Subgroups — #480×C₁₂₇ #22549488

Order 8128.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#121×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#107×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#102×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₁₂₇	—	2	1	no	—
7	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₁₂₇	—	2	1	no	—
9	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₄⋊C₄×C₁₂₇	—	2	1	no	—
12	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
15	C₂³×C₁₂₇	—	2	1	no	—
16	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
17	D₄×C₁₂₇	—	2	2	no	—
18	D₄×C₁₂₇	—	4	2	no	—
19	C₂×C₄×C₁₂₇	—	2	2	no	—
20	C₂×C₄×C₁₂₇	—	2	1	no	—
21	C₂×C₄×C₁₂₇	—	2	2	no	—
22	C₂×C₄×C₁₂₇	—	4	2	no	—
23	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₈×C₁₂₇	—	2	2	no	—
26	C₂²×C₁₂₇	—	2	1	no	—
27	C₂²×C₁₂₇	—	2	1	no	—
28	C₂²×C₁₂₇	—	4	1	no	—
29	C₂²×C₁₂₇	—	4	1	no	—
30	C₂²×C₁₂₇	—	4	1	no	—
31	C₂²×C₁₂₇	D₈	1	2	yes	—
32	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₂₇	—	2	1	no	—
34	C₄×C₁₂₇	—	4	2	no	—
35	C₄×C₁₂₇	—	4	1	no	—
36	C₄×C₁₂₇	#41	1	2	yes	—
37	C₂×C₁₂₇	—	2	2	no	—
38	C₂×C₁₂₇	—	8	1	no	—
39	C₂×C₁₂₇	#123	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂×D₈	1	1	yes	characteristic
41	C₂×C₁₂₇	#137	1	1	yes	characteristic
42	C₁₂₇	#480	1	1	yes	characteristic, direct product factor, Sylow
43	#480	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	#121	C₂×C₁₂₇	1	1	yes	characteristic
45	C₂×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
46	#107	C₂×C₁₂₇	1	2	yes	complemented
47	#102	C₂×C₁₂₇	1	2	yes	complemented
48	C₂²⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
49	C₂×D₄	—	2	1	no	—
50	C₂×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
53	C₂×C₈	C₂²×C₁₂₇	1	2	yes	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
56	C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₂³	—	2	1	no	—
59	C₂³	D₄×C₁₂₇	1	1	yes	characteristic
60	D₄	—	2	2	no	—
61	D₄	—	4	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×C₁₂₇	1	1	yes	characteristic
67	C₂×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	D₈×C₁₂₇	1	2	yes	—
75	C₂²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	4	1	no	—
79	C₄	#41×C₁₂₇	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	8	1	no	—
82	C₂	#123×C₁₂₇	1	1	yes	characteristic
83	C₂	C₂×D₈×C₁₂₇	1	1	yes	characteristic
84	C₂	#137×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₂₇
prime radical	#121×C₁₂₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₄×C₁₂₇	C₂	semidirect
3. #107×C₁₂₇	C₂	semidirect
4. #102×C₁₂₇	C₂	semidirect
5. C₂²⋊C₈×C₁₂₇	C₂	semidirect
42. C₁₂₇	#480	direct (N × H)
43. #480	C₁₂₇	direct (N × H)
45. C₂×C₄⋊C₄	C₂×C₁₂₇	semidirect
46. #107	C₂×C₁₂₇	semidirect
47. #102	C₂×C₁₂₇	semidirect
48. C₂²⋊C₈	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#121×C₁₂₇	2
C₂×C₄⋊C₄×C₁₂₇	2
#107×C₁₂₇	2
#102×C₁₂₇	2
C₂²⋊C₈×C₁₂₇	2
#480	127

Subgroups — #480×C127 #22549488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #480×C₁₂₇ #22549488