Subgroups — M₁₆×#89137 #20295986

Order 6976.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#89137	C₂	1	1	yes	maximal, characteristic
3	#16812753	C₂	1	2	yes	maximal, complemented
4	C₈×#89137	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#89137	C₄	1	2	yes	—
7	#13182039	C₄	1	2	yes	complemented
8	#13182038	C₄	1	2	yes	complemented
9	D₁₀₉⋊C₈	C₂²	1	2	yes	—
10	C₄×#89137	C₄	1	2	yes	—
11	C₄×#89137	C₂²	1	1	yes	characteristic
12	C₄×#89137	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	#13182013	C₄	1	1	yes	characteristic, complemented
15	#13182012	C₄	1	2	yes	complemented
16	C₈×D₁₀₉	C₂²	1	2	yes	—
17	M₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
18	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×#89137	—	2	4	no	—
20	C₂×#89137	C₂×C₄	1	2	yes	—
21	C₂×#89137	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×D₁₀₉	C₂³	1	1	yes	characteristic
28	#11724786	C₂×C₄	1	4	yes	complemented
29	C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
30	#11724784	C₂×C₄	1	2	yes	complemented
31	C₂²×C₁₀₉	C₄²	1	1	yes	characteristic
32	C₂×D₁₀₉	—	2	2	no	—
33	C₂×D₁₀₉	C₄²	1	1	yes	characteristic
34	C₂×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	#89137	—	2	2	no	—
36	#89137	M₁₆	1	4	yes	direct product factor
37	C₄×C₁₀₉	C₄²	1	1	yes	characteristic
38	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	Dic₁₀₉	C₄²	1	1	yes	characteristic
40	Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₁₀₉	—	2	1	no	—
42	C₂×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
43	D₁₀₉	—	2	1	no	—
44	D₁₀₉	C₂×M₁₆	1	2	yes	—
45	C₁₀₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×M₁₆	—	109	1	no	maximal, Sylow
47	C₂×M₁₆	—	109	1	no	—
48	C₂×M₁₆	—	109	1	no	—
49	C₂×C₄²	—	109	1	no	—
50	#97	—	109	2	no	—
51	C₄×C₈	—	109	2	no	—
52	C₂²×C₄	—	109	1	no	—
53	C₂²×C₄	—	109	2	no	—
54	M₁₆	—	109	2	no	—
55	M₁₆	—	109	2	no	—
56	M₁₆	—	109	1	no	—
57	M₁₆	—	109	2	no	—
58	M₁₆	#89137	1	1	yes	characteristic, direct product factor
59	C₂×C₈	—	109	2	no	—
60	C₂×C₈	—	109	2	no	—
61	C₄²	—	109	2	no	—
62	C₄²	—	109	1	no	—
63	C₄²	—	109	1	no	—
64	C₂³	—	109	1	no	—
65	C₂×C₄	—	109	2	no	—
66	C₂×C₄	—	109	1	no	—
67	C₂×C₄	—	109	1	no	—
68	C₂×C₄	—	109	1	no	—
69	C₂×C₄	—	109	1	no	—
70	C₂×C₄	—	109	1	no	—
71	C₂×C₄	—	109	2	no	—
72	C₂×C₄	—	218	4	no	—
73	C₂×C₄	C₂×#89137	1	1	yes	characteristic
74	C₈	—	109	4	no	—
75	C₈	—	109	2	no	—
76	C₈	C₂×#89137	1	2	yes	complemented
77	C₂²	—	109	1	no	—
78	C₂²	—	109	1	no	—
79	C₂²	—	218	2	no	—
80	C₂²	C₄×#89137	1	1	yes	characteristic
81	C₄	—	109	4	no	—
82	C₄	—	109	1	no	—
83	C₄	—	109	1	no	—
84	C₄	—	218	2	no	—
85	C₄	C₄×#89137	1	1	yes	characteristic
86	C₄	C₂²×#89137	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	109	2	no	—
89	C₂	—	218	1	no	—
90	C₂	C₂×C₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#89137
commutator, socle	C₂×C₁₀₉	C₂×C₄²
Fitting	M₁₆×C₁₀₉	C₄
hypercenter	M₁₆	#89137
hypocenter	C₁₀₉	C₄×M₁₆
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #16812753	C₂	semidirect
4. C₈×#89137	C₂	semidirect
7. #13182039	C₄	semidirect
8. #13182038	C₄	semidirect
14. #13182013	C₄	semidirect
15. #13182012	C₄	semidirect
17. M₁₆×C₁₀₉	C₄	semidirect
28. #11724786	C₂×C₄	semidirect
29. C₈×C₁₀₉	C₂×C₄	semidirect
30. #11724784	C₂×C₄	semidirect
36. #89137	M₁₆	direct (N × H)
45. C₁₀₉	C₄×M₁₆	semidirect
58. M₁₆	#89137	direct (N × H)
76. C₈	C₂×#89137	semidirect

Subgroups — M16×#89137 #20295986

Conjugacy classes of subgroups

Complements

Subgroups — M₁₆×#89137 #20295986