Subgroups — C₂×C₁₂₇×#1712 #44583274

Order 32512.

1342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₂₇	C₂	1	6	yes	maximal, complemented
2	C₁₂₇×#1712	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₈×C₁₂₇	—	2	12	no	—
5	C₂×#127×C₁₂₇	C₂²	1	4	yes	—
6	#493×C₁₂₇	—	2	12	no	—
7	#493×C₁₂₇	C₂²	1	24	yes	complemented
8	C₂×C₄×C₈×C₁₂₇	C₂²	1	3	yes	—
9	C₈²×C₁₂₇	C₂²	1	4	yes	complemented
10	C₂²×D₄×C₁₂₇	—	2	12	no	—
11	C₂×D₈×C₁₂₇	—	2	48	no	—
12	C₂×D₈×C₁₂₇	—	4	48	no	—
13	C₂²×C₈×C₁₂₇	D₄	1	6	yes	—
14	#127×C₁₂₇	—	2	12	no	—
15	#127×C₁₂₇	C₂³	1	8	yes	—
16	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₁₂₇	D₄	1	6	yes	—
18	C₄×C₈×C₁₂₇	C₂³	1	6	yes	—
19	C₂⁴×C₁₂₇	—	4	4	no	—
20	C₂×D₄×C₁₂₇	—	2	24	no	—
21	C₂×D₄×C₁₂₇	—	4	12	no	—
22	C₂×D₄×C₁₂₇	—	4	48	no	—
23	C₂²×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
24	D₈×C₁₂₇	—	4	96	no	—
25	C₂×C₈×C₁₂₇	D₈	1	24	yes	complemented
26	C₂×C₈×C₁₂₇	C₂×D₄	1	12	yes	—
27	C₄²×C₁₂₇	C₂×D₄	1	3	yes	—
28	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
29	C₂³×C₁₂₇	—	4	8	no	—
30	C₂³×C₁₂₇	—	8	24	no	—
31	C₂³×C₁₂₇	#127	1	1	yes	characteristic
32	D₄×C₁₂₇	—	4	48	no	—
33	D₄×C₁₂₇	—	8	24	no	—
34	C₂×C₄×C₁₂₇	#127	1	3	yes	—
35	C₂×C₄×C₁₂₇	C₂×D₈	1	12	yes	—
36	C₂×C₄×C₁₂₇	C₂²×D₄	1	3	yes	—
37	C₈×C₁₂₇	C₂×D₈	1	24	yes	complemented
38	C₂²×C₁₂₇	—	8	24	no	—
39	C₂²×C₁₂₇	—	16	16	no	—
40	C₂²×C₁₂₇	#493	1	6	yes	—
41	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
42	C₄×C₁₂₇	#493	1	6	yes	—
43	C₄×C₁₂₇	C₂²×D₈	1	6	yes	—
44	C₂×#1712	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×C₁₂₇	—	16	8	no	—
46	C₂×C₁₂₇	#1712	1	4	yes	direct product factor
47	C₂×C₁₂₇	C₂×#493	1	3	yes	—
48	C₂×#493	C₂×C₁₂₇	1	6	yes	complemented
49	#1712	C₂×C₁₂₇	1	8	yes	direct product factor
50	C₂×C₈²	C₂×C₁₂₇	1	1	yes	characteristic, complemented
51	C₁₂₇	C₂×#1712	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×D₈	—	2	12	no	—
53	C₂×#127	C₂²×C₁₂₇	1	4	yes	—
54	#493	—	2	12	no	—
55	#493	C₂²×C₁₂₇	1	24	yes	complemented
56	C₂×C₄×C₈	C₂²×C₁₂₇	1	3	yes	—
57	C₈²	C₂²×C₁₂₇	1	4	yes	complemented
58	C₂²×D₄	—	2	12	no	—
59	C₂×D₈	—	2	48	no	—
60	C₂×D₈	—	4	48	no	—
61	C₂²×C₈	D₄×C₁₂₇	1	6	yes	—
62	#127	—	2	12	no	—
63	#127	C₂³×C₁₂₇	1	8	yes	—
64	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
65	C₄×C₈	D₄×C₁₂₇	1	6	yes	—
66	C₄×C₈	C₂³×C₁₂₇	1	6	yes	—
67	C₂⁴	—	4	4	no	—
68	C₂×D₄	—	2	24	no	—
69	C₂×D₄	—	4	12	no	—
70	C₂×D₄	—	4	48	no	—
71	C₂²×C₄	C₂×D₄×C₁₂₇	1	3	yes	—
72	D₈	—	4	96	no	—
73	C₂×C₈	D₈×C₁₂₇	1	24	yes	complemented
74	C₂×C₈	C₂×D₄×C₁₂₇	1	12	yes	—
75	C₄²	C₂×D₄×C₁₂₇	1	3	yes	—
76	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
77	C₂³	—	4	8	no	—
78	C₂³	—	8	24	no	—
79	C₂³	#127×C₁₂₇	1	1	yes	characteristic
80	D₄	—	4	48	no	—
81	D₄	—	8	24	no	—
82	C₂×C₄	#127×C₁₂₇	1	3	yes	—
83	C₂×C₄	C₂×D₈×C₁₂₇	1	12	yes	—
84	C₂×C₄	C₂²×D₄×C₁₂₇	1	3	yes	—
85	C₈	C₂×D₈×C₁₂₇	1	24	yes	complemented
86	C₂²	—	8	24	no	—
87	C₂²	—	16	16	no	—
88	C₂²	#493×C₁₂₇	1	6	yes	—
89	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
90	C₄	#493×C₁₂₇	1	6	yes	—
91	C₄	C₂²×D₈×C₁₂₇	1	6	yes	—
92	C₂	—	16	8	no	—
93	C₂	C₁₂₇×#1712	1	4	yes	direct product factor
94	C₂	C₂×#493×C₁₂₇	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#493×C₁₂₇	C₂	semidirect
2. C₁₂₇×#1712	C₂	direct (N × H)
3. C₂×C₈²×C₁₂₇	C₂	semidirect
7. #493×C₁₂₇	C₂²	semidirect
9. C₈²×C₁₂₇	C₂²	semidirect
25. C₂×C₈×C₁₂₇	D₈	semidirect
37. C₈×C₁₂₇	C₂×D₈	semidirect
44. C₂×#1712	C₁₂₇	direct (N × H)
46. C₂×C₁₂₇	#1712	direct (N × H)
48. C₂×#493	C₂×C₁₂₇	semidirect
49. #1712	C₂×C₁₂₇	direct (N × H)
50. C₂×C₈²	C₂×C₁₂₇	semidirect
51. C₁₂₇	C₂×#1712	direct (N × H)
55. #493	C₂²×C₁₂₇	semidirect
57. C₈²	C₂²×C₁₂₇	semidirect
73. C₂×C₈	D₈×C₁₂₇	semidirect
85. C₈	C₂×D₈×C₁₂₇	semidirect
93. C₂	C₁₂₇×#1712	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₂₇	2
C₁₂₇×#1712	2
C₂×C₈²×C₁₂₇	2
C₂×#1712	127

Subgroups — C2×C127×#1712 #44583274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1712 #44583274