Subgroups — C₂²×#106×C₁₂₇ #30635859

Order 16256.

600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×#106×C₁₂₇	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₁₂₇	—	2	2	no	—
5	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₁₂₇	C₄	1	8	yes	complemented
7	C₂²×C₈×C₁₂₇	C₂²	1	6	yes	—
8	C₂×C₄⋊C₄×C₁₂₇	—	2	6	no	—
9	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	12	yes	—
10	#106×C₁₂₇	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	—	2	12	no	—
13	C₂²×C₄×C₁₂₇	—	4	8	no	—
14	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
15	C₂²×C₄×C₁₂₇	D₄	1	3	yes	—
16	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
17	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
18	C₂×C₈×C₁₂₇	C₂×C₄	1	24	yes	complemented
19	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
20	C₄⋊C₄×C₁₂₇	—	2	12	no	—
21	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
22	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
23	C₂³×C₁₂₇	SD₁₆	1	8	yes	—
24	C₂³×C₁₂₇	C₂×D₄	1	3	yes	—
25	C₂×C₄×C₁₂₇	—	2	8	no	—
26	C₂×C₄×C₁₂₇	—	4	24	no	—
27	C₂×C₄×C₁₂₇	C₄⋊C₄	1	12	yes	—
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
30	C₂×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
31	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
32	C₈×C₁₂₇	C₂²×C₄	1	8	yes	complemented
33	C₂²×C₁₂₇	#106	1	16	yes	direct product factor
34	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	6	yes	—
35	C₂²×C₁₂₇	C₂×SD₁₆	1	12	yes	—
36	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
37	C₄×C₁₂₇	—	4	8	no	—
38	C₄×C₁₂₇	C₂×C₄⋊C₄	1	6	yes	—
39	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
40	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
41	C₂×C₁₂₇	C₂×#106	1	12	yes	direct product factor
42	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
44	C₂²×#106	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₁₂₇	C₂²×#106	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₈	C₂×C₁₂₇	1	1	yes	characteristic
47	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	2	yes	—
48	C₂×#106	C₂×C₁₂₇	1	12	yes	direct product factor
49	C₂³×C₄	—	2	2	no	—
50	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
51	C₂²×C₈	C₄×C₁₂₇	1	8	yes	complemented
52	C₂²×C₈	C₂²×C₁₂₇	1	6	yes	—
53	C₂×C₄⋊C₄	—	2	6	no	—
54	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	12	yes	—
55	#106	C₂²×C₁₂₇	1	16	yes	direct product factor
56	C₂⁴	D₄×C₁₂₇	1	1	yes	characteristic
57	C₂²×C₄	—	2	12	no	—
58	C₂²×C₄	—	4	8	no	—
59	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
60	C₂²×C₄	D₄×C₁₂₇	1	3	yes	—
61	C₂²×C₄	Q₈×C₁₂₇	1	4	yes	—
62	C₂²×C₄	C₂³×C₁₂₇	1	3	yes	—
63	C₂×C₈	C₂×C₄×C₁₂₇	1	24	yes	complemented
64	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
65	C₄⋊C₄	—	2	12	no	—
66	C₄⋊C₄	C₂³×C₁₂₇	1	8	yes	—
67	C₂³	C₄⋊C₄×C₁₂₇	1	4	yes	—
68	C₂³	SD₁₆×C₁₂₇	1	8	yes	—
69	C₂³	C₂×D₄×C₁₂₇	1	3	yes	—
70	C₂×C₄	—	2	8	no	—
71	C₂×C₄	—	4	24	no	—
72	C₂×C₄	C₄⋊C₄×C₁₂₇	1	12	yes	—
73	C₂×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
74	C₂×C₄	C₂×D₄×C₁₂₇	1	3	yes	—
75	C₂×C₄	C₂×Q₈×C₁₂₇	1	6	yes	—
76	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
77	C₈	C₂²×C₄×C₁₂₇	1	8	yes	complemented
78	C₂²	#106×C₁₂₇	1	16	yes	direct product factor
79	C₂²	C₂×C₄⋊C₄×C₁₂₇	1	6	yes	—
80	C₂²	C₂×SD₁₆×C₁₂₇	1	12	yes	—
81	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
82	C₄	—	4	8	no	—
83	C₄	C₂×C₄⋊C₄×C₁₂₇	1	6	yes	—
84	C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
85	C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
86	C₂	C₂×#106×C₁₂₇	1	12	yes	direct product factor
87	C₂	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
88	C₂	C₂²×SD₁₆×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₂₇	D₄
commutator	C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#106×C₁₂₇	C₂	direct (N × H)
6. C₂²×C₈×C₁₂₇	C₄	semidirect
10. #106×C₁₂₇	C₂²	direct (N × H)
18. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
32. C₈×C₁₂₇	C₂²×C₄	semidirect
33. C₂²×C₁₂₇	#106	direct (N × H)
41. C₂×C₁₂₇	C₂×#106	direct (N × H)
44. C₂²×#106	C₁₂₇	direct (N × H)
45. C₁₂₇	C₂²×#106	direct (N × H)
48. C₂×#106	C₂×C₁₂₇	direct (N × H)
51. C₂²×C₈	C₄×C₁₂₇	semidirect
55. #106	C₂²×C₁₂₇	direct (N × H)
63. C₂×C₈	C₂×C₄×C₁₂₇	semidirect
77. C₈	C₂²×C₄×C₁₂₇	semidirect
78. C₂²	#106×C₁₂₇	direct (N × H)
86. C₂	C₂×#106×C₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₁₂₇	2
C₂²×C₄⋊C₄×C₁₂₇	2
C₂×#106×C₁₂₇	2
C₂²×#106	127

Subgroups — C22×#106×C127 #30635859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#106×C₁₂₇ #30635859