Subgroups — #30647701

Order 16256.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550404	C₂	1	1	yes	maximal, characteristic
2	#22550004	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549952	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549949	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₂²⋊D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₂²⋊D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#22549938	C₂	1	1	yes	maximal, characteristic, complemented
11	#22549937	C₂	1	1	yes	maximal, characteristic, complemented
12	#22549931	C₂	1	1	yes	maximal, characteristic, complemented
13	#22549830	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
15	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523231	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
20	#16861204	C₄	1	1	yes	characteristic
21	#16861202	C₄	1	1	yes	characteristic
22	#16861199	C₄	1	1	yes	characteristic
23	#16861195	C₄	1	1	yes	characteristic
24	#16861194	C₄	1	1	yes	characteristic
25	#16861191	C₄	1	1	yes	characteristic
26	#16861187	C₄	1	1	yes	characteristic
27	#16861182	C₄	1	1	yes	characteristic
28	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
29	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₂₇	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₂₇	C₂²	1	1	yes	characteristic
32	#16861166	C₂²	1	1	yes	characteristic
33	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
34	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
36	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
42	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
43	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
46	#16861120	C₂²	1	1	yes	characteristic
47	#16861119	C₂²	1	1	yes	characteristic
48	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
54	#117×C₁₂₇	C₂²	1	1	yes	characteristic
55	#97×C₁₂₇	C₂²	1	1	yes	characteristic
56	#97×C₁₂₇	C₂²	1	1	yes	characteristic
57	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
60	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
67	#14523231	—	2	2	no	—
68	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₈×D₁₂₇	—	2	2	no	—
73	C₈×D₁₂₇	—	2	2	no	—
74	C₈×D₁₂₇	—	2	2	no	—
75	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
76	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
77	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
78	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
79	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
80	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
81	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
82	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
89	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
90	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
96	M₁₆×C₁₂₇	—	2	2	no	—
97	C₂×C₈×C₁₂₇	—	2	2	no	—
98	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
106	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
107	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
108	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
110	Dic₂₅₄	—	2	2	no	—
111	D₅₀₈	—	2	2	no	—
112	C₂²⋊D₁₂₇	—	2	2	no	—
113	C₂²⋊D₁₂₇	—	2	2	no	—
114	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
120	C₄×D₁₂₇	—	2	2	no	—
121	C₄×D₁₂₇	—	2	1	no	—
122	C₄×D₁₂₇	—	2	1	no	—
123	C₄×D₁₂₇	—	2	1	no	—
124	C₄×D₁₂₇	—	2	1	no	—
125	C₄×D₁₂₇	—	2	2	no	—
126	C₁₂₇⋊C₈	—	2	1	no	—
127	C₁₂₇⋊C₈	—	2	1	no	—
128	C₁₂₇⋊C₈	—	2	1	no	—
129	C₁₂₇⋊C₈	—	2	1	no	—
130	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₂₇	—	2	1	no	—
132	C₂×C₄×C₁₂₇	—	2	1	no	—
133	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₂₇	—	2	1	no	—
140	C₈×C₁₂₇	—	2	1	no	—
141	C₈×C₁₂₇	#41	1	2	yes	—
142	C₈×C₁₂₇	#41	1	2	yes	—
143	C₂²×C₁₂₇	—	2	1	no	—
144	C₂²×C₁₂₇	—	2	1	no	—
145	C₂²×C₁₂₇	—	2	1	no	—
146	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₁₂₇	—	2	1	no	—
148	C₂×D₁₂₇	—	2	1	no	—
149	C₂×D₁₂₇	—	2	1	no	—
150	C₂×D₁₂₇	—	2	1	no	—
151	C₂×D₁₂₇	—	2	1	no	—
152	C₂×D₁₂₇	#131	1	2	yes	—
153	C₄×C₁₂₇	—	2	1	no	—
154	C₄×C₁₂₇	—	2	1	no	—
155	C₄×C₁₂₇	—	2	1	no	—
156	C₄×C₁₂₇	—	2	1	no	—
157	C₄×C₁₂₇	—	2	1	no	—
158	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
160	Dic₁₂₇	—	2	1	no	—
161	Dic₁₂₇	—	2	1	no	—
162	Dic₁₂₇	—	2	1	no	—
163	Dic₁₂₇	—	2	1	no	—
164	Dic₁₂₇	—	2	1	no	—
165	Dic₁₂₇	#131	1	2	yes	—
166	C₂×C₁₂₇	—	4	1	no	—
167	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
168	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
169	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
170	D₁₂₇	—	2	2	no	—
171	D₁₂₇	—	4	1	no	—
172	#2966	—	127	1	no	maximal, Sylow
173	C₁₂₇	#2966	1	1	yes	characteristic, complemented, Sylow
174	#529	—	127	1	no	—
175	#446	—	127	1	no	—
176	C₈×Q₈	—	127	1	no	—
177	#436	—	127	1	no	—
178	#435	—	127	1	no	—
179	#435	—	127	1	no	—
180	C₈×D₄	—	127	1	no	—
181	C₈×D₄	—	127	1	no	—
182	C₈×D₄	—	127	1	no	—
183	#433	—	127	1	no	—
184	#433	—	127	1	no	—
185	#432	—	127	1	no	—
186	#405	—	127	1	no	—
187	#405	D₁₂₇	1	1	yes	characteristic, complemented
188	C₂×C₄×C₈	—	127	1	no	—
189	C₂×M₁₆	—	127	1	no	—
190	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
191	C₂²×C₈	—	127	1	no	—
192	C₂²×C₈	—	127	1	no	—
193	C₂²×C₈	—	127	1	no	—
194	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
195	#126	—	127	1	no	—
196	#126	—	127	1	no	—
197	#125	—	127	1	no	—
198	#124	—	127	1	no	—
199	#123	—	127	1	no	—
200	#123	—	127	1	no	—
201	C₂²⋊Q₈	—	127	1	no	—
202	#121	—	127	1	no	—
203	C₄×Q₈	—	127	1	no	—
204	C₄×D₄	—	127	1	no	—
205	C₄×D₄	—	127	1	no	—
206	C₄×D₄	—	127	1	no	—
207	#117	—	127	1	no	—
208	#117	C₂×D₁₂₇	1	1	yes	characteristic
209	C₂×C₄²	—	127	1	no	—
210	C₄⋊C₈	—	127	1	no	—
211	C₄⋊C₈	—	127	1	no	—
212	C₄⋊C₈	—	127	1	no	—
213	C₄⋊C₈	—	127	1	no	—
214	C₄⋊C₈	—	127	1	no	—
215	C₄⋊C₈	—	127	1	no	—
216	C₂²⋊C₈	—	127	1	no	—
217	C₂²⋊C₈	—	127	1	no	—
218	C₂²⋊C₈	—	127	1	no	—
219	C₂²⋊C₈	—	127	1	no	—
220	C₂²⋊C₈	—	127	1	no	—
221	C₂²⋊C₈	—	127	1	no	—
222	#97	—	127	1	no	—
223	#97	—	127	1	no	—
224	#97	C₂×D₁₂₇	1	1	yes	characteristic
225	#97	C₂×D₁₂₇	1	1	yes	characteristic
226	C₄×C₈	—	127	1	no	—
227	C₄×C₈	—	127	1	no	—
228	C₄×C₈	—	127	1	no	—
229	C₄×C₈	—	127	1	no	—
230	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
231	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
232	C₂×Q₈	—	127	1	no	—
233	C₂×D₄	—	127	1	no	—
234	C₂×D₄	—	127	1	no	—
235	C₂×D₄	—	127	1	no	—
236	C₂²×C₄	—	127	1	no	—
237	C₂²×C₄	—	127	1	no	—
238	C₂²×C₄	—	127	1	no	—
239	C₂²×C₄	—	127	1	no	—
240	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
241	M₁₆	—	2	2	no	—
242	M₁₆	—	254	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	127	1	no	—
245	C₂×C₈	—	127	1	no	—
246	C₂×C₈	—	127	1	no	—
247	C₂×C₈	—	127	1	no	—
248	C₂×C₈	—	254	2	no	—
249	C₂×C₈	—	254	2	no	—
250	C₂×C₈	—	254	2	no	—
251	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
255	C₄⋊C₄	—	127	1	no	—
256	C₄⋊C₄	—	127	1	no	—
257	C₄⋊C₄	—	127	1	no	—
258	C₄⋊C₄	—	127	1	no	—
259	C₄⋊C₄	—	127	1	no	—
260	C₄⋊C₄	—	127	1	no	—
261	C₄⋊C₄	—	127	1	no	—
262	C₄⋊C₄	—	127	1	no	—
263	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	127	1	no	—
266	C₂²⋊C₄	—	127	1	no	—
267	C₂²⋊C₄	—	127	1	no	—
268	C₂²⋊C₄	—	127	1	no	—
269	C₂²⋊C₄	—	127	1	no	—
270	C₂²⋊C₄	—	127	1	no	—
271	C₂²⋊C₄	—	127	1	no	—
272	C₂²⋊C₄	—	127	1	no	—
273	C₂²⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
275	C₄²	—	127	1	no	—
276	C₄²	—	127	1	no	—
277	C₄²	—	127	1	no	—
278	C₄²	—	127	1	no	—
279	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
280	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
281	C₂³	—	127	1	no	—
282	C₂³	—	127	1	no	—
283	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
284	Q₈	—	254	2	no	—
285	D₄	—	254	2	no	—
286	D₄	—	254	2	no	—
287	D₄	—	254	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	127	1	no	—
291	C₂×C₄	—	127	1	no	—
292	C₂×C₄	—	127	1	no	—
293	C₂×C₄	—	127	1	no	—
294	C₂×C₄	—	127	1	no	—
295	C₂×C₄	—	127	1	no	—
296	C₂×C₄	—	254	2	no	—
297	C₂×C₄	—	254	2	no	—
298	C₂×C₄	—	254	1	no	—
299	C₂×C₄	—	254	1	no	—
300	C₂×C₄	—	254	1	no	—
301	C₂×C₄	—	254	1	no	—
302	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	254	1	no	—
311	C₈	—	254	1	no	—
312	C₈	—	254	1	no	—
313	C₈	—	254	1	no	—
314	C₈	#14523248	1	2	yes	—
315	C₈	#14523248	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	127	2	no	—
320	C₂²	—	254	1	no	—
321	C₂²	—	254	1	no	—
322	C₂²	—	254	1	no	—
323	C₂²	—	254	1	no	—
324	C₂²	—	254	1	no	—
325	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	127	2	no	—
332	C₄	—	254	1	no	—
333	C₄	—	254	1	no	—
334	C₄	—	254	1	no	—
335	C₄	—	254	1	no	—
336	C₄	—	254	1	no	—
337	C₄	C₂×#14523248	1	1	yes	characteristic
338	C₄	C₂×#14523248	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	508	1	no	—
341	C₂	—	254	2	no	—
342	C₂	C₄×#14523248	1	1	yes	characteristic
343	C₂	#131×D₁₂₇	1	1	yes	characteristic
344	C₂	#131×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#405×C₁₂₇	C₂
prime radical	#16861182	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#405	D₁₂₇
hypocenter	C₁₂₇	#2966
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22550004	C₂	semidirect
3. C₈×Dic₂₅₄	C₂	semidirect
4. #22549952	C₂	semidirect
5. #22549949	C₂	semidirect
6. #22549945	C₂	semidirect
7. C₈×D₅₀₈	C₂	semidirect
8. C₈×C₂²⋊D₁₂₇	C₂	semidirect
9. C₈×C₂²⋊D₁₂₇	C₂	semidirect
10. #22549938	C₂	semidirect
11. #22549937	C₂	semidirect
12. #22549931	C₂	semidirect
13. #22549830	C₂	semidirect
14. C₄×C₈×D₁₂₇	C₂	semidirect
15. #405×C₁₂₇	C₂	semidirect
173. C₁₂₇	#2966	semidirect
187. #405	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550404	2
#22550004	2
C₈×Dic₂₅₄	2
#22549952	2
#22549949	2
#22549945	2
C₈×D₅₀₈	2
C₈×C₂²⋊D₁₂₇	2
#22549938	2
#22549937	2
#22549931	2
#22549830	2
C₄×C₈×D₁₂₇	2
#405×C₁₂₇	2
#2966	127