Subgroups — C₈²×D₉₇ #27140008

Order 12416.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×D₉₇	C₂	1	3	yes	maximal
2	C₈×#11715895	C₂	1	3	yes	maximal, complemented
3	C₈²×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₉₇	C₄	1	6	yes	—
5	C₄²×D₉₇	C₂²	1	1	yes	characteristic
6	C₄×#11715895	C₂²	1	3	yes	—
7	C₈×Dic₉₇	C₄	1	6	yes	—
8	C₄×C₈×C₉₇	C₂²	1	3	yes	—
9	C₂×C₄×D₉₇	C₂×C₄	1	3	yes	—
10	C₂×#11715895	C₂×C₄	1	6	yes	—
11	C₈×D₉₇	C₈	1	24	yes	direct product factor
12	C₄×Dic₉₇	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₉₇	C₂×C₄	1	6	yes	—
14	C₄²×C₉₇	C₂³	1	1	yes	characteristic
15	C₂²×D₉₇	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
17	C₂×Dic₉₇	C₄²	1	3	yes	—
18	C₄×D₉₇	C₂×C₈	1	12	yes	—
19	C₈×C₉₇	C₂×C₈	1	12	yes	complemented
20	#11715895	C₂×C₈	1	12	yes	complemented
21	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
22	C₂×D₉₇	C₄×C₈	1	6	yes	—
23	C₄×C₉₇	C₂²×C₈	1	6	yes	—
24	Dic₉₇	C₄×C₈	1	6	yes	—
25	C₂×C₉₇	C₂×C₄×C₈	1	3	yes	—
26	D₉₇	C₈²	1	4	yes	direct product factor
27	C₂×C₈²	—	97	1	no	maximal, Sylow
28	C₉₇	C₂×C₈²	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₄×C₈	—	97	3	no	—
30	C₈²	—	97	3	no	—
31	C₈²	D₉₇	1	1	yes	characteristic, direct product factor
32	C₂²×C₈	—	97	6	no	—
33	C₂×C₄²	—	97	1	no	—
34	C₄×C₈	—	97	6	no	—
35	C₄×C₈	—	97	3	no	—
36	C₄×C₈	C₂×D₉₇	1	3	yes	—
37	C₂²×C₄	—	97	3	no	—
38	C₂×C₈	—	97	6	no	—
39	C₂×C₈	—	97	24	no	—
40	C₂×C₈	C₄×D₉₇	1	6	yes	—
41	C₄²	—	97	3	no	—
42	C₄²	C₂²×D₉₇	1	1	yes	characteristic
43	C₂³	—	97	1	no	—
44	C₂×C₄	—	97	3	no	—
45	C₂×C₄	—	97	12	no	—
46	C₂×C₄	C₂×C₄×D₉₇	1	3	yes	—
47	C₈	—	97	12	no	—
48	C₈	C₈×D₉₇	1	12	yes	direct product factor
49	C₂²	—	97	6	no	—
50	C₂²	C₄²×D₉₇	1	1	yes	characteristic
51	C₄	—	97	6	no	—
52	C₄	C₂×C₈×D₉₇	1	6	yes	—
53	C₂	—	97	4	no	—
54	C₂	C₄×C₈×D₉₇	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₈²	D₉₇
commutator, hypocenter	C₉₇	C₂×C₈²
Frattini	C₄²	C₂²×D₉₇
Fitting	C₈²×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄²
prime radical	C₂²×D₉₇	C₄²
cosocle	C₄²×C₉₇	C₂³
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. C₈×#11715895	C₂	semidirect
3. C₈²×C₉₇	C₂	semidirect
11. C₈×D₉₇	C₈	direct (N × H)
19. C₈×C₉₇	C₂×C₈	semidirect
20. #11715895	C₂×C₈	semidirect
26. D₉₇	C₈²	direct (N × H)
28. C₉₇	C₂×C₈²	semidirect
31. C₈²	D₉₇	direct (N × H)
48. C₈	C₈×D₉₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₈×D₉₇	2
C₈×#11715895	2
C₈²×C₉₇	2
C₂×C₈²	97

Subgroups — C82×D97 #27140008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈²×D₉₇ #27140008