Subgroups — #22550151

Order 8128.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861239	C₂	1	2	yes	maximal
2	C₂×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861145	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861127	C₂	1	2	yes	maximal, complemented
5	#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₅₄	C₂²	1	2	yes	—
7	Dic₅₀₈	—	2	1	no	—
8	Dic₅₀₈	—	2	1	no	—
9	#14523238	—	2	2	no	—
10	#14523236	—	2	2	no	—
11	#14523232	C₂²	1	2	yes	—
12	#41×C₁₂₇	C₂²	1	1	yes	characteristic
13	M₁₆×C₁₂₇	—	2	1	no	—
14	M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×C₁₂₇	—	2	1	no	—
16	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	Dic₂₅₄	—	2	2	no	—
18	Dic₂₅₄	—	2	2	no	—
19	C₂×Dic₁₂₇	—	2	2	no	—
20	C₁₂₇⋊C₈	—	2	2	no	—
21	Q₈×C₁₂₇	D₄	1	1	yes	characteristic
22	D₄×C₁₂₇	—	2	1	no	—
23	D₄×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	—	2	1	no	—
25	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₈×C₁₂₇	—	2	1	no	—
27	C₈×C₁₂₇	D₄	1	1	yes	characteristic
28	C₈×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂²×C₁₂₇	—	2	1	no	—
30	C₂²×C₁₂₇	#41	1	1	yes	characteristic
31	C₄×C₁₂₇	—	2	1	no	—
32	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	Dic₁₂₇	—	4	2	no	—
35	C₂×C₁₂₇	—	2	1	no	—
36	C₂×C₁₂₇	—	4	1	no	—
37	C₂×C₁₂₇	#121	1	1	yes	characteristic
38	C₁₂₇	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	127	1	no	maximal, Sylow
40	#137	—	127	2	no	—
41	C₂×Q₁₆	—	127	1	no	—
42	#131	D₁₂₇	1	1	yes	characteristic
43	#108	—	127	1	no	—
44	#101	—	127	2	no	—
45	#41	C₂×D₁₂₇	1	1	yes	characteristic
46	C₂×Q₈	—	127	2	no	—
47	Q₁₆	—	254	2	no	—
48	Q₁₆	—	254	1	no	—
49	Q₁₆	—	254	1	no	—
50	SD₁₆	—	254	2	no	—
51	M₁₆	—	2	1	no	—
52	M₁₆	—	127	2	no	—
53	M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	Q₈	—	254	2	no	—
57	Q₈	—	254	2	no	—
58	Q₈	D₅₀₈	1	1	yes	characteristic
59	D₄	—	2	1	no	—
60	D₄	D₅₀₈	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	254	2	no	—
63	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	—	254	2	no	—
66	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
67	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₂²	—	2	1	no	—
69	C₂²	#14523248	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	508	2	no	—
72	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
73	C₄	C₂×D₅₀₈	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	#16861183	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16861183
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#131×C₁₂₇	C₂
socle	C₂×C₁₂₇	#121
hypercenter	#131	D₁₂₇
hypocenter	C₁₂₇	#473
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₅₀₈	C₂	semidirect
3. #16861145	C₂	semidirect
4. #16861127	C₂	semidirect
38. C₁₂₇	#473	semidirect