Subgroups — C₂⁴×#11715906 #27158992

Order 12416.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11715906	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×#87604	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715906	C₂²	1	140	yes	direct product factor
4	C₂³×#87604	C₄	1	15	yes	—
5	C₂³×#87604	C₂²	1	15	yes	—
6	C₂⁴×D₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715906	C₂³	1	120	yes	direct product factor
8	C₂²×#87604	C₂×C₄	1	105	yes	—
9	C₂²×#87604	C₂³	1	35	yes	—
10	C₂³×D₉₇	C₈	1	15	yes	complemented
11	C₂³×D₉₇	C₂×C₄	1	15	yes	—
12	C₂⁴×C₉₇	C₈	1	1	yes	characteristic, complemented
13	C₂³×C₉₇	C₂×C₈	1	15	yes	complemented
14	C₂²×D₉₇	C₂×C₈	1	105	yes	complemented
15	C₂²×D₉₇	C₂²×C₄	1	35	yes	—
16	C₂×#87604	C₂²×C₄	1	105	yes	—
17	C₂×#87604	C₂⁴	1	15	yes	—
18	#11715906	C₂⁴	1	16	yes	direct product factor
19	C₂²×C₉₇	C₂²×C₈	1	35	yes	complemented
20	C₂×D₉₇	C₂²×C₈	1	105	yes	complemented
21	C₂×D₉₇	C₂³×C₄	1	15	yes	—
22	#87604	C₂³×C₄	1	15	yes	—
23	#87604	C₂⁵	1	1	yes	characteristic
24	C₂×C₉₇	C₂³×C₈	1	15	yes	complemented
25	D₉₇	C₂³×C₈	1	15	yes	complemented
26	D₉₇	C₂⁴×C₄	1	1	yes	characteristic
27	C₂⁴×C₈	—	97	1	no	maximal, Sylow
28	C₉₇	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
29	C₂⁴×C₄	—	97	1	no	—
30	C₂³×C₈	—	97	30	no	—
31	C₂⁵	—	97	1	no	—
32	C₂³×C₄	—	97	15	no	—
33	C₂³×C₄	—	97	15	no	—
34	C₂²×C₈	—	97	140	no	—
35	C₂⁴	—	97	15	no	—
36	C₂⁴	—	97	15	no	—
37	C₂⁴	#11715906	1	1	yes	characteristic, direct product factor
38	C₂²×C₄	—	97	105	no	—
39	C₂²×C₄	—	97	35	no	—
40	C₂×C₈	—	97	120	no	—
41	C₂³	—	97	35	no	—
42	C₂³	—	97	105	no	—
43	C₂³	C₂×#11715906	1	15	yes	direct product factor
44	C₂×C₄	—	97	15	no	—
45	C₂×C₄	—	97	105	no	—
46	C₈	—	97	16	no	—
47	C₂²	—	97	15	no	—
48	C₂²	—	97	105	no	—
49	C₂²	C₂²×#11715906	1	35	yes	direct product factor
50	C₄	—	97	1	no	—
51	C₄	—	97	15	no	—
52	C₂	—	97	1	no	—
53	C₂	—	97	15	no	—
54	C₂	C₂³×#11715906	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	#11715906
commutator, hypocenter	C₉₇	C₂⁴×C₈
Frattini	C₁	G
Fitting, socle	C₂⁴×C₉₇	C₈
prime radical	C₂⁴×D₉₇	C₄
cosocle	#87604	C₂⁵
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂³×#11715906	C₂	direct (N × H)
3. C₂²×#11715906	C₂²	direct (N × H)
7. C₂×#11715906	C₂³	direct (N × H)
10. C₂³×D₉₇	C₈	semidirect
12. C₂⁴×C₉₇	C₈	semidirect
13. C₂³×C₉₇	C₂×C₈	semidirect
14. C₂²×D₉₇	C₂×C₈	semidirect
18. #11715906	C₂⁴	direct (N × H)
19. C₂²×C₉₇	C₂²×C₈	semidirect
20. C₂×D₉₇	C₂²×C₈	semidirect
24. C₂×C₉₇	C₂³×C₈	semidirect
25. D₉₇	C₂³×C₈	semidirect
28. C₉₇	C₂⁴×C₈	semidirect
37. C₂⁴	#11715906	direct (N × H)
43. C₂³	C₂×#11715906	direct (N × H)
49. C₂²	C₂²×#11715906	direct (N × H)
54. C₂	C₂³×#11715906	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11715906	2
C₂⁴×#87604	2
C₂⁴×C₈	97

Subgroups — C24×#11715906 #27158992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#11715906 #27158992