Subgroups — C₁₅₁×#11886 #50316811

Order 38656.

470 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#1808×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1347×C₁₅₁	C₂	1	4	yes	maximal, complemented
5	C₂²×D₈×C₁₅₁	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
7	C₂×#121×C₁₅₁	—	2	1	no	—
8	C₂²×C₁₆×C₁₅₁	—	2	1	no	—
9	#466×C₁₅₁	—	2	1	no	—
10	#466×C₁₅₁	—	2	1	no	—
11	#466×C₁₅₁	C₄	1	2	yes	complemented
12	#466×C₁₅₁	C₂²	1	2	yes	—
13	C₂×#107×C₁₅₁	—	2	1	no	—
14	C₂×#107×C₁₅₁	C₄	1	1	yes	characteristic, complemented
15	C₂×#102×C₁₅₁	—	2	1	no	—
16	#380×C₁₅₁	—	2	1	no	—
17	C₂²⋊C₁₆×C₁₅₁	—	2	1	no	—
18	C₂²⋊C₁₆×C₁₅₁	C₂²	1	2	yes	complemented
19	#340×C₁₅₁	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₅₁	—	2	1	no	—
21	C₂²×D₄×C₁₅₁	—	4	1	no	—
22	C₂³×C₄×C₁₅₁	D₄	1	1	yes	characteristic
23	C₂×D₈×C₁₅₁	—	2	1	no	—
24	C₂×D₈×C₁₅₁	—	4	2	no	—
25	C₂×D₈×C₁₅₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₅₁	—	4	2	no	—
27	C₂²×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
33	#121×C₁₅₁	—	2	2	no	—
34	#121×C₁₅₁	—	2	2	no	—
35	#121×C₁₅₁	—	4	1	no	—
36	#121×C₁₅₁	—	4	1	no	—
37	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₁₅₁	—	4	1	no	—
40	C₂×C₂²⋊C₄×C₁₅₁	—	4	1	no	—
41	C₂×C₁₆×C₁₅₁	—	2	2	no	—
42	C₂×C₁₆×C₁₅₁	—	4	2	no	—
43	#107×C₁₅₁	—	2	1	no	—
44	#107×C₁₅₁	—	4	2	no	—
45	#107×C₁₅₁	C₂×C₄	1	2	yes	—
46	#102×C₁₅₁	—	2	2	no	—
47	#102×C₁₅₁	—	2	2	no	—
48	#95×C₁₅₁	—	4	1	no	—
49	C₂⁴×C₁₅₁	—	4	1	no	—
50	C₂⁴×C₁₅₁	D₈	1	1	yes	characteristic
51	C₂×D₄×C₁₅₁	—	2	2	no	—
52	C₂×D₄×C₁₅₁	—	4	2	no	—
53	C₂×D₄×C₁₅₁	—	4	2	no	—
54	C₂×D₄×C₁₅₁	—	4	2	no	—
55	C₂×D₄×C₁₅₁	—	4	2	no	—
56	C₂×D₄×C₁₅₁	—	4	1	no	—
57	C₂×D₄×C₁₅₁	—	8	2	no	—
58	C₂×D₄×C₁₅₁	—	8	2	no	—
59	C₂²×C₄×C₁₅₁	—	4	1	no	—
60	C₂²×C₄×C₁₅₁	—	4	1	no	—
61	C₂²×C₄×C₁₅₁	—	4	1	no	—
62	C₂²×C₄×C₁₅₁	—	4	2	no	—
63	C₂²×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₅₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₅₁	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₁₅₁	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
69	D₈×C₁₅₁	—	4	2	no	—
70	D₈×C₁₅₁	—	4	2	no	—
71	C₂×C₈×C₁₅₁	—	2	2	no	—
72	C₂×C₈×C₁₅₁	—	2	2	no	—
73	C₂×C₈×C₁₅₁	—	2	2	no	—
74	C₂×C₈×C₁₅₁	—	4	1	no	—
75	C₂×C₈×C₁₅₁	—	4	1	no	—
76	C₂×C₈×C₁₅₁	—	4	1	no	—
77	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₅₁	—	2	2	no	—
82	C₄⋊C₄×C₁₅₁	—	4	2	no	—
83	C₄⋊C₄×C₁₅₁	—	4	2	no	—
84	C₄⋊C₄×C₁₅₁	—	4	1	no	—
85	C₂²⋊C₄×C₁₅₁	—	4	2	no	—
86	C₂²⋊C₄×C₁₅₁	—	4	2	no	—
87	C₂²⋊C₄×C₁₅₁	—	8	2	no	—
88	C₁₆×C₁₅₁	—	4	2	no	—
89	C₂³×C₁₅₁	—	4	1	no	—
90	C₂³×C₁₅₁	—	4	1	no	—
91	C₂³×C₁₅₁	—	4	2	no	—
92	C₂³×C₁₅₁	—	8	2	no	—
93	C₂³×C₁₅₁	—	8	2	no	—
94	C₂³×C₁₅₁	—	8	2	no	—
95	C₂³×C₁₅₁	#102	1	1	yes	characteristic
96	C₂³×C₁₅₁	#102	1	1	yes	characteristic
97	C₂³×C₁₅₁	D₁₆	1	2	yes	—
98	C₂³×C₁₅₁	SD₃₂	1	2	yes	—
99	C₂³×C₁₅₁	C₂×D₈	1	1	yes	characteristic
100	D₄×C₁₅₁	—	4	2	no	—
101	D₄×C₁₅₁	—	4	2	no	—
102	D₄×C₁₅₁	—	8	2	no	—
103	D₄×C₁₅₁	—	8	4	no	—
104	D₄×C₁₅₁	—	8	2	no	—
105	D₄×C₁₅₁	—	8	2	no	—
106	C₂×C₄×C₁₅₁	—	2	1	no	—
107	C₂×C₄×C₁₅₁	—	2	1	no	—
108	C₂×C₄×C₁₅₁	—	2	1	no	—
109	C₂×C₄×C₁₅₁	—	2	1	no	—
110	C₂×C₄×C₁₅₁	—	2	1	no	—
111	C₂×C₄×C₁₅₁	—	2	1	no	—
112	C₂×C₄×C₁₅₁	—	4	1	no	—
113	C₂×C₄×C₁₅₁	—	4	1	no	—
114	C₂×C₄×C₁₅₁	—	4	1	no	—
115	C₂×C₄×C₁₅₁	—	4	2	no	—
116	C₂×C₄×C₁₅₁	—	8	2	no	—
117	C₂×C₄×C₁₅₁	—	8	2	no	—
118	C₂×C₄×C₁₅₁	—	8	2	no	—
119	C₂×C₄×C₁₅₁	—	8	2	no	—
120	C₂×C₄×C₁₅₁	#102	1	1	yes	characteristic
121	C₂×C₄×C₁₅₁	#102	1	1	yes	characteristic
122	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₈×C₁₅₁	—	2	1	no	—
125	C₈×C₁₅₁	—	2	1	no	—
126	C₈×C₁₅₁	—	2	1	no	—
127	C₈×C₁₅₁	#99	1	2	yes	—
128	C₂²×C₁₅₁	—	2	1	no	—
129	C₂²×C₁₅₁	—	2	1	no	—
130	C₂²×C₁₅₁	—	2	1	no	—
131	C₂²×C₁₅₁	—	2	1	no	—
132	C₂²×C₁₅₁	—	2	1	no	—
133	C₂²×C₁₅₁	—	2	1	no	—
134	C₂²×C₁₅₁	—	4	2	no	—
135	C₂²×C₁₅₁	—	4	2	no	—
136	C₂²×C₁₅₁	—	8	2	no	—
137	C₂²×C₁₅₁	—	8	2	no	—
138	C₂²×C₁₅₁	—	8	2	no	—
139	C₂²×C₁₅₁	—	16	4	no	—
140	C₂²×C₁₅₁	#357	1	2	yes	—
141	C₂²×C₁₅₁	#357	1	2	yes	—
142	C₂²×C₁₅₁	C₂×#102	1	1	yes	characteristic
143	C₂²×C₁₅₁	C₂×D₁₆	1	1	yes	characteristic
144	C₂²×C₁₅₁	C₂×SD₃₂	1	1	yes	characteristic
145	C₄×C₁₅₁	—	2	1	no	—
146	C₄×C₁₅₁	—	2	1	no	—
147	C₄×C₁₅₁	—	2	1	no	—
148	C₄×C₁₅₁	—	8	2	no	—
149	C₄×C₁₅₁	—	16	2	no	—
150	C₄×C₁₅₁	C₂×#99	1	1	yes	characteristic
151	C₄×C₁₅₁	#421	1	1	yes	characteristic
152	C₂×C₁₅₁	—	2	1	no	—
153	C₂×C₁₅₁	—	2	2	no	—
154	C₂×C₁₅₁	—	2	2	no	—
155	C₂×C₁₅₁	—	2	1	no	—
156	C₂×C₁₅₁	—	16	2	no	—
157	C₂×C₁₅₁	#1519	1	1	yes	characteristic
158	C₂×C₁₅₁	C₂×#357	1	1	yes	characteristic
159	C₂×C₁₅₁	#2145	1	1	yes	characteristic
160	#11886	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₁₅₁	#11886	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#466	C₂×C₁₅₁	1	1	yes	characteristic
163	C₂×C₂²⋊C₁₆	C₂×C₁₅₁	1	1	yes	characteristic, complemented
164	#1808	C₂×C₁₅₁	1	1	yes	characteristic, complemented
165	#1347	C₂×C₁₅₁	1	4	yes	complemented
166	C₂²×D₈	C₄×C₁₅₁	1	1	yes	characteristic, complemented
167	C₂³×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
168	C₂×#121	—	2	1	no	—
169	C₂²×C₁₆	—	2	1	no	—
170	#466	—	2	1	no	—
171	#466	—	2	1	no	—
172	#466	C₄×C₁₅₁	1	2	yes	complemented
173	#466	C₂²×C₁₅₁	1	2	yes	—
174	C₂×#107	—	2	1	no	—
175	C₂×#107	C₄×C₁₅₁	1	1	yes	characteristic, complemented
176	C₂×#102	—	2	1	no	—
177	#380	—	2	1	no	—
178	C₂²⋊C₁₆	—	2	1	no	—
179	C₂²⋊C₁₆	C₂²×C₁₅₁	1	2	yes	complemented
180	#340	C₂²×C₁₅₁	1	2	yes	complemented
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	—	4	1	no	—
183	C₂³×C₄	D₄×C₁₅₁	1	1	yes	characteristic
184	C₂×D₈	—	2	1	no	—
185	C₂×D₈	—	4	2	no	—
186	C₂×D₈	C₂×C₄×C₁₅₁	1	2	yes	—
187	C₂²×C₈	—	4	2	no	—
188	C₂²×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
189	C₂²×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
190	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
191	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
192	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
194	#121	—	2	2	no	—
195	#121	—	2	2	no	—
196	#121	—	4	1	no	—
197	#121	—	4	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	4	1	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	4	2	no	—
204	#107	—	2	1	no	—
205	#107	—	4	2	no	—
206	#107	C₂×C₄×C₁₅₁	1	2	yes	—
207	#102	—	2	2	no	—
208	#102	—	2	2	no	—
209	#95	—	4	1	no	—
210	C₂⁴	—	4	1	no	—
211	C₂⁴	D₈×C₁₅₁	1	1	yes	characteristic
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	8	2	no	—
219	C₂×D₄	—	8	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
226	C₂²×C₄	D₈×C₁₅₁	1	1	yes	characteristic
227	C₂²×C₄	SD₁₆×C₁₅₁	1	1	yes	characteristic
228	C₂²×C₄	SD₁₆×C₁₅₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
230	D₈	—	4	2	no	—
231	D₈	—	4	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	8	2	no	—
249	C₁₆	—	4	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	8	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	2	no	—
256	C₂³	#102×C₁₅₁	1	1	yes	characteristic
257	C₂³	#102×C₁₅₁	1	1	yes	characteristic
258	C₂³	D₁₆×C₁₅₁	1	2	yes	—
259	C₂³	SD₃₂×C₁₅₁	1	2	yes	—
260	C₂³	C₂×D₈×C₁₅₁	1	1	yes	characteristic
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	4	no	—
265	D₄	—	8	2	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	#102×C₁₅₁	1	1	yes	characteristic
282	C₂×C₄	#102×C₁₅₁	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₁₅₁	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	#99×C₁₅₁	1	2	yes	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	16	4	no	—
301	C₂²	#357×C₁₅₁	1	2	yes	—
302	C₂²	#357×C₁₅₁	1	2	yes	—
303	C₂²	C₂×#102×C₁₅₁	1	1	yes	characteristic
304	C₂²	C₂×D₁₆×C₁₅₁	1	1	yes	characteristic
305	C₂²	C₂×SD₃₂×C₁₅₁	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	16	2	no	—
311	C₄	C₂×#99×C₁₅₁	1	1	yes	characteristic
312	C₄	#421×C₁₅₁	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#1519×C₁₅₁	1	1	yes	characteristic
319	C₂	C₂×#357×C₁₅₁	1	1	yes	characteristic
320	C₂	#2145×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
prime radical	C₂×#466×C₁₅₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂³×C₄×C₁₅₁ ◁ C₂³×C₈×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊C₁₆×C₁₅₁	C₂	semidirect
3. #1808×C₁₅₁	C₂	semidirect
4. #1347×C₁₅₁	C₂	semidirect
5. C₂²×D₈×C₁₅₁	C₄	semidirect
11. #466×C₁₅₁	C₄	semidirect
14. C₂×#107×C₁₅₁	C₄	semidirect
18. C₂²⋊C₁₆×C₁₅₁	C₂²	semidirect
19. #340×C₁₅₁	C₂²	semidirect
160. #11886	C₁₅₁	direct (N × H)
161. C₁₅₁	#11886	direct (N × H)
163. C₂×C₂²⋊C₁₆	C₂×C₁₅₁	semidirect
164. #1808	C₂×C₁₅₁	semidirect
165. #1347	C₂×C₁₅₁	semidirect
166. C₂²×D₈	C₄×C₁₅₁	semidirect
172. #466	C₄×C₁₅₁	semidirect
175. C₂×#107	C₄×C₁₅₁	semidirect
179. C₂²⋊C₁₆	C₂²×C₁₅₁	semidirect
180. #340	C₂²×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#466×C₁₅₁	2
C₂×C₂²⋊C₁₆×C₁₅₁	2
#1808×C₁₅₁	2
#1347×C₁₅₁	2
#11886	151

Subgroups — C151×#11886 #50316811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#11886 #50316811