Subgroups — D₇×M₁₆ #6469

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	#1131	C₂	1	2	yes	maximal, complemented
5	C₈×D₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₇	C₄	1	1	yes	characteristic
9	C₄×D₇	C₄	1	1	yes	characteristic
10	C₄×D₇	C₄	1	1	yes	characteristic
11	C₄×D₇	C₂²	1	1	yes	characteristic
12	C₄×D₇	C₂²	1	1	yes	characteristic
13	C₇×C₈	C₂²	1	2	yes	complemented
14	C₇⋊C₈	C₂²	1	2	yes	complemented
15	C₂×M₁₆	—	7	1	no	maximal, Sylow
16	C₂²×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×D₇	—	2	2	no	—
18	C₂×D₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₇	C₂³	1	1	yes	characteristic
22	Dic₇	C₂×C₄	1	1	yes	characteristic
23	Dic₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄	—	7	1	no	—
25	M₁₆	—	7	1	no	—
26	M₁₆	—	7	2	no	—
27	M₁₆	D₇	1	1	yes	characteristic, direct product factor
28	C₂×C₈	—	7	2	no	—
29	C₂×C₇	—	2	1	no	—
30	C₂×C₇	C₂²×C₄	1	1	yes	characteristic
31	D₇	—	2	1	no	—
32	D₇	M₁₆	1	2	yes	direct product factor
33	C₂³	—	7	1	no	—
34	C₂×C₄	—	7	1	no	—
35	C₂×C₄	—	7	1	no	—
36	C₂×C₄	—	7	1	no	—
37	C₂×C₄	—	7	1	no	—
38	C₂×C₄	—	7	1	no	—
39	C₂×C₄	C₂×D₇	1	1	yes	characteristic
40	C₈	—	7	2	no	—
41	C₈	C₂×D₇	1	2	yes	complemented
42	C₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
43	C₂²	—	7	1	no	—
44	C₂²	—	7	1	no	—
45	C₂²	—	14	2	no	—
46	C₂²	C₄×D₇	1	1	yes	characteristic
47	C₄	—	7	1	no	—
48	C₄	—	7	1	no	—
49	C₄	C₄×D₇	1	1	yes	characteristic
50	C₄	C₂²×D₇	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	—	7	2	no	—
53	C₂	—	14	1	no	—
54	C₂	C₂×C₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₇
commutator, socle	C₂×C₇	C₂²×C₄
Fitting	C₇×M₁₆	C₂
prime radical	C₂²×D₇	C₄
cosocle	C₄×C₇	C₂³
hypercenter	M₁₆	D₇
hypocenter	C₇	C₂×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₇×M₁₆	C₂	semidirect
3. #1136	C₂	semidirect
4. #1131	C₂	semidirect
5. C₈×D₇	C₂	semidirect
13. C₇×C₈	C₂²	semidirect
14. C₇⋊C₈	C₂²	semidirect
27. M₁₆	D₇	direct (N × H)
32. D₇	M₁₆	direct (N × H)
41. C₈	C₂×D₇	semidirect
42. C₇	C₂×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇	2
C₇×M₁₆	2
#1136	2
#1131	2
C₈×D₇	2
C₂×M₁₆	7

Subgroups — D7×M16 #6469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×M₁₆ #6469