Subgroups — #27139939

Order 12416.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690716	C₂	1	1	yes	maximal, characteristic
2	#20260565	C₂	1	1	yes	maximal, characteristic
3	#368×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690746	—	2	1	no	—
5	#15690716	C₄	1	2	yes	—
6	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
7	#108×C₉₇	—	2	1	no	—
8	#13095495	—	4	1	no	—
9	M₁₆×C₉₇	—	4	1	no	—
10	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
11	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₉₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
14	C₈×C₉₇	—	8	1	no	—
15	C₈×C₉₇	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₉₇	Q₁₆	1	1	yes	characteristic
17	#11715895	—	8	1	no	—
18	C₂²×C₉₇	Q₃₂	1	1	yes	characteristic
19	C₄×C₉₇	#107	1	1	yes	characteristic
20	C₄×C₉₇	D₁₆	1	1	yes	characteristic
21	C₂×C₉₇	—	2	1	no	—
22	C₂×C₉₇	#366	1	1	yes	characteristic
23	#1425	—	97	1	no	maximal, Sylow
24	C₉₇	#1425	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃₂	—	97	1	no	—
26	#368	—	97	1	no	—
27	#368	D₉₇	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
29	#108	—	2	1	no	—
30	#108	—	194	1	no	—
31	C₃₂	—	97	2	no	—
32	M₁₆	—	4	1	no	—
33	M₁₆	—	388	1	no	—
34	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
35	C₁₆	Dic₁₉₄	1	1	yes	characteristic
36	C₁₆	C₄×D₉₇	1	1	yes	characteristic
37	C₂×C₄	#13095496	1	1	yes	characteristic
38	C₈	—	8	1	no	—
39	C₈	—	776	1	no	—
40	C₈	#13095490	1	1	yes	characteristic
41	C₈	#13095501	1	1	yes	characteristic
42	C₂²	#15690756	1	1	yes	characteristic
43	C₄	#15690743	1	1	yes	characteristic
44	C₄	#15690751	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#20260560	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690751
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	#368×C₉₇	C₂
socle	C₂×C₉₇	#366
prime radical	C₂²×C₉₇	Q₃₂
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	#368	D₉₇
hypocenter	C₉₇	#1425
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
24. C₉₇	#1425	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690716	2
#20260565	2
#368×C₉₇	2
#1425	97