Subgroups — #16845317

Order 3968.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521429	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521426	C₂	1	1	yes	maximal, characteristic
3	#14521396	C₂	1	4	yes	maximal, complemented
4	C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₁₂₄	—	2	2	no	—
6	C₂×D₂₄₈	—	2	2	no	—
7	#11758451	C₂²	1	2	yes	—
8	#11758446	C₂²	1	2	yes	—
9	C₄×C₈×C₃₁	C₂²	1	2	yes	—
10	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	#11758409	—	2	8	no	—
12	C₂×D₁₂₄	—	2	4	no	—
13	C₂×D₁₂₄	—	2	2	no	—
14	C₂×Dic₆₂	—	2	2	no	—
15	C₂×C₈×C₃₁	D₄	1	4	yes	—
16	C₂×C₈×C₃₁	D₄	1	2	yes	—
17	C₄²×C₃₁	C₂³	1	1	yes	characteristic
18	C₄⋊Dic₃₁	—	2	4	no	—
19	Dic₁₂₄	—	4	4	no	—
20	D₂₄₈	—	4	4	no	—
21	C₂²×D₃₁	—	4	2	no	—
22	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
24	C₂×Dic₃₁	—	4	2	no	—
25	D₁₂₄	—	4	4	no	—
26	D₁₂₄	—	4	8	no	—
27	Dic₆₂	—	4	4	no	—
28	C₈×C₃₁	—	2	4	no	—
29	C₈×C₃₁	D₈	1	4	yes	—
30	#1709	—	31	1	no	maximal, Sylow
31	C₂²×C₃₁	#127	1	1	yes	characteristic
32	C₂×D₃₁	—	8	2	no	—
33	C₂×D₃₁	—	8	4	no	—
34	C₄×C₃₁	C₂×D₈	1	2	yes	—
35	C₄×C₃₁	#135	1	4	yes	—
36	Dic₃₁	—	8	2	no	—
37	#494	—	31	1	no	—
38	#493	—	31	1	no	—
39	#486	—	31	4	no	—
40	C₈²	D₃₁	1	1	yes	characteristic, complemented
41	C₂×C₃₁	#493	1	1	yes	characteristic
42	C₂×C₃₁	#495	1	2	yes	—
43	D₃₁	—	16	2	no	—
44	C₂×Q₁₆	—	62	2	no	—
45	C₂×D₈	—	62	2	no	—
46	C₄⋊Q₈	—	31	2	no	—
47	#127	—	31	2	no	—
48	#102	—	62	8	no	—
49	C₄×C₈	C₂×D₃₁	1	2	yes	—
50	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
51	C₃₁	#1709	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₈	—	62	2	no	—
53	C₂×D₄	—	62	4	no	—
54	C₂×D₄	—	62	2	no	—
55	Q₁₆	—	124	4	no	—
56	D₈	—	124	4	no	—
57	C₂×C₈	D₁₂₄	1	4	yes	—
58	C₂×C₈	D₁₂₄	1	2	yes	—
59	C₄⋊C₄	—	62	4	no	—
60	C₄²	C₂²×D₃₁	1	1	yes	characteristic
61	C₂³	—	124	2	no	—
62	Q₈	—	124	4	no	—
63	D₄	—	124	4	no	—
64	D₄	—	124	8	no	—
65	C₂×C₄	—	124	2	no	—
66	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
67	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
68	C₈	—	2	4	no	—
69	C₈	D₂₄₈	1	4	yes	—
70	C₂²	—	248	2	no	—
71	C₂²	—	248	4	no	—
72	C₂²	#11758451	1	1	yes	characteristic
73	C₄	—	248	2	no	—
74	C₄	C₂×D₂₄₈	1	2	yes	—
75	C₄	#11758481	1	4	yes	—
76	C₂	—	496	2	no	—
77	C₂	#14521426	1	1	yes	characteristic
78	C₂	#14521433	1	2	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	#14521426	C₂
hypercenter	C₈²	D₃₁
hypocenter	C₃₁	#1709
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. #14521429	C₂	semidirect
3. #14521396	C₂	semidirect
4. C₈²×C₃₁	C₂	semidirect
40. C₈²	D₃₁	semidirect
51. C₃₁	#1709	semidirect

Maximal subgroups

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

subgroup	index
#14521429	2
#14521426	2
#14521396	2
C₈²×C₃₁	2
#1709	31