Subgroups — #31823206

Order 17984.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582876	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#14555886	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₁₂₄	—	2	1	no	—
5	C₂×D₁₁₂₄	—	2	1	no	—
6	C₂×D₁₁₂₄	C₄	1	2	yes	—
7	C₂×#14555886	—	2	1	no	—
8	C₂×C₈×C₂₈₁	—	2	1	no	—
9	C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
10	C₂²×D₂₈₁	—	2	2	no	—
11	D₁₁₂₄	—	4	2	no	—
12	D₁₁₂₄	—	4	1	no	—
13	D₁₁₂₄	—	4	1	no	—
14	D₁₁₂₄	—	4	1	no	—
15	D₁₁₂₄	—	4	1	no	—
16	#14555886	—	4	1	no	—
17	C₂×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₈₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₈₁	D₄	1	1	yes	characteristic
20	C₈×C₂₈₁	—	4	1	no	—
21	C₂²×C₂₈₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₂₈₁	—	4	2	no	—
23	C₂×D₂₈₁	—	4	2	no	—
24	C₂×D₂₈₁	—	4	2	no	—
25	C₄×C₂₈₁	—	2	1	no	—
26	C₄×C₂₈₁	D₈	1	1	yes	characteristic
27	C₄×C₂₈₁	D₈	1	1	yes	characteristic
28	C₄×C₂₈₁	SD₁₆	1	1	yes	characteristic
29	C₄×C₂₈₁	SD₁₆	1	1	yes	characteristic
30	C₂×C₂₈₁	#100	1	1	yes	characteristic
31	C₂×C₂₈₁	#102	1	1	yes	characteristic
32	C₂×C₂₈₁	#102	1	1	yes	characteristic
33	D₂₈₁	—	8	2	no	—
34	C₂₈₁	#331	1	1	yes	characteristic, complemented, Sylow
35	#331	—	281	1	no	maximal, Sylow
36	#127	—	281	1	no	—
37	C₄⋊C₈	—	281	1	no	—
38	C₄⋊C₈	D₂₈₁	1	1	yes	characteristic, complemented
39	C₂×D₄	—	281	2	no	—
40	C₂×D₄	—	562	1	no	—
41	C₂×D₄	—	562	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	562	1	no	—
44	C₄²	C₂×D₂₈₁	1	1	yes	characteristic
45	C₂³	—	562	2	no	—
46	D₄	—	1124	2	no	—
47	D₄	—	1124	1	no	—
48	D₄	—	1124	1	no	—
49	D₄	—	1124	1	no	—
50	D₄	—	1124	1	no	—
51	C₂×C₄	C₄×D₂₈₁	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊D₂₈₁	1	1	yes	characteristic
53	C₂×C₄	D₁₁₂₄	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	1124	1	no	—
56	C₂²	—	1124	2	no	—
57	C₂²	—	1124	2	no	—
58	C₂²	—	1124	2	no	—
59	C₂²	#17985115	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#17985123	1	1	yes	characteristic
62	C₄	D₂₂₄₈	1	1	yes	characteristic
63	C₄	#17985125	1	1	yes	characteristic
64	C₄	#17985127	1	1	yes	characteristic
65	C₂	—	2248	2	no	—
66	C₂	#22582795	1	1	yes	characteristic
67	C₂	#22582799	1	1	yes	characteristic
68	C₂	#22582801	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985115
commutator	C₂×C₄×C₂₈₁	C₂×C₄
Frattini	C₄²	C₂×D₂₈₁
Fitting	C₄⋊C₈×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂²⋊C₄
prime radical	#22582876	C₂
cosocle	C₄²×C₂₈₁	C₂²
hypercenter	C₄⋊C₈	D₂₈₁
hypocenter	C₂₈₁	#331
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊#14555886	C₂	semidirect
3. C₄⋊C₈×C₂₈₁	C₂	semidirect
34. C₂₈₁	#331	semidirect
38. C₄⋊C₈	D₂₈₁	semidirect

Maximal subgroups

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

subgroup	index
#22582876	2
C₄⋊#14555886	2
C₄⋊C₈×C₂₈₁	2
#331	281