Subgroups — C₂×#22549744 #30641120

Order 16256.

624 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549744	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₁₂₇⋊#95	C₂	1	6	yes	maximal
3	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₂₇	C₂²	1	4	yes	—
5	C₁₂₇⋊#95	C₂²	1	24	yes	—
6	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
7	C₂×C₄²×C₁₂₇	C₄	1	8	yes	complemented
8	C₂×C₄²×C₁₂₇	C₂²	1	4	yes	—
9	C₂²×Dic₁₂₇	—	2	24	no	—
10	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
11	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	—	2	12	no	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
14	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
15	C₄²×C₁₂₇	C₂×C₄	1	16	yes	complemented
16	C₂×Dic₁₂₇	—	2	24	no	—
17	C₂×Dic₁₂₇	—	4	16	no	—
18	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
19	C₂³×C₁₂₇	#41	1	12	yes	—
20	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
21	C₂×C₄×C₁₂₇	—	2	24	no	—
22	C₂×C₄×C₁₂₇	—	2	12	no	—
23	C₂×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
24	C₂²×C₁₂₇	#117	1	12	yes	—
25	C₂²×C₁₂₇	#126	1	16	yes	—
26	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
27	C₂²×C₁₂₇	C₂×#41	1	6	yes	—
28	C₄×C₁₂₇	—	2	12	no	—
29	Dic₁₂₇	—	4	8	no	—
30	C₂×C₁₂₇	#383	1	8	yes	complemented
31	C₂×C₁₂₇	C₂×#117	1	3	yes	—
32	C₂×C₁₂₇	C₂×#126	1	4	yes	—
33	C₂×#383	—	127	1	no	maximal, Sylow
34	C₁₂₇	C₂×#383	1	1	yes	characteristic, complemented, Sylow
35	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
36	#383	—	127	8	no	—
37	C₂×#95	—	127	6	no	—
38	C₂³×C₄	—	127	4	no	—
39	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
40	C₂×C₄²	Dic₁₂₇	1	8	yes	complemented
41	C₂×C₄²	C₂×D₁₂₇	1	4	yes	—
42	#95	—	127	24	no	—
43	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
44	C₂²×C₄	—	2	12	no	—
45	C₂²×C₄	—	127	8	no	—
46	C₂²×C₄	—	254	24	no	—
47	C₂²×C₄	C₂×Dic₁₂₇	1	12	yes	—
48	C₂²×C₄	C₂²×D₁₂₇	1	6	yes	—
49	C₄²	C₂×Dic₁₂₇	1	16	yes	complemented
50	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
51	C₂³	#14523248	1	12	yes	—
52	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
53	C₂×C₄	—	2	12	no	—
54	C₂×C₄	—	2	24	no	—
55	C₂×C₄	—	508	16	no	—
56	C₂×C₄	—	254	24	no	—
57	C₂×C₄	C₂²×Dic₁₂₇	1	12	yes	—
58	C₂²	#16861168	1	12	yes	—
59	C₂²	#16861203	1	16	yes	—
60	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
61	C₂²	C₂×#14523248	1	6	yes	—
62	C₄	—	2	12	no	—
63	C₄	—	508	8	no	—
64	C₂	#22549744	1	8	yes	direct product factor
65	C₂	C₂×#16861168	1	3	yes	—
66	C₂	C₂×#16861203	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	C₂³
cosocle	C₂³×C₁₂₇	C₂⁴
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₂×#383
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #22549744	C₂	direct (N × H)
7. C₂×C₄²×C₁₂₇	C₄	semidirect
15. C₄²×C₁₂₇	C₂×C₄	semidirect
30. C₂×C₁₂₇	#383	semidirect
34. C₁₂₇	C₂×#383	semidirect
40. C₂×C₄²	Dic₁₂₇	semidirect
49. C₄²	C₂×Dic₁₂₇	semidirect
64. C₂	#22549744	direct (N × H)

Maximal subgroups

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

subgroup	index
#22549744	2
C₂×C₁₂₇⋊#95	2
C₂²×C₄²×C₁₂₇	2
C₂×#383	127

Subgroups — C2×#22549744 #30641120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549744 #30641120