Subgroups — #30637029

Order 16256.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549927	C₂	1	3	yes	maximal
2	#22549598	C₂	1	3	yes	maximal, complemented
3	C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
5	Dic₁₂₇⋊C₈	C₄	1	6	yes	—
6	(C₂×D₁₂₇)⋊C₈	C₄	1	6	yes	—
7	C₄×C₁₂₇⋊C₈	C₂²	1	3	yes	—
8	C₄×C₈×C₁₂₇	C₂²	1	3	yes	—
9	C₂×C₄×D₁₂₇	C₂×C₄	1	3	yes	—
10	C₂×C₁₂₇⋊C₈	C₂×C₄	1	6	yes	—
11	C₄×Dic₁₂₇	C₂×C₄	1	3	yes	—
12	C₂×C₈×C₁₂₇	C₂×C₄	1	6	yes	—
13	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
14	C₂²×D₁₂₇	C₄²	1	1	yes	characteristic
15	C₂×Dic₁₂₇	C₄²	1	3	yes	—
16	C₄×D₁₂₇	—	2	6	no	—
17	C₁₂₇⋊C₈	—	2	6	no	—
18	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
19	C₈×C₁₂₇	—	2	6	no	—
20	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
21	C₂×D₁₂₇	—	2	3	no	—
22	C₄×C₁₂₇	#131	1	6	yes	—
23	Dic₁₂₇	—	2	3	no	—
24	C₂×C₁₂₇	#405	1	3	yes	—
25	D₁₂₇	—	4	1	no	—
26	#1454	—	127	1	no	maximal, Sylow
27	C₁₂₇	#1454	1	1	yes	characteristic, complemented, Sylow
28	#431	—	127	3	no	—
29	#322	—	127	3	no	—
30	C₈²	D₁₂₇	1	1	yes	characteristic, complemented
31	C₂×C₄²	—	127	1	no	—
32	C₄⋊C₈	—	127	6	no	—
33	C₂²⋊C₈	—	127	6	no	—
34	C₄×C₈	—	127	3	no	—
35	C₄×C₈	C₂×D₁₂₇	1	3	yes	—
36	C₂²×C₄	—	127	3	no	—
37	C₂×C₈	—	127	6	no	—
38	C₂×C₈	C₄×D₁₂₇	1	6	yes	—
39	C₄²	—	127	3	no	—
40	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
41	C₂³	—	127	1	no	—
42	C₂×C₄	—	127	3	no	—
43	C₂×C₄	—	254	6	no	—
44	C₂×C₄	C₂×C₄×D₁₂₇	1	3	yes	—
45	C₈	—	2	6	no	—
46	C₈	—	254	6	no	—
47	C₂²	—	254	3	no	—
48	C₂²	C₄²×D₁₂₇	1	1	yes	characteristic
49	C₄	—	254	3	no	—
50	C₄	#16861215	1	6	yes	—
51	C₂	—	508	1	no	—
52	C₂	#22549830	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂×C₄²
Fitting	C₈²×C₁₂₇	C₂
prime radical	C₂²×D₁₂₇	C₄²
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	C₈²	D₁₂₇
hypocenter	C₁₂₇	#1454
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #22549598	C₂	semidirect
3. C₈²×C₁₂₇	C₂	semidirect
27. C₁₂₇	#1454	semidirect
30. C₈²	D₁₂₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22549927	2
#22549598	2
C₈²×C₁₂₇	2
#1454	127