Subgroups — #30640617

Order 16256.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550308	C₂	1	1	yes	maximal, characteristic
2	#22549994	C₂	1	2	yes	maximal, complemented
3	#22549706	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549684	C₂	1	2	yes	maximal, complemented
5	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861231	C₂²	1	2	yes	—
7	#16861215	—	2	2	no	—
8	Dic₁₀₁₆	C₄	1	1	yes	characteristic, complemented
9	#16861154	C₄	1	2	yes	complemented
10	D₂₀₃₂	C₄	1	1	yes	characteristic, complemented
11	#16861145	C₂²	1	2	yes	—
12	#16861137	—	2	2	no	—
13	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	#14523248	—	2	2	no	—
17	Dic₅₀₈	C₂×C₄	1	2	yes	—
18	#14523237	—	2	2	no	—
19	D₁₀₁₆	C₂×C₄	1	2	yes	—
20	#14523232	—	2	2	no	—
21	#14523231	—	4	2	no	—
22	C₈×D₁₂₇	—	4	2	no	—
23	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
26	C₁₆×C₁₂₇	—	2	1	no	—
27	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	Dic₂₅₄	—	2	2	no	—
30	D₅₀₈	—	2	2	no	—
31	C₂²⋊D₁₂₇	—	4	2	no	—
32	C₄×D₁₂₇	—	4	2	no	—
33	C₁₂₇⋊C₈	—	4	2	no	—
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₂₇	D₈	1	2	yes	—
37	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₂₇	#41	1	1	yes	characteristic
39	C₂²×C₁₂₇	#135	1	1	yes	characteristic
40	C₂×D₁₂₇	—	4	2	no	—
41	C₄×C₁₂₇	—	4	1	no	—
42	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
44	Dic₁₂₇	—	4	2	no	—
45	C₂×C₁₂₇	—	2	1	no	—
46	C₂×C₁₂₇	C₄×D₈	1	1	yes	characteristic
47	D₁₂₇	—	8	2	no	—
48	#2179	—	127	1	no	maximal, Sylow
49	C₁₂₇	#2179	1	1	yes	characteristic, complemented, Sylow
50	#508	—	127	1	no	—
51	#443	—	127	2	no	—
52	#368	—	127	1	no	—
53	#359	—	127	2	no	—
54	#346	D₁₂₇	1	1	yes	characteristic, complemented
55	#135	—	127	2	no	—
56	#131	—	254	2	no	—
57	Q₃₂	—	127	1	no	—
58	SD₃₂	—	127	2	no	—
59	D₁₆	—	127	1	no	—
60	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
62	#108	—	127	2	no	—
63	#104	—	254	2	no	—
64	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
65	#41	—	254	2	no	—
66	Q₁₆	—	127	2	no	—
67	SD₁₆	—	254	2	no	—
68	D₈	—	127	2	no	—
69	M₁₆	—	508	2	no	—
70	M₁₆	—	254	2	no	—
71	C₂×C₈	—	508	2	no	—
72	C₂×C₈	D₅₀₈	1	1	yes	characteristic
73	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₄²	D₅₀₈	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
77	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
78	Q₈	—	254	2	no	—
79	D₄	—	508	2	no	—
80	D₄	—	254	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	508	2	no	—
83	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
84	C₈	—	508	2	no	—
85	C₈	D₁₀₁₆	1	2	yes	—
86	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
87	C₈	#14523248	1	1	yes	characteristic
88	C₂²	—	508	2	no	—
89	C₂²	#16861231	1	1	yes	characteristic
90	C₄	—	4	1	no	—
91	C₄	—	508	2	no	—
92	C₄	C₄×D₅₀₈	1	1	yes	characteristic
93	C₄	C₂×D₁₀₁₆	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	1016	2	no	—
96	C₂	C₄×D₁₀₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₀₁₆
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#346×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₄×D₈
prime radical	#22550308	C₂
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#346	D₁₂₇
hypocenter	C₁₂₇	#2179
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #22549994	C₂	semidirect
3. #22549706	C₂	semidirect
4. #22549684	C₂	semidirect
5. #346×C₁₂₇	C₂	semidirect
8. Dic₁₀₁₆	C₄	semidirect
9. #16861154	C₄	semidirect
10. D₂₀₃₂	C₄	semidirect
49. C₁₂₇	#2179	semidirect
54. #346	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550308	2
#22549994	2
#22549706	2
#22549684	2
#346×C₁₂₇	2
#2179	127