Subgroups — #42122858

Order 27712.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357307	C₂	1	2	yes	maximal, complemented
2	#28357290	C₂	1	1	yes	maximal, characteristic, complemented
3	#28357284	C₂	1	2	yes	maximal, complemented
4	#28357274	C₂	1	1	yes	maximal, characteristic, complemented
5	#28357239	C₂	1	1	yes	maximal, characteristic
6	#20294107	C₂²	1	2	yes	—
7	#20294106	—	2	2	no	—
8	#20294105	C₂²	1	1	yes	characteristic
9	C₂×#16812365	—	2	2	no	—
10	C₄×#13181884	C₂²	1	1	yes	characteristic
11	#20294096	C₂²	1	2	yes	—
12	Dic₁₇₃₂	C₄	1	1	yes	characteristic, complemented
13	#20294085	C₄	1	2	yes	complemented
14	C₈×D₄₃₃	C₂²	1	1	yes	characteristic
15	D₈×C₄₃₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#13181884	—	2	1	no	—
17	#16812365	—	2	2	no	—
18	#16812365	D₄	1	2	yes	—
19	Dic₈₆₆	C₂×C₄	1	2	yes	—
20	C₂²⋊D₄₃₃	—	2	2	no	—
21	C₂×Dic₄₃₃	—	2	2	no	—
22	C₄×D₄₃₃	C₂³	1	1	yes	characteristic
23	#16812358	C₂×C₄	1	1	yes	characteristic
24	D₄×C₄₃₃	C₂×C₄	1	2	yes	—
25	C₈×C₄₃₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₃₃	—	2	2	no	—
27	C₂×D₄₃₃	#41	1	1	yes	characteristic
28	#13181884	—	4	1	no	—
29	C₄×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
30	Dic₄₃₃	—	2	2	no	—
31	Dic₄₃₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄₃₃	—	4	2	no	—
33	C₂×C₄₃₃	C₄×D₄	1	1	yes	characteristic
34	D₄₃₃	—	2	1	no	—
35	C₄₃₃	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	433	1	no	maximal, Sylow
37	#135	—	433	1	no	—
38	#131	—	433	2	no	—
39	#108	—	433	1	no	—
40	#104	—	433	2	no	—
41	#97	—	433	1	no	—
42	#41	—	433	2	no	—
43	Q₁₆	—	433	1	no	—
44	SD₁₆	—	433	2	no	—
45	D₈	#13181884	1	1	yes	characteristic, complemented
46	M₁₆	—	866	2	no	—
47	M₁₆	—	433	2	no	—
48	C₂×C₈	—	866	2	no	—
49	C₂×C₈	—	433	1	no	—
50	C₂×C₈	—	433	1	no	—
51	C₄²	—	433	1	no	—
52	Q₈	—	433	2	no	—
53	D₄	—	866	2	no	—
54	D₄	C₂×#13181884	1	2	yes	—
55	C₂×C₄	—	866	2	no	—
56	C₂×C₄	—	866	1	no	—
57	C₂×C₄	—	433	1	no	—
58	C₈	—	866	2	no	—
59	C₈	—	433	2	no	—
60	C₈	—	433	1	no	—
61	C₈	C₂×#13181884	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	—	433	1	no	—
64	C₄	—	866	2	no	—
65	C₄	—	433	1	no	—
66	C₄	—	1732	1	no	—
67	C₄	C₂²×#13181884	1	1	yes	characteristic
68	C₂	—	4	2	no	—
69	C₂	—	866	1	no	—
70	C₂	D₄×#13181884	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13181884
commutator	C₄×C₄₃₃	C₂²×C₄
Frattini	C₄	C₂²×#13181884
Fitting	D₈×C₄₃₃	C₄
socle	C₂×C₄₃₃	C₄×D₄
prime radical	#28357239	C₂
cosocle	C₄×D₄₃₃	C₂³
hypercenter	D₈	#13181884
hypocenter	C₄₃₃	#444
LCS	G ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #28357307	C₂	semidirect
2. #28357290	C₂	semidirect
3. #28357284	C₂	semidirect
4. #28357274	C₂	semidirect
12. Dic₁₇₃₂	C₄	semidirect
13. #20294085	C₄	semidirect
15. D₈×C₄₃₃	C₄	semidirect
35. C₄₃₃	#444	semidirect
45. D₈	#13181884	semidirect

Maximal subgroups

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

subgroup	index
#28357307	2
#28357290	2
#28357284	2
#28357274	2
#28357239	2
#444	433