Subgroups — #42121620

Order 27712.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357242	C₂	1	1	yes	maximal, characteristic
2	#28357160	C₂	1	1	yes	maximal, characteristic, complemented
3	#106×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294097	—	2	1	no	—
5	Dic₁₇₃₂	C₄	1	1	yes	characteristic, complemented
6	#20294086	—	2	1	no	—
7	D₃₄₆₄	C₄	1	1	yes	characteristic, complemented
8	#20294072	—	2	1	no	—
9	C₂×C₈×C₄₃₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₄₃₃	—	2	1	no	—
11	Dic₈₆₆	—	2	1	no	—
12	D₁₇₃₂	—	2	1	no	—
13	C₂²⋊D₄₃₃	—	4	1	no	—
14	C₄×D₄₃₃	—	4	1	no	—
15	C₂×C₄×C₄₃₃	—	4	1	no	—
16	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
17	C₈×C₄₃₃	C₂×C₄	1	1	yes	characteristic
18	C₈×C₄₃₃	D₄	1	1	yes	characteristic
19	C₂²×C₄₃₃	D₈	1	1	yes	characteristic
20	C₂×D₄₃₃	—	4	1	no	—
21	C₄×C₄₃₃	—	8	1	no	—
22	C₄×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₄₃₃	SD₁₆	1	1	yes	characteristic
24	Dic₄₃₃	—	4	1	no	—
25	C₂×C₄₃₃	—	2	1	no	—
26	C₂×C₄₃₃	#102	1	1	yes	characteristic
27	D₄₃₃	—	8	1	no	—
28	C₄₃₃	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	433	1	no	maximal, Sylow
30	#135	—	433	1	no	—
31	M₃₂	—	433	1	no	—
32	#106	D₄₃₃	1	1	yes	characteristic, complemented
33	#41	—	866	1	no	—
34	Q₁₆	—	433	1	no	—
35	SD₁₆	—	866	1	no	—
36	D₈	—	433	1	no	—
37	C₂×C₈	C₂×D₄₃₃	1	1	yes	characteristic
38	C₄⋊C₄	—	2	1	no	—
39	C₁₆	—	866	1	no	—
40	Q₈	—	866	1	no	—
41	D₄	—	866	1	no	—
42	D₄	—	1732	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	1732	1	no	—
45	C₂×C₄	C₂²⋊D₄₃₃	1	1	yes	characteristic
46	C₈	C₄×D₄₃₃	1	1	yes	characteristic
47	C₈	D₁₇₃₂	1	1	yes	characteristic
48	C₂²	—	1732	1	no	—
49	C₂²	#20294082	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	—	1732	1	no	—
52	C₄	#20294074	1	1	yes	characteristic
53	C₄	#20294084	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	3464	1	no	—
56	C₂	#28357140	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28357140
commutator	C₈×C₄₃₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₄₃₃
Fitting	#106×C₄₃₃	C₂
socle	C₂×C₄₃₃	#102
prime radical	#28357242	C₂
cosocle	C₂×C₈×C₄₃₃	C₂²
hypercenter	#106	D₄₃₃
hypocenter	C₄₃₃	#360
LCS	G ▷ C₈×C₄₃₃ ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

Complements

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

N	H ≅ G/N	type
2. #28357160	C₂	semidirect
3. #106×C₄₃₃	C₂	semidirect
5. Dic₁₇₃₂	C₄	semidirect
7. D₃₄₆₄	C₄	semidirect
28. C₄₃₃	#360	semidirect
32. #106	D₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
#28357242	2
#28357160	2
#106×C₄₃₃	2
#360	433