Subgroups — #27156652

Order 12416.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261624	C₂	1	1	yes	maximal, characteristic
2	#20261576	C₂	1	1	yes	maximal, characteristic
3	#20260510	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	C₄	1	2	yes	—
5	C₄×#11715897	C₄	1	2	yes	—
6	C₂×#13095486	C₄	1	1	yes	characteristic
7	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
9	C₂×#11715897	—	2	2	no	—
10	#13095514	—	2	2	no	—
11	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
12	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
13	#13095486	C₈	1	2	yes	complemented
14	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₉₇	C₈	1	2	yes	complemented
16	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
17	C₂×Dic₉₇	M₁₆	1	2	yes	—
18	#11715897	—	4	2	no	—
19	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
20	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
21	#11715895	M₁₆	1	2	yes	—
22	C₂²×C₉₇	#97	1	1	yes	characteristic
23	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
24	C₄×C₉₇	#97	1	1	yes	characteristic
25	Dic₉₇	—	2	2	no	—
26	C₂×C₉₇	#322	1	1	yes	characteristic
27	C₂×C₉₇	#347	1	2	yes	—
28	#1313	—	97	1	no	maximal, Sylow
29	C₉₇	#1313	1	1	yes	characteristic, complemented, Sylow
30	#346	—	97	1	no	—
31	#346	—	97	1	no	—
32	#322	—	97	1	no	—
33	C₂×C₁₆	—	97	1	no	—
34	C₂×C₁₆	—	97	2	no	—
35	C₂×C₁₆	#87604	1	1	yes	characteristic
36	C₄×C₈	—	97	1	no	—
37	C₄×C₈	—	97	2	no	—
38	C₂×C₈	—	97	1	no	—
39	C₂×C₈	—	194	2	no	—
40	C₂×C₈	C₂×#87604	1	1	yes	characteristic
41	C₄²	—	97	1	no	—
42	C₁₆	—	97	2	no	—
43	C₁₆	—	194	2	no	—
44	C₁₆	#11715897	1	2	yes	complemented
45	C₂×C₄	—	97	2	no	—
46	C₂×C₄	C₄×#87604	1	1	yes	characteristic
47	C₈	—	97	2	no	—
48	C₈	—	388	2	no	—
49	C₈	C₂×#11715897	1	1	yes	characteristic
50	C₈	#13095519	1	1	yes	characteristic
51	C₂²	#15690864	1	1	yes	characteristic
52	C₄	—	194	2	no	—
53	C₄	C₄×#11715897	1	1	yes	characteristic
54	C₄	#15690864	1	1	yes	characteristic
55	C₂	#20261576	1	1	yes	characteristic
56	C₂	#20261625	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×#87604
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₂×C₈	C₂×#87604
Fitting	C₂×C₁₆×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	#97
cosocle	C₈×Dic₉₇	C₂²
hypercenter	C₂×C₁₆	#87604
hypocenter	C₉₇	#1313
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
13. #13095486	C₈	semidirect
15. C₁₆×C₉₇	C₈	semidirect
29. C₉₇	#1313	semidirect
44. C₁₆	#11715897	semidirect

Maximal subgroups

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

subgroup	index
#20261624	2
#20261576	2
#20260510	2
#1313	97