Subgroups — #42121838

Order 27712.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20294081	C₂	1	1	yes	maximal, characteristic
2	C₂²×#16812358	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#16812358	C₂	1	4	yes	maximal, complemented
4	#117×C₄₃₃	C₂	1	1	yes	maximal, characteristic
5	#20294081	—	2	2	no	—
6	C₂×#16812358	—	2	2	no	—
7	C₂×#16812358	C₂²	1	2	yes	—
8	C₂×#16812358	C₂²	1	2	yes	—
9	C₂²×C₄×C₄₃₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₄₃₃	C₄	1	2	yes	—
11	C₂²⋊C₄×C₄₃₃	C₄	1	2	yes	—
12	C₄²×C₄₃₃	C₂²	1	2	yes	—
13	#16812358	—	2	2	no	—
14	#16812358	—	2	2	no	—
15	C₂³×C₄₃₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₄₃₃	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₃₃	D₄	1	2	yes	—
19	C₂×C₄×C₄₃₃	Q₈	1	2	yes	—
20	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄₃₃	—	2	2	no	—
22	C₂²×C₄₃₃	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₄₃₃	—	2	4	no	—
25	C₄×C₄₃₃	C₄⋊C₄	1	2	yes	—
26	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
27	C₄×C₄₃₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄₃₃	—	2	2	no	—
29	C₂×C₄₃₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄₃₃	#131	1	2	yes	—
31	C₄₃₃	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	433	1	no	maximal, Sylow
33	C₂×M₁₆	—	433	1	no	—
34	C₂²×C₈	—	433	1	no	—
35	#117	D₄₃₃	1	1	yes	characteristic
36	C₄⋊C₈	—	433	4	no	—
37	C₂²×C₄	C₂×D₄₃₃	1	1	yes	characteristic
38	M₁₆	—	866	2	no	—
39	C₂×C₈	—	866	2	no	—
40	C₂×C₈	—	433	2	no	—
41	C₂×C₈	—	433	2	no	—
42	C₄⋊C₄	Dic₄₃₃	1	2	yes	—
43	C₂²⋊C₄	Dic₄₃₃	1	2	yes	—
44	C₄²	C₂×D₄₃₃	1	2	yes	—
45	C₂³	C₂×Dic₄₃₃	1	1	yes	characteristic
46	C₂×C₄	C₂×Dic₄₃₃	1	4	yes	—
47	C₂×C₄	C₂×Dic₄₃₃	1	1	yes	characteristic
48	C₂×C₄	D₁₇₃₂	1	2	yes	—
49	C₂×C₄	Dic₈₆₆	1	2	yes	—
50	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
51	C₈	—	866	2	no	—
52	C₈	—	866	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊Dic₄₃₃	1	2	yes	—
55	C₂²	C₂²×Dic₄₃₃	1	1	yes	characteristic
56	C₄	—	2	4	no	—
57	C₄	C₄⋊Dic₄₃₃	1	2	yes	—
58	C₄	C₂×D₁₇₃₂	1	1	yes	characteristic
59	C₄	C₂×Dic₈₆₆	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₄₃₃	1	1	yes	characteristic
62	C₂	#28357227	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₃₃
commutator, socle	C₂²×C₄₃₃	C₂²×C₄
Fitting	#117×C₄₃₃	C₂
prime radical	C₂³×C₄₃₃	C₂×C₄
cosocle	C₂×C₄×C₄₃₃	C₂³
hypercenter	#117	D₄₃₃
hypocenter	C₄₃₃	#424
LCS	G ▷ C₂²×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊#16812358	C₂	semidirect
31. C₄₃₃	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20294081	2
C₂²×#16812358	2
C₄⋊#16812358	2
#117×C₄₃₃	2
#424	433