Subgroups — #22577521

Order 8768.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984110	C₂	1	1	yes	maximal, characteristic
2	#17984094	C₂	1	2	yes	maximal, complemented
3	#17984024	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984016	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14555510	C₂²	1	2	yes	—
7	Dic₅₄₈	C₄	1	1	yes	characteristic, complemented
8	#14555499	C₄	1	2	yes	complemented
9	D₁₀₉₆	C₄	1	1	yes	characteristic, complemented
10	#14555494	C₂²	1	2	yes	—
11	#14555493	—	2	2	no	—
12	C₈×D₁₃₇	—	2	2	no	—
13	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	Dic₂₇₄	C₂×C₄	1	2	yes	—
17	D₅₄₈	C₂×C₄	1	2	yes	—
18	C₂²⋊D₁₃₇	—	2	2	no	—
19	C₄×D₁₃₇	—	2	2	no	—
20	#11764040	—	2	2	no	—
21	C₂×C₄×C₁₃₇	—	2	1	no	—
22	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
23	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₃₇	D₄	1	1	yes	characteristic
26	C₈×C₁₃₇	D₄	1	1	yes	characteristic
27	C₂²×C₁₃₇	#41	1	1	yes	characteristic
28	C₂×D₁₃₇	—	2	2	no	—
29	C₄×C₁₃₇	—	2	2	no	—
30	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
32	Dic₁₃₇	—	2	2	no	—
33	C₂×C₁₃₇	—	2	1	no	—
34	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
35	D₁₃₇	—	4	2	no	—
36	C₁₃₇	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	137	1	no	maximal, Sylow
38	#135	—	137	1	no	—
39	#131	—	137	2	no	—
40	#108	—	137	1	no	—
41	#104	—	137	2	no	—
42	C₄×C₈	D₁₃₇	1	1	yes	characteristic, complemented
43	#41	—	137	2	no	—
44	Q₁₆	—	137	1	no	—
45	SD₁₆	—	137	2	no	—
46	D₈	—	137	1	no	—
47	M₁₆	—	274	2	no	—
48	M₁₆	—	137	2	no	—
49	C₂×C₈	—	274	2	no	—
50	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
51	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
52	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
53	Q₈	—	137	2	no	—
54	D₄	—	274	2	no	—
55	D₄	—	137	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	274	2	no	—
58	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
59	C₈	—	274	2	no	—
60	C₈	C₄×D₁₃₇	1	1	yes	characteristic
61	C₈	C₄×D₁₃₇	1	1	yes	characteristic
62	C₈	D₅₄₈	1	1	yes	characteristic
63	C₈	D₅₄₈	1	1	yes	characteristic
64	C₂²	—	274	2	no	—
65	C₂²	#14555510	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	274	2	no	—
68	C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
69	C₄	C₂×D₅₄₈	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	548	2	no	—
72	C₂	C₄×D₅₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₅₄₈
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₄×C₈×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₄×D₄
hypercenter	C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	#443
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #17984094	C₂	semidirect
3. #17984024	C₂	semidirect
4. #17984016	C₂	semidirect
5. C₄×C₈×C₁₃₇	C₂	semidirect
7. Dic₅₄₈	C₄	semidirect
8. #14555499	C₄	semidirect
9. D₁₀₉₆	C₄	semidirect
36. C₁₃₇	#443	semidirect
42. C₄×C₈	D₁₃₇	semidirect