Subgroups — #22549859

Order 8128.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523249	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₂₇	C₂	1	2	yes	maximal, complemented
3	#16861128	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861125	C₂	1	2	yes	maximal, complemented
5	#101×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523249	—	2	2	no	—
7	#14523249	—	2	2	no	—
8	C₂×D₅₀₈	C₄	1	2	yes	—
9	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₂₇	C₄	1	2	yes	—
11	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
12	#14523232	C₂²	1	2	yes	—
13	#14523231	—	2	2	no	—
14	C₈×D₁₂₇	—	2	2	no	—
15	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	M₁₆×C₁₂₇	C₂²	1	2	yes	—
17	C₂²×D₁₂₇	C₂×C₄	1	2	yes	—
18	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
19	D₅₀₈	—	2	4	no	—
20	D₅₀₈	—	2	2	no	—
21	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₂₇	—	2	2	no	—
23	C₄×D₁₂₇	—	2	2	no	—
24	C₄×D₁₂₇	D₄	1	2	yes	—
25	C₄×D₁₂₇	D₄	1	2	yes	—
26	C₁₂₇⋊C₈	—	2	2	no	—
27	Q₈×C₁₂₇	—	2	2	no	—
28	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₈×C₁₂₇	—	2	2	no	—
31	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×D₁₂₇	—	2	2	no	—
33	C₂×D₁₂₇	—	4	1	no	—
34	C₂×D₁₂₇	—	4	2	no	—
35	C₂×D₁₂₇	C₂²⋊C₄	1	2	yes	—
36	C₄×C₁₂₇	—	2	2	no	—
37	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
38	Dic₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₁₂₇	—	2	1	no	—
41	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	D₁₂₇	—	2	2	no	—
43	D₁₂₇	—	4	2	no	—
44	C₁₂₇	#413	1	1	yes	characteristic, complemented, Sylow
45	#413	—	127	1	no	maximal, Sylow
46	C₂×#41	—	127	1	no	—
47	C₂×M₁₆	—	127	2	no	—
48	#101	—	127	1	no	—
49	#101	D₁₂₇	1	1	yes	characteristic, complemented
50	#100	—	127	2	no	—
51	#41	—	254	2	no	—
52	#41	—	254	2	no	—
53	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
54	C₂×D₄	—	127	2	no	—
55	C₂×D₄	—	127	1	no	—
56	C₂²×C₄	—	127	1	no	—
57	C₂²×C₄	—	127	2	no	—
58	M₁₆	—	127	2	no	—
59	M₁₆	—	254	2	no	—
60	M₁₆	C₂×D₁₂₇	1	2	yes	—
61	C₂×C₈	—	254	2	no	—
62	C₂³	—	127	1	no	—
63	C₂³	—	127	2	no	—
64	Q₈	—	2	2	no	—
65	D₄	—	254	4	no	—
66	D₄	—	254	2	no	—
67	C₂×C₄	—	127	1	no	—
68	C₂×C₄	—	127	2	no	—
69	C₂×C₄	—	127	2	no	—
70	C₂×C₄	—	254	2	no	—
71	C₂×C₄	—	254	2	no	—
72	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
73	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	254	2	no	—
76	C₂²	—	127	2	no	—
77	C₂²	—	508	2	no	—
78	C₂²	—	508	1	no	—
79	C₂²	—	254	2	no	—
80	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	127	1	no	—
83	C₄	—	127	1	no	—
84	C₄	D₄×D₁₂₇	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	508	2	no	—
87	C₂	—	254	2	no	—
88	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#101×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₂²⋊C₄
hypercenter	#101	D₁₂₇
hypocenter	C₁₂₇	#413
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₁₂₇	C₂	semidirect
3. #16861128	C₂	semidirect
4. #16861125	C₂	semidirect
5. #101×C₁₂₇	C₂	semidirect
44. C₁₂₇	#413	semidirect
49. #101	D₁₂₇	semidirect