Subgroups — #27156841

Order 12416.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690860	C₂	1	2	yes	maximal
2	C₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690860	C₄	1	4	yes	complemented
4	C₂×#13095486	C₂²	1	1	yes	characteristic
5	C₈×Dic₉₇	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715895	C₈	1	1	yes	characteristic
8	C₄×Dic₉₇	C₈	1	1	yes	characteristic
9	#13095486	C₂×C₄	1	2	yes	—
10	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
11	C₁₆×C₉₇	C₈	1	2	yes	—
12	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
13	C₂×Dic₉₇	C₁₆	1	2	yes	—
14	C₈×C₉₇	C₄²	1	1	yes	characteristic
15	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
16	#11715895	C₁₆	1	2	yes	—
17	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
18	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
19	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
20	Dic₉₇	C₃₂	1	4	yes	complemented
21	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
22	C₂×C₉₇	C₂×C₃₂	1	2	yes	—
23	C₄×C₃₂	—	97	1	no	maximal, Sylow
24	C₉₇	C₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃₂	—	97	2	no	—
26	C₄×C₁₆	—	97	1	no	—
27	C₂×C₁₆	—	97	1	no	—
28	C₂×C₁₆	#87604	1	1	yes	characteristic
29	C₄×C₈	—	97	1	no	—
30	C₃₂	—	97	4	no	—
31	C₂×C₈	—	97	1	no	—
32	C₂×C₈	C₂×#87604	1	1	yes	characteristic
33	C₄²	—	97	1	no	—
34	C₁₆	—	97	2	no	—
35	C₁₆	#11715897	1	2	yes	—
36	C₂×C₄	—	97	2	no	—
37	C₂×C₄	#13095519	1	1	yes	characteristic
38	C₈	—	97	2	no	—
39	C₈	C₄×#87604	1	1	yes	characteristic
40	C₈	C₂×#11715897	1	1	yes	characteristic
41	C₂²	#15690883	1	1	yes	characteristic
42	C₄	—	97	4	no	—
43	C₄	C₄×#11715897	1	1	yes	characteristic
44	C₄	#15690883	1	1	yes	characteristic
45	C₂	#20261621	1	1	yes	characteristic
46	C₂	#20261666	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₁₆	#87604
commutator, hypocenter	C₉₇	C₄×C₃₂
Frattini	C₂×C₈	C₂×#87604
Fitting	C₂×C₁₆×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₂×C₁₆
cosocle	C₂×#13095486	C₂²
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
3. #15690860	C₄	semidirect
20. Dic₉₇	C₃₂	semidirect
24. C₉₇	C₄×C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690860	2
C₁₆×Dic₉₇	2
C₄×C₃₂	97