Subgroups — #22550148

Order 8128.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861240	C₂	1	2	yes	maximal
2	#16861215	C₂	1	1	yes	maximal, characteristic
3	#16861145	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861128	C₂	1	2	yes	maximal, complemented
5	C₂×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523248	C₂²	1	1	yes	characteristic
7	#14523238	—	2	2	no	—
8	#14523235	—	2	2	no	—
9	#14523232	C₂²	1	1	yes	characteristic
10	#14523232	C₂²	1	2	yes	—
11	#14523231	—	2	1	no	—
12	C₈×D₁₂₇	—	2	1	no	—
13	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
14	Q₁₆×C₁₂₇	—	2	1	no	—
15	Q₁₆×C₁₂₇	—	2	1	no	—
16	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	Dic₂₅₄	D₄	1	1	yes	characteristic
18	D₅₀₈	D₄	1	1	yes	characteristic
19	C₂²⋊D₁₂₇	—	2	1	no	—
20	C₄×D₁₂₇	—	2	1	no	—
21	C₁₂₇⋊C₈	—	2	1	no	—
22	C₁₂₇⋊C₈	—	2	2	no	—
23	Q₈×C₁₂₇	—	2	2	no	—
24	Q₈×C₁₂₇	—	2	2	no	—
25	C₂×C₄×C₁₂₇	—	2	2	no	—
26	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₈×C₁₂₇	D₄	1	1	yes	characteristic
28	C₈×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂²×C₁₂₇	#41	1	1	yes	characteristic
30	C₂×D₁₂₇	—	2	1	no	—
31	C₄×C₁₂₇	—	4	2	no	—
32	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	Dic₁₂₇	—	2	1	no	—
35	C₂×C₁₂₇	—	2	1	no	—
36	C₂×C₁₂₇	#121	1	1	yes	characteristic
37	D₁₂₇	—	4	1	no	—
38	C₁₂₇	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	127	1	no	maximal, Sylow
40	#137	—	127	2	no	—
41	C₂×Q₁₆	D₁₂₇	1	1	yes	characteristic, complemented
42	#131	—	127	1	no	—
43	#108	—	127	1	no	—
44	#101	—	127	2	no	—
45	#41	—	127	1	no	—
46	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	2	1	no	—
49	Q₁₆	—	254	2	no	—
50	SD₁₆	—	254	2	no	—
51	M₁₆	—	127	1	no	—
52	M₁₆	—	127	2	no	—
53	M₁₆	—	254	1	no	—
54	C₂×C₈	—	254	1	no	—
55	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	Q₈	—	127	1	no	—
59	D₄	—	127	1	no	—
60	D₄	—	254	1	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	254	1	no	—
63	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
64	C₈	—	254	2	no	—
65	C₈	—	254	1	no	—
66	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
67	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₂²	—	254	1	no	—
69	C₂²	#14523247	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	254	1	no	—
72	C₄	D₄×D₁₂₇	1	1	yes	characteristic
73	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	508	1	no	—
76	C₂	#16861180	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16861180
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂×Q₁₆×C₁₂₇	C₂
socle	C₂×C₁₂₇	#121
hypercenter	C₂×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	#473
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #16861145	C₂	semidirect
4. #16861128	C₂	semidirect
5. C₂×Q₁₆×C₁₂₇	C₂	semidirect
38. C₁₂₇	#473	semidirect
41. C₂×Q₁₆	D₁₂₇	semidirect