Subgroups — #42122693

Order 27712.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄₃₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#28357299	C₂	1	1	yes	maximal, characteristic
3	C₂×#20294080	C₂	1	1	yes	maximal, characteristic
4	#20294107	—	2	1	no	—
5	#20294106	C₄	1	2	yes	complemented
6	#20294105	—	2	1	no	—
7	C₄⋊#13181884	—	2	1	no	—
8	C₂²⋊#13181884	C₄	1	2	yes	—
9	C₄×#13181884	—	2	1	no	—
10	C₂×C₄×D₄₃₃	C₂²	1	1	yes	characteristic
11	#20294080	—	2	1	no	—
12	#20294080	—	2	1	no	—
13	C₂×#16812358	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×C₄₃₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#13181884	—	2	2	no	—
16	#16812365	—	2	2	no	—
17	C₂²×D₄₃₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₄₃₃	C₂×C₄	1	1	yes	characteristic
19	C₄×D₄₃₃	D₄	1	1	yes	characteristic
20	C₄×D₄₃₃	D₄	1	1	yes	characteristic
21	C₄×D₄₃₃	D₄	1	1	yes	characteristic
22	C₄×D₄₃₃	Q₈	1	1	yes	characteristic
23	#16812358	—	2	1	no	—
24	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₄₃₃	—	2	1	no	—
26	C₂²×C₄₃₃	C₄²	1	1	yes	characteristic
27	C₂×D₄₃₃	—	4	1	no	—
28	C₂×D₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₄₃₃	C₄⋊C₄	1	1	yes	characteristic
30	#13181884	—	4	2	no	—
31	C₄×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₄₃₃	C₄⋊C₄	1	1	yes	characteristic
33	Dic₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₄₃₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄₃₃	—	2	1	no	—
36	C₂×C₄₃₃	#95	1	1	yes	characteristic
37	D₄₃₃	—	2	1	no	—
38	D₄₃₃	—	2	1	no	—
39	C₄₃₃	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	433	1	no	maximal, Sylow
41	C₂×M₁₆	—	433	1	no	—
42	C₂×M₁₆	—	433	1	no	—
43	#117	—	433	1	no	—
44	C₂²×C₄	—	433	1	no	—
45	M₁₆	—	866	1	no	—
46	M₁₆	—	866	1	no	—
47	M₁₆	—	866	1	no	—
48	M₁₆	—	433	2	no	—
49	C₂×C₈	—	866	1	no	—
50	C₂×C₈	—	433	1	no	—
51	C₂×C₈	#13181884	1	1	yes	characteristic, complemented
52	C₄⋊C₄	—	866	1	no	—
53	C₂²⋊C₄	—	433	2	no	—
54	C₄²	—	866	1	no	—
55	C₂³	—	433	1	no	—
56	C₂×C₄	—	866	2	no	—
57	C₂×C₄	—	433	1	no	—
58	C₂×C₄	—	433	1	no	—
59	C₂×C₄	—	433	1	no	—
60	C₂×C₄	—	433	1	no	—
61	C₂×C₄	—	433	1	no	—
62	C₂×C₄	C₂×#13181884	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	866	2	no	—
65	C₈	—	866	1	no	—
66	C₂²	—	433	1	no	—
67	C₂²	—	433	1	no	—
68	C₂²	—	1732	1	no	—
69	C₂²	C₄×#13181884	1	1	yes	characteristic
70	C₄	—	433	1	no	—
71	C₄	—	433	1	no	—
72	C₄	—	1732	2	no	—
73	C₄	C₂²⋊#13181884	1	1	yes	characteristic
74	C₄	C₄⋊#13181884	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	866	1	no	—
77	C₂	—	866	1	no	—
78	C₂	(C₂×#13181884)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#13181884
commutator	C₂²×C₄₃₃	C₄²
Frattini	C₂×C₄	C₂×#13181884
Fitting	C₂×C₈×C₄₃₃	C₄
socle	C₂×C₄₃₃	#95
prime radical	C₂²×D₄₃₃	C₂×C₄
cosocle	C₂×C₄×D₄₃₃	C₂²
hypercenter	C₂×C₈	#13181884
hypocenter	C₄₃₃	#344
LCS	G ▷ C₂²×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. #20294106	C₄	semidirect
13. C₂×#16812358	C₄	semidirect
14. C₂×C₈×C₄₃₃	C₄	semidirect
39. C₄₃₃	#344	semidirect
51. C₂×C₈	#13181884	semidirect

Maximal subgroups

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

subgroup	index
D₄₃₃⋊M₁₆	2
#28357299	2
C₂×#20294080	2
#344	433