Subgroups — M₁₆×#10587739 #22578363

Order 8768.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10587739	C₂	1	1	yes	maximal, characteristic
3	#17984142	C₂	1	2	yes	maximal, complemented
4	C₈×#10587739	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10587739	C₄	1	2	yes	—
7	#14555520	C₄	1	2	yes	complemented
8	#14555519	C₄	1	2	yes	complemented
9	#14555518	C₂²	1	2	yes	—
10	C₄×#10587739	C₄	1	2	yes	—
11	C₄×#10587739	C₂²	1	1	yes	characteristic
12	C₄×#10587739	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	#14555494	C₄	1	1	yes	characteristic, complemented
15	#14555493	C₄	1	2	yes	complemented
16	C₈×D₁₃₇	C₂²	1	2	yes	—
17	M₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
18	C₂×#10587739	—	2	4	no	—
19	C₂×#10587739	C₂×C₄	1	2	yes	—
20	C₂×#10587739	C₂×C₄	1	2	yes	—
21	#11764047	C₂×C₄	1	4	yes	complemented
22	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×D₁₃₇	C₂³	1	1	yes	characteristic
28	#11764040	C₂×C₄	1	2	yes	complemented
29	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₈×C₁₃₇	C₂×C₄	1	2	yes	complemented
31	C₂²×C₁₃₇	C₄²	1	1	yes	characteristic
32	C₂×D₁₃₇	—	2	2	no	—
33	C₂×D₁₃₇	C₄²	1	1	yes	characteristic
34	C₂×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	#10587739	—	2	2	no	—
36	#10587739	M₁₆	1	4	yes	direct product factor
37	C₄×C₁₃₇	C₄²	1	1	yes	characteristic
38	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	Dic₁₃₇	C₄²	1	1	yes	characteristic
40	Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₁₃₇	—	2	1	no	—
42	C₂×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
43	D₁₃₇	—	2	1	no	—
44	D₁₃₇	C₂×M₁₆	1	2	yes	—
45	C₁₃₇	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×M₁₆	—	137	1	no	maximal, Sylow
47	C₂×M₁₆	—	137	1	no	—
48	C₂×M₁₆	—	137	1	no	—
49	C₂×C₄²	—	137	1	no	—
50	#97	—	137	2	no	—
51	C₄×C₈	—	137	2	no	—
52	C₂²×C₄	—	137	2	no	—
53	C₂²×C₄	—	137	1	no	—
54	M₁₆	—	137	2	no	—
55	M₁₆	—	137	2	no	—
56	M₁₆	—	137	1	no	—
57	M₁₆	—	137	2	no	—
58	M₁₆	#10587739	1	1	yes	characteristic, direct product factor
59	C₂×C₈	—	137	2	no	—
60	C₂×C₈	—	137	2	no	—
61	C₄²	—	137	1	no	—
62	C₄²	—	137	1	no	—
63	C₄²	—	137	2	no	—
64	C₂³	—	137	1	no	—
65	C₂×C₄	—	274	4	no	—
66	C₂×C₄	—	137	2	no	—
67	C₂×C₄	—	137	2	no	—
68	C₂×C₄	—	137	1	no	—
69	C₂×C₄	—	137	1	no	—
70	C₂×C₄	—	137	1	no	—
71	C₂×C₄	—	137	1	no	—
72	C₂×C₄	—	137	1	no	—
73	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
74	C₈	—	137	4	no	—
75	C₈	—	137	2	no	—
76	C₈	C₂×#10587739	1	2	yes	complemented
77	C₂²	—	274	2	no	—
78	C₂²	—	137	1	no	—
79	C₂²	—	137	1	no	—
80	C₂²	C₄×#10587739	1	1	yes	characteristic
81	C₄	—	274	2	no	—
82	C₄	—	137	4	no	—
83	C₄	—	137	1	no	—
84	C₄	—	137	1	no	—
85	C₄	C₄×#10587739	1	1	yes	characteristic
86	C₄	C₂²×#10587739	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	274	1	no	—
89	C₂	—	137	2	no	—
90	C₂	C₂×C₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#10587739
commutator, socle	C₂×C₁₃₇	C₂×C₄²
Fitting	M₁₆×C₁₃₇	C₄
hypercenter	M₁₆	#10587739
hypocenter	C₁₃₇	C₄×M₁₆
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #17984142	C₂	semidirect
4. C₈×#10587739	C₂	semidirect
7. #14555520	C₄	semidirect
8. #14555519	C₄	semidirect
14. #14555494	C₄	semidirect
15. #14555493	C₄	semidirect
17. M₁₆×C₁₃₇	C₄	semidirect
21. #11764047	C₂×C₄	semidirect
28. #11764040	C₂×C₄	semidirect
30. C₈×C₁₃₇	C₂×C₄	semidirect
36. #10587739	M₁₆	direct (N × H)
45. C₁₃₇	C₄×M₁₆	semidirect
58. M₁₆	#10587739	direct (N × H)
76. C₈	C₂×#10587739	semidirect

Subgroups — M16×#10587739 #22578363

Conjugacy classes of subgroups

Complements

Subgroups — M₁₆×#10587739 #22578363