Subgroups — #27141492

Order 12416.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260693	C₂	1	3	yes	maximal
2	#20260492	C₂	1	3	yes	maximal
3	#338×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715895	C₂²	1	1	yes	characteristic
5	#15690769	C₂²	1	3	yes	—
6	#15690720	C₄	1	6	yes	complemented
7	C₈×Dic₉₇	C₄	1	6	yes	complemented
8	C₂×M₁₆×C₉₇	C₂²	1	3	yes	—
9	C₂×#11715895	C₂×C₄	1	3	yes	—
10	C₂×#11715895	C₂×C₄	1	3	yes	—
11	C₄⋊Dic₉₇	—	2	3	no	—
12	C₂²⋊Dic₉₇	—	2	3	no	—
13	C₄×Dic₉₇	C₂×C₄	1	6	yes	—
14	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
15	M₁₆×C₉₇	—	2	6	no	—
16	C₂×C₈×C₉₇	C₂×C₄	1	6	yes	—
17	C₂³×C₉₇	#41	1	1	yes	characteristic
18	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
19	C₂×C₄×C₉₇	#41	1	3	yes	—
20	C₂×Dic₉₇	—	2	6	no	—
21	C₈×C₉₇	—	2	6	no	—
22	#11715895	C₄²	1	1	yes	characteristic
23	#11715895	C₄²	1	3	yes	—
24	C₂²×C₉₇	—	4	1	no	—
25	C₂²×C₉₇	#117	1	3	yes	—
26	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
27	C₄×C₉₇	#117	1	3	yes	—
28	Dic₉₇	—	4	6	no	—
29	C₂×C₉₇	—	2	3	no	—
30	C₂×C₉₇	#376	1	1	yes	characteristic
31	#1747	—	97	1	no	maximal, Sylow
32	C₉₇	#1747	1	1	yes	characteristic, complemented, Sylow
33	#405	—	97	3	no	—
34	#338	D₉₇	1	1	yes	characteristic
35	H(3,ℤ/4)	—	97	3	no	—
36	C₂×M₁₆	C₂×D₉₇	1	3	yes	—
37	C₂²×C₈	—	97	1	no	—
38	#117	—	97	3	no	—
39	#97	—	97	6	no	—
40	C₄×C₈	—	97	6	no	—
41	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
42	M₁₆	—	2	6	no	—
43	C₂×C₈	—	97	3	no	—
44	C₂×C₈	—	97	3	no	—
45	C₂×C₈	C₄×D₉₇	1	6	yes	—
46	C₄⋊C₄	—	194	3	no	—
47	C₂²⋊C₄	—	194	3	no	—
48	C₄²	—	97	6	no	—
49	C₂³	#13095512	1	1	yes	characteristic
50	C₂×C₄	—	194	6	no	—
51	C₂×C₄	C₂×C₄×D₉₇	1	3	yes	—
52	C₂×C₄	#13095510	1	3	yes	—
53	C₈	—	2	6	no	—
54	C₈	—	97	1	no	—
55	C₈	—	97	3	no	—
56	C₂²	—	4	1	no	—
57	C₂²	#15690768	1	3	yes	—
58	C₄	—	388	6	no	—
59	C₄	C₄²×D₉₇	1	1	yes	characteristic
60	C₄	#15690766	1	3	yes	—
61	C₂	—	2	3	no	—
62	C₂	#20260580	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×D₉₇
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#338×C₉₇	C₂
socle	C₂×C₉₇	#376
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#338	D₉₇
hypocenter	C₉₇	#1747
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
6. #15690720	C₄	semidirect
7. C₈×Dic₉₇	C₄	semidirect
32. C₉₇	#1747	semidirect

Maximal subgroups

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

subgroup	index
#20260693	2
#20260492	2
#338×C₉₇	2
#1747	97