Subgroups — C₂²×#16861203 #30651986

Order 16256.

1320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861203	C₂	1	24	yes	maximal, direct product factor
2	C₂²×Dic₁₂₇⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂²×(C₂×D₁₂₇)⋊C₄	C₂	1	3	yes	maximal
4	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₂₇	C₂²	1	1	yes	characteristic
6	C₂³×Dic₁₂₇	C₂²	1	3	yes	—
7	#16861203	C₂²	1	64	yes	direct product factor
8	C₂×Dic₁₂₇⋊C₄	C₂²	1	36	yes	complemented
9	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	36	yes	—
10	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	12	yes	complemented
12	C₂³×D₁₂₇	—	2	12	no	—
13	C₂³×D₁₂₇	C₂³	1	6	yes	—
14	C₂²×Dic₁₂₇	—	2	12	no	—
15	C₂²×Dic₁₂₇	C₂³	1	18	yes	—
16	Dic₁₂₇⋊C₄	C₂³	1	48	yes	complemented
17	(C₂×D₁₂₇)⋊C₄	C₂³	1	48	yes	—
18	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	—	2	12	no	—
20	C₂²×C₄×C₁₂₇	C₂³	1	18	yes	—
21	C₄²×C₁₂₇	C₂³	1	16	yes	complemented
22	C₂²×D₁₂₇	—	2	36	no	—
23	C₂²×D₁₂₇	—	4	16	no	—
24	C₂²×D₁₂₇	C₂⁴	1	4	yes	—
25	C₂×Dic₁₂₇	—	2	36	no	—
26	C₂×Dic₁₂₇	C₂⁴	1	12	yes	—
27	C₂³×C₁₂₇	#41	1	12	yes	—
28	C₂³×C₁₂₇	C₂⁴	1	3	yes	—
29	C₂×C₄×C₁₂₇	—	2	36	no	—
30	C₂×C₄×C₁₂₇	C₂⁴	1	12	yes	—
31	C₂²×C₁₂₇	#126	1	16	yes	complemented
32	C₂²×C₁₂₇	C₂×#41	1	18	yes	—
33	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
34	C₂×D₁₂₇	—	2	12	no	—
35	C₂×D₁₂₇	—	4	24	no	—
36	C₄×C₁₂₇	—	2	12	no	—
37	Dic₁₂₇	—	2	12	no	—
38	C₂×C₁₂₇	C₂×#126	1	12	yes	complemented
39	C₂×C₁₂₇	C₂²×#41	1	3	yes	—
40	D₁₂₇	—	4	4	no	—
41	C₂²×#126	—	127	1	no	maximal, Sylow
42	C₁₂₇	C₂²×#126	1	1	yes	characteristic, complemented, Sylow
43	C₂×#126	—	127	24	no	—
44	C₂²×C₄⋊C₄	—	127	3	no	—
45	C₂²×C₂²⋊C₄	—	127	3	no	—
46	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic, complemented
47	C₂⁵	—	127	1	no	—
48	C₂³×C₄	—	127	3	no	—
49	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
50	#126	—	127	64	no	—
51	C₂×C₄⋊C₄	—	127	36	no	—
52	C₂×C₂²⋊C₄	—	127	36	no	—
53	C₂×C₄²	C₂×D₁₂₇	1	12	yes	complemented
54	C₂⁴	—	127	6	no	—
55	C₂⁴	—	254	12	no	—
56	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
57	C₂²×C₄	—	2	12	no	—
58	C₂²×C₄	—	127	18	no	—
59	C₂²×C₄	—	254	12	no	—
60	C₂²×C₄	C₂²×D₁₂₇	1	18	yes	—
61	C₄⋊C₄	—	127	48	no	—
62	C₂²⋊C₄	—	127	48	no	—
63	C₄²	C₂²×D₁₂₇	1	16	yes	complemented
64	C₂³	—	127	4	no	—
65	C₂³	—	508	16	no	—
66	C₂³	—	254	36	no	—
67	C₂³	#14523248	1	12	yes	—
68	C₂³	C₂³×D₁₂₇	1	3	yes	—
69	C₂×C₄	—	2	36	no	—
70	C₂×C₄	—	127	12	no	—
71	C₂×C₄	—	254	36	no	—
72	C₂×C₄	C₂³×D₁₂₇	1	12	yes	—
73	C₂²	—	508	24	no	—
74	C₂²	—	254	12	no	—
75	C₂²	#16861203	1	16	yes	direct product factor
76	C₂²	C₂×#14523248	1	18	yes	—
77	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
78	C₄	—	2	12	no	—
79	C₄	—	254	12	no	—
80	C₂	—	508	4	no	—
81	C₂	C₂×#16861203	1	12	yes	direct product factor
82	C₂	C₂²×#14523248	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle	C₂⁴×C₁₂₇	C₂³
prime radical	C₂⁴×D₁₂₇	C₂²
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₂²×#126
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#16861203	C₂	direct (N × H)
2. C₂²×Dic₁₂₇⋊C₄	C₂	semidirect
4. C₂²×C₄²×C₁₂₇	C₂	semidirect
7. #16861203	C₂²	direct (N × H)
8. C₂×Dic₁₂₇⋊C₄	C₂²	semidirect
11. C₂×C₄²×C₁₂₇	C₂²	semidirect
16. Dic₁₂₇⋊C₄	C₂³	semidirect
21. C₄²×C₁₂₇	C₂³	semidirect
31. C₂²×C₁₂₇	#126	semidirect
38. C₂×C₁₂₇	C₂×#126	semidirect
42. C₁₂₇	C₂²×#126	semidirect
46. C₂²×C₄²	D₁₂₇	semidirect
53. C₂×C₄²	C₂×D₁₂₇	semidirect
63. C₄²	C₂²×D₁₂₇	semidirect
75. C₂²	#16861203	direct (N × H)
81. C₂	C₂×#16861203	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×#16861203	2
C₂²×Dic₁₂₇⋊C₄	2
C₂²×(C₂×D₁₂₇)⋊C₄	2
C₂²×C₄²×C₁₂₇	2
C₂²×#126	127

Subgroups — C22×#16861203 #30651986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16861203 #30651986