Subgroups — C₂³×C₄⋊C₄×C₂₂₇ #44397376

Order 29056.

2488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₂₂₇	C₂	1	2	yes	maximal
2	C₂⁴×C₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₂₂₇	C₂	1	28	yes	maximal, direct product factor
4	C₂⁵×C₂₂₇	C₂²	1	1	yes	characteristic
5	C₂³×C₄×C₂₂₇	—	2	16	no	—
6	C₂³×C₄×C₂₂₇	C₄	1	16	yes	complemented
7	C₂³×C₄×C₂₂₇	C₂²	1	28	yes	—
8	C₂³×C₄×C₂₂₇	C₂²	1	14	yes	—
9	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	112	yes	direct product factor
10	C₂⁴×C₂₂₇	C₂×C₄	1	8	yes	—
11	C₂⁴×C₂₂₇	D₄	1	8	yes	—
12	C₂⁴×C₂₂₇	Q₈	1	8	yes	—
13	C₂⁴×C₂₂₇	C₂³	1	7	yes	—
14	C₂²×C₄×C₂₂₇	—	2	112	no	—
15	C₂²×C₄×C₂₂₇	C₂×C₄	1	112	yes	complemented
16	C₂²×C₄×C₂₂₇	C₂³	1	56	yes	—
17	C₂²×C₄×C₂₂₇	C₂³	1	28	yes	—
18	C₄⋊C₄×C₂₂₇	C₂³	1	64	yes	direct product factor
19	C₂³×C₂₂₇	C₄⋊C₄	1	64	yes	direct product factor
20	C₂³×C₂₂₇	C₂²×C₄	1	28	yes	—
21	C₂³×C₂₂₇	C₂×D₄	1	28	yes	—
22	C₂³×C₂₂₇	C₂×Q₈	1	28	yes	—
23	C₂³×C₂₂₇	C₂⁴	1	7	yes	—
24	C₂×C₄×C₂₂₇	—	2	112	no	—
25	C₂×C₄×C₂₂₇	C₂²×C₄	1	112	yes	complemented
26	C₂×C₄×C₂₂₇	C₂⁴	1	16	yes	—
27	C₂×C₄×C₂₂₇	C₂⁴	1	8	yes	—
28	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	112	yes	direct product factor
29	C₂²×C₂₂₇	C₂³×C₄	1	14	yes	—
30	C₂²×C₂₂₇	C₂²×D₄	1	14	yes	—
31	C₂²×C₂₂₇	C₂²×Q₈	1	14	yes	—
32	C₂²×C₂₂₇	C₂⁵	1	1	yes	characteristic
33	C₄×C₂₂₇	—	2	16	no	—
34	C₄×C₂₂₇	C₂³×C₄	1	16	yes	complemented
35	C₂×C₂₂₇	C₂²×C₄⋊C₄	1	28	yes	direct product factor
36	C₂×C₂₂₇	C₂⁴×C₄	1	1	yes	characteristic
37	C₂×C₂₂₇	C₂³×D₄	1	1	yes	characteristic
38	C₂×C₂₂₇	C₂³×Q₈	1	1	yes	characteristic
39	C₂₂₇	C₂³×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
40	C₂³×C₄⋊C₄	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂⁴×C₄	C₂×C₂₂₇	1	2	yes	—
42	C₂⁴×C₄	C₂×C₂₂₇	1	1	yes	characteristic
43	C₂²×C₄⋊C₄	C₂×C₂₂₇	1	28	yes	direct product factor
44	C₂⁵	C₂²×C₂₂₇	1	1	yes	characteristic
45	C₂³×C₄	—	2	16	no	—
46	C₂³×C₄	C₄×C₂₂₇	1	16	yes	complemented
47	C₂³×C₄	C₂²×C₂₂₇	1	28	yes	—
48	C₂³×C₄	C₂²×C₂₂₇	1	14	yes	—
49	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	112	yes	direct product factor
50	C₂⁴	C₂×C₄×C₂₂₇	1	8	yes	—
51	C₂⁴	D₄×C₂₂₇	1	8	yes	—
52	C₂⁴	Q₈×C₂₂₇	1	8	yes	—
53	C₂⁴	C₂³×C₂₂₇	1	7	yes	—
54	C₂²×C₄	—	2	112	no	—
55	C₂²×C₄	C₂×C₄×C₂₂₇	1	112	yes	complemented
56	C₂²×C₄	C₂³×C₂₂₇	1	56	yes	—
57	C₂²×C₄	C₂³×C₂₂₇	1	28	yes	—
58	C₄⋊C₄	C₂³×C₂₂₇	1	64	yes	direct product factor
59	C₂³	C₄⋊C₄×C₂₂₇	1	64	yes	direct product factor
60	C₂³	C₂²×C₄×C₂₂₇	1	28	yes	—
61	C₂³	C₂×D₄×C₂₂₇	1	28	yes	—
62	C₂³	C₂×Q₈×C₂₂₇	1	28	yes	—
63	C₂³	C₂⁴×C₂₂₇	1	7	yes	—
64	C₂×C₄	—	2	112	no	—
65	C₂×C₄	C₂²×C₄×C₂₂₇	1	112	yes	complemented
66	C₂×C₄	C₂⁴×C₂₂₇	1	16	yes	—
67	C₂×C₄	C₂⁴×C₂₂₇	1	8	yes	—
68	C₂²	C₂×C₄⋊C₄×C₂₂₇	1	112	yes	direct product factor
69	C₂²	C₂³×C₄×C₂₂₇	1	14	yes	—
70	C₂²	C₂²×D₄×C₂₂₇	1	14	yes	—
71	C₂²	C₂²×Q₈×C₂₂₇	1	14	yes	—
72	C₂²	C₂⁵×C₂₂₇	1	1	yes	characteristic
73	C₄	—	2	16	no	—
74	C₄	C₂³×C₄×C₂₂₇	1	16	yes	complemented
75	C₂	C₂²×C₄⋊C₄×C₂₂₇	1	28	yes	direct product factor
76	C₂	C₂⁴×C₄×C₂₂₇	1	1	yes	characteristic
77	C₂	C₂³×D₄×C₂₂₇	1	1	yes	characteristic
78	C₂	C₂³×Q₈×C₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₂₂₇	C₂²
commutator	C₂	C₂⁴×C₄×C₂₂₇
Frattini, cosocle	C₂²	C₂⁵×C₂₂₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊C₄×C₂₂₇	C₂	direct (N × H)
6. C₂³×C₄×C₂₂₇	C₄	semidirect
9. C₂×C₄⋊C₄×C₂₂₇	C₂²	direct (N × H)
15. C₂²×C₄×C₂₂₇	C₂×C₄	semidirect
18. C₄⋊C₄×C₂₂₇	C₂³	direct (N × H)
19. C₂³×C₂₂₇	C₄⋊C₄	direct (N × H)
25. C₂×C₄×C₂₂₇	C₂²×C₄	semidirect
28. C₂²×C₂₂₇	C₂×C₄⋊C₄	direct (N × H)
34. C₄×C₂₂₇	C₂³×C₄	semidirect
35. C₂×C₂₂₇	C₂²×C₄⋊C₄	direct (N × H)
39. C₂₂₇	C₂³×C₄⋊C₄	direct (N × H)
40. C₂³×C₄⋊C₄	C₂₂₇	direct (N × H)
43. C₂²×C₄⋊C₄	C₂×C₂₂₇	direct (N × H)
46. C₂³×C₄	C₄×C₂₂₇	semidirect
49. C₂×C₄⋊C₄	C₂²×C₂₂₇	direct (N × H)
55. C₂²×C₄	C₂×C₄×C₂₂₇	semidirect
58. C₄⋊C₄	C₂³×C₂₂₇	direct (N × H)
59. C₂³	C₄⋊C₄×C₂₂₇	direct (N × H)
65. C₂×C₄	C₂²×C₄×C₂₂₇	semidirect
68. C₂²	C₂×C₄⋊C₄×C₂₂₇	direct (N × H)
74. C₄	C₂³×C₄×C₂₂₇	semidirect
75. C₂	C₂²×C₄⋊C₄×C₂₂₇	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×C₄×C₂₂₇	2
C₂²×C₄⋊C₄×C₂₂₇	2
C₂³×C₄⋊C₄	227

Subgroups — C23×C4⋊C4×C227 #44397376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊C₄×C₂₂₇ #44397376