Subgroups — #578×D₁₂₇ #30653492

Order 16256.

1408 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×D₁₂₇	C₂	1	2	yes	maximal, complemented
2	#22550693	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550692	C₂	1	6	yes	maximal, complemented
4	#22550691	C₂	1	2	yes	maximal, complemented
5	#22550690	C₂	1	3	yes	maximal, complemented
6	#22550689	C₂	1	3	yes	maximal, complemented
7	#137×D₁₂₇	C₂	1	6	yes	maximal, complemented
8	#135×D₁₂₇	C₂	1	3	yes	maximal, complemented
9	C₂×Q₁₆×D₁₂₇	C₂	1	3	yes	maximal, complemented
10	#131×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#578×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16861257	—	2	2	no	—
13	#16861257	C₂²	1	6	yes	complemented
14	#16861256	—	2	6	no	—
15	#16861256	C₂²	1	2	yes	complemented
16	#16861256	C₂²	1	6	yes	complemented
17	#41×D₁₂₇	—	2	6	no	—
18	#41×D₁₂₇	C₂²	1	6	yes	—
19	#41×D₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×Q₈×D₁₂₇	—	2	2	no	—
21	C₂×Q₈×D₁₂₇	C₂²	1	6	yes	—
22	#16861242	C₂²	1	6	yes	complemented
23	#16861241	C₂²	1	12	yes	complemented
24	#16861240	C₂²	1	6	yes	complemented
25	#16861239	C₂²	1	6	yes	complemented
26	#16861238	C₂²	1	12	yes	complemented
27	#16861232	C₂²	1	3	yes	complemented
28	#16861231	C₂²	1	3	yes	complemented
29	#16861230	C₂²	1	6	yes	complemented
30	#16861229	C₂²	1	6	yes	complemented
31	#16861228	C₂²	1	3	yes	complemented
32	C₂×Dic₅₀₈	C₂²	1	3	yes	complemented
33	C₂×#14523238	C₂²	1	6	yes	complemented
34	Q₁₆×D₁₂₇	C₂²	1	3	yes	—
35	Q₁₆×D₁₂₇	C₂²	1	6	yes	—
36	Q₁₆×D₁₂₇	C₂²	1	6	yes	complemented
37	Q₁₆×D₁₂₇	C₂²	1	6	yes	complemented
38	SD₁₆×D₁₂₇	C₂²	1	6	yes	—
39	D₈×D₁₂₇	C₂²	1	1	yes	characteristic
40	#16861217	C₂²	1	3	yes	complemented
41	#16861216	C₂²	1	1	yes	characteristic, complemented
42	#16861215	C₂²	1	3	yes	complemented
43	M₁₆×D₁₂₇	C₂²	1	3	yes	—
44	C₂×C₈×D₁₂₇	C₂²	1	3	yes	—
45	2₋¹⁺⁴×C₁₂₇	C₂²	1	2	yes	complemented
46	#137×C₁₂₇	C₂²	1	6	yes	complemented
47	#135×C₁₂₇	C₂²	1	3	yes	complemented
48	C₂×Q₁₆×C₁₂₇	C₂²	1	3	yes	complemented
49	#131×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
50	#14523249	—	2	6	no	—
51	#14523249	D₄	1	1	yes	characteristic
52	#14523249	C₂³	1	6	yes	—
53	#14523248	—	2	6	no	—
54	#14523248	—	2	12	no	—
55	#14523248	—	2	6	no	—
56	#14523248	C₂³	1	6	yes	—
57	#14523248	C₂³	1	3	yes	—
58	#14523247	—	2	12	no	—
59	#14523247	—	2	6	no	—
60	#14523247	—	2	6	no	—
61	#14523247	D₄	1	3	yes	—
62	#14523247	C₂³	1	6	yes	—
63	C₂×Dic₂₅₄	—	2	6	no	—
64	C₂×Dic₂₅₄	—	2	6	no	—
65	C₂×Dic₂₅₄	C₂³	1	6	yes	—
66	Q₈×D₁₂₇	—	2	6	no	—
67	Q₈×D₁₂₇	—	2	6	no	—
68	Q₈×D₁₂₇	—	2	6	no	—
69	Q₈×D₁₂₇	—	2	2	no	—
70	Q₈×D₁₂₇	D₄	1	1	yes	characteristic
71	Q₈×D₁₂₇	C₂³	1	6	yes	—
72	Q₈×D₁₂₇	C₂³	1	12	yes	—
73	D₄×D₁₂₇	—	2	6	no	—
74	D₄×D₁₂₇	D₄	1	3	yes	—
75	D₄×D₁₂₇	C₂³	1	2	yes	—
76	C₂×C₄×D₁₂₇	—	2	6	no	—
77	C₂×C₄×D₁₂₇	—	2	6	no	—
78	C₂×C₄×D₁₂₇	C₂³	1	3	yes	—
79	Dic₅₀₈	C₂³	1	3	yes	—
80	Dic₅₀₈	C₂³	1	6	yes	—
81	#14523238	C₂³	1	6	yes	—
82	#14523238	C₂³	1	12	yes	—
83	#14523237	C₂³	1	6	yes	—
84	#14523236	C₂³	1	6	yes	—
85	#14523235	C₂³	1	6	yes	—
86	D₁₀₁₆	C₂³	1	1	yes	characteristic
87	#14523233	C₂³	1	2	yes	—
88	#14523232	C₂³	1	3	yes	—
89	#14523231	C₂³	1	6	yes	—
90	C₂×C₁₂₇⋊C₈	C₂³	1	3	yes	—
91	C₈×D₁₂₇	C₂³	1	1	yes	characteristic
92	C₈×D₁₂₇	C₂³	1	3	yes	—
93	C₈×D₁₂₇	C₂³	1	3	yes	—
94	C₈×D₁₂₇	C₂³	1	3	yes	—
95	#41×C₁₂₇	—	2	6	no	—
96	#41×C₁₂₇	C₂³	1	6	yes	—
97	#41×C₁₂₇	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₂₇	—	2	2	no	—
99	C₂×Q₈×C₁₂₇	C₂³	1	6	yes	—
100	Q₁₆×C₁₂₇	C₂³	1	3	yes	—
101	Q₁₆×C₁₂₇	C₂³	1	6	yes	—
102	SD₁₆×C₁₂₇	C₂³	1	6	yes	—
103	D₈×C₁₂₇	C₂³	1	1	yes	characteristic
104	M₁₆×C₁₂₇	C₂³	1	3	yes	—
105	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
106	C₂²×D₁₂₇	—	2	2	no	—
107	C₂²×D₁₂₇	C₂×D₄	1	3	yes	—
108	Dic₂₅₄	—	2	6	no	—
109	Dic₂₅₄	—	2	12	no	—
110	Dic₂₅₄	—	2	6	no	—
111	Dic₂₅₄	C₂×D₄	1	3	yes	—
112	Dic₂₅₄	C₂⁴	1	6	yes	—
113	D₅₀₈	—	2	6	no	—
114	D₅₀₈	C₂×D₄	1	3	yes	—
115	D₅₀₈	C₂⁴	1	2	yes	—
116	C₂²⋊D₁₂₇	—	2	6	no	—
117	C₂²⋊D₁₂₇	—	2	6	no	—
118	C₂²⋊D₁₂₇	—	2	2	no	—
119	C₂²⋊D₁₂₇	C₂×D₄	1	6	yes	—
120	C₂×Dic₁₂₇	—	2	6	no	—
121	C₂×Dic₁₂₇	—	2	6	no	—
122	C₂×Dic₁₂₇	C₂×D₄	1	3	yes	—
123	C₄×D₁₂₇	—	2	6	no	—
124	C₄×D₁₂₇	—	2	12	no	—
125	C₄×D₁₂₇	—	2	6	no	—
126	C₄×D₁₂₇	—	2	6	no	—
127	C₄×D₁₂₇	C₂×D₄	1	3	yes	—
128	C₄×D₁₂₇	C₂×D₄	1	3	yes	—
129	C₄×D₁₂₇	C₂⁴	1	1	yes	characteristic
130	C₄×D₁₂₇	C₂⁴	1	3	yes	—
131	C₁₂₇⋊C₈	C₂⁴	1	1	yes	characteristic
132	C₁₂₇⋊C₈	C₂⁴	1	3	yes	—
133	Q₈×C₁₂₇	—	2	6	no	—
134	Q₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
135	Q₈×C₁₂₇	C₂⁴	1	6	yes	—
136	D₄×C₁₂₇	—	2	6	no	—
137	D₄×C₁₂₇	C₂×D₄	1	3	yes	—
138	D₄×C₁₂₇	C₂⁴	1	2	yes	—
139	C₂×C₄×C₁₂₇	—	2	6	no	—
140	C₂×C₄×C₁₂₇	—	2	6	no	—
141	C₂×C₄×C₁₂₇	C₂⁴	1	3	yes	—
142	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₂₇	C₂⁴	1	3	yes	—
144	C₂²×C₁₂₇	—	2	2	no	—
145	C₂²×C₁₂₇	C₂²×D₄	1	3	yes	—
146	C₂×D₁₂₇	—	2	6	no	—
147	C₂×D₁₂₇	—	2	2	no	—
148	C₂×D₁₂₇	—	4	4	no	—
149	C₂×D₁₂₇	C₂²×D₄	1	3	yes	—
150	C₂×D₁₂₇	C₂²×D₄	1	1	yes	characteristic
151	C₄×C₁₂₇	—	2	6	no	—
152	C₄×C₁₂₇	C₂²×D₄	1	3	yes	—
153	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
154	Dic₁₂₇	—	2	6	no	—
155	Dic₁₂₇	C₂²×D₄	1	3	yes	—
156	Dic₁₂₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₁₂₇	—	2	3	no	—
158	C₂×C₁₂₇	—	4	2	no	—
159	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
160	D₁₂₇	—	2	3	no	—
161	D₁₂₇	—	4	2	no	—
162	D₁₂₇	#578	1	2	yes	direct product factor
163	C₂×#578	—	127	1	no	maximal, Sylow
164	C₁₂₇	C₂×#578	1	1	yes	characteristic, complemented, Sylow
165	C₂×2₋¹⁺⁴	—	127	2	no	—
166	#578	—	127	2	no	—
167	#578	—	127	6	no	—
168	#578	—	127	3	no	—
169	#578	—	127	3	no	—
170	#578	—	127	1	no	—
171	#578	D₁₂₇	1	1	yes	characteristic, direct product factor
172	C₂×#137	—	127	6	no	—
173	C₂×#135	—	127	3	no	—
174	C₂²×Q₁₆	—	127	3	no	—
175	C₂×#131	—	127	1	no	—
176	2₋¹⁺⁴	—	127	6	no	—
177	2₋¹⁺⁴	—	127	2	no	—
178	2₋¹⁺⁴	—	127	6	no	—
179	2₋¹⁺⁴	—	254	2	no	—
180	2₋¹⁺⁴	—	254	6	no	—
181	2₋¹⁺⁴	C₂×D₁₂₇	1	2	yes	complemented
182	C₂×#41	—	127	1	no	—
183	C₂×#41	—	127	6	no	—
184	C₂×#41	—	254	6	no	—
185	C₂²×Q₈	—	127	6	no	—
186	C₂²×Q₈	—	254	2	no	—
187	#137	—	127	6	no	—
188	#137	—	127	6	no	—
189	#137	—	127	12	no	—
190	#137	—	127	12	no	—
191	#137	—	127	6	no	—
192	#137	C₂×D₁₂₇	1	6	yes	complemented
193	#135	—	127	6	no	—
194	#135	—	127	3	no	—
195	#135	—	127	6	no	—
196	#135	—	127	3	no	—
197	#135	—	127	3	no	—
198	#135	C₂×D₁₂₇	1	3	yes	complemented
199	C₂×Q₁₆	—	127	6	no	—
200	C₂×Q₁₆	—	127	3	no	—
201	C₂×Q₁₆	—	127	6	no	—
202	C₂×Q₁₆	—	127	6	no	—
203	C₂×Q₁₆	—	127	6	no	—
204	C₂×Q₁₆	—	127	3	no	—
205	C₂×Q₁₆	C₂×D₁₂₇	1	3	yes	complemented
206	C₂×SD₁₆	—	127	6	no	—
207	C₂×D₈	—	127	1	no	—
208	#131	—	127	1	no	—
209	#131	—	127	3	no	—
210	#131	—	127	3	no	—
211	#131	C₂×D₁₂₇	1	1	yes	characteristic, complemented
212	C₂×M₁₆	—	127	3	no	—
213	C₂²×C₈	—	127	3	no	—
214	#41	—	2	6	no	—
215	#41	—	127	1	no	—
216	#41	—	127	3	no	—
217	#41	—	127	3	no	—
218	#41	—	127	6	no	—
219	#41	—	127	6	no	—
220	#41	—	127	6	no	—
221	#41	—	254	6	no	—
222	#41	—	254	12	no	—
223	#41	—	254	6	no	—
224	#41	—	254	6	no	—
225	#41	—	254	12	no	—
226	#41	—	254	6	no	—
227	#41	—	254	6	no	—
228	#41	C₂²×D₁₂₇	1	6	yes	—
229	#41	C₂²×D₁₂₇	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	127	1	no	—
232	C₂×Q₈	—	127	6	no	—
233	C₂×Q₈	—	127	12	no	—
234	C₂×Q₈	—	127	6	no	—
235	C₂×Q₈	—	254	6	no	—
236	C₂×Q₈	—	254	6	no	—
237	C₂×Q₈	—	254	6	no	—
238	C₂×Q₈	—	254	6	no	—
239	C₂×Q₈	—	254	6	no	—
240	C₂×Q₈	—	254	2	no	—
241	C₂×Q₈	C₂²×D₁₂₇	1	6	yes	—
242	C₂×D₄	—	127	3	no	—
243	C₂×D₄	—	127	2	no	—
244	C₂×D₄	—	254	6	no	—
245	C₂²×C₄	—	127	3	no	—
246	C₂²×C₄	—	254	6	no	—
247	C₂²×C₄	—	254	6	no	—
248	Q₁₆	—	127	6	no	—
249	Q₁₆	—	127	12	no	—
250	Q₁₆	—	127	3	no	—
251	Q₁₆	—	127	6	no	—
252	Q₁₆	C₂²×D₁₂₇	1	3	yes	—
253	Q₁₆	C₂²×D₁₂₇	1	6	yes	—
254	SD₁₆	—	127	6	no	—
255	SD₁₆	—	127	6	no	—
256	SD₁₆	—	127	6	no	—
257	SD₁₆	C₂²×D₁₂₇	1	6	yes	—
258	D₈	—	127	2	no	—
259	D₈	—	127	1	no	—
260	D₈	C₂²×D₁₂₇	1	1	yes	characteristic
261	M₁₆	—	127	3	no	—
262	M₁₆	—	127	6	no	—
263	M₁₆	C₂²×D₁₂₇	1	3	yes	—
264	C₂×C₈	—	127	3	no	—
265	C₂×C₈	—	127	1	no	—
266	C₂×C₈	—	127	3	no	—
267	C₂×C₈	—	127	3	no	—
268	C₂×C₈	—	127	3	no	—
269	C₂×C₈	C₂²×D₁₂₇	1	3	yes	—
270	C₂³	—	127	3	no	—
271	C₂³	—	254	2	no	—
272	Q₈	—	2	6	no	—
273	Q₈	—	127	3	no	—
274	Q₈	—	127	6	no	—
275	Q₈	—	254	6	no	—
276	Q₈	—	254	12	no	—
277	Q₈	—	254	6	no	—
278	Q₈	D₄×D₁₂₇	1	1	yes	characteristic
279	Q₈	C₂³×D₁₂₇	1	6	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	127	6	no	—
282	D₄	—	127	3	no	—
283	D₄	—	127	2	no	—
284	D₄	—	254	6	no	—
285	D₄	—	254	2	no	—
286	D₄	—	254	6	no	—
287	D₄	—	254	6	no	—
288	D₄	D₄×D₁₂₇	1	3	yes	—
289	D₄	C₂³×D₁₂₇	1	2	yes	—
290	C₂×C₄	—	2	6	no	—
291	C₂×C₄	—	2	6	no	—
292	C₂×C₄	—	127	3	no	—
293	C₂×C₄	—	127	3	no	—
294	C₂×C₄	—	127	3	no	—
295	C₂×C₄	—	127	1	no	—
296	C₂×C₄	—	127	3	no	—
297	C₂×C₄	—	254	6	no	—
298	C₂×C₄	—	254	6	no	—
299	C₂×C₄	—	254	12	no	—
300	C₂×C₄	—	254	6	no	—
301	C₂×C₄	—	254	6	no	—
302	C₂×C₄	—	254	6	no	—
303	C₂×C₄	C₂³×D₁₂₇	1	3	yes	—
304	C₈	—	127	1	no	—
305	C₈	—	127	3	no	—
306	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
307	C₈	C₂³×D₁₂₇	1	3	yes	—
308	C₂²	—	2	2	no	—
309	C₂²	—	127	1	no	—
310	C₂²	—	127	3	no	—
311	C₂²	—	508	4	no	—
312	C₂²	—	254	6	no	—
313	C₂²	—	254	2	no	—
314	C₂²	C₂×D₄×D₁₂₇	1	3	yes	—
315	C₄	—	2	6	no	—
316	C₄	—	127	1	no	—
317	C₄	—	127	3	no	—
318	C₄	—	254	6	no	—
319	C₄	C₂×D₄×D₁₂₇	1	3	yes	—
320	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
321	C₂	—	2	3	no	—
322	C₂	—	4	2	no	—
323	C₂	—	127	2	no	—
324	C₂	—	508	2	no	—
325	C₂	—	254	3	no	—
326	C₂	C₂²×D₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₂₇
commutator, cosocle	C₄×C₁₂₇	C₂⁵
Frattini	C₄	C₂⁴×D₁₂₇
Fitting	#578×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂³×D₄
hypercenter	#578	D₁₂₇
hypocenter	C₁₂₇	C₂×#578
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×D₁₂₇	C₂	semidirect
2. #22550693	C₂	semidirect
3. #22550692	C₂	semidirect
4. #22550691	C₂	semidirect
5. #22550690	C₂	semidirect
6. #22550689	C₂	semidirect
7. #137×D₁₂₇	C₂	semidirect
8. #135×D₁₂₇	C₂	semidirect
9. C₂×Q₁₆×D₁₂₇	C₂	semidirect
10. #131×D₁₂₇	C₂	semidirect
11. #578×C₁₂₇	C₂	semidirect
13. #16861257	C₂²	semidirect
15. #16861256	C₂²	semidirect
16. #16861256	C₂²	semidirect
22. #16861242	C₂²	semidirect
23. #16861241	C₂²	semidirect
24. #16861240	C₂²	semidirect
25. #16861239	C₂²	semidirect
26. #16861238	C₂²	semidirect
27. #16861232	C₂²	semidirect
28. #16861231	C₂²	semidirect
29. #16861230	C₂²	semidirect
30. #16861229	C₂²	semidirect
31. #16861228	C₂²	semidirect
32. C₂×Dic₅₀₈	C₂²	semidirect
33. C₂×#14523238	C₂²	semidirect
36. Q₁₆×D₁₂₇	C₂²	semidirect
37. Q₁₆×D₁₂₇	C₂²	semidirect
40. #16861217	C₂²	semidirect
41. #16861216	C₂²	semidirect
42. #16861215	C₂²	semidirect
45. 2₋¹⁺⁴×C₁₂₇	C₂²	semidirect
46. #137×C₁₂₇	C₂²	semidirect
47. #135×C₁₂₇	C₂²	semidirect
48. C₂×Q₁₆×C₁₂₇	C₂²	semidirect
49. #131×C₁₂₇	C₂²	semidirect
162. D₁₂₇	#578	direct (N × H)
164. C₁₂₇	C₂×#578	semidirect
171. #578	D₁₂₇	direct (N × H)
181. 2₋¹⁺⁴	C₂×D₁₂₇	semidirect
192. #137	C₂×D₁₂₇	semidirect
198. #135	C₂×D₁₂₇	semidirect
205. C₂×Q₁₆	C₂×D₁₂₇	semidirect
211. #131	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×D₁₂₇	2
#22550693	2
#22550692	2
#22550691	2
#22550690	2
#22550689	2
#137×D₁₂₇	2
#135×D₁₂₇	2
C₂×Q₁₆×D₁₂₇	2
#131×D₁₂₇	2
#578×C₁₂₇	2
C₂×#578	127

Subgroups — #578×D127 #30653492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #578×D₁₂₇ #30653492