Subgroups — C₁₂₇×#13726 #44577399

Order 32512.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄×C₁₂₇	C₂	1	4	yes	maximal
2	C₁₂₇×#2247	C₂	1	2	yes	maximal
3	C₄×C₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₁₂₇	—	2	2	no	—
5	C₂×Q₃₂×C₁₂₇	C₂²	1	4	yes	—
6	#494×C₁₂₇	—	2	2	no	—
7	Q₆₄×C₁₂₇	—	2	8	no	—
8	C₂×C₃₂×C₁₂₇	C₂²	1	2	yes	—
9	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
11	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
12	C₄⋊Q₈×C₁₂₇	—	4	2	no	—
13	Q₃₂×C₁₂₇	—	2	8	no	—
14	Q₃₂×C₁₂₇	—	4	4	no	—
15	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₃₂×C₁₂₇	D₄	1	4	yes	—
19	C₂×Q₈×C₁₂₇	—	4	4	no	—
20	Q₁₆×C₁₂₇	—	4	4	no	—
21	Q₁₆×C₁₂₇	—	4	4	no	—
22	Q₁₆×C₁₂₇	—	8	4	no	—
23	C₂×C₈×C₁₂₇	D₈	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
25	C₄⋊C₄×C₁₂₇	—	8	4	no	—
26	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
27	C₁₆×C₁₂₇	D₈	1	2	yes	—
28	C₁₆×C₁₂₇	C₂×D₄	1	2	yes	—
29	Q₈×C₁₂₇	—	8	4	no	—
30	Q₈×C₁₂₇	—	8	4	no	—
31	C₂×C₄×C₁₂₇	—	8	4	no	—
32	C₂×C₄×C₁₂₇	D₁₆	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
34	C₈×C₁₂₇	D₁₆	1	2	yes	—
35	C₈×C₁₂₇	#127	1	1	yes	characteristic
36	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
37	C₂²×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
38	C₄×C₁₂₇	—	16	4	no	—
39	C₄×C₁₂₇	Q₆₄	1	4	yes	complemented
40	C₄×C₁₂₇	#493	1	1	yes	characteristic
41	C₄×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
42	#13726	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂×C₁₂₇	#2246	1	1	yes	characteristic
44	C₂×C₁₂₇	C₂×Q₆₄	1	2	yes	—
45	C₂×Q₆₄	C₂×C₁₂₇	1	4	yes	—
46	#2247	C₂×C₁₂₇	1	2	yes	—
47	C₄×C₃₂	C₂×C₁₂₇	1	1	yes	characteristic
48	C₁₂₇	#13726	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×Q₃₂	—	2	2	no	—
50	C₂×Q₃₂	C₂²×C₁₂₇	1	4	yes	—
51	#494	—	2	2	no	—
52	Q₆₄	—	2	8	no	—
53	C₂×C₃₂	C₂²×C₁₂₇	1	2	yes	—
54	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
55	C₂×Q₁₆	—	2	4	no	—
56	C₂×Q₁₆	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	Q₃₂	—	2	8	no	—
59	Q₃₂	—	4	4	no	—
60	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
61	C₂×C₁₆	C₂³×C₁₂₇	1	1	yes	characteristic
62	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
63	C₃₂	D₄×C₁₂₇	1	4	yes	—
64	C₂×Q₈	—	4	4	no	—
65	Q₁₆	—	4	4	no	—
66	Q₁₆	—	4	4	no	—
67	Q₁₆	—	8	4	no	—
68	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
69	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
70	C₄⋊C₄	—	8	4	no	—
71	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
72	C₁₆	D₈×C₁₂₇	1	2	yes	—
73	C₁₆	C₂×D₄×C₁₂₇	1	2	yes	—
74	Q₈	—	8	4	no	—
75	Q₈	—	8	4	no	—
76	C₂×C₄	—	8	4	no	—
77	C₂×C₄	D₁₆×C₁₂₇	1	2	yes	—
78	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
79	C₈	D₁₆×C₁₂₇	1	2	yes	—
80	C₈	#127×C₁₂₇	1	1	yes	characteristic
81	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
82	C₂²	C₂×D₁₆×C₁₂₇	1	1	yes	characteristic
83	C₄	—	16	4	no	—
84	C₄	Q₆₄×C₁₂₇	1	4	yes	complemented
85	C₄	#493×C₁₂₇	1	1	yes	characteristic
86	C₄	C₂×D₁₆×C₁₂₇	1	1	yes	characteristic
87	C₂	C₁₂₇×#2246	1	1	yes	characteristic
88	C₂	C₂×Q₆₄×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₂₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
39. C₄×C₁₂₇	Q₆₄	semidirect
42. #13726	C₁₂₇	direct (N × H)
48. C₁₂₇	#13726	direct (N × H)
84. C₄	Q₆₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₆₄×C₁₂₇	2
C₁₂₇×#2247	2
C₄×C₃₂×C₁₂₇	2
#13726	127

Subgroups — C127×#13726 #44577399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13726 #44577399