Subgroups — M₁₆×Dic₇ #89889

Order 448.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#1136	C₂	1	1	yes	maximal, characteristic
4	#6389	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₇	C₄	1	2	yes	—
8	C₇×M₁₆	C₄	1	4	yes	complemented
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₄×Dic₇	C₄	1	2	yes	—
11	C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	#1136	C₄	1	4	yes	complemented
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	C₄×M₁₆	—	7	1	no	maximal, Sylow
16	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₇	—	2	4	no	—
22	C₂×Dic₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₇	C₂×C₄	1	2	yes	—
24	C₇×C₈	C₂×C₄	1	4	yes	complemented
25	C₇⋊C₈	C₂×C₄	1	4	yes	complemented
26	C₂×M₁₆	—	7	1	no	—
27	C₂×M₁₆	D₇	1	1	yes	characteristic
28	C₂×C₄²	—	7	1	no	—
29	#97	—	7	2	no	—
30	C₄×C₈	—	7	2	no	—
31	C₂²×C₇	—	2	2	no	—
32	C₂²×C₇	C₄²	1	2	yes	—
33	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₇	C₄²	1	2	yes	—
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
37	Dic₇	—	2	2	no	—
38	Dic₇	M₁₆	1	4	yes	direct product factor
39	C₂²×C₄	—	7	2	no	—
40	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
41	M₁₆	—	7	4	no	—
42	M₁₆	Dic₇	1	4	yes	direct product factor
43	C₂×C₈	—	7	2	no	—
44	C₂×C₈	C₂×D₇	1	2	yes	—
45	C₄²	—	7	2	no	—
46	C₄²	—	7	1	no	—
47	C₄²	—	7	1	no	—
48	C₂×C₇	—	2	2	no	—
49	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₇	C₂×M₁₆	1	2	yes	—
51	C₂³	C₄×D₇	1	1	yes	characteristic
52	C₂×C₄	—	7	2	no	—
53	C₂×C₄	—	7	2	no	—
54	C₂×C₄	—	14	4	no	—
55	C₂×C₄	C₄×D₇	1	1	yes	characteristic
56	C₂×C₄	C₄×D₇	1	2	yes	—
57	C₂×C₄	C₂×Dic₇	1	2	yes	—
58	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
59	C₈	—	7	4	no	—
60	C₈	C₂×Dic₇	1	4	yes	complemented
61	C₇	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	2	2	no	—
63	C₂²	C₄×Dic₇	1	2	yes	—
64	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
65	C₄	—	7	4	no	—
66	C₄	—	14	2	no	—
67	C₄	C₄×Dic₇	1	2	yes	—
68	C₄	C₂×C₄×D₇	1	1	yes	characteristic
69	C₄	C₂²×Dic₇	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	D₇×M₁₆	1	2	yes	—
72	C₂	C₂×C₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂×C₄²
Fitting	C₂×C₇×M₁₆	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	C₄×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #6389	C₂	semidirect
5. C₈×Dic₇	C₂	semidirect
8. C₇×M₁₆	C₄	semidirect
13. #1136	C₄	semidirect
24. C₇×C₈	C₂×C₄	semidirect
25. C₇⋊C₈	C₂×C₄	semidirect
38. Dic₇	M₁₆	direct (N × H)
42. M₁₆	Dic₇	direct (N × H)
60. C₈	C₂×Dic₇	semidirect
61. C₇	C₄×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₁₆	2
C₂×C₄×Dic₇	2
C₂×#1136	2
#6389	2
C₈×Dic₇	2
C₄×M₁₆	7

Subgroups — M16×Dic7 #89889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₇ #89889