Subgroups — C₂³×C₄⋊Dic₁₂₇ #30651891

Order 16256.

2488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₂₇	C₂	1	2	yes	maximal
2	C₂²×C₄⋊Dic₁₂₇	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₂₇	—	2	16	no	—
5	C₂³×Dic₁₂₇	C₂²	1	28	yes	—
6	C₂×C₄⋊Dic₁₂₇	C₂²	1	112	yes	direct product factor
7	C₂⁵×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₂₇	C₄	1	16	yes	complemented
9	C₂³×C₄×C₁₂₇	C₂²	1	14	yes	—
10	C₂²×Dic₁₂₇	—	2	112	no	—
11	C₂²×Dic₁₂₇	C₂³	1	56	yes	—
12	C₄⋊Dic₁₂₇	C₂³	1	64	yes	direct product factor
13	C₂⁴×C₁₂₇	C₂×C₄	1	8	yes	—
14	C₂⁴×C₁₂₇	D₄	1	8	yes	—
15	C₂⁴×C₁₂₇	Q₈	1	8	yes	—
16	C₂⁴×C₁₂₇	C₂³	1	7	yes	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	112	yes	complemented
18	C₂²×C₄×C₁₂₇	C₂³	1	28	yes	—
19	C₂×Dic₁₂₇	—	2	112	no	—
20	C₂×Dic₁₂₇	C₂⁴	1	16	yes	—
21	C₂³×C₁₂₇	C₄⋊C₄	1	64	yes	complemented
22	C₂³×C₁₂₇	C₂²×C₄	1	28	yes	—
23	C₂³×C₁₂₇	C₂×D₄	1	28	yes	—
24	C₂³×C₁₂₇	C₂×Q₈	1	28	yes	—
25	C₂³×C₁₂₇	C₂⁴	1	7	yes	—
26	C₂×C₄×C₁₂₇	C₂²×C₄	1	112	yes	complemented
27	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
28	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	112	yes	complemented
29	C₂²×C₁₂₇	C₂³×C₄	1	14	yes	—
30	C₂²×C₁₂₇	C₂²×D₄	1	14	yes	—
31	C₂²×C₁₂₇	C₂²×Q₈	1	14	yes	—
32	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
33	C₄×C₁₂₇	C₂³×C₄	1	16	yes	complemented
34	Dic₁₂₇	—	2	16	no	—
35	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	28	yes	complemented
36	C₂×C₁₂₇	C₂⁴×C₄	1	1	yes	characteristic
37	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
38	C₂×C₁₂₇	C₂³×Q₈	1	1	yes	characteristic
39	C₂³×C₄⋊C₄	—	127	1	no	maximal, Sylow
40	C₁₂₇	C₂³×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴×C₄	—	127	2	no	—
42	C₂⁴×C₄	D₁₂₇	1	1	yes	characteristic
43	C₂²×C₄⋊C₄	—	127	28	no	—
44	C₂⁵	C₂×D₁₂₇	1	1	yes	characteristic
45	C₂³×C₄	—	127	28	no	—
46	C₂³×C₄	—	254	16	no	—
47	C₂³×C₄	Dic₁₂₇	1	16	yes	complemented
48	C₂³×C₄	C₂×D₁₂₇	1	14	yes	—
49	C₂×C₄⋊C₄	—	127	112	no	—
50	C₂⁴	C₂×Dic₁₂₇	1	8	yes	—
51	C₂⁴	D₅₀₈	1	8	yes	—
52	C₂⁴	Dic₂₅₄	1	8	yes	—
53	C₂⁴	C₂²×D₁₂₇	1	7	yes	—
54	C₂²×C₄	—	127	56	no	—
55	C₂²×C₄	—	254	112	no	—
56	C₂²×C₄	C₂×Dic₁₂₇	1	112	yes	complemented
57	C₂²×C₄	C₂²×D₁₂₇	1	28	yes	—
58	C₄⋊C₄	—	127	64	no	—
59	C₂³	C₄⋊Dic₁₂₇	1	64	yes	direct product factor
60	C₂³	C₂²×Dic₁₂₇	1	28	yes	—
61	C₂³	C₂×D₅₀₈	1	28	yes	—
62	C₂³	C₂×Dic₂₅₄	1	28	yes	—
63	C₂³	C₂³×D₁₂₇	1	7	yes	—
64	C₂×C₄	—	127	16	no	—
65	C₂×C₄	—	254	112	no	—
66	C₂×C₄	C₂²×Dic₁₂₇	1	112	yes	complemented
67	C₂×C₄	C₂³×D₁₂₇	1	8	yes	—
68	C₂²	C₂×C₄⋊Dic₁₂₇	1	112	yes	direct product factor
69	C₂²	C₂³×Dic₁₂₇	1	14	yes	—
70	C₂²	C₂²×D₅₀₈	1	14	yes	—
71	C₂²	C₂²×Dic₂₅₄	1	14	yes	—
72	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
73	C₄	—	254	16	no	—
74	C₄	C₂³×Dic₁₂₇	1	16	yes	complemented
75	C₂	C₂²×C₄⋊Dic₁₂₇	1	28	yes	direct product factor
76	C₂	C₂⁴×Dic₁₂₇	1	1	yes	characteristic
77	C₂	C₂³×D₅₀₈	1	1	yes	characteristic
78	C₂	C₂³×Dic₂₅₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂⁴×C₄×C₁₂₇	C₂
socle, prime radical	C₂⁵×C₁₂₇	C₂²
cosocle	C₂²×C₁₂₇	C₂⁵
hypercenter	C₂⁴×C₄	D₁₂₇
hypocenter	C₁₂₇	C₂³×C₄⋊C₄
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊Dic₁₂₇	C₂	direct (N × H)
6. C₂×C₄⋊Dic₁₂₇	C₂²	direct (N × H)
8. C₂³×C₄×C₁₂₇	C₄	semidirect
12. C₄⋊Dic₁₂₇	C₂³	direct (N × H)
17. C₂²×C₄×C₁₂₇	C₂×C₄	semidirect
21. C₂³×C₁₂₇	C₄⋊C₄	semidirect
26. C₂×C₄×C₁₂₇	C₂²×C₄	semidirect
28. C₂²×C₁₂₇	C₂×C₄⋊C₄	semidirect
33. C₄×C₁₂₇	C₂³×C₄	semidirect
35. C₂×C₁₂₇	C₂²×C₄⋊C₄	semidirect
40. C₁₂₇	C₂³×C₄⋊C₄	semidirect
47. C₂³×C₄	Dic₁₂₇	semidirect
56. C₂²×C₄	C₂×Dic₁₂₇	semidirect
59. C₂³	C₄⋊Dic₁₂₇	direct (N × H)
66. C₂×C₄	C₂²×Dic₁₂₇	semidirect
68. C₂²	C₂×C₄⋊Dic₁₂₇	direct (N × H)
74. C₄	C₂³×Dic₁₂₇	semidirect
75. C₂	C₂²×C₄⋊Dic₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₂₇	2
C₂²×C₄⋊Dic₁₂₇	2
C₂⁴×C₄×C₁₂₇	2
C₂³×C₄⋊C₄	127

Subgroups — C23×C4⋊Dic127 #30651891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊Dic₁₂₇ #30651891