Subgroups — C₂×#22549651 #30640319

Order 16256.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549651	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₁₆	C₄	1	4	yes	complemented
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₁₂₇	C₄	1	2	yes	—
10	C₈×Dic₁₂₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₂₇	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₂₇	C₈	1	2	yes	—
15	C₂×C₁₂₇⋊C₈	C₈	1	4	yes	—
16	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₂₇	C₈	1	2	yes	—
18	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
19	C₁₂₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
20	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
26	C₂×Dic₁₂₇	—	2	4	no	—
27	C₂×Dic₁₂₇	C₂×C₈	1	4	yes	—
28	C₁₂₇⋊C₈	C₂×C₈	1	4	yes	—
29	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
31	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₂₇	C₄²	1	2	yes	—
35	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₁₂₇	C₄×C₈	1	2	yes	—
38	C₂²×C₁₂₇	M₃₂	1	4	yes	—
39	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₁₂₇	C₄×C₈	1	2	yes	—
41	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
42	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
43	Dic₁₂₇	—	2	4	no	—
44	C₂×C₁₂₇	#346	1	4	yes	complemented
45	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
46	C₂×C₁₂₇	C₂×M₃₂	1	2	yes	—
47	C₂×#346	—	127	1	no	maximal, Sylow
48	C₁₂₇	C₂×#346	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₁₆	—	127	1	no	—
50	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic
51	C₂×C₄×C₈	—	127	1	no	—
52	#346	—	127	4	no	—
53	C₂²×C₈	—	127	1	no	—
54	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
55	C₂×C₄²	—	127	1	no	—
56	C₂×C₁₆	—	127	4	no	—
57	C₂×C₁₆	—	127	2	no	—
58	C₂×C₁₆	Dic₁₂₇	1	4	yes	complemented
59	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
60	C₄×C₈	—	127	2	no	—
61	C₄×C₈	—	127	2	no	—
62	C₂²×C₄	—	127	2	no	—
63	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
64	C₂×C₈	—	127	4	no	—
65	C₂×C₈	—	127	2	no	—
66	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
67	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
68	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
69	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
70	C₄²	—	127	2	no	—
71	C₄²	—	127	2	no	—
72	C₁₆	—	127	4	no	—
73	C₁₆	C₂×Dic₁₂₇	1	4	yes	complemented
74	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
75	C₂×C₄	—	127	4	no	—
76	C₂×C₄	—	254	4	no	—
77	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
78	C₂×C₄	C₈×D₁₂₇	1	2	yes	—
79	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
80	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
81	C₈	—	127	4	no	—
82	C₈	C₄×Dic₁₂₇	1	2	yes	—
83	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
84	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
85	C₂²	C₈×Dic₁₂₇	1	2	yes	—
86	C₂²	#16861148	1	4	yes	—
87	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
88	C₄	—	254	4	no	—
89	C₄	C₈×Dic₁₂₇	1	2	yes	—
90	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
91	C₄	C₂×C₈×D₁₂₇	1	1	yes	characteristic
92	C₂	#22549651	1	4	yes	direct product factor
93	C₂	C₂×C₈×Dic₁₂₇	1	1	yes	characteristic
94	C₂	C₂×#16861148	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂²×C₁₆×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂²×C₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×#346
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22549651	C₂	direct (N × H)
7. C₂×C₁₂₇⋊C₁₆	C₄	semidirect
12. C₂×C₁₆×C₁₂₇	C₄	semidirect
19. C₁₂₇⋊C₁₆	C₂×C₄	semidirect
25. C₁₆×C₁₂₇	C₂×C₄	semidirect
44. C₂×C₁₂₇	#346	semidirect
48. C₁₂₇	C₂×#346	semidirect
58. C₂×C₁₆	Dic₁₂₇	semidirect
73. C₁₆	C₂×Dic₁₂₇	semidirect
92. C₂	#22549651	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×C₁₂₇⋊C₁₆	2
C₂×C₈×Dic₁₂₇	2
#22549651	2
C₂²×C₁₆×C₁₂₇	2
C₂×#346	127

Subgroups — C2×#22549651 #30640319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549651 #30640319