Subgroups — C₂×#22549746 #30641131

Order 16256.

888 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₂₇	C₂	1	6	yes	maximal
2	#22549746	C₂	1	8	yes	maximal, direct product factor
3	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₂₇	C₂²	1	4	yes	—
5	C₂×C₄⋊Dic₁₂₇	—	2	24	no	—
6	C₂×C₄⋊Dic₁₂₇	C₂²	1	24	yes	—
7	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
8	C₂×C₄²×C₁₂₇	C₄	1	8	yes	complemented
9	C₂×C₄²×C₁₂₇	C₂²	1	4	yes	—
10	C₂²×Dic₁₂₇	—	2	24	no	—
11	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
12	C₄⋊Dic₁₂₇	—	2	48	no	—
13	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
15	C₂²×C₄×C₁₂₇	D₄	1	12	yes	—
16	C₂²×C₄×C₁₂₇	Q₈	1	12	yes	—
17	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
18	C₄²×C₁₂₇	C₂×C₄	1	16	yes	complemented
19	C₂×Dic₁₂₇	—	2	24	no	—
20	C₂×Dic₁₂₇	—	4	16	no	—
21	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
22	C₂³×C₁₂₇	C₂×D₄	1	6	yes	—
23	C₂³×C₁₂₇	C₂×Q₈	1	6	yes	—
24	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
25	C₂×C₄×C₁₂₇	C₄⋊C₄	1	48	yes	complemented
26	C₂×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
27	C₂×C₄×C₁₂₇	C₂×D₄	1	12	yes	—
28	C₂×C₄×C₁₂₇	C₂×Q₈	1	12	yes	—
29	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	12	yes	—
30	C₂²×C₁₂₇	#127	1	4	yes	—
31	C₂²×C₁₂₇	C₄⋊Q₈	1	12	yes	—
32	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
33	C₂²×C₁₂₇	C₂²×D₄	1	3	yes	—
34	C₂²×C₁₂₇	C₂²×Q₈	1	3	yes	—
35	C₄×C₁₂₇	C₂×C₄⋊C₄	1	24	yes	complemented
36	Dic₁₂₇	—	4	8	no	—
37	C₂×C₁₂₇	#384	1	8	yes	complemented
38	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	3	yes	—
39	C₂×C₁₂₇	C₂×#127	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	3	yes	—
41	C₂×#384	—	127	1	no	maximal, Sylow
42	C₁₂₇	C₂×#384	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₄⋊C₄	—	127	6	no	—
44	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
45	#384	—	127	8	no	—
46	C₂³×C₄	—	127	4	no	—
47	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
48	C₂×C₄⋊C₄	—	127	24	no	—
49	C₂×C₄⋊C₄	—	254	24	no	—
50	C₂×C₄²	Dic₁₂₇	1	8	yes	complemented
51	C₂×C₄²	C₂×D₁₂₇	1	4	yes	—
52	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
53	C₂²×C₄	—	127	8	no	—
54	C₂²×C₄	—	254	24	no	—
55	C₂²×C₄	C₂×Dic₁₂₇	1	12	yes	—
56	C₂²×C₄	D₅₀₈	1	12	yes	—
57	C₂²×C₄	Dic₂₅₄	1	12	yes	—
58	C₂²×C₄	C₂²×D₁₂₇	1	6	yes	—
59	C₄⋊C₄	—	254	48	no	—
60	C₄²	C₂×Dic₁₂₇	1	16	yes	complemented
61	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
62	C₂³	C₂×D₅₀₈	1	6	yes	—
63	C₂³	C₂×Dic₂₅₄	1	6	yes	—
64	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
65	C₂×C₄	—	508	16	no	—
66	C₂×C₄	—	254	24	no	—
67	C₂×C₄	C₄⋊Dic₁₂₇	1	48	yes	complemented
68	C₂×C₄	C₂²×Dic₁₂₇	1	12	yes	—
69	C₂×C₄	C₂×D₅₀₈	1	12	yes	—
70	C₂×C₄	C₂×Dic₂₅₄	1	12	yes	—
71	C₂²	C₂×C₄⋊Dic₁₂₇	1	12	yes	—
72	C₂²	#16861206	1	4	yes	—
73	C₂²	#16861209	1	12	yes	—
74	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
75	C₂²	C₂²×D₅₀₈	1	3	yes	—
76	C₂²	C₂²×Dic₂₅₄	1	3	yes	—
77	C₄	—	508	8	no	—
78	C₄	C₂×C₄⋊Dic₁₂₇	1	24	yes	complemented
79	C₂	#22549746	1	8	yes	direct product factor
80	C₂	C₂²×C₄⋊Dic₁₂₇	1	3	yes	—
81	C₂	C₂×#16861206	1	1	yes	characteristic
82	C₂	C₂×#16861209	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	C₂³
cosocle	C₂³×C₁₂₇	C₂⁴
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₂×#384
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
2. #22549746	C₂	direct (N × H)
8. C₂×C₄²×C₁₂₇	C₄	semidirect
18. C₄²×C₁₂₇	C₂×C₄	semidirect
25. C₂×C₄×C₁₂₇	C₄⋊C₄	semidirect
35. C₄×C₁₂₇	C₂×C₄⋊C₄	semidirect
37. C₂×C₁₂₇	#384	semidirect
42. C₁₂₇	C₂×#384	semidirect
50. C₂×C₄²	Dic₁₂₇	semidirect
60. C₄²	C₂×Dic₁₂₇	semidirect
67. C₂×C₄	C₄⋊Dic₁₂₇	semidirect
78. C₄	C₂×C₄⋊Dic₁₂₇	semidirect
79. C₂	#22549746	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₂₇	2
#22549746	2
C₂²×C₄²×C₁₂₇	2
C₂×#384	127

Subgroups — C2×#22549746 #30641131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549746 #30641131