Subgroups — #444×C₁₉₉ #27163594

Order 12736.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×C₁₉₉	C₂	1	1	yes	maximal, characteristic
2	#131×C₁₉₉	C₂	1	2	yes	maximal, complemented
3	#108×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#104×C₁₉₉	C₂	1	2	yes	maximal, complemented
5	#97×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₁₉₉	C₂²	1	2	yes	—
7	Q₁₆×C₁₉₉	C₄	1	1	yes	characteristic, complemented
8	SD₁₆×C₁₉₉	C₄	1	2	yes	complemented
9	D₈×C₁₉₉	C₄	1	1	yes	characteristic, complemented
10	M₁₆×C₁₉₉	—	2	2	no	—
11	M₁₆×C₁₉₉	C₂²	1	2	yes	—
12	C₂×C₈×C₁₉₉	—	2	2	no	—
13	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	Q₈×C₁₉₉	C₂×C₄	1	2	yes	—
17	D₄×C₁₉₉	—	2	2	no	—
18	D₄×C₁₉₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₉₉	—	2	2	no	—
20	C₂×C₄×C₁₉₉	—	2	1	no	—
21	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
22	C₈×C₁₉₉	—	2	2	no	—
23	C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
24	C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₉₉	D₄	1	2	yes	—
26	C₂²×C₁₉₉	—	2	2	no	—
27	C₂²×C₁₉₉	#41	1	1	yes	characteristic
28	C₄×C₁₉₉	—	2	2	no	—
29	C₄×C₁₉₉	—	4	1	no	—
30	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₁₉₉	—	2	1	no	—
33	C₂×C₁₉₉	—	4	2	no	—
34	C₂×C₁₉₉	C₄×D₄	1	1	yes	characteristic
35	C₁₉₉	#444	1	1	yes	characteristic, direct product factor, Sylow
36	#444	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	#135	C₂×C₁₉₉	1	1	yes	characteristic
38	#131	C₂×C₁₉₉	1	2	yes	complemented
39	#108	C₂×C₁₉₉	1	1	yes	characteristic, complemented
40	#104	C₂×C₁₉₉	1	2	yes	complemented
41	#97	C₂×C₁₉₉	1	1	yes	characteristic, complemented
42	#41	C₂²×C₁₉₉	1	2	yes	—
43	Q₁₆	C₄×C₁₉₉	1	1	yes	characteristic, complemented
44	SD₁₆	C₄×C₁₉₉	1	2	yes	complemented
45	D₈	C₄×C₁₉₉	1	1	yes	characteristic, complemented
46	M₁₆	—	2	2	no	—
47	M₁₆	C₂²×C₁₉₉	1	2	yes	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	C₂²×C₁₉₉	1	1	yes	characteristic
50	C₂×C₈	C₂²×C₁₉₉	1	1	yes	characteristic
51	C₄²	C₂²×C₁₉₉	1	1	yes	characteristic
52	Q₈	C₂×C₄×C₁₉₉	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×C₄×C₁₉₉	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	C₂³×C₁₉₉	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	C₂×C₄×C₁₉₉	1	1	yes	characteristic
60	C₈	C₂×C₄×C₁₉₉	1	1	yes	characteristic
61	C₈	D₄×C₁₉₉	1	2	yes	—
62	C₂²	—	2	2	no	—
63	C₂²	#41×C₁₉₉	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	C₂²×C₄×C₁₉₉	1	1	yes	characteristic
67	C₄	C₂×D₄×C₁₉₉	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	C₄×D₄×C₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉₉	C₂×D₄
commutator	C₄	C₂²×C₄×C₁₉₉
Frattini, cosocle	C₂×C₄	C₂³×C₁₉₉
socle	C₂×C₁₉₉	C₄×D₄
prime radical	#135×C₁₉₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉₉ ◁ C₂×C₈×C₁₉₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #131×C₁₉₉	C₂	semidirect
3. #108×C₁₉₉	C₂	semidirect
4. #104×C₁₉₉	C₂	semidirect
5. #97×C₁₉₉	C₂	semidirect
7. Q₁₆×C₁₉₉	C₄	semidirect
8. SD₁₆×C₁₉₉	C₄	semidirect
9. D₈×C₁₉₉	C₄	semidirect
35. C₁₉₉	#444	direct (N × H)
36. #444	C₁₉₉	direct (N × H)
38. #131	C₂×C₁₉₉	semidirect
39. #108	C₂×C₁₉₉	semidirect
40. #104	C₂×C₁₉₉	semidirect
41. #97	C₂×C₁₉₉	semidirect
43. Q₁₆	C₄×C₁₉₉	semidirect
44. SD₁₆	C₄×C₁₉₉	semidirect
45. D₈	C₄×C₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#135×C₁₉₉	2
#131×C₁₉₉	2
#108×C₁₉₉	2
#104×C₁₉₉	2
#97×C₁₉₉	2
#444	199

Subgroups — #444×C199 #27163594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #444×C₁₉₉ #27163594