Subgroups — C₂²×#16861141 #30647302

Order 16256.

600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₂₇	C₂	1	2	yes	maximal
2	C₂×#16861141	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₂₇	—	2	2	no	—
5	C₂×C₄⋊Dic₁₂₇	—	2	6	no	—
6	C₂×C₄⋊Dic₁₂₇	C₂²	1	12	yes	—
7	#16861141	C₂²	1	16	yes	direct product factor
8	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₂₇	C₄	1	8	yes	complemented
10	C₂²×C₈×C₁₂₇	C₂²	1	6	yes	—
11	C₂²×Dic₁₂₇	—	2	12	no	—
12	C₂²×Dic₁₂₇	—	4	8	no	—
13	C₄⋊Dic₁₂₇	—	2	12	no	—
14	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
15	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₂₇	D₄	1	3	yes	—
18	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
20	C₂×C₈×C₁₂₇	C₂×C₄	1	24	yes	complemented
21	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
22	C₂×Dic₁₂₇	—	2	8	no	—
23	C₂×Dic₁₂₇	—	4	24	no	—
24	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
25	C₂³×C₁₂₇	SD₁₆	1	8	yes	—
26	C₂³×C₁₂₇	C₂×D₄	1	3	yes	—
27	C₂×C₄×C₁₂₇	C₄⋊C₄	1	12	yes	—
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
30	C₂×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
31	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
32	C₈×C₁₂₇	C₂²×C₄	1	8	yes	complemented
33	C₂²×C₁₂₇	#106	1	16	yes	complemented
34	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	6	yes	—
35	C₂²×C₁₂₇	C₂×SD₁₆	1	12	yes	—
36	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
37	C₄×C₁₂₇	C₂×C₄⋊C₄	1	6	yes	—
38	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
39	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
40	Dic₁₂₇	—	4	8	no	—
41	C₂×C₁₂₇	C₂×#106	1	12	yes	complemented
42	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
44	C₂²×#106	—	127	1	no	maximal, Sylow
45	C₁₂₇	C₂²×#106	1	1	yes	characteristic, complemented, Sylow
46	C₂³×C₈	D₁₂₇	1	1	yes	characteristic
47	C₂²×C₄⋊C₄	—	127	2	no	—
48	C₂×#106	—	127	12	no	—
49	C₂³×C₄	—	254	2	no	—
50	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂²×C₈	Dic₁₂₇	1	8	yes	complemented
52	C₂²×C₈	C₂×D₁₂₇	1	6	yes	—
53	C₂×C₄⋊C₄	—	127	12	no	—
54	C₂×C₄⋊C₄	—	254	6	no	—
55	#106	—	127	16	no	—
56	C₂⁴	D₅₀₈	1	1	yes	characteristic
57	C₂²×C₄	—	508	8	no	—
58	C₂²×C₄	—	254	12	no	—
59	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
60	C₂²×C₄	D₅₀₈	1	3	yes	—
61	C₂²×C₄	Dic₂₅₄	1	4	yes	—
62	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
63	C₂×C₈	C₂×Dic₁₂₇	1	24	yes	complemented
64	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
65	C₄⋊C₄	—	127	8	no	—
66	C₄⋊C₄	—	254	12	no	—
67	C₂³	C₄⋊Dic₁₂₇	1	4	yes	—
68	C₂³	#14523237	1	8	yes	—
69	C₂³	C₂×D₅₀₈	1	3	yes	—
70	C₂×C₄	—	508	24	no	—
71	C₂×C₄	—	254	8	no	—
72	C₂×C₄	C₄⋊Dic₁₂₇	1	12	yes	—
73	C₂×C₄	C₂²×Dic₁₂₇	1	6	yes	—
74	C₂×C₄	C₂×D₅₀₈	1	3	yes	—
75	C₂×C₄	C₂×Dic₂₅₄	1	6	yes	—
76	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
77	C₈	C₂²×Dic₁₂₇	1	8	yes	complemented
78	C₂²	#16861141	1	16	yes	direct product factor
79	C₂²	C₂×C₄⋊Dic₁₂₇	1	6	yes	—
80	C₂²	C₂×#14523237	1	12	yes	—
81	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
82	C₄	—	508	8	no	—
83	C₄	C₂×C₄⋊Dic₁₂₇	1	6	yes	—
84	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
85	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
86	C₂	C₂×#16861141	1	12	yes	direct product factor
87	C₂	C₂²×C₄⋊Dic₁₂₇	1	1	yes	characteristic
88	C₂	C₂²×#14523237	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₅₀₈
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂³×C₈×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	D₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂³×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂²×#106
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16861141	C₂	direct (N × H)
7. #16861141	C₂²	direct (N × H)
9. C₂²×C₈×C₁₂₇	C₄	semidirect
20. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
32. C₈×C₁₂₇	C₂²×C₄	semidirect
33. C₂²×C₁₂₇	#106	semidirect
41. C₂×C₁₂₇	C₂×#106	semidirect
45. C₁₂₇	C₂²×#106	semidirect
51. C₂²×C₈	Dic₁₂₇	semidirect
63. C₂×C₈	C₂×Dic₁₂₇	semidirect
77. C₈	C₂²×Dic₁₂₇	semidirect
78. C₂²	#16861141	direct (N × H)
86. C₂	C₂×#16861141	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
C₂×#16861141	2
C₂³×C₈×C₁₂₇	2
C₂²×#106	127

Subgroups — C22×#16861141 #30647302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16861141 #30647302