Subgroups — #42122932

Order 27712.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357317	C₂	1	1	yes	maximal, characteristic, complemented
2	#28357316	C₂	1	2	yes	maximal, complemented
3	#28357315	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#20294109	C₂	1	2	yes	maximal, complemented
5	#28357307	C₂	1	1	yes	maximal, characteristic
6	#20294109	C₂²	1	1	yes	characteristic
7	#20294109	C₂²	1	2	yes	—
8	#20294109	C₂²	1	1	yes	characteristic
9	#20294107	C₄	1	1	yes	characteristic
10	#20294106	C₄	1	2	yes	—
11	#20294105	C₂²	1	1	yes	characteristic
12	C₂×#16812365	C₂²	1	2	yes	—
13	#20294096	C₄	1	1	yes	characteristic
14	#16812365	C₂×C₄	1	2	yes	—
15	#16812365	C₂×C₄	1	1	yes	characteristic
16	#16812365	C₂³	1	1	yes	characteristic
17	Dic₈₆₆	C₈	1	1	yes	characteristic
18	C₂²⋊D₄₃₃	C₈	1	2	yes	—
19	C₂×Dic₄₃₃	C₂×C₄	1	2	yes	—
20	C₄×D₄₃₃	C₂×C₄	1	1	yes	characteristic
21	D₄×C₄₃₃	C₈	1	1	yes	characteristic
22	C₂²×C₄₃₃	C₂×C₈	1	2	yes	—
23	C₂×D₄₃₃	C₂×C₈	1	1	yes	characteristic
24	C₄×C₄₃₃	C₂×C₈	1	1	yes	characteristic
25	Dic₄₃₃	C₂×C₈	1	2	yes	—
26	Dic₄₃₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄₃₃	—	2	2	no	—
28	C₂×C₄₃₃	C₂²×C₈	1	1	yes	characteristic
29	D₄₃₃	—	2	1	no	—
30	C₄₃₃	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	433	1	no	maximal, Sylow
32	#131	—	433	1	no	—
33	M₃₂	—	433	2	no	—
34	M₃₂	—	433	1	no	—
35	C₂×C₁₆	—	433	2	no	—
36	C₂×C₁₆	—	433	1	no	—
37	#41	—	433	1	no	—
38	M₁₆	—	433	2	no	—
39	M₁₆	—	433	1	no	—
40	C₂×C₈	—	433	2	no	—
41	C₂×C₈	—	433	1	no	—
42	C₁₆	—	433	1	no	—
43	C₁₆	—	433	2	no	—
44	C₁₆	—	433	1	no	—
45	Q₈	—	433	1	no	—
46	D₄	—	433	2	no	—
47	D₄	#16812367	1	1	yes	characteristic
48	C₂×C₄	—	433	2	no	—
49	C₂×C₄	—	433	1	no	—
50	C₈	—	433	1	no	—
51	C₈	—	433	2	no	—
52	C₈	—	433	1	no	—
53	C₂²	—	433	1	no	—
54	C₂²	C₂×#16812367	1	2	yes	—
55	C₄	—	433	1	no	—
56	C₄	—	433	2	no	—
57	C₄	C₂×#16812367	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	—	866	1	no	—
60	C₂	C₂²×#16812367	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#16812367
commutator, socle	C₂×C₄₃₃	C₂²×C₈
Fitting	D₄×C₄₃₃	C₈
prime radical	#20294096	C₄
cosocle	#16812365	C₂³
hypercenter	D₄	#16812367
hypocenter	C₄₃₃	#504
LCS	G ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #28357317	C₂	semidirect
2. #28357316	C₂	semidirect
3. #28357315	C₂	semidirect
4. C₂×#20294109	C₂	semidirect
30. C₄₃₃	#504	semidirect

Maximal subgroups

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

subgroup	index
#28357317	2
#28357316	2
#28357315	2
C₂×#20294109	2
#28357307	2
#504	433