Subgroups — #27163759

Order 12736.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263947	C₂	1	1	yes	maximal, characteristic
2	#20263866	C₂	1	1	yes	maximal, characteristic
3	#97×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉₉⋊C₈	—	2	1	no	—
5	C₄⋊Dic₁₉₉	C₄	1	1	yes	characteristic
6	C₄⋊Dic₁₉₉	C₄	1	1	yes	characteristic
7	Dic₁₉₉⋊C₄	—	2	1	no	—
8	Dic₁₉₉⋊C₄	—	2	1	no	—
9	C₂×C₈×C₁₉₉	—	2	1	no	—
10	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₉	—	2	1	no	—
12	C₂×Dic₁₉₉	—	2	1	no	—
13	C₁₉₉⋊C₈	—	2	2	no	—
14	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
17	C₈×C₁₉₉	—	2	2	no	—
18	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₉	—	2	1	no	—
20	C₄×C₁₉₉	—	2	1	no	—
21	C₄×C₁₉₉	—	2	1	no	—
22	Dic₁₉₉	—	4	1	no	—
23	Dic₁₉₉	—	4	1	no	—
24	C₂×C₁₉₉	#101	1	1	yes	characteristic
25	C₂×C₁₉₉	#104	1	1	yes	characteristic
26	C₂×C₁₉₉	#104	1	1	yes	characteristic
27	C₁₉₉	#330	1	1	yes	characteristic, complemented, Sylow
28	#330	—	199	1	no	maximal, Sylow
29	#125	—	199	1	no	—
30	#97	—	199	1	no	—
31	#97	D₁₉₉	1	1	yes	characteristic
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	398	1	no	—
34	C₄⋊C₄	—	398	1	no	—
35	C₄⋊C₄	—	398	1	no	—
36	C₄⋊C₄	—	199	1	no	—
37	C₄⋊C₄	—	199	1	no	—
38	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
39	C₂×C₄	—	398	1	no	—
40	C₂×C₄	—	398	1	no	—
41	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
42	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
43	C₂×C₄	D₇₉₆	1	1	yes	characteristic
44	C₈	—	2	2	no	—
45	C₈	—	398	2	no	—
46	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	796	1	no	—
51	C₄	—	796	1	no	—
52	C₂	#20263873	1	1	yes	characteristic
53	C₂	#20263881	1	1	yes	characteristic
54	C₂	#20263883	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₉₉)⋊C₄
commutator	C₂×C₄×C₁₉₉	C₂×C₄
Frattini	C₄²	C₂×D₁₉₉
Fitting	#97×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂²⋊C₄
cosocle	C₄²×C₁₉₉	C₂²
hypercenter	#97	D₁₉₉
hypocenter	C₁₉₉	#330
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
27. C₁₉₉	#330	semidirect

Maximal subgroups

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

subgroup	index
#20263947	2
#20263866	2
#97×C₁₉₉	2
#330	199