Subgroups — #22578278

Order 8768.

90 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×#14555518	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#14555520	C₄	1	2	yes	—
5	#14555519	—	2	1	no	—
6	#14555518	—	2	1	no	—
7	#14555518	—	2	1	no	—
8	#14555518	C₄	1	2	yes	—
9	C₂×#11764047	—	2	1	no	—
10	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
11	#14555494	C₄	1	1	yes	characteristic
12	#14555493	—	2	1	no	—
13	C₈×D₁₃₇	—	2	1	no	—
14	M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
15	#11764047	—	2	2	no	—
16	#11764047	—	2	2	no	—
17	C₂²×D₁₃₇	Q₈	1	1	yes	characteristic
18	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
19	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄×D₁₃₇	D₄	1	1	yes	characteristic
22	C₄×D₁₃₇	D₄	1	1	yes	characteristic
23	#11764040	—	2	1	no	—
24	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₃₇	—	2	1	no	—
26	C₂²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂×D₁₃₇	—	2	2	no	—
28	C₂×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₁₃₇	C₄²	1	1	yes	characteristic
31	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₁₃₇	—	2	1	no	—
35	C₂×C₁₃₇	#95	1	1	yes	characteristic
36	D₁₃₇	—	2	1	no	—
37	D₁₃₇	#108	1	2	yes	—
38	C₁₃₇	#341	1	1	yes	characteristic, complemented, Sylow
39	#341	—	137	1	no	maximal, Sylow
40	C₂×M₁₆	—	137	1	no	—
41	C₂×M₁₆	—	137	1	no	—
42	C₂²×C₈	—	137	1	no	—
43	C₂²×C₄	—	137	1	no	—
44	M₁₆	—	274	1	no	—
45	M₁₆	—	274	1	no	—
46	M₁₆	—	137	2	no	—
47	M₁₆	—	137	1	no	—
48	M₁₆	#10587739	1	1	yes	characteristic
49	C₂×C₈	—	274	1	no	—
50	C₂×C₈	—	274	1	no	—
51	C₂×C₈	—	274	1	no	—
52	C₂×C₈	—	274	1	no	—
53	C₂×C₈	—	137	2	no	—
54	C₂³	—	137	1	no	—
55	C₂×C₄	—	137	1	no	—
56	C₂×C₄	—	137	1	no	—
57	C₂×C₄	—	137	1	no	—
58	C₂×C₄	—	137	1	no	—
59	C₂×C₄	—	137	1	no	—
60	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	274	2	no	—
63	C₈	—	274	2	no	—
64	C₈	—	274	1	no	—
65	C₂²	—	274	2	no	—
66	C₂²	—	137	1	no	—
67	C₂²	—	137	1	no	—
68	C₂²	C₄⋊#10587739	1	1	yes	characteristic
69	C₄	—	137	1	no	—
70	C₄	—	137	1	no	—
71	C₄	C₄×#10587739	1	1	yes	characteristic
72	C₄	C₂²⋊#10587739	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	274	1	no	—
75	C₂	—	137	2	no	—
76	C₂	(C₂×#10587739)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₄²
Frattini	C₂×C₄	C₂×#10587739
Fitting	M₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	#95
hypercenter	M₁₆	#10587739
hypocenter	C₁₃₇	#341
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
38. C₁₃₇	#341	semidirect