Subgroups — #30640616

Order 16256.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549993	C₂	1	2	yes	maximal, complemented
2	#22549706	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549683	C₂	1	2	yes	maximal, complemented
4	#22549651	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861216	—	2	2	no	—
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16861145	C₂²	1	2	yes	—
9	#16861136	—	2	2	no	—
10	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	#135×C₁₂₇	C₂²	1	2	yes	—
12	Q₃₂×C₁₂₇	C₄	1	1	yes	characteristic, complemented
13	SD₃₂×C₁₂₇	C₄	1	2	yes	complemented
14	D₁₆×C₁₂₇	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
16	#14523232	—	2	2	no	—
17	#14523232	—	4	2	no	—
18	C₂×C₁₂₇⋊C₈	—	4	2	no	—
19	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
20	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
21	C₁₂₇⋊C₁₆	—	2	1	no	—
22	#41×C₁₂₇	—	2	2	no	—
23	Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	SD₁₆×C₁₂₇	—	2	2	no	—
25	D₈×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂×Dic₁₂₇	—	2	1	no	—
30	C₁₂₇⋊C₈	—	4	2	no	—
31	C₁₂₇⋊C₈	D₈	1	2	yes	—
32	Q₈×C₁₂₇	—	2	2	no	—
33	D₄×C₁₂₇	—	2	2	no	—
34	D₄×C₁₂₇	—	4	2	no	—
35	C₂×C₄×C₁₂₇	—	4	2	no	—
36	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₂₇	#41	1	1	yes	characteristic
39	C₂²×C₁₂₇	—	4	2	no	—
40	C₂²×C₁₂₇	#135	1	1	yes	characteristic
41	C₄×C₁₂₇	—	4	2	no	—
42	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
44	Dic₁₂₇	—	4	1	no	—
45	C₂×C₁₂₇	—	2	1	no	—
46	C₂×C₁₂₇	—	8	2	no	—
47	C₂×C₁₂₇	C₄×D₈	1	1	yes	characteristic
48	#2179	—	127	1	no	maximal, Sylow
49	C₁₂₇	#2179	1	1	yes	characteristic, complemented, Sylow
50	#508	D₁₂₇	1	1	yes	characteristic
51	#443	—	127	2	no	—
52	#368	—	127	1	no	—
53	#359	—	127	2	no	—
54	#346	—	127	1	no	—
55	#135	C₂×D₁₂₇	1	2	yes	—
56	#131	—	254	2	no	—
57	Q₃₂	Dic₁₂₇	1	1	yes	characteristic, complemented
58	SD₃₂	Dic₁₂₇	1	2	yes	complemented
59	D₁₆	Dic₁₂₇	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	127	1	no	—
61	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
62	#108	—	127	2	no	—
63	#104	—	254	2	no	—
64	C₄×C₈	—	127	1	no	—
65	#41	—	2	2	no	—
66	Q₁₆	C₂×Dic₁₂₇	1	2	yes	—
67	SD₁₆	—	2	2	no	—
68	D₈	C₂×Dic₁₂₇	1	2	yes	—
69	M₁₆	—	508	2	no	—
70	M₁₆	—	254	2	no	—
71	C₂×C₈	—	127	1	no	—
72	C₂×C₈	—	508	2	no	—
73	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₄²	—	127	1	no	—
75	C₁₆	—	254	1	no	—
76	C₁₆	C₂×Dic₁₂₇	1	1	yes	characteristic
77	C₁₆	C₂×Dic₁₂₇	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	254	1	no	—
83	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
84	C₈	—	127	2	no	—
85	C₈	—	508	2	no	—
86	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
87	C₈	#14523247	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	#16861228	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	508	1	no	—
92	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
93	C₄	D₈×D₁₂₇	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	8	2	no	—
96	C₂	D₈×Dic₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₂₇
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting, prime radical	#508×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₄×D₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#508	D₁₂₇
hypocenter	C₁₂₇	#2179
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #22549993	C₂	semidirect
2. #22549706	C₂	semidirect
3. #22549683	C₂	semidirect
4. #22549651	C₂	semidirect
12. Q₃₂×C₁₂₇	C₄	semidirect
13. SD₃₂×C₁₂₇	C₄	semidirect
14. D₁₆×C₁₂₇	C₄	semidirect
49. C₁₂₇	#2179	semidirect
57. Q₃₂	Dic₁₂₇	semidirect
58. SD₃₂	Dic₁₂₇	semidirect
59. D₁₆	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549993	2
#22549706	2
#22549683	2
#22549651	2
#508×C₁₂₇	2
#2179	127