Subgroups — #42121559

Order 27712.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357220	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#16812358	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₄₃₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₆₆	C₄	1	2	yes	—
5	C₂×#16812358	—	2	1	no	—
6	C₄⋊Dic₄₃₃	—	2	1	no	—
7	C₄⋊Dic₄₃₃	—	2	1	no	—
8	C₂×C₈×C₄₃₃	—	2	1	no	—
9	C₄²×C₄₃₃	C₂²	1	1	yes	characteristic
10	Dic₈₆₆	—	4	2	no	—
11	C₂×Dic₄₃₃	—	2	2	no	—
12	#16812358	—	4	1	no	—
13	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
15	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
16	C₈×C₄₃₃	—	4	1	no	—
17	C₂²×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
18	C₄×C₄₃₃	—	2	1	no	—
19	C₄×C₄₃₃	SD₁₆	1	1	yes	characteristic
20	C₄×C₄₃₃	SD₁₆	1	1	yes	characteristic
21	C₄×C₄₃₃	Q₁₆	1	1	yes	characteristic
22	C₄×C₄₃₃	Q₁₆	1	1	yes	characteristic
23	Dic₄₃₃	—	4	2	no	—
24	C₂×C₄₃₃	#100	1	1	yes	characteristic
25	C₂×C₄₃₃	#103	1	1	yes	characteristic
26	C₂×C₄₃₃	#103	1	1	yes	characteristic
27	C₄₃₃	#333	1	1	yes	characteristic, complemented, Sylow
28	#333	—	433	1	no	maximal, Sylow
29	C₄⋊Q₈	—	433	1	no	—
30	C₄⋊C₈	—	433	1	no	—
31	C₄⋊C₈	D₄₃₃	1	1	yes	characteristic
32	C₂×Q₈	—	433	2	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	866	1	no	—
35	C₄⋊C₄	—	866	1	no	—
36	C₄⋊C₄	—	866	1	no	—
37	C₄²	C₂×D₄₃₃	1	1	yes	characteristic
38	Q₈	—	1732	2	no	—
39	C₂×C₄	—	866	2	no	—
40	C₂×C₄	C₄×D₄₃₃	1	1	yes	characteristic
41	C₂×C₄	C₂²⋊D₄₃₃	1	1	yes	characteristic
42	C₂×C₄	D₁₇₃₂	1	1	yes	characteristic
43	C₈	—	4	1	no	—
44	C₈	—	1732	1	no	—
45	C₂²	#20294074	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	1732	2	no	—
48	C₄	#20294085	1	1	yes	characteristic
49	C₄	#20294086	1	1	yes	characteristic
50	C₄	#20294087	1	1	yes	characteristic
51	C₄	Dic₁₇₃₂	1	1	yes	characteristic
52	C₂	#28357136	1	1	yes	characteristic
53	C₂	#28357143	1	1	yes	characteristic
54	C₂	#28357145	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20294074
commutator	C₂×C₄×C₄₃₃	C₂×C₄
Frattini	C₄²	C₂×D₄₃₃
Fitting	C₄⋊C₈×C₄₃₃	C₂
socle, prime radical	C₂²×C₄₃₃	C₂²⋊C₄
cosocle	C₄²×C₄₃₃	C₂²
hypercenter	C₄⋊C₈	D₄₃₃
hypocenter	C₄₃₃	#333
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂²×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. C₄₃₃	#333	semidirect

Maximal subgroups

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

subgroup	index
#28357220	2
C₄⋊#16812358	2
C₄⋊C₈×C₄₃₃	2
#333	433