Subgroups — #16848021

Order 3968.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521482	C₂	1	2	yes	maximal, complemented
2	#14520886	C₂	1	2	yes	maximal, complemented
3	D₉₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758496	—	2	2	no	—
7	C₃₁×Q₃₂	C₂²	1	2	yes	—
8	#11758416	—	2	2	no	—
9	D₄₉₆	C₂²	1	2	yes	—
10	C₁₆×D₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×C₃₂	C₂²	1	1	yes	characteristic
12	C₃₁⋊C₃₂	C₂²	1	1	yes	characteristic
13	#92499	—	4	2	no	—
14	Q₁₆×C₃₁	—	2	2	no	—
15	#92481	—	4	2	no	—
16	D₂₄₈	—	2	2	no	—
17	C₈×D₃₁	D₄	1	1	yes	characteristic
18	C₁₆×C₃₁	C₂³	1	1	yes	characteristic
19	C₃₁⋊C₁₆	D₄	1	1	yes	characteristic
20	Q₈×C₃₁	—	4	2	no	—
21	D₁₂₄	—	4	2	no	—
22	D₁₂₄	—	8	2	no	—
23	C₄×D₃₁	—	8	2	no	—
24	C₄×D₃₁	D₈	1	1	yes	characteristic
25	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
26	C₃₁⋊C₈	D₈	1	1	yes	characteristic
27	#2262	—	31	1	no	maximal, Sylow
28	C₂×D₃₁	—	8	2	no	—
29	C₂×D₃₁	D₁₆	1	1	yes	characteristic
30	C₄×C₃₁	—	8	2	no	—
31	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
32	Dic₃₁	D₁₆	1	1	yes	characteristic
33	#508	—	31	2	no	—
34	Q₆₄	D₃₁	1	1	yes	characteristic, complemented
35	SD₆₄	—	31	2	no	—
36	D₃₂	—	31	1	no	—
37	C₂×C₃₂	—	31	1	no	—
38	C₂×C₃₁	C₂×D₁₆	1	1	yes	characteristic
39	D₃₁	—	2	1	no	—
40	D₃₁	—	16	2	no	—
41	#135	—	62	2	no	—
42	Q₃₂	C₂×D₃₁	1	2	yes	—
43	SD₃₂	—	62	2	no	—
44	D₁₆	—	31	2	no	—
45	C₂×C₁₆	—	31	1	no	—
46	C₃₂	—	31	1	no	—
47	C₃₂	C₂×D₃₁	1	1	yes	characteristic
48	C₃₁	#2262	1	1	yes	characteristic, complemented, Sylow
49	#41	—	124	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	124	2	no	—
52	D₈	—	62	2	no	—
53	C₂×C₈	—	31	1	no	—
54	C₁₆	—	31	1	no	—
55	C₁₆	C₂²×D₃₁	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	124	2	no	—
58	D₄	—	248	2	no	—
59	C₂×C₄	—	31	1	no	—
60	C₂×C₄	—	248	2	no	—
61	C₈	—	31	1	no	—
62	C₈	D₄×D₃₁	1	1	yes	characteristic
63	C₂²	—	31	1	no	—
64	C₂²	—	248	2	no	—
65	C₄	—	8	2	no	—
66	C₄	—	31	1	no	—
67	C₄	D₈×D₃₁	1	1	yes	characteristic
68	C₂	—	62	1	no	—
69	C₂	—	496	2	no	—
70	C₂	D₁₆×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₃₁
commutator, cosocle	C₁₆×C₃₁	C₂³
Frattini	C₁₆	C₂²×D₃₁
Fitting	C₃₁×Q₆₄	C₂
socle	C₂×C₃₁	C₂×D₁₆
hypercenter	Q₆₄	D₃₁
hypocenter	C₃₁	#2262
LCS	G ▷ C₁₆×C₃₁ ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #14521482	C₂	semidirect
2. #14520886	C₂	semidirect
3. D₉₉₂	C₂	semidirect
4. C₃₂×D₃₁	C₂	semidirect
5. C₃₁×Q₆₄	C₂	semidirect
34. Q₆₄	D₃₁	semidirect
48. C₃₁	#2262	semidirect

Maximal subgroups

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

subgroup	index
#14521482	2
#14520886	2
D₉₉₂	2
C₃₂×D₃₁	2
C₃₁×Q₆₄	2
#2262	31