Subgroups — #405×C₁₂₇ #22549413

Order 8128.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#97×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
7	M₁₆×C₁₂₇	C₄	1	4	yes	complemented
8	C₂×C₈×C₁₂₇	C₄	1	4	yes	complemented
9	C₂×C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
14	C₄²×C₁₂₇	C₂²	1	2	yes	—
15	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₈×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₈×C₁₂₇	C₂×C₄	1	4	yes	—
24	C₂²×C₁₂₇	—	2	2	no	—
25	C₂²×C₁₂₇	C₄²	1	2	yes	—
26	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₁₂₇	—	2	4	no	—
28	C₄×C₁₂₇	C₄²	1	2	yes	—
29	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₁₂₇	—	2	2	no	—
32	C₂×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
33	C₂×C₁₂₇	#131	1	2	yes	—
34	C₁₂₇	#405	1	1	yes	characteristic, direct product factor, Sylow
35	#405	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
37	C₂²×C₈	C₂×C₁₂₇	1	1	yes	characteristic
38	#117	C₂×C₁₂₇	1	1	yes	characteristic
39	#97	C₂×C₁₂₇	1	2	yes	complemented
40	C₄×C₈	C₂×C₁₂₇	1	2	yes	complemented
41	C₂²×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
42	M₁₆	C₄×C₁₂₇	1	4	yes	complemented
43	C₂×C₈	C₄×C₁₂₇	1	4	yes	complemented
44	C₂×C₈	C₂²×C₁₂₇	1	2	yes	—
45	C₂×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
46	C₂×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
47	C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
48	C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
49	C₄²	C₂²×C₁₂₇	1	2	yes	—
50	C₂³	C₂×C₄×C₁₂₇	1	1	yes	characteristic
51	C₂×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
52	C₂×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
53	C₂×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
54	C₂×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
55	C₂×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
56	C₈	C₂×C₄×C₁₂₇	1	2	yes	—
57	C₈	C₂×C₄×C₁₂₇	1	2	yes	—
58	C₈	C₂×C₄×C₁₂₇	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	C₄²×C₁₂₇	1	2	yes	—
61	C₂²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
62	C₄	—	2	4	no	—
63	C₄	C₄²×C₁₂₇	1	2	yes	—
64	C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
65	C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
68	C₂	#131×C₁₂₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #97×C₁₂₇	C₂	semidirect
5. C₄×C₈×C₁₂₇	C₂	semidirect
7. M₁₆×C₁₂₇	C₄	semidirect
8. C₂×C₈×C₁₂₇	C₄	semidirect
34. C₁₂₇	#405	direct (N × H)
35. #405	C₁₂₇	direct (N × H)
39. #97	C₂×C₁₂₇	semidirect
40. C₄×C₈	C₂×C₁₂₇	semidirect
42. M₁₆	C₄×C₁₂₇	semidirect
43. C₂×C₈	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₁₂₇	2
C₂²×C₈×C₁₂₇	2
#117×C₁₂₇	2
#97×C₁₂₇	2
C₄×C₈×C₁₂₇	2
#405	127

Subgroups — #405×C127 #22549413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #405×C₁₂₇ #22549413