Subgroups — #27156842

Order 12416.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261666	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#87604	C₂	1	1	yes	maximal, characteristic
3	C₃₂×D₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690883	C₄	1	1	yes	characteristic
5	C₈×#87604	C₄	1	1	yes	characteristic
6	#15690860	C₄	1	2	yes	complemented
7	C₁₆×D₉₇	C₂²	1	1	yes	characteristic
8	#15690716	C₄	1	1	yes	characteristic, complemented
9	C₃₂×C₉₇	C₄	1	1	yes	characteristic, complemented
10	#13095519	C₈	1	1	yes	characteristic
11	C₄×#87604	C₈	1	1	yes	characteristic
12	#13095514	C₈	1	2	yes	—
13	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
14	#13095486	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#87604	C₁₆	1	2	yes	—
17	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
18	#11715897	C₁₆	1	2	yes	—
19	C₈×C₉₇	C₄²	1	1	yes	characteristic
20	#11715895	C₂×C₈	1	1	yes	characteristic
21	C₂×D₉₇	C₂×C₁₆	1	1	yes	characteristic
22	#87604	—	2	2	no	—
23	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
24	Dic₉₇	C₂×C₁₆	1	1	yes	characteristic
25	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
26	D₉₇	M₆₄	1	2	yes	—
27	#1397	—	97	1	no	maximal, Sylow
28	C₉₇	#1397	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃₂	—	97	1	no	—
30	C₂×C₃₂	—	97	1	no	—
31	C₄×C₁₆	—	97	1	no	—
32	C₂×C₁₆	—	97	1	no	—
33	C₂×C₁₆	—	97	1	no	—
34	C₄×C₈	—	97	1	no	—
35	C₃₂	—	97	2	no	—
36	C₃₂	—	97	1	no	—
37	C₃₂	#87604	1	1	yes	characteristic, complemented
38	C₂×C₈	—	97	1	no	—
39	C₂×C₈	—	97	1	no	—
40	C₄²	—	97	1	no	—
41	C₁₆	—	97	2	no	—
42	C₁₆	—	97	1	no	—
43	C₁₆	C₂×#87604	1	1	yes	characteristic
44	C₂×C₄	—	97	1	no	—
45	C₂×C₄	—	97	2	no	—
46	C₈	—	97	2	no	—
47	C₈	—	97	1	no	—
48	C₈	C₄×#87604	1	1	yes	characteristic
49	C₂²	—	97	1	no	—
50	C₄	—	97	1	no	—
51	C₄	—	194	2	no	—
52	C₄	C₈×#87604	1	1	yes	characteristic
53	C₂	—	97	2	no	—
54	C₂	C₁₆×#87604	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂×#87604
commutator, socle	C₂×C₉₇	C₄×C₁₆
Fitting	C₃₂×C₉₇	C₄
prime radical	C₂×D₉₇	C₂×C₁₆
cosocle	C₁₆×D₉₇	C₂²
hypercenter	C₃₂	#87604
hypocenter	C₉₇	#1397
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
6. #15690860	C₄	semidirect
8. #15690716	C₄	semidirect
9. C₃₂×C₉₇	C₄	semidirect
28. C₉₇	#1397	semidirect
37. C₃₂	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261666	2
C₁₆×#87604	2
C₃₂×D₉₇	2
#1397	97