Subgroups — #16845457

Order 3968.

296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃₁⋊C₈	C₂	1	2	yes	maximal
2	#14520804	C₂	1	4	yes	maximal
3	C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₃₁	C₄	1	2	yes	—
5	C₂×C₄²×C₃₁	C₄	1	2	yes	—
6	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×C₃₁⋊C₈	C₂²	1	4	yes	—
9	C₄×C₃₁⋊C₈	C₄	1	8	yes	complemented
10	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
11	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
14	C₄²×C₃₁	—	2	4	no	—
15	C₄²×C₃₁	—	2	4	no	—
16	C₄²×C₃₁	C₈	1	8	yes	complemented
17	C₄²×C₃₁	C₂×C₄	1	4	yes	—
18	C₂×C₃₁⋊C₈	—	2	8	no	—
19	C₂×C₃₁⋊C₈	C₂×C₄	1	8	yes	—
20	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₁	—	2	4	no	—
22	C₂×C₄×C₃₁	—	2	4	no	—
23	C₂×C₄×C₃₁	C₄²	1	2	yes	—
24	C₂×C₄×C₃₁	C₄²	1	2	yes	—
25	C₂×C₄×C₃₁	C₂×C₈	1	8	yes	—
26	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₃₁	M₁₆	1	4	yes	—
28	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
29	C₂×C₄×C₃₁	#41	1	4	yes	—
30	C₃₁⋊C₈	—	2	8	no	—
31	#1744	—	31	1	no	maximal, Sylow
32	C₂²×C₃₁	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₃₁	#117	1	2	yes	—
34	C₂²×C₃₁	C₂²×C₈	1	2	yes	—
35	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
36	C₄×C₃₁	—	2	8	no	—
37	C₄×C₃₁	C₄×C₈	1	4	yes	—
38	C₄×C₃₁	#97	1	4	yes	—
39	C₄×C₃₁	#117	1	4	yes	—
40	C₂×C₄×C₈	—	31	2	no	—
41	C₄³	D₃₁	1	1	yes	characteristic
42	#336	—	31	4	no	—
43	C₂×C₃₁	#376	1	1	yes	characteristic
44	C₂×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
45	C₂×C₃₁	C₂×#97	1	1	yes	characteristic
46	C₂×C₃₁	#431	1	4	yes	—
47	C₂²×C₈	—	31	4	no	—
48	C₂×C₄²	Dic₃₁	1	2	yes	—
49	C₂×C₄²	Dic₃₁	1	2	yes	—
50	C₂×C₄²	C₂×D₃₁	1	2	yes	—
51	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
52	C₄×C₈	—	31	8	no	—
53	C₃₁	#1744	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
55	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
57	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
58	C₂×C₈	—	31	8	no	—
59	C₂×C₈	—	62	8	no	—
60	C₄²	—	2	4	no	—
61	C₄²	—	2	4	no	—
62	C₄²	C₃₁⋊C₈	1	8	yes	complemented
63	C₄²	C₄×D₃₁	1	4	yes	—
64	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₂×C₃₁⋊C₈	1	8	yes	—
68	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	—
69	C₂×C₄	#92474	1	4	yes	—
70	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
71	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
72	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
73	C₂×C₄	#92495	1	4	yes	—
74	C₈	—	62	8	no	—
75	C₂²	C₂²×C₃₁⋊C₈	1	2	yes	—
76	C₂²	C₂×#92474	1	2	yes	—
77	C₂²	C₂×C₄×Dic₃₁	1	1	yes	characteristic
78	C₂²	#11758503	1	2	yes	—
79	C₄	—	2	8	no	—
80	C₄	C₄×C₃₁⋊C₈	1	4	yes	—
81	C₄	#11758391	1	4	yes	—
82	C₄	#11758449	1	4	yes	—
83	C₂	#14520895	1	1	yes	characteristic
84	C₂	C₂×C₄×C₃₁⋊C₈	1	1	yes	characteristic
85	C₂	C₂×#11758391	1	1	yes	characteristic
86	C₂	#14521102	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₃₁
commutator	C₂×C₃₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₄³×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₄³	D₃₁
hypocenter	C₃₁	#1744
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
9. C₄×C₃₁⋊C₈	C₄	semidirect
16. C₄²×C₃₁	C₈	semidirect
53. C₃₁	#1744	semidirect
62. C₄²	C₃₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃₁⋊C₈	2
#14520804	2
C₄³×C₃₁	2
#1744	31