Subgroups — C₄×C₂²⋊C₄×C₁₂₇ #22549385

Order 8128.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₄×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×C₄²×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#95×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₁₂₇	—	2	4	no	—
7	C₂²×C₄×C₁₂₇	C₄	1	4	yes	complemented
8	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₁₂₇	C₂²	1	4	yes	—
10	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²⋊C₄×C₁₂₇	C₄	1	8	yes	direct product factor
12	C₄²×C₁₂₇	—	2	4	no	—
13	C₂³×C₁₂₇	—	2	2	no	—
14	C₂³×C₁₂₇	—	2	2	no	—
15	C₂³×C₁₂₇	C₂×C₄	1	4	yes	—
16	C₂³×C₁₂₇	C₂×C₄	1	2	yes	—
17	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	—	2	4	no	—
19	C₂×C₄×C₁₂₇	—	2	2	no	—
20	C₂×C₄×C₁₂₇	—	2	4	no	—
21	C₂×C₄×C₁₂₇	—	2	8	no	—
22	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₁₂₇	C₂×C₄	1	8	yes	complemented
25	C₂×C₄×C₁₂₇	D₄	1	4	yes	—
26	C₂²×C₁₂₇	—	2	2	no	—
27	C₂²×C₁₂₇	—	2	2	no	—
28	C₂²×C₁₂₇	—	2	8	no	—
29	C₂²×C₁₂₇	C₄²	1	4	yes	complemented
30	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
32	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
33	C₂²×C₁₂₇	#41	1	2	yes	—
34	C₄×C₁₂₇	—	2	2	no	—
35	C₄×C₁₂₇	—	2	8	no	—
36	C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	direct product factor
37	C₂×C₁₂₇	—	2	4	no	—
38	C₂×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₁₂₇	#117	1	1	yes	characteristic
41	C₂×C₁₂₇	C₄×D₄	1	4	yes	—
42	C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
43	C₄×C₂²⋊C₄	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₄	C₂×C₁₂₇	1	1	yes	characteristic
45	C₂×C₂²⋊C₄	C₂×C₁₂₇	1	2	yes	—
46	C₂×C₄²	C₂×C₁₂₇	1	2	yes	complemented
47	#95	C₂×C₁₂₇	1	2	yes	complemented
48	C₂⁴	C₂²×C₁₂₇	1	1	yes	characteristic
49	C₂²×C₄	—	2	4	no	—
50	C₂²×C₄	C₄×C₁₂₇	1	4	yes	complemented
51	C₂²×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
52	C₂²×C₄	C₂²×C₁₂₇	1	4	yes	—
53	C₂²×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
54	C₂²⋊C₄	C₄×C₁₂₇	1	8	yes	direct product factor
55	C₄²	—	2	4	no	—
56	C₂³	—	2	2	no	—
57	C₂³	—	2	2	no	—
58	C₂³	C₂×C₄×C₁₂₇	1	4	yes	—
59	C₂³	C₂×C₄×C₁₂₇	1	2	yes	—
60	C₂³	C₂³×C₁₂₇	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	2	8	no	—
65	C₂×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
66	C₂×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
67	C₂×C₄	C₂×C₄×C₁₂₇	1	8	yes	complemented
68	C₂×C₄	D₄×C₁₂₇	1	4	yes	—
69	C₂²	—	2	2	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	2	8	no	—
72	C₂²	C₄²×C₁₂₇	1	4	yes	complemented
73	C₂²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
74	C₂²	C₂²×C₄×C₁₂₇	1	2	yes	—
75	C₂²	C₂×D₄×C₁₂₇	1	2	yes	—
76	C₂²	#41×C₁₂₇	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	—	2	8	no	—
79	C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	direct product factor
80	C₂	—	2	4	no	—
81	C₂	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
82	C₂	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
83	C₂	#117×C₁₂₇	1	1	yes	characteristic
84	C₂	C₄×D₄×C₁₂₇	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂²
commutator	C₂	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂³	C₂³×C₁₂₇
socle	C₂³×C₁₂₇	C₂³
prime radical	C₂⁴×C₁₂₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄²×C₁₂₇	C₂	semidirect
4. #95×C₁₂₇	C₂	semidirect
7. C₂²×C₄×C₁₂₇	C₄	semidirect
11. C₂²⋊C₄×C₁₂₇	C₄	direct (N × H)
24. C₂×C₄×C₁₂₇	C₂×C₄	semidirect
29. C₂²×C₁₂₇	C₄²	semidirect
36. C₄×C₁₂₇	C₂²⋊C₄	direct (N × H)
42. C₁₂₇	C₄×C₂²⋊C₄	direct (N × H)
43. C₄×C₂²⋊C₄	C₁₂₇	direct (N × H)
46. C₂×C₄²	C₂×C₁₂₇	semidirect
47. #95	C₂×C₁₂₇	semidirect
50. C₂²×C₄	C₄×C₁₂₇	semidirect
54. C₂²⋊C₄	C₄×C₁₂₇	direct (N × H)
67. C₂×C₄	C₂×C₄×C₁₂₇	semidirect
72. C₂²	C₄²×C₁₂₇	semidirect
79. C₄	C₂²⋊C₄×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×C₄×C₁₂₇	2
C₂×C₂²⋊C₄×C₁₂₇	2
C₂×C₄²×C₁₂₇	2
#95×C₁₂₇	2
C₄×C₂²⋊C₄	127

Subgroups — C4×C22⋊C4×C127 #22549385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊C₄×C₁₂₇ #22549385