Subgroups — #16845328

Order 3968.

192 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	#14521452	C₂	1	2	yes	maximal
3	#14521428	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	C₂×#92482	—	2	4	no	—
8	#11758446	C₂²	1	2	yes	—
9	C₃₁×#107	—	2	2	no	—
10	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	#11758406	—	2	2	no	—
12	#11758395	—	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	#92482	—	4	8	no	—
22	C₄⋊Dic₃₁	—	2	2	no	—
23	C₄⋊Dic₃₁	—	2	2	no	—
24	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
26	Q₈×C₃₁	—	4	4	no	—
27	C₂×C₄×C₃₁	—	4	2	no	—
28	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₃₁	—	4	2	no	—
32	Dic₆₂	—	4	4	no	—
33	C₈×C₃₁	Q₁₆	1	4	yes	complemented
34	C₃₁⋊C₈	D₈	1	4	yes	—
35	C₃₁⋊C₈	Q₁₆	1	4	yes	complemented
36	#1713	—	31	1	no	maximal, Sylow
37	C₂²×C₃₁	#127	1	1	yes	characteristic
38	C₄×C₃₁	—	8	2	no	—
39	C₄×C₃₁	C₂×D₈	1	2	yes	—
40	C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
41	C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
42	Dic₃₁	—	8	2	no	—
43	#500	—	31	2	no	—
44	#500	—	31	1	no	—
45	#500	D₃₁	1	1	yes	characteristic
46	#494	—	31	2	no	—
47	C₈²	—	31	1	no	—
48	C₂×C₃₁	#493	1	1	yes	characteristic
49	C₂×C₃₁	#494	1	1	yes	characteristic
50	C₂×C₃₁	#494	1	1	yes	characteristic
51	C₂×Q₁₆	—	62	4	no	—
52	C₄⋊Q₈	—	31	2	no	—
53	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
54	#107	—	2	2	no	—
55	#107	—	62	4	no	—
56	#107	—	62	2	no	—
57	C₄×C₈	—	31	1	no	—
58	C₄×C₈	—	31	1	no	—
59	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
60	C₃₁	#1713	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	62	2	no	—
63	Q₁₆	—	124	8	no	—
64	C₂×C₈	—	31	2	no	—
65	C₂×C₈	—	31	2	no	—
66	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	62	2	no	—
70	C₄⋊C₄	—	62	2	no	—
71	C₄²	C₂²×D₃₁	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	124	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	124	2	no	—
76	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
77	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
79	C₈	—	31	4	no	—
80	C₈	—	31	4	no	—
81	C₈	#92482	1	4	yes	complemented
82	C₂²	#11758517	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	248	2	no	—
85	C₄	D₈×D₃₁	1	2	yes	—
86	C₄	Q₁₆×D₃₁	1	2	yes	—
87	C₄	C₂×#92482	1	2	yes	—
88	C₂	#14521424	1	1	yes	characteristic
89	C₂	#14521427	1	1	yes	characteristic
90	C₂	#14521428	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	#1713
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
33. C₈×C₃₁	Q₁₆	semidirect
35. C₃₁⋊C₈	Q₁₆	semidirect
60. C₃₁	#1713	semidirect
81. C₈	#92482	semidirect

Maximal subgroups

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

subgroup	index
#14521454	2
#14521452	2
#14521428	2
C₈×C₃₁⋊C₈	2
C₃₁×#500	2
#1713	31