Finite Group Databasebeta

Subgroups — #30640630

Order 16256.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550314	C₂	1	1	yes	maximal, characteristic
2	#22549999	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549989	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549699	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549690	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549687	C₂	1	1	yes	maximal, characteristic, complemented
7	#347×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#16861231	C₂²	1	1	yes	characteristic
9	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
10	#16861215	—	2	1	no	—
11	C₄×D₅₀₈	—	2	1	no	—
12	#16861154	C₄	1	2	yes	complemented
13	D₂₀₃₂	C₄	1	2	yes	complemented
14	#16861145	C₂²	1	1	yes	characteristic
15	#16861141	C₂²	1	1	yes	characteristic
16	#16861137	—	2	1	no	—
17	#16861131	—	2	1	no	—
18	M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
19	M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
20	#97×C₁₂₇	C₂²	1	1	yes	characteristic
21	#14523248	—	2	1	no	—
22	C₂×D₅₀₈	—	2	1	no	—
23	C₂×C₄×D₁₂₇	—	4	1	no	—
24	Dic₅₀₈	C₂×C₄	1	1	yes	characteristic, complemented
25	#14523237	—	2	1	no	—
26	D₁₀₁₆	—	2	1	no	—
27	D₁₀₁₆	C₂×C₄	1	2	yes	—
28	D₁₀₁₆	C₂×C₄	1	1	yes	characteristic, complemented
29	#14523232	—	2	1	no	—
30	#14523231	—	4	1	no	—
31	C₈×D₁₂₇	—	4	1	no	—
32	C₄⋊Dic₁₂₇	—	2	1	no	—
33	(C₂×D₁₂₇)⋊C₄	—	4	1	no	—
34	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
37	C₁₆×C₁₂₇	—	2	1	no	—
38	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	complemented
39	C₂²×D₁₂₇	—	4	1	no	—
40	Dic₂₅₄	—	2	1	no	—
41	D₅₀₈	—	2	2	no	—
42	D₅₀₈	—	2	1	no	—
43	D₅₀₈	—	4	1	no	—
44	C₂²⋊D₁₂₇	—	4	1	no	—
45	C₂×Dic₁₂₇	—	4	1	no	—
46	C₄×D₁₂₇	—	4	1	no	—
47	C₄×D₁₂₇	—	8	2	no	—
48	C₁₂₇⋊C₈	—	4	1	no	—
49	C₂×C₄×C₁₂₇	—	2	1	no	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₂₇	—	2	1	no	—
52	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₁₂₇	#41	1	1	yes	characteristic
54	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₁₂₇	—	4	1	no	—
56	C₂×D₁₂₇	—	4	2	no	—
57	C₂×D₁₂₇	—	8	1	no	—
58	C₂×D₁₂₇	—	8	1	no	—
59	C₄×C₁₂₇	—	4	1	no	—
60	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
61	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₂₇	—	4	1	no	—
63	Dic₁₂₇	—	8	1	no	—
64	C₂×C₁₂₇	—	2	1	no	—
65	C₂×C₁₂₇	#442	1	1	yes	characteristic
66	D₁₂₇	—	8	1	no	—
67	D₁₂₇	—	8	2	no	—
68	AGL(1,ℤ/16)	—	127	1	no	maximal, Sylow
69	C₁₂₇	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
70	#509	—	127	1	no	—
71	#444	—	127	1	no	—
72	#442	—	127	1	no	—
73	#365	—	127	1	no	—
74	#361	—	127	1	no	—
75	#360	—	127	1	no	—
76	#347	D₁₂₇	1	1	yes	characteristic, complemented
77	#135	—	127	1	no	—
78	C₂×D₈	—	127	1	no	—
79	#131	—	254	1	no	—
80	C₄×D₄	—	254	1	no	—
81	SD₃₂	—	127	2	no	—
82	D₁₆	—	127	2	no	—
83	M₃₂	C₂×D₁₂₇	1	1	yes	characteristic
84	M₃₂	C₂×D₁₂₇	1	1	yes	characteristic
85	#108	—	127	1	no	—
86	#106	—	127	1	no	—
87	#104	—	254	1	no	—
88	#102	—	254	1	no	—
89	#97	C₂×D₁₂₇	1	1	yes	characteristic
90	#41	—	254	1	no	—
91	C₂×D₄	—	254	1	no	—
92	C₂²×C₄	—	508	1	no	—
93	Q₁₆	—	127	1	no	—
94	SD₁₆	—	254	1	no	—
95	D₈	—	127	2	no	—
96	D₈	—	127	1	no	—
97	D₈	—	254	1	no	—
98	M₁₆	—	508	1	no	—
99	M₁₆	—	254	1	no	—
100	C₂×C₈	—	508	1	no	—
101	C₂×C₈	D₅₀₈	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
103	C₄⋊C₄	—	254	1	no	—
104	C₂²⋊C₄	—	508	1	no	—
105	C₄²	D₅₀₈	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	C₄×D₁₂₇	1	2	yes	complemented
108	C₂³	—	508	1	no	—
109	Q₈	—	254	1	no	—
110	D₄	—	508	1	no	—
111	D₄	—	508	1	no	—
112	D₄	—	254	2	no	—
113	D₄	—	254	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	1016	2	no	—
116	C₂×C₄	—	508	1	no	—
117	C₂×C₄	—	508	1	no	—
118	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	508	1	no	—
121	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
122	C₈	#14523248	1	1	yes	characteristic
123	C₂²	—	1016	1	no	—
124	C₂²	—	1016	1	no	—
125	C₂²	—	508	1	no	—
126	C₂²	—	508	2	no	—
127	C₂²	#16861237	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	1016	1	no	—
130	C₄	—	508	1	no	—
131	C₄	C₄×D₅₀₈	1	1	yes	characteristic
132	C₄	#16861237	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	1016	2	no	—
135	C₂	—	1016	1	no	—
136	C₂	#22549989	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549989
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#347×C₁₂₇	C₂
socle	C₂×C₁₂₇	#442
prime radical	#22550314	C₂
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#347	D₁₂₇
hypocenter	C₁₂₇	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #22549999	C₂	semidirect
3. #22549989	C₂	semidirect
4. #22549699	C₂	semidirect
5. #22549690	C₂	semidirect
6. #22549687	C₂	semidirect
7. #347×C₁₂₇	C₂	semidirect
12. #16861154	C₄	semidirect
13. D₂₀₃₂	C₄	semidirect
24. Dic₅₀₈	C₂×C₄	semidirect
28. D₁₀₁₆	C₂×C₄	semidirect
38. C₁₆×C₁₂₇	C₂×C₄	semidirect
69. C₁₂₇	AGL(1,ℤ/16)	semidirect
76. #347	D₁₂₇	semidirect
107. C₁₆	C₄×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550314	2
#22549999	2
#22549989	2
#22549699	2
#22549690	2
#22549687	2
#347×C₁₂₇	2
AGL(1,ℤ/16)	127