Subgroups — #42122853

Order 27712.

106 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	#28357291	C₂	1	1	yes	maximal, characteristic, complemented
3	#28357284	C₂	1	2	yes	maximal, complemented
4	C₈×#13181884	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	1	yes	characteristic
19	#16812365	D₄	1	1	yes	characteristic
20	Dic₈₆₆	C₂×C₄	1	2	yes	—
21	C₂²⋊D₄₃₃	—	2	2	no	—
22	C₂×Dic₄₃₃	—	2	2	no	—
23	C₄×D₄₃₃	C₂³	1	1	yes	characteristic
24	#16812358	C₂×C₄	1	1	yes	characteristic
25	D₄×C₄₃₃	C₂×C₄	1	2	yes	—
26	C₈×C₄₃₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄₃₃	—	2	2	no	—
28	C₂×D₄₃₃	#41	1	1	yes	characteristic
29	#13181884	—	2	2	no	—
30	C₄×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
31	Dic₄₃₃	—	2	2	no	—
32	Dic₄₃₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄₃₃	—	4	2	no	—
34	C₂×C₄₃₃	C₄×D₄	1	1	yes	characteristic
35	D₄₃₃	—	2	1	no	—
36	C₄₃₃	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	433	1	no	maximal, Sylow
38	#135	—	433	1	no	—
39	#131	—	433	2	no	—
40	#108	—	433	1	no	—
41	#104	—	433	2	no	—
42	C₄×C₈	—	433	1	no	—
43	#41	—	433	2	no	—
44	Q₁₆	—	433	1	no	—
45	SD₁₆	—	433	2	no	—
46	D₈	#13181884	1	1	yes	characteristic, complemented
47	M₁₆	—	866	2	no	—
48	M₁₆	—	433	2	no	—
49	C₂×C₈	—	866	2	no	—
50	C₂×C₈	—	433	1	no	—
51	C₂×C₈	—	433	1	no	—
52	C₄²	—	433	1	no	—
53	Q₈	—	433	2	no	—
54	D₄	—	866	2	no	—
55	D₄	C₂×#13181884	1	2	yes	—
56	C₂×C₄	—	866	1	no	—
57	C₂×C₄	—	866	2	no	—
58	C₂×C₄	—	433	1	no	—
59	C₈	—	866	2	no	—
60	C₈	—	433	1	no	—
61	C₈	—	433	1	no	—
62	C₈	—	433	1	no	—
63	C₈	C₂×#13181884	1	1	yes	characteristic
64	C₂²	—	2	2	no	—
65	C₂²	—	433	1	no	—
66	C₄	—	866	2	no	—
67	C₄	—	866	2	no	—
68	C₄	—	433	1	no	—
69	C₄	C₂²×#13181884	1	1	yes	characteristic
70	C₂	—	4	2	no	—
71	C₂	—	866	1	no	—
72	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	#443
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. #28357291	C₂	semidirect
3. #28357284	C₂	semidirect
4. C₈×#13181884	C₂	semidirect
12. Dic₁₇₃₂	C₄	semidirect
13. #20294085	C₄	semidirect
15. D₈×C₄₃₃	C₄	semidirect
36. C₄₃₃	#443	semidirect
46. 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
#28357291	2
#28357284	2
C₈×#13181884	2
#28357239	2
#443	433