Subgroups — #27158802

Order 12416.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261825	C₂	1	2	yes	maximal
2	C₁₆×#87604	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715906	—	2	2	no	—
4	#15690883	C₄	1	1	yes	characteristic
5	C₈×#87604	C₂²	1	1	yes	characteristic
6	C₁₆×D₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715906	—	4	2	no	—
8	#13095519	Q₈	1	1	yes	characteristic
9	C₄×#87604	D₄	1	1	yes	characteristic
10	#13095514	—	2	1	no	—
11	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
12	#13095486	C₈	1	1	yes	characteristic, complemented
13	C₁₆×C₉₇	C₈	1	1	yes	characteristic, complemented
14	C₂×#87604	D₈	1	1	yes	characteristic
15	C₂×#87604	Q₁₆	1	1	yes	characteristic
16	#11715906	—	8	2	no	—
17	C₄×D₉₇	C₄⋊C₄	1	1	yes	characteristic
18	#11715897	—	2	1	no	—
19	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
20	#11715895	M₁₆	1	1	yes	characteristic
21	C₂×D₉₇	#107	1	1	yes	characteristic
22	#87604	—	2	1	no	—
23	#87604	D₁₆	1	1	yes	characteristic
24	#87604	Q₃₂	1	1	yes	characteristic
25	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
26	Dic₉₇	#108	1	1	yes	characteristic
27	C₂×C₉₇	#335	1	1	yes	characteristic
28	D₉₇	#366	1	1	yes	characteristic
29	D₉₇	#368	1	1	yes	characteristic
30	#1371	—	97	1	no	maximal, Sylow
31	C₉₇	#1371	1	1	yes	characteristic, complemented, Sylow
32	C₄×C₁₆	—	97	1	no	—
33	#335	—	97	2	no	—
34	C₂×C₁₆	—	97	1	no	—
35	C₂×C₁₆	—	97	1	no	—
36	C₄⋊C₈	—	194	2	no	—
37	C₄×C₈	—	97	1	no	—
38	C₂×C₈	—	97	1	no	—
39	C₂×C₈	—	97	1	no	—
40	C₂×C₈	—	388	2	no	—
41	C₄²	—	97	1	no	—
42	C₁₆	—	97	1	no	—
43	C₁₆	—	194	1	no	—
44	C₁₆	#11715906	1	1	yes	characteristic, complemented
45	C₂×C₄	—	97	1	no	—
46	C₂×C₄	—	97	1	no	—
47	C₂×C₄	—	97	1	no	—
48	C₈	—	776	2	no	—
49	C₈	—	97	1	no	—
50	C₈	—	194	1	no	—
51	C₈	C₂×#11715906	1	1	yes	characteristic
52	C₂²	—	97	1	no	—
53	C₄	—	97	1	no	—
54	C₄	—	97	1	no	—
55	C₄	—	97	1	no	—
56	C₄	—	194	1	no	—
57	C₄	C₄⋊#11715906	1	1	yes	characteristic
58	C₂	—	97	1	no	—
59	C₂	—	97	1	no	—
60	C₂	#20261825	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261825
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₈	C₂×#11715906
Fitting	C₁₆×C₉₇	C₈
socle	C₂×C₉₇	#335
prime radical	C₂×D₉₇	#107
cosocle	C₈×#87604	C₂²
hypercenter	C₁₆	#11715906
hypocenter	C₉₇	#1371
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
12. #13095486	C₈	semidirect
13. C₁₆×C₉₇	C₈	semidirect
31. C₉₇	#1371	semidirect
44. C₁₆	#11715906	semidirect

Maximal subgroups

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

subgroup	index
#20261825	2
C₁₆×#87604	2
#1371	97