Subgroups — M₁₆×D₃₉₇ #27163099

Order 12704.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic
2	#20263715	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263714	C₂	1	2	yes	maximal, complemented
4	C₈×D₃₉₇	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	C₄	1	1	yes	characteristic
7	C₂×Dic₃₉₇	C₄	1	1	yes	characteristic
8	C₄×D₃₉₇	C₄	1	1	yes	characteristic
9	C₄×D₃₉₇	C₄	1	1	yes	characteristic
10	C₄×D₃₉₇	C₂²	1	1	yes	characteristic
11	C₄×D₃₉₇	C₂²	1	1	yes	characteristic
12	#15691787	C₂²	1	2	yes	complemented
13	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
14	C₈×C₃₉₇	C₂²	1	2	yes	complemented
15	C₂²×C₃₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×D₃₉₇	—	2	2	no	—
17	C₂×D₃₉₇	C₂×C₄	1	1	yes	characteristic
18	C₂×D₃₉₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
21	Dic₃₉₇	C₂×C₄	1	1	yes	characteristic
22	Dic₃₉₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃₉₇	—	2	1	no	—
24	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
25	D₃₉₇	—	2	1	no	—
26	D₃₉₇	M₁₆	1	2	yes	direct product factor
27	C₃₉₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₂×M₁₆	—	397	1	no	maximal, Sylow
29	C₂²×C₄	—	397	1	no	—
30	M₁₆	—	397	1	no	—
31	M₁₆	—	397	2	no	—
32	M₁₆	D₃₉₇	1	1	yes	characteristic, direct product factor
33	C₂×C₈	—	397	2	no	—
34	C₂³	—	397	1	no	—
35	C₂×C₄	—	397	1	no	—
36	C₂×C₄	—	397	1	no	—
37	C₂×C₄	—	397	1	no	—
38	C₂×C₄	—	397	1	no	—
39	C₂×C₄	—	397	1	no	—
40	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
41	C₈	—	397	2	no	—
42	C₈	C₂×D₃₉₇	1	2	yes	complemented
43	C₂²	—	794	2	no	—
44	C₂²	—	397	1	no	—
45	C₂²	—	397	1	no	—
46	C₂²	C₄×D₃₉₇	1	1	yes	characteristic
47	C₄	—	397	1	no	—
48	C₄	—	397	1	no	—
49	C₄	C₄×D₃₉₇	1	1	yes	characteristic
50	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	—	794	1	no	—
53	C₂	—	397	2	no	—
54	C₂	C₂×C₄×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₃₉₇
commutator, socle	C₂×C₃₉₇	C₂²×C₄
Fitting	M₁₆×C₃₉₇	C₂
prime radical	C₂²×D₃₉₇	C₄
cosocle	C₄×C₃₉₇	C₂³
hypercenter	M₁₆	D₃₉₇
hypocenter	C₃₉₇	C₂×M₁₆
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. #20263715	C₂	semidirect
3. #20263714	C₂	semidirect
4. C₈×D₃₉₇	C₂	semidirect
5. M₁₆×C₃₉₇	C₂	semidirect
12. #15691787	C₂²	semidirect
14. C₈×C₃₉₇	C₂²	semidirect
26. D₃₉₇	M₁₆	direct (N × H)
27. C₃₉₇	C₂×M₁₆	semidirect
32. M₁₆	D₃₉₇	direct (N × H)
42. C₈	C₂×D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₉₇	2
#20263715	2
#20263714	2
C₈×D₃₉₇	2
M₁₆×C₃₉₇	2
C₂×M₁₆	397

Subgroups — M16×D397 #27163099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×D₃₉₇ #27163099