Subgroups — #443×D₁₂₇ #30647616

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#131×D₁₂₇	C₂	1	2	yes	maximal, complemented
3	#22549994	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549993	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549992	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549991	C₂	1	2	yes	maximal, complemented
7	#22549990	C₂	1	2	yes	maximal, complemented
8	#108×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₁₂₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₂₇	C₂²	1	2	yes	—
13	#16861232	C₄	1	1	yes	characteristic, complemented
14	#16861231	C₂²	1	1	yes	characteristic
15	#16861230	C₄	1	2	yes	complemented
16	#16861229	C₂²	1	2	yes	—
17	#16861228	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₂₇	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₁₂₇	C₄	1	2	yes	complemented
20	D₈×D₁₂₇	C₄	1	1	yes	characteristic, complemented
21	#16861217	—	2	2	no	—
22	#16861217	C₂²	1	2	yes	complemented
23	#16861216	C₂²	1	2	yes	complemented
24	#16861215	—	2	2	no	—
25	#16861215	C₂²	1	2	yes	complemented
26	M₁₆×D₁₂₇	—	2	2	no	—
27	M₁₆×D₁₂₇	C₂²	1	2	yes	—
28	C₂×C₈×D₁₂₇	—	2	2	no	—
29	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
31	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
32	#16861145	C₂²	1	1	yes	characteristic, complemented
33	#16861144	C₂²	1	2	yes	complemented
34	#16861137	C₂²	1	2	yes	complemented
35	#16861136	C₂²	1	2	yes	complemented
36	#16861135	C₂²	1	2	yes	complemented
37	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
40	#135×C₁₂₇	C₂²	1	1	yes	characteristic
41	#131×C₁₂₇	C₂²	1	2	yes	complemented
42	#108×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
43	#104×C₁₂₇	C₂²	1	2	yes	complemented
44	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
45	#14523249	C₂×C₄	1	2	yes	—
46	#14523248	—	2	2	no	—
47	#14523248	C₂³	1	2	yes	—
48	#14523247	—	2	2	no	—
49	#14523247	C₂×C₄	1	2	yes	—
50	Q₈×D₁₂₇	C₂×C₄	1	2	yes	—
51	D₄×D₁₂₇	—	2	2	no	—
52	D₄×D₁₂₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₂₇	—	2	1	no	—
54	C₂×C₄×D₁₂₇	—	2	2	no	—
55	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
56	Dic₅₀₈	C₂×C₄	1	1	yes	characteristic, complemented
57	#14523238	C₂×C₄	1	2	yes	complemented
58	#14523237	C₂×C₄	1	2	yes	complemented
59	#14523236	C₂×C₄	1	2	yes	complemented
60	#14523235	C₂×C₄	1	2	yes	complemented
61	D₁₀₁₆	C₂×C₄	1	1	yes	characteristic, complemented
62	#14523233	C₂×C₄	1	2	yes	complemented
63	#14523232	—	2	2	no	—
64	#14523232	C₂³	1	2	yes	—
65	#14523231	—	2	2	no	—
66	#14523231	—	2	2	no	—
67	#14523231	—	2	2	no	—
68	C₂×C₁₂₇⋊C₈	—	2	2	no	—
69	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×D₁₂₇	—	2	2	no	—
72	C₈×D₁₂₇	—	2	2	no	—
73	C₈×D₁₂₇	—	2	2	no	—
74	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
75	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
76	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
77	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
78	C₈×D₁₂₇	D₄	1	1	yes	characteristic
79	C₈×D₁₂₇	D₄	1	1	yes	characteristic
80	C₈×D₁₂₇	D₄	1	1	yes	characteristic
81	C₈×D₁₂₇	D₄	1	1	yes	characteristic
82	C₄×Dic₁₂₇	—	2	1	no	—
83	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
84	#41×C₁₂₇	C₂³	1	2	yes	—
85	Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic, complemented
86	SD₁₆×C₁₂₇	C₂×C₄	1	2	yes	complemented
87	D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic, complemented
88	M₁₆×C₁₂₇	—	2	2	no	—
89	M₁₆×C₁₂₇	C₂³	1	2	yes	—
90	C₂×C₈×C₁₂₇	—	2	2	no	—
91	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
93	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
94	C₂²×D₁₂₇	—	2	2	no	—
95	C₂²×D₁₂₇	#41	1	1	yes	characteristic
96	Dic₂₅₄	—	2	2	no	—
97	Dic₂₅₄	C₂²×C₄	1	2	yes	—
98	D₅₀₈	—	2	2	no	—
99	D₅₀₈	C₂²×C₄	1	2	yes	—
100	C₂²⋊D₁₂₇	—	2	2	no	—
101	C₂²⋊D₁₂₇	—	2	2	no	—
102	C₂²⋊D₁₂₇	—	2	2	no	—
103	C₂×Dic₁₂₇	—	2	1	no	—
104	C₂×Dic₁₂₇	—	2	2	no	—
105	C₂×Dic₁₂₇	#41	1	1	yes	characteristic
106	C₄×D₁₂₇	—	2	4	no	—
107	C₄×D₁₂₇	—	2	2	no	—
108	C₄×D₁₂₇	—	2	2	no	—
109	C₄×D₁₂₇	—	2	2	no	—
110	C₄×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₄×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
112	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
113	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
114	C₁₂₇⋊C₈	—	2	2	no	—
115	C₁₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₁₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
117	C₁₂₇⋊C₈	C₂×D₄	1	1	yes	characteristic
118	C₁₂₇⋊C₈	C₂×D₄	1	1	yes	characteristic
119	Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
120	D₄×C₁₂₇	—	2	2	no	—
121	D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
122	C₂×C₄×C₁₂₇	—	2	1	no	—
123	C₂×C₄×C₁₂₇	—	2	2	no	—
124	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
125	C₈×C₁₂₇	—	2	2	no	—
126	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
127	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
128	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
129	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₁₂₇	—	2	2	no	—
131	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
132	C₂×D₁₂₇	—	2	2	no	—
133	C₂×D₁₂₇	—	2	2	no	—
134	C₂×D₁₂₇	—	4	4	no	—
135	C₂×D₁₂₇	C₄×D₄	1	1	yes	characteristic
136	C₂×D₁₂₇	C₄×D₄	1	1	yes	characteristic
137	C₄×C₁₂₇	—	2	2	no	—
138	C₄×C₁₂₇	—	2	2	no	—
139	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
141	Dic₁₂₇	—	2	2	no	—
142	Dic₁₂₇	—	2	2	no	—
143	Dic₁₂₇	C₄×D₄	1	1	yes	characteristic
144	Dic₁₂₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₁₂₇	—	2	1	no	—
146	C₂×C₁₂₇	—	4	2	no	—
147	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
148	D₁₂₇	—	2	1	no	—
149	D₁₂₇	—	4	2	no	—
150	D₁₂₇	#443	1	2	yes	direct product factor
151	C₂×#443	—	127	1	no	maximal, Sylow
152	C₁₂₇	C₂×#443	1	1	yes	characteristic, complemented, Sylow
153	C₂×#135	—	127	1	no	—
154	C₂×#131	—	127	2	no	—
155	#443	—	127	1	no	—
156	#443	—	127	2	no	—
157	#443	—	127	1	no	—
158	#443	—	127	2	no	—
159	#443	—	127	1	no	—
160	#443	D₁₂₇	1	1	yes	characteristic, direct product factor
161	C₂×#108	—	127	1	no	—
162	C₂×#104	—	127	2	no	—
163	C₂×C₄×C₈	—	127	1	no	—
164	C₂×#41	—	127	2	no	—
165	#135	—	127	2	no	—
166	#135	—	127	1	no	—
167	#135	—	127	2	no	—
168	#135	—	127	1	no	—
169	#135	—	127	1	no	—
170	#135	C₂×D₁₂₇	1	1	yes	characteristic
171	C₂×Q₁₆	—	127	1	no	—
172	C₂×SD₁₆	—	127	2	no	—
173	C₂×D₈	—	127	1	no	—
174	#131	—	127	2	no	—
175	#131	—	127	2	no	—
176	#131	—	127	2	no	—
177	#131	—	254	2	no	—
178	#131	—	254	2	no	—
179	#131	C₂×D₁₂₇	1	2	yes	complemented
180	C₂×M₁₆	—	127	2	no	—
181	C₂×M₁₆	—	254	2	no	—
182	C₂²×C₈	—	127	1	no	—
183	C₂²×C₈	—	127	1	no	—
184	C₂²×C₈	—	254	2	no	—
185	C₂×C₄²	—	127	1	no	—
186	#108	—	127	2	no	—
187	#108	—	127	1	no	—
188	#108	C₂×D₁₂₇	1	1	yes	characteristic, complemented
189	#104	—	127	2	no	—
190	#104	—	127	2	no	—
191	#104	—	127	2	no	—
192	#104	C₂×D₁₂₇	1	2	yes	complemented
193	C₄×C₈	—	127	1	no	—
194	C₄×C₈	—	127	1	no	—
195	C₄×C₈	—	127	1	no	—
196	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
197	#41	—	127	2	no	—
198	#41	—	127	2	no	—
199	#41	—	127	2	no	—
200	#41	—	254	2	no	—
201	#41	—	254	2	no	—
202	#41	C₂²×D₁₂₇	1	2	yes	—
203	C₂×Q₈	—	127	2	no	—
204	C₂×D₄	—	127	2	no	—
205	C₂×D₄	—	254	2	no	—
206	C₂²×C₄	—	127	1	no	—
207	C₂²×C₄	—	254	2	no	—
208	C₂²×C₄	—	254	1	no	—
209	Q₁₆	—	127	2	no	—
210	Q₁₆	—	127	1	no	—
211	Q₁₆	C₄×D₁₂₇	1	1	yes	characteristic, complemented
212	SD₁₆	—	127	2	no	—
213	SD₁₆	—	127	2	no	—
214	SD₁₆	—	127	2	no	—
215	SD₁₆	C₄×D₁₂₇	1	2	yes	complemented
216	D₈	—	127	2	no	—
217	D₈	—	127	1	no	—
218	D₈	C₄×D₁₂₇	1	1	yes	characteristic, complemented
219	M₁₆	—	2	2	no	—
220	M₁₆	—	127	2	no	—
221	M₁₆	—	254	2	no	—
222	M₁₆	—	254	2	no	—
223	M₁₆	—	254	2	no	—
224	M₁₆	—	254	2	no	—
225	M₁₆	C₂²×D₁₂₇	1	2	yes	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	127	1	no	—
228	C₂×C₈	—	127	1	no	—
229	C₂×C₈	—	127	1	no	—
230	C₂×C₈	—	127	1	no	—
231	C₂×C₈	—	127	1	no	—
232	C₂×C₈	—	127	1	no	—
233	C₂×C₈	—	127	1	no	—
234	C₂×C₈	—	127	1	no	—
235	C₂×C₈	—	127	1	no	—
236	C₂×C₈	—	127	1	no	—
237	C₂×C₈	—	254	2	no	—
238	C₂×C₈	—	254	2	no	—
239	C₂×C₈	—	254	2	no	—
240	C₂×C₈	—	254	2	no	—
241	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
243	C₄²	—	127	1	no	—
244	C₄²	—	254	1	no	—
245	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
246	C₂³	—	127	1	no	—
247	C₂³	—	254	2	no	—
248	Q₈	—	127	2	no	—
249	Q₈	—	254	2	no	—
250	Q₈	C₂×C₄×D₁₂₇	1	2	yes	—
251	D₄	—	2	2	no	—
252	D₄	—	127	2	no	—
253	D₄	—	254	2	no	—
254	D₄	—	254	2	no	—
255	D₄	—	254	2	no	—
256	D₄	—	254	2	no	—
257	D₄	C₂×C₄×D₁₂₇	1	2	yes	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	127	1	no	—
261	C₂×C₄	—	127	1	no	—
262	C₂×C₄	—	127	1	no	—
263	C₂×C₄	—	127	1	no	—
264	C₂×C₄	—	127	1	no	—
265	C₂×C₄	—	254	2	no	—
266	C₂×C₄	—	254	2	no	—
267	C₂×C₄	—	254	4	no	—
268	C₂×C₄	—	254	2	no	—
269	C₂×C₄	—	254	1	no	—
270	C₂×C₄	—	254	2	no	—
271	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	127	1	no	—
274	C₈	—	127	1	no	—
275	C₈	—	127	1	no	—
276	C₈	—	127	1	no	—
277	C₈	—	254	2	no	—
278	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
279	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
280	C₈	D₄×D₁₂₇	1	1	yes	characteristic
281	C₈	D₄×D₁₂₇	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	127	1	no	—
284	C₂²	—	127	1	no	—
285	C₂²	—	508	4	no	—
286	C₂²	—	254	2	no	—
287	C₂²	—	254	2	no	—
288	C₂²	#41×D₁₂₇	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	127	1	no	—
292	C₄	—	127	1	no	—
293	C₄	—	254	2	no	—
294	C₄	—	254	2	no	—
295	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
296	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	4	2	no	—
299	C₂	—	127	2	no	—
300	C₂	—	508	2	no	—
301	C₂	—	254	1	no	—
302	C₂	C₄×D₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#443×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₄×D₄
prime radical	#135×D₁₂₇	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#443	D₁₂₇
hypocenter	C₁₂₇	C₂×#443
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₁₂₇	C₂	semidirect
3. #22549994	C₂	semidirect
4. #22549993	C₂	semidirect
5. #22549992	C₂	semidirect
6. #22549991	C₂	semidirect
7. #22549990	C₂	semidirect
8. #108×D₁₂₇	C₂	semidirect
9. #104×D₁₂₇	C₂	semidirect
10. C₄×C₈×D₁₂₇	C₂	semidirect
11. #443×C₁₂₇	C₂	semidirect
13. #16861232	C₄	semidirect
15. #16861230	C₄	semidirect
17. #16861228	C₄	semidirect
18. Q₁₆×D₁₂₇	C₄	semidirect
19. SD₁₆×D₁₂₇	C₄	semidirect
20. D₈×D₁₂₇	C₄	semidirect
22. #16861217	C₂²	semidirect
23. #16861216	C₂²	semidirect
25. #16861215	C₂²	semidirect
32. #16861145	C₂²	semidirect
33. #16861144	C₂²	semidirect
34. #16861137	C₂²	semidirect
35. #16861136	C₂²	semidirect
36. #16861135	C₂²	semidirect
37. C₄×C₁₂₇⋊C₈	C₂²	semidirect
38. C₈×Dic₁₂₇	C₂²	semidirect
39. C₈×Dic₁₂₇	C₂²	semidirect
41. #131×C₁₂₇	C₂²	semidirect
42. #108×C₁₂₇	C₂²	semidirect
43. #104×C₁₂₇	C₂²	semidirect
44. C₄×C₈×C₁₂₇	C₂²	semidirect
56. Dic₅₀₈	C₂×C₄	semidirect
57. #14523238	C₂×C₄	semidirect
58. #14523237	C₂×C₄	semidirect
59. #14523236	C₂×C₄	semidirect
60. #14523235	C₂×C₄	semidirect
61. D₁₀₁₆	C₂×C₄	semidirect
62. #14523233	C₂×C₄	semidirect
85. Q₁₆×C₁₂₇	C₂×C₄	semidirect
86. SD₁₆×C₁₂₇	C₂×C₄	semidirect
87. D₈×C₁₂₇	C₂×C₄	semidirect
150. D₁₂₇	#443	direct (N × H)
152. C₁₂₇	C₂×#443	semidirect
160. #443	D₁₂₇	direct (N × H)
179. #131	C₂×D₁₂₇	semidirect
188. #108	C₂×D₁₂₇	semidirect
192. #104	C₂×D₁₂₇	semidirect
196. C₄×C₈	C₂×D₁₂₇	semidirect
211. Q₁₆	C₄×D₁₂₇	semidirect
215. SD₁₆	C₄×D₁₂₇	semidirect
218. D₈	C₄×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#135×D₁₂₇	2
#131×D₁₂₇	2
#22549994	2
#22549993	2
#22549992	2
#22549991	2
#22549990	2
#108×D₁₂₇	2
#104×D₁₂₇	2
C₄×C₈×D₁₂₇	2
#443×C₁₂₇	2
C₂×#443	127

Subgroups — #443×D127 #30647616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #443×D₁₂₇ #30647616