Subgroups — #473×D₁₂₇ #30648439

Order 16256.

448 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#137×D₁₂₇	C₂	1	2	yes	maximal
2	C₂×Q₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×D₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#22550152	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550151	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550150	C₂	1	2	yes	maximal, complemented
7	#22550149	C₂	1	2	yes	maximal, complemented
8	#22550148	C₂	1	1	yes	maximal, characteristic, complemented
9	#108×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#101×D₁₂₇	C₂	1	2	yes	maximal, complemented
11	#473×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₂₇	C₂²	1	2	yes	—
14	#16861242	C₂²	1	2	yes	—
15	#16861241	—	2	2	no	—
16	#16861240	C₂²	1	2	yes	—
17	#16861239	C₂²	1	2	yes	—
18	#16861238	—	2	2	no	—
19	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×#14523238	C₂²	1	2	yes	complemented
21	Q₁₆×D₁₂₇	—	2	2	no	—
22	Q₁₆×D₁₂₇	—	2	1	no	—
23	Q₁₆×D₁₂₇	—	2	1	no	—
24	Q₁₆×D₁₂₇	—	2	1	no	—
25	Q₁₆×D₁₂₇	—	2	1	no	—
26	SD₁₆×D₁₂₇	—	2	2	no	—
27	#16861217	—	2	1	no	—
28	#16861217	C₂²	1	1	yes	characteristic
29	#16861216	C₂²	1	1	yes	characteristic
30	#16861215	—	2	1	no	—
31	#16861215	C₂²	1	1	yes	characteristic
32	M₁₆×D₁₂₇	—	2	1	no	—
33	M₁₆×D₁₂₇	C₂²	1	1	yes	characteristic
34	M₁₆×D₁₂₇	C₂²	1	2	yes	—
35	C₂×C₈×D₁₂₇	—	2	1	no	—
36	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
37	#16861145	C₂²	1	1	yes	characteristic, complemented
38	#16861144	C₂²	1	2	yes	complemented
39	#16861128	C₂²	1	2	yes	complemented
40	#16861127	C₂²	1	2	yes	complemented
41	#16861127	C₂²	1	2	yes	complemented
42	#137×C₁₂₇	C₂²	1	2	yes	—
43	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
44	#131×C₁₂₇	C₂²	1	1	yes	characteristic
45	#108×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
46	#101×C₁₂₇	C₂²	1	2	yes	complemented
47	#14523249	D₄	1	1	yes	characteristic
48	#14523248	—	2	1	no	—
49	#14523248	C₂³	1	1	yes	characteristic
50	#14523247	—	2	1	no	—
51	#14523247	D₄	1	1	yes	characteristic
52	C₂×Dic₂₅₄	—	2	2	no	—
53	C₂×Dic₂₅₄	C₂³	1	2	yes	—
54	Q₈×D₁₂₇	—	2	2	no	—
55	Q₈×D₁₂₇	—	2	2	no	—
56	Q₈×D₁₂₇	—	2	2	no	—
57	Q₈×D₁₂₇	—	2	2	no	—
58	Q₈×D₁₂₇	D₄	1	1	yes	characteristic
59	D₄×D₁₂₇	—	2	1	no	—
60	D₄×D₁₂₇	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₂₇	—	2	1	no	—
62	C₂×C₄×D₁₂₇	—	2	2	no	—
63	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
64	Dic₅₀₈	—	2	2	no	—
65	Dic₅₀₈	—	2	1	no	—
66	Dic₅₀₈	—	2	1	no	—
67	#14523238	—	2	2	no	—
68	#14523238	—	2	2	no	—
69	#14523238	—	2	2	no	—
70	#14523238	—	2	2	no	—
71	#14523237	—	2	2	no	—
72	#14523236	—	2	2	no	—
73	#14523235	—	2	2	no	—
74	#14523232	—	2	1	no	—
75	#14523232	C₂³	1	1	yes	characteristic
76	#14523232	C₂³	1	2	yes	—
77	#14523231	—	2	1	no	—
78	#14523231	—	2	2	no	—
79	#14523231	—	2	1	no	—
80	#14523231	—	2	1	no	—
81	C₂×C₁₂₇⋊C₈	—	2	1	no	—
82	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
83	C₈×D₁₂₇	—	2	1	no	—
84	C₈×D₁₂₇	—	2	2	no	—
85	C₈×D₁₂₇	—	2	1	no	—
86	C₈×D₁₂₇	—	2	1	no	—
87	C₈×D₁₂₇	D₄	1	1	yes	characteristic
88	C₈×D₁₂₇	D₄	1	1	yes	characteristic
89	C₈×D₁₂₇	D₄	1	1	yes	characteristic
90	C₈×D₁₂₇	D₄	1	1	yes	characteristic
91	#41×C₁₂₇	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
93	Q₁₆×C₁₂₇	—	2	2	no	—
94	Q₁₆×C₁₂₇	—	2	1	no	—
95	Q₁₆×C₁₂₇	—	2	1	no	—
96	SD₁₆×C₁₂₇	—	2	2	no	—
97	M₁₆×C₁₂₇	—	2	1	no	—
98	M₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
99	M₁₆×C₁₂₇	C₂³	1	2	yes	—
100	C₂×C₈×C₁₂₇	—	2	1	no	—
101	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
102	C₂²×D₁₂₇	—	2	1	no	—
103	C₂²×D₁₂₇	#41	1	1	yes	characteristic
104	Dic₂₅₄	—	2	1	no	—
105	Dic₂₅₄	—	2	2	no	—
106	Dic₂₅₄	—	2	2	no	—
107	Dic₂₅₄	—	4	2	no	—
108	Dic₂₅₄	—	4	2	no	—
109	Dic₂₅₄	C₂×D₄	1	1	yes	characteristic
110	D₅₀₈	—	2	1	no	—
111	D₅₀₈	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊D₁₂₇	—	2	1	no	—
113	C₂²⋊D₁₂₇	—	2	1	no	—
114	C₂²⋊D₁₂₇	—	2	1	no	—
115	C₂×Dic₁₂₇	—	2	1	no	—
116	C₂×Dic₁₂₇	—	2	2	no	—
117	C₂×Dic₁₂₇	#41	1	1	yes	characteristic
118	C₄×D₁₂₇	—	2	1	no	—
119	C₄×D₁₂₇	—	2	1	no	—
120	C₄×D₁₂₇	—	2	1	no	—
121	C₄×D₁₂₇	—	4	2	no	—
122	C₄×D₁₂₇	—	4	2	no	—
123	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
124	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
127	C₁₂₇⋊C₈	—	2	1	no	—
128	C₁₂₇⋊C₈	—	2	2	no	—
129	C₁₂₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₁₂₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	Q₈×C₁₂₇	—	2	2	no	—
132	Q₈×C₁₂₇	—	2	2	no	—
133	Q₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
134	D₄×C₁₂₇	—	2	1	no	—
135	D₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₂₇	—	2	1	no	—
137	C₂×C₄×C₁₂₇	—	2	2	no	—
138	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₂₇	—	2	1	no	—
140	C₈×C₁₂₇	—	2	2	no	—
141	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
142	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₁₂₇	—	2	1	no	—
144	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
145	C₂×D₁₂₇	—	2	2	no	—
146	C₂×D₁₂₇	—	2	1	no	—
147	C₂×D₁₂₇	—	4	2	no	—
148	C₂×D₁₂₇	#121	1	1	yes	characteristic
149	C₂×D₁₂₇	#121	1	1	yes	characteristic
150	C₄×C₁₂₇	—	2	1	no	—
151	C₄×C₁₂₇	—	4	2	no	—
152	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
153	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
154	Dic₁₂₇	—	2	1	no	—
155	Dic₁₂₇	—	4	2	no	—
156	Dic₁₂₇	#121	1	1	yes	characteristic
157	Dic₁₂₇	#121	1	1	yes	characteristic
158	C₂×C₁₂₇	—	2	1	no	—
159	C₂×C₁₂₇	—	4	1	no	—
160	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
161	D₁₂₇	—	2	1	no	—
162	D₁₂₇	—	4	1	no	—
163	D₁₂₇	#473	1	2	yes	direct product factor
164	C₂×#473	—	127	1	no	maximal, Sylow
165	C₁₂₇	C₂×#473	1	1	yes	characteristic, complemented, Sylow
166	C₂×#137	—	127	2	no	—
167	C₂²×Q₁₆	—	127	1	no	—
168	C₂×#131	—	127	1	no	—
169	#473	—	127	2	no	—
170	#473	—	127	1	no	—
171	#473	—	127	1	no	—
172	#473	—	127	1	no	—
173	#473	—	127	2	no	—
174	#473	D₁₂₇	1	1	yes	characteristic, direct product factor
175	C₂×#108	—	127	1	no	—
176	C₂×#101	—	127	2	no	—
177	C₂×#41	—	127	1	no	—
178	C₂²×Q₈	—	127	2	no	—
179	#137	—	127	2	no	—
180	#137	—	127	2	no	—
181	#137	—	127	2	no	—
182	#137	—	254	2	no	—
183	#137	—	254	2	no	—
184	#137	C₂×D₁₂₇	1	2	yes	—
185	C₂×Q₁₆	—	127	2	no	—
186	C₂×Q₁₆	—	127	1	no	—
187	C₂×Q₁₆	—	254	1	no	—
188	C₂×Q₁₆	—	254	1	no	—
189	C₂×Q₁₆	—	254	1	no	—
190	C₂×Q₁₆	—	254	1	no	—
191	C₂×Q₁₆	—	254	2	no	—
192	C₂×Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic, complemented
193	C₂×SD₁₆	—	254	2	no	—
194	#131	—	127	1	no	—
195	#131	—	127	1	no	—
196	#131	—	127	1	no	—
197	#131	—	254	1	no	—
198	#131	—	254	1	no	—
199	#131	C₂×D₁₂₇	1	1	yes	characteristic
200	C₂×M₁₆	—	127	1	no	—
201	C₂×M₁₆	—	127	2	no	—
202	C₂×M₁₆	—	254	1	no	—
203	C₂²×C₈	—	127	1	no	—
204	C₂²×C₈	—	254	1	no	—
205	#108	—	127	2	no	—
206	#108	—	127	1	no	—
207	#108	C₂×D₁₂₇	1	1	yes	characteristic, complemented
208	#101	—	127	2	no	—
209	#101	—	127	2	no	—
210	#101	—	127	2	no	—
211	#101	C₂×D₁₂₇	1	2	yes	complemented
212	#41	—	127	1	no	—
213	#41	—	127	1	no	—
214	#41	—	127	1	no	—
215	#41	—	254	1	no	—
216	#41	—	254	1	no	—
217	#41	C₂²×D₁₂₇	1	1	yes	characteristic
218	C₂×Q₈	—	127	1	no	—
219	C₂×Q₈	—	127	2	no	—
220	C₂×Q₈	—	254	2	no	—
221	C₂×Q₈	—	254	2	no	—
222	C₂×Q₈	—	254	2	no	—
223	C₂×Q₈	—	254	2	no	—
224	C₂×Q₈	—	254	2	no	—
225	C₂×Q₈	C₂²×D₁₂₇	1	2	yes	—
226	C₂×D₄	—	127	1	no	—
227	C₂×D₄	—	254	1	no	—
228	C₂²×C₄	—	127	1	no	—
229	C₂²×C₄	—	254	1	no	—
230	C₂²×C₄	—	254	2	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	1	no	—
233	Q₁₆	—	2	1	no	—
234	Q₁₆	—	254	2	no	—
235	Q₁₆	—	254	2	no	—
236	Q₁₆	—	254	2	no	—
237	Q₁₆	—	254	2	no	—
238	Q₁₆	—	254	2	no	—
239	Q₁₆	—	254	1	no	—
240	Q₁₆	—	254	1	no	—
241	SD₁₆	—	2	2	no	—
242	SD₁₆	—	254	2	no	—
243	SD₁₆	—	254	2	no	—
244	SD₁₆	—	254	2	no	—
245	M₁₆	—	2	1	no	—
246	M₁₆	—	127	1	no	—
247	M₁₆	—	127	2	no	—
248	M₁₆	—	254	1	no	—
249	M₁₆	—	254	1	no	—
250	M₁₆	—	254	2	no	—
251	M₁₆	—	254	1	no	—
252	M₁₆	—	254	1	no	—
253	M₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
254	M₁₆	C₂²×D₁₂₇	1	2	yes	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	127	1	no	—
257	C₂×C₈	—	127	1	no	—
258	C₂×C₈	—	127	1	no	—
259	C₂×C₈	—	127	1	no	—
260	C₂×C₈	—	127	1	no	—
261	C₂×C₈	—	254	1	no	—
262	C₂×C₈	—	254	1	no	—
263	C₂×C₈	—	254	2	no	—
264	C₂×C₈	—	254	1	no	—
265	C₂×C₈	—	254	1	no	—
266	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
267	C₂³	—	127	1	no	—
268	C₂³	—	254	1	no	—
269	Q₈	—	2	2	no	—
270	Q₈	—	2	2	no	—
271	Q₈	—	127	1	no	—
272	Q₈	—	508	2	no	—
273	Q₈	—	508	2	no	—
274	Q₈	—	254	1	no	—
275	Q₈	—	254	2	no	—
276	Q₈	—	254	2	no	—
277	Q₈	D₄×D₁₂₇	1	1	yes	characteristic
278	D₄	—	2	1	no	—
279	D₄	—	127	1	no	—
280	D₄	—	254	1	no	—
281	D₄	—	254	1	no	—
282	D₄	—	254	1	no	—
283	D₄	—	254	1	no	—
284	D₄	D₄×D₁₂₇	1	1	yes	characteristic
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	127	1	no	—
288	C₂×C₄	—	127	1	no	—
289	C₂×C₄	—	127	1	no	—
290	C₂×C₄	—	127	1	no	—
291	C₂×C₄	—	127	1	no	—
292	C₂×C₄	—	508	2	no	—
293	C₂×C₄	—	508	2	no	—
294	C₂×C₄	—	254	1	no	—
295	C₂×C₄	—	254	1	no	—
296	C₂×C₄	—	254	1	no	—
297	C₂×C₄	—	254	2	no	—
298	C₂×C₄	—	254	1	no	—
299	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	2	no	—
302	C₈	—	127	1	no	—
303	C₈	—	127	1	no	—
304	C₈	—	254	1	no	—
305	C₈	—	254	2	no	—
306	C₈	D₄×D₁₂₇	1	1	yes	characteristic
307	C₈	D₄×D₁₂₇	1	1	yes	characteristic
308	C₂²	—	2	1	no	—
309	C₂²	—	127	1	no	—
310	C₂²	—	127	1	no	—
311	C₂²	—	508	2	no	—
312	C₂²	—	254	2	no	—
313	C₂²	—	254	1	no	—
314	C₂²	#41×D₁₂₇	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	127	1	no	—
318	C₄	—	127	1	no	—
319	C₄	—	508	2	no	—
320	C₄	—	254	1	no	—
321	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
322	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	4	1	no	—
325	C₂	—	127	2	no	—
326	C₂	—	508	1	no	—
327	C₂	—	254	1	no	—
328	C₂	#121×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#121×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#473×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×#121
prime radical	#41×D₁₂₇	C₂²
hypercenter	#473	D₁₂₇
hypocenter	C₁₂₇	C₂×#473
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #473

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×D₁₂₇	C₂	semidirect
4. #22550152	C₂	semidirect
5. #22550151	C₂	semidirect
6. #22550150	C₂	semidirect
7. #22550149	C₂	semidirect
8. #22550148	C₂	semidirect
9. #108×D₁₂₇	C₂	semidirect
10. #101×D₁₂₇	C₂	semidirect
11. #473×C₁₂₇	C₂	semidirect
19. C₂×Dic₅₀₈	C₂²	semidirect
20. C₂×#14523238	C₂²	semidirect
37. #16861145	C₂²	semidirect
38. #16861144	C₂²	semidirect
39. #16861128	C₂²	semidirect
40. #16861127	C₂²	semidirect
41. #16861127	C₂²	semidirect
43. C₂×Q₁₆×C₁₂₇	C₂²	semidirect
45. #108×C₁₂₇	C₂²	semidirect
46. #101×C₁₂₇	C₂²	semidirect
163. D₁₂₇	#473	direct (N × H)
165. C₁₂₇	C₂×#473	semidirect
174. #473	D₁₂₇	direct (N × H)
192. C₂×Q₁₆	C₂×D₁₂₇	semidirect
207. #108	C₂×D₁₂₇	semidirect
211. #101	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#137×D₁₂₇	2
C₂×Q₁₆×D₁₂₇	2
#131×D₁₂₇	2
#22550152	2
#22550151	2
#22550150	2
#22550149	2
#22550148	2
#108×D₁₂₇	2
#101×D₁₂₇	2
#473×C₁₂₇	2
C₂×#473	127

Subgroups — #473×D127 #30648439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #473×D₁₂₇ #30648439