Subgroups — #15690842

Order 3104.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13095512	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#13095501	C₂	1	1	yes	maximal, characteristic, complemented
4	#13095500	C₂	1	1	yes	maximal, characteristic, complemented
5	#13095498	C₂	1	1	yes	maximal, characteristic, complemented
6	#13095494	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×C₉₇	C₂²	1	1	yes	characteristic
9	Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	D₃₈₈	—	2	1	no	—
11	D₃₈₈	C₂²	1	1	yes	characteristic
12	C₄×D₉₇	—	2	1	no	—
13	C₄×D₉₇	—	2	1	no	—
14	C₄×D₉₇	C₂²	1	1	yes	characteristic
15	Dic₁₉₄	—	2	1	no	—
16	Dic₁₉₄	C₂²	1	1	yes	characteristic
17	C₈×C₉₇	C₂²	1	1	yes	characteristic
18	#11715895	C₂²	1	1	yes	characteristic
19	C₂×D₉₇	—	2	1	no	—
20	C₂×D₉₇	D₄	1	1	yes	characteristic
21	C₄×C₉₇	—	2	1	no	—
22	C₄×C₉₇	—	2	1	no	—
23	C₄×C₉₇	C₂³	1	1	yes	characteristic
24	Dic₉₇	—	2	1	no	—
25	Dic₉₇	D₄	1	1	yes	characteristic
26	C₂×C₉₇	C₂×D₄	1	1	yes	characteristic
27	D₉₇	—	2	1	no	—
28	D₉₇	—	4	1	no	—
29	C₉₇	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	97	1	no	maximal, Sylow
31	#41	—	97	1	no	—
32	C₂×Q₈	—	97	1	no	—
33	Q₁₆	—	97	1	no	—
34	Q₁₆	D₉₇	1	1	yes	characteristic, complemented
35	SD₁₆	—	97	1	no	—
36	SD₁₆	—	97	1	no	—
37	M₁₆	—	97	1	no	—
38	Q₈	—	97	1	no	—
39	Q₈	—	194	1	no	—
40	Q₈	C₂×D₉₇	1	1	yes	characteristic
41	Q₈	C₂×D₉₇	1	1	yes	characteristic
42	D₄	—	97	1	no	—
43	D₄	—	194	1	no	—
44	C₂×C₄	—	97	1	no	—
45	C₂×C₄	—	194	1	no	—
46	C₂×C₄	—	194	1	no	—
47	C₈	—	97	1	no	—
48	C₈	C₂×D₉₇	1	1	yes	characteristic
49	C₂²	—	97	1	no	—
50	C₂²	—	194	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	97	1	no	—
54	C₄	—	194	1	no	—
55	C₄	C₂²×D₉₇	1	1	yes	characteristic
56	C₂	—	388	1	no	—
57	C₂	—	194	1	no	—
58	C₂	D₄×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₉₇
commutator, cosocle	C₄×C₉₇	C₂³
Frattini	C₄	C₂²×D₉₇
Fitting	Q₁₆×C₉₇	C₂
socle	C₂×C₉₇	C₂×D₄
prime radical	#13095512	C₂
hypercenter	Q₁₆	D₉₇
hypocenter	C₉₇	#137
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₉₇	C₂	semidirect
3. #13095501	C₂	semidirect
4. #13095500	C₂	semidirect
5. #13095498	C₂	semidirect
6. #13095494	C₂	semidirect
7. Q₁₆×C₉₇	C₂	semidirect
29. C₉₇	#137	semidirect
34. Q₁₆	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#13095512	2
Q₈×D₉₇	2
#13095501	2
#13095500	2
#13095498	2
#13095494	2
Q₁₆×C₉₇	2
#137	97