Subgroups — #22549615

Order 8128.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861197	C₂	1	1	yes	maximal, characteristic
2	#16861120	C₂	1	1	yes	maximal, characteristic, complemented
3	#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₂₅₄	C₄	1	1	yes	characteristic
5	C₂×D₅₀₈	C₄	1	1	yes	characteristic
6	C₂×C₁₂₇⋊C₈	—	2	1	no	—
7	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
8	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
9	C₂×C₈×C₁₂₇	—	2	1	no	—
10	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×D₁₂₇	—	2	1	no	—
12	Dic₂₅₄	—	4	1	no	—
13	D₅₀₈	—	4	1	no	—
14	C₂×Dic₁₂₇	—	2	1	no	—
15	C₁₂₇⋊C₈	—	2	2	no	—
16	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
19	C₈×C₁₂₇	—	2	2	no	—
20	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×D₁₂₇	—	4	1	no	—
22	C₂×D₁₂₇	—	4	1	no	—
23	C₂×D₁₂₇	—	4	1	no	—
24	C₄×C₁₂₇	—	2	1	no	—
25	C₄×C₁₂₇	—	2	1	no	—
26	C₄×C₁₂₇	—	2	1	no	—
27	Dic₁₂₇	—	4	1	no	—
28	C₂×C₁₂₇	#100	1	1	yes	characteristic
29	C₂×C₁₂₇	#104	1	1	yes	characteristic
30	C₂×C₁₂₇	#104	1	1	yes	characteristic
31	D₁₂₇	—	8	1	no	—
32	C₁₂₇	#329	1	1	yes	characteristic, complemented, Sylow
33	#329	—	127	1	no	maximal, Sylow
34	#124	—	127	1	no	—
35	#97	—	127	1	no	—
36	#97	D₁₂₇	1	1	yes	characteristic, complemented
37	C₂×Q₈	—	127	1	no	—
38	C₂×D₄	—	127	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	254	1	no	—
41	C₂²⋊C₄	—	254	1	no	—
42	C₂²⋊C₄	—	254	1	no	—
43	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
44	C₂³	—	254	1	no	—
45	Q₈	—	508	1	no	—
46	D₄	—	508	1	no	—
47	C₂×C₄	—	254	1	no	—
48	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
50	C₂×C₄	D₅₀₈	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	254	2	no	—
53	C₂²	—	508	1	no	—
54	C₂²	—	508	1	no	—
55	C₂²	—	508	1	no	—
56	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	508	1	no	—
61	C₂	—	1016	1	no	—
62	C₂	#16861125	1	1	yes	characteristic
63	C₂	#16861135	1	1	yes	characteristic
64	C₂	#16861137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₂₇)⋊C₄
commutator	C₂×C₄×C₁₂₇	C₂×C₄
Frattini	C₄²	C₂×D₁₂₇
Fitting	#97×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	#97	D₁₂₇
hypocenter	C₁₂₇	#329
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. #16861120	C₂	semidirect
3. #97×C₁₂₇	C₂	semidirect
32. C₁₂₇	#329	semidirect
36. #97	D₁₂₇	semidirect