Subgroups — C₂²×#16861209 #30652006

Order 16256.

1624 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₅₄	C₂	1	2	yes	maximal
2	C₂×#16861209	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊Dic₁₂₇	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	—	2	16	no	—
6	C₂²×Dic₂₅₄	C₂²	1	24	yes	—
7	C₂³×Dic₁₂₇	C₂²	1	4	yes	—
8	#16861209	C₂²	1	64	yes	direct product factor
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	48	yes	—
10	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	12	yes	—
13	C₂×Dic₂₅₄	—	2	96	no	—
14	C₂×Dic₂₅₄	C₂³	1	32	yes	—
15	C₂²×Dic₁₂₇	—	2	16	no	—
16	C₂²×Dic₁₂₇	C₂³	1	24	yes	—
17	C₄⋊Dic₁₂₇	C₂³	1	64	yes	—
18	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	D₄	1	8	yes	—
20	C₂²×C₄×C₁₂₇	Q₈	1	16	yes	complemented
21	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
22	C₂²×C₄×C₁₂₇	C₂³	1	12	yes	—
23	C₄²×C₁₂₇	C₂³	1	16	yes	—
24	Dic₂₅₄	—	2	64	no	—
25	C₂×Dic₁₂₇	—	2	48	no	—
26	C₂×Dic₁₂₇	C₂⁴	1	16	yes	—
27	C₂³×C₁₂₇	C₂×D₄	1	4	yes	—
28	C₂³×C₁₂₇	C₂×Q₈	1	8	yes	—
29	C₂³×C₁₂₇	C₂⁴	1	3	yes	—
30	C₂×C₄×C₁₂₇	C₂×D₄	1	24	yes	—
31	C₂×C₄×C₁₂₇	C₂×Q₈	1	48	yes	complemented
32	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
33	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
34	C₂²×C₁₂₇	C₄⋊Q₈	1	16	yes	complemented
35	C₂²×C₁₂₇	C₂²×D₄	1	6	yes	—
36	C₂²×C₁₂₇	C₂²×Q₈	1	12	yes	—
37	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
38	C₄×C₁₂₇	C₂²×D₄	1	8	yes	—
39	C₄×C₁₂₇	C₂²×Q₈	1	16	yes	complemented
40	Dic₁₂₇	—	2	16	no	—
41	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	12	yes	complemented
42	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂³×Q₈	1	2	yes	—
44	C₂²×C₄⋊Q₈	—	127	1	no	maximal, Sylow
45	C₁₂₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
46	C₂³×Q₈	—	127	2	no	—
47	C₂×C₄⋊Q₈	—	127	24	no	—
48	C₂²×C₄⋊C₄	—	127	4	no	—
49	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
50	C₂²×Q₈	—	127	24	no	—
51	C₂²×Q₈	—	254	16	no	—
52	C₂³×C₄	—	127	4	no	—
53	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
54	C₂³×C₄	C₂×D₁₂₇	1	2	yes	—
55	C₄⋊Q₈	—	127	64	no	—
56	C₂×C₄⋊C₄	—	127	48	no	—
57	C₂×C₄²	C₂×D₁₂₇	1	12	yes	—
58	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
59	C₂×Q₈	—	127	32	no	—
60	C₂×Q₈	—	254	96	no	—
61	C₂²×C₄	—	127	24	no	—
62	C₂²×C₄	—	254	16	no	—
63	C₂²×C₄	D₅₀₈	1	8	yes	—
64	C₂²×C₄	Dic₂₅₄	1	16	yes	complemented
65	C₂²×C₄	C₂²×D₁₂₇	1	6	yes	—
66	C₂²×C₄	C₂²×D₁₂₇	1	12	yes	—
67	C₄⋊C₄	—	127	64	no	—
68	C₄²	C₂²×D₁₂₇	1	16	yes	—
69	C₂³	C₂×D₅₀₈	1	4	yes	—
70	C₂³	C₂×Dic₂₅₄	1	8	yes	—
71	C₂³	C₂³×D₁₂₇	1	3	yes	—
72	Q₈	—	254	64	no	—
73	C₂×C₄	—	127	16	no	—
74	C₂×C₄	—	254	48	no	—
75	C₂×C₄	C₂×D₅₀₈	1	24	yes	—
76	C₂×C₄	C₂×Dic₂₅₄	1	48	yes	complemented
77	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
78	C₂×C₄	C₂³×D₁₂₇	1	8	yes	—
79	C₂²	#16861209	1	16	yes	direct product factor
80	C₂²	C₂²×D₅₀₈	1	6	yes	—
81	C₂²	C₂²×Dic₂₅₄	1	12	yes	—
82	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
83	C₄	—	254	16	no	—
84	C₄	C₂²×D₅₀₈	1	8	yes	—
85	C₄	C₂²×Dic₂₅₄	1	16	yes	complemented
86	C₂	C₂×#16861209	1	12	yes	direct product factor
87	C₂	C₂³×D₅₀₈	1	1	yes	characteristic
88	C₂	C₂³×Dic₂₅₄	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	C₂³
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₂²×C₄⋊Q₈
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. C₂×#16861209	C₂	direct (N × H)
8. #16861209	C₂²	direct (N × H)
20. C₂²×C₄×C₁₂₇	Q₈	semidirect
31. C₂×C₄×C₁₂₇	C₂×Q₈	semidirect
34. C₂²×C₁₂₇	C₄⋊Q₈	semidirect
39. C₄×C₁₂₇	C₂²×Q₈	semidirect
41. C₂×C₁₂₇	C₂×C₄⋊Q₈	semidirect
45. C₁₂₇	C₂²×C₄⋊Q₈	semidirect
64. C₂²×C₄	Dic₂₅₄	semidirect
76. C₂×C₄	C₂×Dic₂₅₄	semidirect
79. C₂²	#16861209	direct (N × H)
85. C₄	C₂²×Dic₂₅₄	semidirect
86. C₂	C₂×#16861209	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×Dic₂₅₄	2
C₂×#16861209	2
C₂²×C₄⋊Dic₁₂₇	2
C₂²×C₄²×C₁₂₇	2
C₂²×C₄⋊Q₈	127

Subgroups — C22×#16861209 #30652006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16861209 #30652006