Subgroups — #16845320

Order 3968.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521454	C₂	1	1	yes	maximal, characteristic
2	#14521401	C₂	1	2	yes	maximal
3	#14521399	C₂	1	2	yes	maximal
4	C₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#500	C₂	1	1	yes	maximal, characteristic
6	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	#11758446	C₂²	1	2	yes	—
8	C₃₁×#107	—	2	2	no	—
9	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	#11758423	—	2	4	no	—
11	#11758406	—	2	2	no	—
12	#11758398	—	2	4	no	—
13	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₁	—	2	2	no	—
16	C₂×Dic₆₂	—	2	2	no	—
17	C₂×C₈×C₃₁	D₄	1	2	yes	—
18	C₄⋊C₄×C₃₁	—	2	2	no	—
19	C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₄²×C₃₁	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₃₁	—	2	2	no	—
22	C₄⋊Dic₃₁	—	2	2	no	—
23	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
24	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
25	Q₈×C₃₁	—	4	4	no	—
26	C₂×C₄×C₃₁	—	4	2	no	—
27	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₃₁	—	4	2	no	—
31	Dic₆₂	—	4	4	no	—
32	C₈×C₃₁	Q₁₆	1	4	yes	—
33	C₃₁⋊C₈	—	2	2	no	—
34	C₃₁⋊C₈	—	2	2	no	—
35	#1710	—	31	1	no	maximal, Sylow
36	C₂²×C₃₁	#127	1	1	yes	characteristic
37	C₄×C₃₁	—	8	2	no	—
38	C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
39	C₄×C₃₁	#135	1	2	yes	—
40	C₄×C₃₁	#135	1	2	yes	—
41	Dic₃₁	—	8	2	no	—
42	#500	—	31	1	no	—
43	#500	D₃₁	1	1	yes	characteristic
44	#487	—	31	2	no	—
45	#487	—	31	2	no	—
46	C₈²	—	31	1	no	—
47	C₂×C₃₁	#494	1	1	yes	characteristic
48	C₂×C₃₁	#495	1	1	yes	characteristic
49	C₂×C₃₁	#495	1	1	yes	characteristic
50	C₄⋊Q₈	—	31	2	no	—
51	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
52	#107	—	2	2	no	—
53	#107	—	62	2	no	—
54	#103	—	62	4	no	—
55	#103	—	62	4	no	—
56	C₄×C₈	—	31	1	no	—
57	C₄×C₈	—	31	1	no	—
58	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
59	C₃₁	#1710	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	62	2	no	—
62	C₂×C₈	—	31	2	no	—
63	C₂×C₈	—	31	2	no	—
64	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	62	2	no	—
68	C₄⋊C₄	—	62	2	no	—
69	C₄²	C₂²×D₃₁	1	1	yes	characteristic
70	Q₈	—	4	4	no	—
71	Q₈	—	124	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	124	2	no	—
74	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
75	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
77	C₈	—	62	2	no	—
78	C₈	—	62	2	no	—
79	C₈	#92482	1	4	yes	—
80	C₂²	#11758517	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	248	2	no	—
83	C₄	#11758489	1	2	yes	—
84	C₄	#11758496	1	2	yes	—
85	C₄	C₂×#92482	1	2	yes	—
86	C₂	#14521428	1	1	yes	characteristic
87	C₂	#14521432	1	1	yes	characteristic
88	C₂	#14521434	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#500	C₂
socle, prime radical	C₂²×C₃₁	#127
hypercenter	#500	D₃₁
hypocenter	C₃₁	#1710
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
59. C₃₁	#1710	semidirect

Maximal subgroups

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

subgroup	index
#14521454	2
#14521401	2
#14521399	2
C₈×C₃₁⋊C₈	2
C₃₁×#500	2
#1710	31