Subgroups — #560×D₁₂₇ #30652843

Order 16256.

1344 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550617	C₂	1	2	yes	maximal, complemented
2	#22550616	C₂	1	3	yes	maximal, complemented
3	#22550615	C₂	1	3	yes	maximal, complemented
4	#22550614	C₂	1	6	yes	maximal, complemented
5	#22550613	C₂	1	1	yes	maximal, characteristic, complemented
6	#127×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#126×D₁₂₇	C₂	1	2	yes	maximal, complemented
8	#123×D₁₂₇	C₂	1	3	yes	maximal, complemented
9	#121×D₁₂₇	C₂	1	6	yes	maximal, complemented
10	#120×D₁₂₇	C₂	1	3	yes	maximal, complemented
11	#560×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×D₁₂₇	C₂²	1	6	yes	—
14	C₂×D₄×D₁₂₇	C₂²	1	3	yes	—
15	C₂×D₄×D₁₂₇	C₂²	1	3	yes	complemented
16	C₂²×C₄×D₁₂₇	C₂²	1	3	yes	complemented
17	#16861206	C₂²	1	1	yes	characteristic, complemented
18	#16861205	C₂²	1	3	yes	complemented
19	#16861205	C₂²	1	3	yes	complemented
20	#16861204	C₂²	1	6	yes	complemented
21	#16861203	C₂²	1	2	yes	complemented
22	#16861202	C₂²	1	6	yes	complemented
23	#16861194	C₂²	1	6	yes	complemented
24	#16861193	C₂²	1	3	yes	complemented
25	#16861192	C₂²	1	3	yes	complemented
26	#16861191	C₂²	1	6	yes	complemented
27	#16861190	C₂²	1	3	yes	complemented
28	#16861184	C₂²	1	6	yes	complemented
29	#16861183	C₂²	1	6	yes	complemented
30	#16861182	C₂²	1	6	yes	complemented
31	#16861182	C₂²	1	6	yes	complemented
32	#16861181	C₂²	1	6	yes	complemented
33	#16861181	C₂²	1	6	yes	complemented
34	#16861180	C₂²	1	6	yes	complemented
35	#16861179	C₂²	1	6	yes	complemented
36	#16861179	C₂²	1	3	yes	complemented
37	#16861178	C₂²	1	3	yes	complemented
38	#16861177	C₂²	1	3	yes	complemented
39	#16861177	C₂²	1	6	yes	complemented
40	C₄⋊C₄×D₁₂₇	C₂²	1	6	yes	complemented
41	C₂²⋊C₄×D₁₂₇	C₂²	1	6	yes	—
42	C₂²⋊C₄×D₁₂₇	C₂²	1	3	yes	—
43	C₂²⋊C₄×D₁₂₇	C₂²	1	3	yes	complemented
44	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic, complemented
45	#127×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
46	#126×C₁₂₇	C₂²	1	2	yes	complemented
47	#123×C₁₂₇	C₂²	1	3	yes	complemented
48	#121×C₁₂₇	C₂²	1	6	yes	complemented
49	#120×C₁₂₇	C₂²	1	3	yes	complemented
50	C₂³×D₁₂₇	—	4	6	no	—
51	C₂³×D₁₂₇	C₂³	1	3	yes	—
52	C₂³×D₁₂₇	C₂³	1	2	yes	—
53	C₂³×D₁₂₇	C₂³	1	1	yes	characteristic
54	C₂³×D₁₂₇	C₂³	1	1	yes	characteristic
55	C₂³×D₁₂₇	C₂³	1	2	yes	—
56	C₂×D₅₀₈	C₂³	1	6	yes	—
57	C₂×D₅₀₈	C₂³	1	3	yes	—
58	C₂×D₅₀₈	C₂³	1	3	yes	complemented
59	C₂×C₂²⋊D₁₂₇	C₂³	1	3	yes	—
60	C₂×C₂²⋊D₁₂₇	C₂³	1	3	yes	—
61	C₂×C₂²⋊D₁₂₇	C₂³	1	6	yes	—
62	C₂×C₂²⋊D₁₂₇	C₂³	1	6	yes	—
63	C₂×C₂²⋊D₁₂₇	C₂³	1	6	yes	complemented
64	D₄×D₁₂₇	—	4	12	no	—
65	D₄×D₁₂₇	—	4	6	no	—
66	D₄×D₁₂₇	—	4	6	no	—
67	C₂²×Dic₁₂₇	C₂³	1	3	yes	complemented
68	C₂×C₄×D₁₂₇	—	4	6	no	—
69	C₂×C₄×D₁₂₇	C₂³	1	6	yes	—
70	C₂×C₄×D₁₂₇	C₂³	1	3	yes	—
71	C₂×C₄×D₁₂₇	C₂³	1	6	yes	complemented
72	C₄⋊Dic₁₂₇	C₂³	1	6	yes	complemented
73	Dic₁₂₇⋊C₄	C₂³	1	6	yes	complemented
74	Dic₁₂₇⋊C₄	C₂³	1	6	yes	complemented
75	C₂²⋊Dic₁₂₇	C₂³	1	3	yes	—
76	C₂²⋊Dic₁₂₇	C₂³	1	6	yes	—
77	C₂²⋊Dic₁₂₇	C₂³	1	3	yes	complemented
78	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
79	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
80	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
81	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	complemented
82	C₄×Dic₁₂₇	C₂³	1	3	yes	complemented
83	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₁₂₇	C₂³	1	3	yes	—
85	C₂×D₄×C₁₂₇	C₂³	1	6	yes	—
86	C₂×D₄×C₁₂₇	C₂³	1	3	yes	complemented
87	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	complemented
88	C₄⋊C₄×C₁₂₇	C₂³	1	6	yes	complemented
89	C₂²⋊C₄×C₁₂₇	C₂³	1	3	yes	—
90	C₂²⋊C₄×C₁₂₇	C₂³	1	6	yes	—
91	C₂²⋊C₄×C₁₂₇	C₂³	1	3	yes	complemented
92	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic, complemented
93	C₂²×D₁₂₇	—	2	6	no	—
94	C₂²×D₁₂₇	—	2	12	no	—
95	C₂²×D₁₂₇	—	2	12	no	—
96	C₂²×D₁₂₇	—	2	6	no	—
97	C₂²×D₁₂₇	—	4	3	no	—
98	C₂²×D₁₂₇	—	4	6	no	—
99	C₂²×D₁₂₇	—	4	16	no	—
100	C₂²×D₁₂₇	C₂⁴	1	2	yes	—
101	C₂²×D₁₂₇	C₂⁴	1	1	yes	characteristic
102	C₂²×D₁₂₇	C₂⁴	1	3	yes	—
103	C₂²×D₁₂₇	C₂⁴	1	1	yes	characteristic
104	D₅₀₈	—	4	6	no	—
105	D₅₀₈	—	4	3	no	—
106	D₅₀₈	—	4	3	no	—
107	C₂²⋊D₁₂₇	—	4	6	no	—
108	C₂²⋊D₁₂₇	—	4	6	no	—
109	C₂²⋊D₁₂₇	—	4	3	no	—
110	C₂²⋊D₁₂₇	—	4	3	no	—
111	C₂²⋊D₁₂₇	—	4	6	no	—
112	C₂×Dic₁₂₇	—	4	3	no	—
113	C₂×Dic₁₂₇	C₂⁴	1	3	yes	—
114	C₂×Dic₁₂₇	C₂⁴	1	6	yes	—
115	C₄×D₁₂₇	—	2	12	no	—
116	C₄×D₁₂₇	—	2	6	no	—
117	C₄×D₁₂₇	—	4	6	no	—
118	C₂³×C₁₂₇	—	4	3	no	—
119	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
120	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
121	C₂³×C₁₂₇	C₂⁴	1	3	yes	—
122	D₄×C₁₂₇	—	4	6	no	—
123	D₄×C₁₂₇	—	4	3	no	—
124	D₄×C₁₂₇	—	4	3	no	—
125	C₂×C₄×C₁₂₇	—	4	3	no	—
126	C₂×C₄×C₁₂₇	C₂⁴	1	3	yes	—
127	C₂×C₄×C₁₂₇	C₂⁴	1	6	yes	—
128	C₂²×C₁₂₇	—	2	3	no	—
129	C₂²×C₁₂₇	—	2	6	no	—
130	C₂²×C₁₂₇	—	2	6	no	—
131	C₂²×C₁₂₇	—	2	3	no	—
132	C₂²×C₁₂₇	—	4	4	no	—
133	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
134	C₂×D₁₂₇	—	2	3	no	—
135	C₂×D₁₂₇	—	2	6	no	—
136	C₂×D₁₂₇	—	2	6	no	—
137	C₂×D₁₂₇	—	2	3	no	—
138	C₂×D₁₂₇	—	4	4	no	—
139	C₂×D₁₂₇	—	4	8	no	—
140	C₂×D₁₂₇	—	4	12	no	—
141	C₂×D₁₂₇	—	4	8	no	—
142	C₂×D₁₂₇	—	4	4	no	—
143	C₂×D₁₂₇	H(5,𝔽₂)	1	6	yes	—
144	C₄×C₁₂₇	—	2	6	no	—
145	C₄×C₁₂₇	—	2	3	no	—
146	Dic₁₂₇	—	2	6	no	—
147	Dic₁₂₇	—	2	3	no	—
148	C₂×C₁₂₇	—	4	2	no	—
149	C₂×C₁₂₇	—	4	1	no	—
150	C₂×C₁₂₇	—	4	3	no	—
151	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	3	yes	—
152	D₁₂₇	—	4	3	no	—
153	D₁₂₇	—	4	2	no	—
154	D₁₂₇	—	4	1	no	—
155	D₁₂₇	#560	1	4	yes	direct product factor
156	C₂×#560	—	127	1	no	maximal, Sylow
157	C₁₂₇	C₂×#560	1	1	yes	characteristic, complemented, Sylow
158	#560	—	127	1	no	—
159	#560	—	127	2	no	—
160	#560	—	127	3	no	—
161	#560	—	127	6	no	—
162	#560	—	127	3	no	—
163	#560	D₁₂₇	1	1	yes	characteristic, direct product factor
164	C₂×#127	—	127	1	no	—
165	C₂×#126	—	127	2	no	—
166	C₂×#123	—	127	3	no	—
167	C₂×#121	—	127	6	no	—
168	C₂×#120	—	127	3	no	—
169	C₂⁵	—	127	1	no	—
170	C₂²×D₄	—	127	6	no	—
171	C₂²×D₄	—	127	3	no	—
172	C₂²×D₄	—	127	3	no	—
173	C₂³×C₄	—	127	3	no	—
174	#127	—	127	3	no	—
175	#127	—	127	3	no	—
176	#127	—	127	1	no	—
177	#127	C₂×D₁₂₇	1	1	yes	characteristic, complemented
178	#126	—	127	6	no	—
179	#126	—	127	6	no	—
180	#126	—	127	2	no	—
181	#126	C₂×D₁₂₇	1	2	yes	complemented
182	#123	—	127	3	no	—
183	#123	—	127	3	no	—
184	#123	—	127	6	no	—
185	#123	—	127	6	no	—
186	#123	—	127	3	no	—
187	#123	C₂×D₁₂₇	1	3	yes	complemented
188	#121	—	127	6	no	—
189	#121	—	127	6	no	—
190	#121	—	127	6	no	—
191	#121	—	127	6	no	—
192	#121	—	127	6	no	—
193	#121	—	127	6	no	—
194	#121	—	127	6	no	—
195	#121	C₂×D₁₂₇	1	6	yes	complemented
196	#120	—	127	3	no	—
197	#120	—	127	6	no	—
198	#120	—	127	3	no	—
199	#120	—	127	3	no	—
200	#120	—	127	6	no	—
201	#120	C₂×D₁₂₇	1	3	yes	complemented
202	C₂×C₄⋊C₄	—	127	6	no	—
203	C₂×C₂²⋊C₄	—	127	6	no	—
204	C₂×C₂²⋊C₄	—	127	3	no	—
205	C₂×C₂²⋊C₄	—	127	3	no	—
206	C₂×C₄²	—	127	1	no	—
207	C₂⁴	—	127	1	no	—
208	C₂⁴	—	127	2	no	—
209	C₂⁴	—	127	2	no	—
210	C₂⁴	—	127	1	no	—
211	C₂⁴	—	127	3	no	—
212	C₂⁴	—	508	6	no	—
213	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
214	C₂×D₄	—	127	6	no	—
215	C₂×D₄	—	127	6	no	—
216	C₂×D₄	—	127	3	no	—
217	C₂×D₄	—	127	3	no	—
218	C₂×D₄	—	127	6	no	—
219	C₂×D₄	—	127	6	no	—
220	C₂×D₄	—	127	3	no	—
221	C₂×D₄	—	127	3	no	—
222	C₂×D₄	—	508	12	no	—
223	C₂×D₄	—	508	6	no	—
224	C₂×D₄	—	508	6	no	—
225	C₂×D₄	C₂²×D₁₂₇	1	6	yes	—
226	C₂×D₄	C₂²×D₁₂₇	1	3	yes	—
227	C₂×D₄	C₂²×D₁₂₇	1	3	yes	complemented
228	C₂²×C₄	—	127	3	no	—
229	C₂²×C₄	—	127	6	no	—
230	C₂²×C₄	—	127	3	no	—
231	C₂²×C₄	—	127	6	no	—
232	C₂²×C₄	—	508	6	no	—
233	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	complemented
234	C₄⋊C₄	—	127	6	no	—
235	C₄⋊C₄	—	127	6	no	—
236	C₄⋊C₄	—	127	6	no	—
237	C₄⋊C₄	C₂²×D₁₂₇	1	6	yes	complemented
238	C₂²⋊C₄	—	127	6	no	—
239	C₂²⋊C₄	—	127	3	no	—
240	C₂²⋊C₄	—	127	3	no	—
241	C₂²⋊C₄	—	127	6	no	—
242	C₂²⋊C₄	—	127	6	no	—
243	C₂²⋊C₄	—	127	6	no	—
244	C₂²⋊C₄	—	127	6	no	—
245	C₂²⋊C₄	C₂²×D₁₂₇	1	6	yes	—
246	C₂²⋊C₄	C₂²×D₁₂₇	1	3	yes	—
247	C₂²⋊C₄	C₂²×D₁₂₇	1	3	yes	complemented
248	C₄²	—	127	3	no	—
249	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic, complemented
250	C₂³	—	4	3	no	—
251	C₂³	—	127	1	no	—
252	C₂³	—	127	2	no	—
253	C₂³	—	127	1	no	—
254	C₂³	—	127	3	no	—
255	C₂³	—	508	6	no	—
256	C₂³	—	508	3	no	—
257	C₂³	—	508	16	no	—
258	C₂³	—	254	6	no	—
259	C₂³	—	254	12	no	—
260	C₂³	—	254	12	no	—
261	C₂³	—	254	6	no	—
262	C₂³	C₂³×D₁₂₇	1	2	yes	—
263	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
264	C₂³	C₂³×D₁₂₇	1	3	yes	—
265	D₄	—	4	6	no	—
266	D₄	—	4	3	no	—
267	D₄	—	4	3	no	—
268	D₄	—	508	6	no	—
269	D₄	—	508	3	no	—
270	D₄	—	508	3	no	—
271	D₄	—	508	6	no	—
272	D₄	—	508	6	no	—
273	D₄	—	508	3	no	—
274	D₄	—	508	6	no	—
275	D₄	—	508	3	no	—
276	C₂×C₄	—	4	3	no	—
277	C₂×C₄	—	127	3	no	—
278	C₂×C₄	—	127	6	no	—
279	C₂×C₄	—	508	6	no	—
280	C₂×C₄	—	508	3	no	—
281	C₂×C₄	—	254	12	no	—
282	C₂×C₄	—	254	6	no	—
283	C₂×C₄	C₂³×D₁₂₇	1	3	yes	—
284	C₂×C₄	C₂³×D₁₂₇	1	6	yes	—
285	C₂²	—	2	3	no	—
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	3	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	127	6	no	—
291	C₂²	—	508	8	no	—
292	C₂²	—	508	4	no	—
293	C₂²	—	508	12	no	—
294	C₂²	—	508	8	no	—
295	C₂²	—	508	4	no	—
296	C₂²	—	254	3	no	—
297	C₂²	—	254	6	no	—
298	C₂²	—	254	6	no	—
299	C₂²	—	254	3	no	—
300	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
301	C₄	—	2	6	no	—
302	C₄	—	2	3	no	—
303	C₄	—	254	6	no	—
304	C₄	—	254	3	no	—
305	C₂	—	4	2	no	—
306	C₂	—	4	1	no	—
307	C₂	—	4	3	no	—
308	C₂	—	127	4	no	—
309	C₂	—	508	2	no	—
310	C₂	—	508	1	no	—
311	C₂	—	508	3	no	—
312	C₂	H(5,𝔽₂)×D₁₂₇	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₂₇
commutator, socle, cosocle	C₂²×C₁₂₇	C₂⁵
Fitting	#560×C₁₂₇	C₂
hypercenter	#560	D₁₂₇
hypocenter	C₁₂₇	C₂×#560
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #560

Complements

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

N	H ≅ G/N	type
1. #22550617	C₂	semidirect
2. #22550616	C₂	semidirect
3. #22550615	C₂	semidirect
4. #22550614	C₂	semidirect
5. #22550613	C₂	semidirect
6. #127×D₁₂₇	C₂	semidirect
7. #126×D₁₂₇	C₂	semidirect
8. #123×D₁₂₇	C₂	semidirect
9. #121×D₁₂₇	C₂	semidirect
10. #120×D₁₂₇	C₂	semidirect
11. #560×C₁₂₇	C₂	semidirect
15. C₂×D₄×D₁₂₇	C₂²	semidirect
16. C₂²×C₄×D₁₂₇	C₂²	semidirect
17. #16861206	C₂²	semidirect
18. #16861205	C₂²	semidirect
19. #16861205	C₂²	semidirect
20. #16861204	C₂²	semidirect
21. #16861203	C₂²	semidirect
22. #16861202	C₂²	semidirect
23. #16861194	C₂²	semidirect
24. #16861193	C₂²	semidirect
25. #16861192	C₂²	semidirect
26. #16861191	C₂²	semidirect
27. #16861190	C₂²	semidirect
28. #16861184	C₂²	semidirect
29. #16861183	C₂²	semidirect
30. #16861182	C₂²	semidirect
31. #16861182	C₂²	semidirect
32. #16861181	C₂²	semidirect
33. #16861181	C₂²	semidirect
34. #16861180	C₂²	semidirect
35. #16861179	C₂²	semidirect
36. #16861179	C₂²	semidirect
37. #16861178	C₂²	semidirect
38. #16861177	C₂²	semidirect
39. #16861177	C₂²	semidirect
40. C₄⋊C₄×D₁₂₇	C₂²	semidirect
43. C₂²⋊C₄×D₁₂₇	C₂²	semidirect
44. C₄²×D₁₂₇	C₂²	semidirect
45. #127×C₁₂₇	C₂²	semidirect
46. #126×C₁₂₇	C₂²	semidirect
47. #123×C₁₂₇	C₂²	semidirect
48. #121×C₁₂₇	C₂²	semidirect
49. #120×C₁₂₇	C₂²	semidirect
58. C₂×D₅₀₈	C₂³	semidirect
63. C₂×C₂²⋊D₁₂₇	C₂³	semidirect
67. C₂²×Dic₁₂₇	C₂³	semidirect
71. C₂×C₄×D₁₂₇	C₂³	semidirect
72. C₄⋊Dic₁₂₇	C₂³	semidirect
73. Dic₁₂₇⋊C₄	C₂³	semidirect
74. Dic₁₂₇⋊C₄	C₂³	semidirect
77. C₂²⋊Dic₁₂₇	C₂³	semidirect
81. (C₂×D₁₂₇)⋊C₄	C₂³	semidirect
82. C₄×Dic₁₂₇	C₂³	semidirect
86. C₂×D₄×C₁₂₇	C₂³	semidirect
87. C₂²×C₄×C₁₂₇	C₂³	semidirect
88. C₄⋊C₄×C₁₂₇	C₂³	semidirect
91. C₂²⋊C₄×C₁₂₇	C₂³	semidirect
92. C₄²×C₁₂₇	C₂³	semidirect
155. D₁₂₇	#560	direct (N × H)
157. C₁₂₇	C₂×#560	semidirect
163. #560	D₁₂₇	direct (N × H)
177. #127	C₂×D₁₂₇	semidirect
181. #126	C₂×D₁₂₇	semidirect
187. #123	C₂×D₁₂₇	semidirect
195. #121	C₂×D₁₂₇	semidirect
201. #120	C₂×D₁₂₇	semidirect
227. C₂×D₄	C₂²×D₁₂₇	semidirect
233. C₂²×C₄	C₂²×D₁₂₇	semidirect
237. C₄⋊C₄	C₂²×D₁₂₇	semidirect
247. C₂²⋊C₄	C₂²×D₁₂₇	semidirect
249. C₄²	C₂²×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550617	2
#22550616	2
#22550615	2
#22550614	2
#22550613	2
#127×D₁₂₇	2
#126×D₁₂₇	2
#123×D₁₂₇	2
#121×D₁₂₇	2
#120×D₁₂₇	2
#560×C₁₂₇	2
C₂×#560	127

Subgroups — #560×D127 #30652843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #560×D₁₂₇ #30652843