Subgroups — #42122718

Order 27712.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357279	C₂	1	2	yes	maximal
2	#28357208	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#13181884	—	2	2	no	—
4	C₂×Dic₈₆₆	C₄	1	1	yes	characteristic, complemented
5	C₂×D₁₇₃₂	C₂²	1	1	yes	characteristic
6	#20294074	—	2	2	no	—
7	C₄²×C₄₃₃	C₄	1	1	yes	characteristic, complemented
8	C₂×#13181884	—	4	2	no	—
9	C₂²×D₄₃₃	D₄	1	2	yes	—
10	Dic₈₆₆	—	4	1	no	—
11	D₁₇₃₂	—	4	1	no	—
12	C₂×Dic₄₃₃	—	2	1	no	—
13	C₂×C₄×C₄₃₃	—	2	1	no	—
14	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
16	C₂×D₄₃₃	—	2	2	no	—
17	C₂×D₄₃₃	—	4	2	no	—
18	#13181884	—	8	2	no	—
19	C₄×C₄₃₃	—	2	1	no	—
20	C₄×C₄₃₃	—	4	1	no	—
21	Dic₄₃₃	—	4	1	no	—
22	C₂×C₄₃₃	—	2	1	no	—
23	C₂×C₄₃₃	#99	1	1	yes	characteristic
24	D₄₃₃	—	4	2	no	—
25	C₄₃₃	#354	1	1	yes	characteristic, complemented, Sylow
26	#354	—	433	1	no	maximal, Sylow
27	#124	—	433	1	no	—
28	#99	—	433	2	no	—
29	C₂×Q₈	—	433	1	no	—
30	C₂×D₄	—	433	1	no	—
31	C₂²⋊C₄	—	866	2	no	—
32	C₂²⋊C₄	—	866	2	no	—
33	C₄²	#13181884	1	1	yes	characteristic, complemented
34	C₂³	—	433	2	no	—
35	Q₈	—	1732	1	no	—
36	D₄	—	1732	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	866	1	no	—
39	C₂×C₄	—	1732	2	no	—
40	C₂×C₄	C₂×#13181884	1	1	yes	characteristic
41	C₂²	—	866	2	no	—
42	C₂²	—	1732	2	no	—
43	C₂²	C₂²⋊#13181884	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	3464	2	no	—
47	C₄	—	1732	1	no	—
48	C₂	—	2	1	no	—
49	C₂	—	1732	2	no	—
50	C₂	#28357279	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28357279
commutator	C₂×C₄×C₄₃₃	C₂×C₄
Frattini	C₂×C₄	C₂×#13181884
Fitting	C₄²×C₄₃₃	C₄
socle	C₂×C₄₃₃	#99
prime radical, cosocle	C₂×D₁₇₃₂	C₂²
hypercenter	C₄²	#13181884
hypocenter	C₄₃₃	#354
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂²×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₈₆₆	C₄	semidirect
7. C₄²×C₄₃₃	C₄	semidirect
25. C₄₃₃	#354	semidirect
33. C₄²	#13181884	semidirect

Maximal subgroups

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

subgroup	index
#28357279	2
#28357208	2
#354	433