Subgroups — #16848020

Order 3968.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521481	C₂	1	2	yes	maximal, complemented
2	Dic₄₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14520888	C₂	1	2	yes	maximal, complemented
4	D₉₉₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758481	—	2	2	no	—
7	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	Dic₂₄₈	C₂²	1	2	yes	—
9	#11758388	—	2	2	no	—
10	D₄₉₆	C₂²	1	2	yes	—
11	C₃₁×C₃₂	C₂²	1	1	yes	characteristic
12	C₃₁×C₃₂	C₂²	1	1	yes	characteristic
13	#92495	—	4	2	no	—
14	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
15	Dic₁₂₄	—	2	2	no	—
16	D₂₄₈	—	2	2	no	—
17	#92470	—	4	2	no	—
18	C₁₆×C₃₁	D₄	1	1	yes	characteristic
19	C₁₆×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₃₁	D₈	1	1	yes	characteristic
21	C₂²⋊D₃₁	—	8	2	no	—
22	D₁₂₄	—	4	2	no	—
23	C₄×D₃₁	—	8	2	no	—
24	Dic₆₂	—	4	2	no	—
25	C₈×C₃₁	D₈	1	1	yes	characteristic
26	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
27	#2262	—	31	1	no	maximal, Sylow
28	C₂²×C₃₁	D₁₆	1	1	yes	characteristic
29	C₂×D₃₁	—	8	2	no	—
30	C₄×C₃₁	D₁₆	1	1	yes	characteristic
31	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
32	Dic₃₁	—	8	2	no	—
33	#508	—	31	2	no	—
34	Q₆₄	—	31	1	no	—
35	SD₆₄	—	31	2	no	—
36	D₃₂	—	31	1	no	—
37	C₂×C₃₂	D₃₁	1	1	yes	characteristic, complemented
38	C₂×C₃₁	—	2	1	no	—
39	C₂×C₃₁	C₂×D₁₆	1	1	yes	characteristic
40	D₃₁	—	16	2	no	—
41	#135	—	62	2	no	—
42	Q₃₂	—	31	2	no	—
43	SD₃₂	—	62	2	no	—
44	D₁₆	—	31	2	no	—
45	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
46	C₃₂	C₂×D₃₁	1	1	yes	characteristic
47	C₃₂	C₂×D₃₁	1	1	yes	characteristic
48	C₃₁	#2262	1	1	yes	characteristic, complemented, Sylow
49	#41	—	124	2	no	—
50	Q₁₆	—	62	2	no	—
51	SD₁₆	—	124	2	no	—
52	D₈	—	62	2	no	—
53	C₂×C₈	D₁₂₄	1	1	yes	characteristic
54	C₁₆	D₁₂₄	1	1	yes	characteristic
55	C₁₆	C₂²×D₃₁	1	1	yes	characteristic
56	Q₈	—	124	2	no	—
57	D₄	—	124	2	no	—
58	D₄	—	248	2	no	—
59	C₂×C₄	—	248	2	no	—
60	C₂×C₄	D₂₄₈	1	1	yes	characteristic
61	C₈	D₂₄₈	1	1	yes	characteristic
62	C₈	C₂×D₁₂₄	1	1	yes	characteristic
63	C₂²	—	248	2	no	—
64	C₂²	D₄₉₆	1	1	yes	characteristic
65	C₄	—	248	2	no	—
66	C₄	D₄₉₆	1	1	yes	characteristic
67	C₄	C₂×D₂₄₈	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	496	2	no	—
70	C₂	C₂×D₄₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₉₆
commutator, cosocle	C₁₆×C₃₁	C₂³
Frattini	C₁₆	C₂²×D₃₁
Fitting	C₂×C₃₁×C₃₂	C₂
socle	C₂×C₃₁	C₂×D₁₆
hypercenter	C₂×C₃₂	D₃₁
hypocenter	C₃₁	#2262
LCS	G ▷ C₁₆×C₃₁ ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #14521481	C₂	semidirect
2. Dic₄₉₆	C₂	semidirect
3. #14520888	C₂	semidirect
4. D₉₉₂	C₂	semidirect
5. C₂×C₃₁×C₃₂	C₂	semidirect
37. C₂×C₃₂	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
#14521481	2
Dic₄₉₆	2
#14520888	2
D₉₉₂	2
C₂×C₃₁×C₃₂	2
#2262	31