Subgroups — #27139802

Order 12416.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260486	C₂	1	2	yes	maximal
2	#346×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715895	—	2	2	no	—
4	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
7	C₂×#11715895	—	4	2	no	—
8	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₉₇	Q₈	1	1	yes	characteristic
10	C₄²×C₉₇	D₄	1	1	yes	characteristic
11	C₁₆×C₉₇	—	2	1	no	—
12	C₁₆×C₉₇	C₈	1	2	yes	complemented
13	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
15	C₈×C₉₇	—	2	1	no	—
16	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
17	C₈×C₉₇	M₁₆	1	1	yes	characteristic
18	#11715895	—	8	2	no	—
19	C₂²×C₉₇	#106	1	1	yes	characteristic
20	C₄×C₉₇	—	2	2	no	—
21	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₉₇	#108	1	1	yes	characteristic
23	C₂×C₉₇	#334	1	1	yes	characteristic
24	C₂×C₉₇	#365	1	2	yes	—
25	#1368	—	97	1	no	maximal, Sylow
26	C₉₇	#1368	1	1	yes	characteristic, complemented, Sylow
27	#346	D₉₇	1	1	yes	characteristic
28	#334	—	97	2	no	—
29	C₂×C₁₆	Dic₉₇	1	1	yes	characteristic
30	C₂×C₁₆	Dic₉₇	1	1	yes	characteristic
31	C₄⋊C₈	—	194	2	no	—
32	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
33	C₂×C₈	—	388	2	no	—
34	C₂×C₈	Dic₁₉₄	1	1	yes	characteristic
35	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
36	C₄²	D₃₈₈	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	#11715895	1	2	yes	complemented
39	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
40	C₂×C₄	#13095500	1	2	yes	—
41	C₈	—	2	1	no	—
42	C₈	—	776	2	no	—
43	C₈	C₂×#11715895	1	1	yes	characteristic
44	C₈	#13095495	1	1	yes	characteristic
45	C₂²	#15690742	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄⋊#11715895	1	1	yes	characteristic
48	C₄	#15690746	1	1	yes	characteristic
49	C₂	#20260486	1	1	yes	characteristic
50	C₂	#20260558	1	2	yes	—

	subgroup	G/H
center	C₂²	#15690742
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#346×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#106
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#346	D₉₇
hypocenter	C₉₇	#1368
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₉₇	C₈	semidirect
26. C₉₇	#1368	semidirect
38. C₁₆	#11715895	semidirect

Maximal subgroups

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

subgroup	index
#20260486	2
#346×C₉₇	2
#1368	97