Subgroups — #30653442

Order 16256.

1424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550643	C₂	1	15	yes	maximal, complemented
2	C₂×#16861216	C₂	1	15	yes	maximal, complemented
3	#585×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861216	C₂²	1	20	yes	complemented
5	#16861216	C₂²	1	60	yes	complemented
6	C₂×#14523232	C₂²	1	45	yes	complemented
7	C₂²×C₁₂₇⋊C₈	C₂²	1	15	yes	complemented
8	2₋¹⁺⁴×C₁₂₇	C₄	1	6	yes	—
9	H(5,𝔽₂)×C₁₂₇	C₄	1	10	yes	—
10	C₂×#41×C₁₂₇	C₂²	1	15	yes	—
11	#14523232	C₂³	1	60	yes	—
12	C₂×C₁₂₇⋊C₈	C₂³	1	15	yes	—
13	C₂×C₁₂₇⋊C₈	C₂³	1	45	yes	—
14	#41×C₁₂₇	C₂×C₄	1	60	yes	—
15	#41×C₁₂₇	C₂³	1	20	yes	—
16	C₂×Q₈×C₁₂₇	C₂×C₄	1	15	yes	—
17	C₂×D₄×C₁₂₇	C₂×C₄	1	45	yes	—
18	C₂²×C₄×C₁₂₇	C₂³	1	15	yes	—
19	C₁₂₇⋊C₈	C₂⁴	1	1	yes	characteristic
20	C₁₂₇⋊C₈	C₂⁴	1	15	yes	—
21	C₂³×C₁₂₇	C₂²×C₄	1	15	yes	—
22	Q₈×C₁₂₇	C₂²×C₄	1	20	yes	—
23	D₄×C₁₂₇	C₂²×C₄	1	60	yes	—
24	C₂×C₄×C₁₂₇	C₂²×C₄	1	45	yes	—
25	C₂×C₄×C₁₂₇	C₂⁴	1	15	yes	—
26	C₂²×C₁₂₇	—	4	15	no	—
27	C₂²×C₁₂₇	C₂³×C₄	1	15	yes	—
28	C₄×C₁₂₇	C₂³×C₄	1	15	yes	—
29	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
30	C₂×C₁₂₇	—	2	15	no	—
31	C₂×C₁₂₇	C₂⁴×C₄	1	1	yes	characteristic
32	#3573	—	127	1	no	maximal, Sylow
33	C₁₂₇	#3573	1	1	yes	characteristic, complemented, Sylow
34	#585	D₁₂₇	1	1	yes	characteristic
35	#568	—	127	15	no	—
36	C₂×#131	—	127	15	no	—
37	2₋¹⁺⁴	Dic₁₂₇	1	6	yes	—
38	H(5,𝔽₂)	Dic₁₂₇	1	10	yes	—
39	C₂×#41	C₂×D₁₂₇	1	15	yes	—
40	#131	—	127	20	no	—
41	#131	—	127	60	no	—
42	C₂×M₁₆	—	127	45	no	—
43	C₂²×C₈	—	127	15	no	—
44	#41	C₂×Dic₁₂₇	1	60	yes	—
45	#41	C₂²×D₁₂₇	1	20	yes	—
46	C₂×Q₈	C₂×Dic₁₂₇	1	15	yes	—
47	C₂×D₄	C₂×Dic₁₂₇	1	45	yes	—
48	C₂²×C₄	C₂²×D₁₂₇	1	15	yes	—
49	M₁₆	—	127	60	no	—
50	C₂×C₈	—	127	15	no	—
51	C₂×C₈	—	127	45	no	—
52	C₂³	C₂²×Dic₁₂₇	1	15	yes	—
53	Q₈	C₂²×Dic₁₂₇	1	20	yes	—
54	D₄	C₂²×Dic₁₂₇	1	60	yes	—
55	C₂×C₄	C₂²×Dic₁₂₇	1	45	yes	—
56	C₂×C₄	C₂³×D₁₂₇	1	15	yes	—
57	C₈	—	127	1	no	—
58	C₈	—	127	15	no	—
59	C₂²	—	4	15	no	—
60	C₂²	C₂³×Dic₁₂₇	1	15	yes	—
61	C₄	C₂³×Dic₁₂₇	1	15	yes	—
62	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
63	C₂	—	2	15	no	—
64	C₂	C₂⁴×Dic₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₁₂₇
commutator, socle	C₂×C₁₂₇	C₂⁴×C₄
Fitting, prime radical	#585×C₁₂₇	C₂
cosocle	C₄×C₁₂₇	C₂⁵
hypercenter	#585	D₁₂₇
hypocenter	C₁₂₇	#3573
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #22550643	C₂	semidirect
2. C₂×#16861216	C₂	semidirect
4. #16861216	C₂²	semidirect
5. #16861216	C₂²	semidirect
6. C₂×#14523232	C₂²	semidirect
7. C₂²×C₁₂₇⋊C₈	C₂²	semidirect
33. C₁₂₇	#3573	semidirect

Maximal subgroups

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

subgroup	index
#22550643	2
C₂×#16861216	2
#585×C₁₂₇	2
#3573	127