Subgroups — #27163804

Order 12736.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263923	C₂	1	1	yes	maximal, characteristic
2	#20263872	C₂	1	1	yes	maximal, characteristic, complemented
3	#99×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₁₉₉	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₁₉₉	—	2	1	no	—
6	#15691870	—	2	1	no	—
7	C₂²⋊Dic₁₉₉	C₄	1	1	yes	characteristic, complemented
8	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
9	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
11	C₂²×D₁₉₉	—	2	1	no	—
12	C₂²×D₁₉₉	—	4	2	no	—
13	C₂²×D₁₉₉	—	4	1	no	—
14	C₂²⋊D₁₉₉	—	4	2	no	—
15	C₂×Dic₁₉₉	—	2	1	no	—
16	C₁₉₉⋊C₈	—	4	1	no	—
17	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
19	D₄×C₁₉₉	—	4	1	no	—
20	C₂×C₄×C₁₉₉	—	4	1	no	—
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂²×C₁₉₉	—	2	1	no	—
23	C₂²×C₁₉₉	—	2	1	no	—
24	C₂²×C₁₉₉	—	4	1	no	—
25	C₂²×C₁₉₉	—	4	1	no	—
26	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₁₉₉	—	2	2	no	—
28	C₂×D₁₉₉	—	4	1	no	—
29	C₂×D₁₉₉	—	4	2	no	—
30	C₂×D₁₉₉	—	4	1	no	—
31	C₂×D₁₉₉	—	4	2	no	—
32	C₄×C₁₉₉	—	2	1	no	—
33	C₄×C₁₉₉	—	8	1	no	—
34	Dic₁₉₉	—	4	1	no	—
35	C₂×C₁₉₉	—	2	1	no	—
36	C₂×C₁₉₉	—	4	1	no	—
37	C₂×C₁₉₉	—	4	1	no	—
38	C₂×C₁₉₉	#99	1	1	yes	characteristic
39	D₁₉₉	—	4	2	no	—
40	C₁₉₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	199	1	no	maximal, Sylow
42	#120	—	199	1	no	—
43	#100	—	199	1	no	—
44	#99	D₁₉₉	1	1	yes	characteristic, complemented
45	C₂⁴	—	199	1	no	—
46	C₂×D₄	—	398	1	no	—
47	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
48	M₁₆	—	398	1	no	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	398	1	no	—
51	C₂²⋊C₄	—	199	1	no	—
52	C₂³	—	796	1	no	—
53	C₂³	—	796	2	no	—
54	C₂³	—	398	1	no	—
55	C₂³	C₄×D₁₉₉	1	1	yes	characteristic
56	C₂³	C₂²⋊D₁₉₉	1	1	yes	characteristic
57	D₄	—	4	1	no	—
58	D₄	—	796	2	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	398	1	no	—
61	C₂×C₄	D₇₉₆	1	1	yes	characteristic
62	C₈	—	796	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	796	1	no	—
68	C₂²	—	796	2	no	—
69	C₂²	—	796	2	no	—
70	C₂²	—	796	1	no	—
71	C₂²	—	398	2	no	—
72	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	8	1	no	—
75	C₄	—	796	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	796	2	no	—
80	C₂	#20263869	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20263869
commutator	C₂³×C₁₉₉	C₂×C₄
Frattini	C₂×D₄	C₂×D₁₉₉
Fitting	#99×C₁₉₉	C₂
socle	C₂×C₁₉₉	#99
prime radical	#20263923	C₂
cosocle	C₂×D₄×C₁₉₉	C₂²
hypercenter	#99	D₁₉₉
hypocenter	C₁₉₉	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
2. #20263872	C₂	semidirect
3. #99×C₁₉₉	C₂	semidirect
4. C₂³×D₁₉₉	C₄	semidirect
7. C₂²⋊Dic₁₉₉	C₄	semidirect
40. C₁₉₉	AΣL(1,𝔽₁₆)	semidirect
44. #99	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263923	2
#20263872	2
#99×C₁₉₉	2
AΣL(1,𝔽₁₆)	199