Subgroups — #31823852

Order 17984.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985127	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₂₄₈	C₂	1	2	yes	maximal, complemented
3	#22582876	C₂	1	1	yes	maximal, characteristic, complemented
4	#22582867	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₆₂	C₂²	1	1	yes	characteristic
7	C₂×D₁₁₂₄	—	2	2	no	—
8	C₂×D₁₁₂₄	C₂²	1	1	yes	characteristic
9	C₂×D₁₁₂₄	C₂²	1	2	yes	—
10	#17985127	—	2	4	no	—
11	D₂₂₄₈	—	2	4	no	—
12	#17985115	—	2	2	no	—
13	C₂×C₈×C₂₈₁	C₂²	1	2	yes	—
14	C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
15	C₂²×D₂₈₁	—	2	1	no	—
16	C₂²×D₂₈₁	—	2	2	no	—
17	Dic₅₆₂	—	2	2	no	—
18	D₁₁₂₄	—	2	2	no	—
19	D₁₁₂₄	—	2	2	no	—
20	D₁₁₂₄	—	2	2	no	—
21	D₁₁₂₄	—	4	4	no	—
22	C₂×Dic₂₈₁	—	2	1	no	—
23	C₂×C₄×C₂₈₁	D₄	1	2	yes	—
24	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
25	C₈×C₂₈₁	D₄	1	4	yes	complemented
26	C₂²×C₂₈₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₈₁	—	4	1	no	—
28	C₂×D₂₈₁	—	4	2	no	—
29	C₂×D₂₈₁	—	4	4	no	—
30	C₂×D₂₈₁	—	4	2	no	—
31	C₄×C₂₈₁	—	2	2	no	—
32	C₄×C₂₈₁	C₂×D₄	1	2	yes	—
33	Dic₂₈₁	—	4	1	no	—
34	C₂×C₂₈₁	#127	1	1	yes	characteristic
35	C₂×C₂₈₁	AGL(1,ℤ/8)	1	2	yes	—
36	D₂₈₁	—	8	2	no	—
37	D₂₈₁	—	8	1	no	—
38	C₂₈₁	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	281	1	no	maximal, Sylow
40	C₂×SD₁₆	—	281	2	no	—
41	C₂×D₈	—	281	2	no	—
42	#127	—	281	1	no	—
43	#124	—	281	1	no	—
44	#97	D₂₈₁	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	281	1	no	—
46	C₂×D₄	—	281	1	no	—
47	C₂×D₄	—	281	2	no	—
48	C₂×D₄	—	562	2	no	—
49	SD₁₆	—	562	4	no	—
50	D₈	—	562	4	no	—
51	C₂×C₈	C₂×D₂₈₁	1	2	yes	—
52	C₂²⋊C₄	—	562	2	no	—
53	C₄²	C₂×D₂₈₁	1	1	yes	characteristic
54	C₂³	—	562	1	no	—
55	C₂³	—	562	2	no	—
56	Q₈	—	562	2	no	—
57	D₄	—	562	2	no	—
58	D₄	—	562	2	no	—
59	D₄	—	562	2	no	—
60	D₄	—	1124	4	no	—
61	C₂×C₄	—	562	1	no	—
62	C₂×C₄	D₁₁₂₄	1	2	yes	—
63	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
64	C₈	D₁₁₂₄	1	4	yes	complemented
65	C₂²	—	1124	1	no	—
66	C₂²	—	1124	2	no	—
67	C₂²	—	1124	4	no	—
68	C₂²	—	1124	2	no	—
69	C₂²	C₂×D₁₁₂₄	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	1124	1	no	—
72	C₄	C₂×D₁₁₂₄	1	2	yes	—
73	C₂	—	2248	2	no	—
74	C₂	—	2248	1	no	—
75	C₂	#22582876	1	1	yes	characteristic
76	C₂	#22582907	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₁₂₄
commutator, cosocle	C₂×C₄×C₂₈₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₈₁
Fitting	#97×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂×D₄
hypercenter	#97	D₂₈₁
hypocenter	C₂₈₁	#496
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×#17985127	C₂	semidirect
2. C₂×D₂₂₄₈	C₂	semidirect
3. #22582876	C₂	semidirect
4. #22582867	C₂	semidirect
5. #97×C₂₈₁	C₂	semidirect
25. C₈×C₂₈₁	D₄	semidirect
38. C₂₈₁	#496	semidirect
44. #97	D₂₈₁	semidirect
64. C₈	D₁₁₂₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985127	2
C₂×D₂₂₄₈	2
#22582876	2
#22582867	2
#97×C₂₈₁	2
#496	281