1	4	yes	—
16	C₂×Q₈×C₃₁	C₈	1	1	yes	characteristic
17	C₂×D₄×C₃₁	C₈	1	3	yes	—
18	C₂²×C₄×C₃₁	C₂×C₄	1	3	yes	—
19	M₁₆×C₃₁	C₂×C₄	1	12	yes	—
20	C₂×C₈×C₃₁	C₂×C₄	1	3	yes	—
21	C₂×C₈×C₃₁	C₂×C₄	1	6	yes	—
22	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₃₁	C₂³	1	6	yes	—
24	C₁₆×C₃₁	C₂³	1	6	yes	—
25	C₁₆×C₃₁	C₂³	1	2	yes	—
26	C₂³×C₃₁	C₂×C₈	1	3	yes	—
27	Q₈×C₃₁	C₂×C₈	1	4	yes	—
28	D₄×C₃₁	C₂×C₈	1	12	yes	—
29	C₂×C₄×C₃₁	C₂×C₈	1	3	yes	—
30	C₂×C₄×C₃₁	C₂×C₈	1	6	yes	—
31	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	C₂²×C₄	1	6	yes	—
33	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₃₁	C₂²×C₄	1	6	yes	—
35	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
36	C₂×#504	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₃₁	—	2	6	no	—
38	C₂²×C₃₁	C₂²×C₈	1	6	yes	—
39	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
41	C₄×C₃₁	C₂²×C₈	1	6	yes	—
42	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
43	C₂×#131	C₂×C₃₁	1	1	yes	characteristic
44	#504	C₂×C₃₁	1	8	yes	direct product factor
45	C₂×M₃₂	C₂×C₃₁	1	3	yes	complemented
46	C₂²×C₁₆	C₂×C₃₁	1	3	yes	complemented
47	C₂×C₃₁	—	2	6	no	—
48	C₂×C₃₁	#504	1	2	yes	direct product factor
49	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
50	C₂×#41	C₄×C₃₁	1	1	yes	characteristic
51	#131	C₄×C₃₁	1	4	yes	—
52	#131	C₂²×C₃₁	1	4	yes	—
53	C₂×M₁₆	C₄×C₃₁	1	3	yes	—
54	C₂²×C₈	C₂²×C₃₁	1	3	yes	—
55	M₃₂	C₂²×C₃₁	1	12	yes	complemented
56	C₂×C₁₆	C₂²×C₃₁	1	3	yes	—
57	C₂×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
58	C₂×C₁₆	C₂²×C₃₁	1	12	yes	complemented
59	C₃₁	C₂×#504	1	1	yes	characteristic, direct product factor, Sylow
60	#41	C₈×C₃₁	1	4	yes	—
61	#41	C₂×C₄×C₃₁	1	4	yes	—
62	C₂×Q₈	C₈×C₃₁	1	1	yes	characteristic
63	C₂×D₄	C₈×C₃₁	1	3	yes	—
64	C₂²×C₄	C₂×C₄×C₃₁	1	3	yes	—
65	M₁₆	C₂×C₄×C₃₁	1	12	yes	—
66	C₂×C₈	C₂×C₄×C₃₁	1	3	yes	—
67	C₂×C₈	C₂×C₄×C₃₁	1	6	yes	—
68	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
69	C₂×C₈	C₂³×C₃₁	1	6	yes	—
70	C₁₆	C₂³×C₃₁	1	2	yes	—
71	C₁₆	C₂³×C₃₁	1	6	yes	—
72	C₂³	C₂×C₈×C₃₁	1	3	yes	—
73	Q₈	C₂×C₈×C₃₁	1	4	yes	—
74	D₄	C₂×C₈×C₃₁	1	12	yes	—
75	C₂×C₄	C₂×C₈×C₃₁	1	3	yes	—
76	C₂×C₄	C₂×C₈×C₃₁	1	6	yes	—
77	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
78	C₂×C₄	C₂²×C₄×C₃₁	1	6	yes	—
79	C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
80	C₈	C₂²×C₄×C₃₁	1	6	yes	—
81	C₈	C₂⁴×C₃₁	1	1	yes	characteristic
82	C₂²	—	2	6	no	—
83	C₂²	C₂²×C₈×C₃₁	1	6	yes	—
84	C₂²	C₂²×C₈×C₃₁	1	1	yes	characteristic
85	C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
86	C₄	C₂²×C₈×C₃₁	1	6	yes	—
87	C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
88	C₂	—	2	6	no	—
89	C₂	C₃₁×#504	1	2	yes	direct product factor
90	C₂	C₂³×C₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₃₁	C₂²
commutator	C₂	C₂³×C₈×C₃₁
Frattini, cosocle	C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₈
prime radical	C₂×#41×C₃₁	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#504	C₂	direct (N × H)
3. C₂×C₃₁×M₃₂	C₂	semidirect
4. C₂²×C₁₆×C₃₁	C₂	semidirect
10. C₃₁×M₃₂	C₂²	semidirect
13. C₂×C₁₆×C₃₁	C₂²	semidirect
36. C₂×#504	C₃₁	direct (N × H)
44. #504	C₂×C₃₁	direct (N × H)
45. C₂×M₃₂	C₂×C₃₁	semidirect
46. C₂²×C₁₆	C₂×C₃₁	semidirect
48. C₂×C₃₁	#504	direct (N × H)
55. M₃₂	C₂²×C₃₁	semidirect
58. C₂×C₁₆	C₂²×C₃₁	semidirect
59. C₃₁	C₂×#504	direct (N × H)
89. C₂	C₃₁×#504	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×C₃₁×#131	2
C₃₁×#504	2
C₂×C₃₁×M₃₂	2
C₂²×C₁₆×C₃₁	2
C₂×#504	31

Subgroups — C2×C31×#504 #16843339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#504 #16843339