Subgroups — #16845323

Order 3968.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521400	C₂	1	3	yes	maximal, complemented
2	#14521396	C₂	1	3	yes	maximal
3	C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758451	C₂²	1	1	yes	characteristic
5	#11758446	C₂²	1	3	yes	—
6	C₄×C₈×C₃₁	C₂²	1	3	yes	—
7	#11758409	—	2	6	no	—
8	#11758404	—	2	6	no	—
9	C₂×D₁₂₄	—	2	3	no	—
10	C₂×Dic₆₂	—	2	3	no	—
11	C₂×C₈×C₃₁	D₄	1	6	yes	—
12	C₄²×C₃₁	C₂³	1	1	yes	characteristic
13	C₄⋊Dic₃₁	—	2	3	no	—
14	C₄⋊Dic₃₁	—	2	3	no	—
15	C₂²×D₃₁	—	4	1	no	—
16	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
17	C₂×Dic₃₁	—	4	3	no	—
18	D₁₂₄	—	4	6	no	—
19	Dic₆₂	—	4	6	no	—
20	C₈×C₃₁	—	2	6	no	—
21	#1711	—	31	1	no	maximal, Sylow
22	C₂²×C₃₁	#127	1	1	yes	characteristic
23	C₂×D₃₁	—	8	3	no	—
24	C₄×C₃₁	#135	1	6	yes	—
25	Dic₃₁	—	8	3	no	—
26	#487	—	31	3	no	—
27	#486	—	31	3	no	—
28	C₈²	D₃₁	1	1	yes	characteristic, complemented
29	C₂×C₃₁	#495	1	3	yes	—
30	D₃₁	—	16	1	no	—
31	C₄⋊Q₈	—	31	3	no	—
32	#127	—	31	1	no	—
33	#103	—	62	6	no	—
34	#102	—	62	6	no	—
35	C₄×C₈	C₂×D₃₁	1	3	yes	—
36	C₃₁	#1711	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₈	—	62	3	no	—
38	C₂×D₄	—	62	3	no	—
39	C₂×C₈	D₁₂₄	1	6	yes	—
40	C₄⋊C₄	—	62	3	no	—
41	C₄⋊C₄	—	62	3	no	—
42	C₄²	C₂²×D₃₁	1	1	yes	characteristic
43	C₂³	—	124	1	no	—
44	Q₈	—	124	6	no	—
45	D₄	—	124	6	no	—
46	C₂×C₄	—	124	3	no	—
47	C₂×C₄	C₂×D₁₂₄	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	248	3	no	—
50	C₂²	#11758451	1	1	yes	characteristic
51	C₄	—	248	3	no	—
52	C₄	#11758481	1	6	yes	—
53	C₂	—	496	1	no	—
54	C₂	#14521433	1	3	yes	—

	subgroup	G/H
center	C₂²	#11758451
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₈²×C₃₁	C₂
socle	C₂²×C₃₁	#127
prime radical	#11758451	C₂²
hypercenter	C₈²	D₃₁
hypocenter	C₃₁	#1711
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #14521400	C₂	semidirect
3. C₈²×C₃₁	C₂	semidirect
28. C₈²	D₃₁	semidirect
36. C₃₁	#1711	semidirect

Maximal subgroups

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

subgroup	index
#14521400	2
#14521396	2
C₈²×C₃₁	2
#1711	31