Subgroups — C₁₅₁×#11241 #50316166

Order 38656.

638 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#344×C₁₅₁	C₂	1	2	yes	maximal
4	C₂×#341×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1733×C₁₅₁	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₅₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₅₁	C₂²	1	2	yes	—
10	C₂×#117×C₁₅₁	C₂²	1	1	yes	characteristic
11	#427×C₁₅₁	—	2	4	no	—
12	C₂×#107×C₁₅₁	—	2	1	no	—
13	C₂×#106×C₁₅₁	—	2	1	no	—
14	#424×C₁₅₁	—	2	1	no	—
15	#424×C₁₅₁	—	2	1	no	—
16	#424×C₁₅₁	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₁₅₁	—	2	2	no	—
18	#344×C₁₅₁	—	2	4	no	—
19	#341×C₁₅₁	—	2	2	no	—
20	#339×C₁₅₁	—	2	4	no	—
21	C₂³×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₅₁	—	2	2	no	—
23	C₂×M₁₆×C₁₅₁	—	2	1	no	—
24	C₂×M₁₆×C₁₅₁	—	2	2	no	—
25	C₂×M₁₆×C₁₅₁	—	2	2	no	—
26	C₂×M₁₆×C₁₅₁	—	2	2	no	—
27	C₂×M₁₆×C₁₅₁	—	2	2	no	—
28	C₂×M₁₆×C₁₅₁	—	2	2	no	—
29	C₂×M₁₆×C₁₅₁	—	2	2	no	—
30	C₂×M₁₆×C₁₅₁	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₁₅₁	Q₈	1	4	yes	—
32	C₂²×C₈×C₁₅₁	—	2	2	no	—
33	C₂²×C₈×C₁₅₁	—	2	2	no	—
34	C₂²×C₈×C₁₅₁	—	2	1	no	—
35	C₂²×C₈×C₁₅₁	—	2	2	no	—
36	C₂²×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₅₁	D₄	1	2	yes	—
38	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
40	#117×C₁₅₁	—	2	1	no	—
41	#117×C₁₅₁	—	2	1	no	—
42	#117×C₁₅₁	—	2	2	no	—
43	#117×C₁₅₁	—	2	2	no	—
44	#117×C₁₅₁	—	2	2	no	—
45	#117×C₁₅₁	C₂×C₄	1	2	yes	—
46	#117×C₁₅₁	D₄	1	4	yes	—
47	#117×C₁₅₁	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₅₁	Q₈	1	2	yes	—
50	C₂×C₂²⋊C₄×C₁₅₁	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₁₅₁	—	2	2	no	—
52	C₂×C₄²×C₁₅₁	—	2	1	no	—
53	C₂×C₄²×C₁₅₁	—	2	2	no	—
54	#107×C₁₅₁	—	2	4	no	—
55	#106×C₁₅₁	—	2	4	no	—
56	C₄⋊C₈×C₁₅₁	—	2	4	no	—
57	C₄⋊C₈×C₁₅₁	—	2	4	no	—
58	C₂⁴×C₁₅₁	#41	1	1	yes	characteristic
59	C₂²×C₄×C₁₅₁	—	2	2	no	—
60	C₂²×C₄×C₁₅₁	—	4	2	no	—
61	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
69	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
70	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
71	M₁₆×C₁₅₁	—	2	4	no	—
72	M₁₆×C₁₅₁	—	2	4	no	—
73	M₁₆×C₁₅₁	—	2	4	no	—
74	M₁₆×C₁₅₁	—	4	1	no	—
75	M₁₆×C₁₅₁	—	4	2	no	—
76	M₁₆×C₁₅₁	—	4	2	no	—
77	M₁₆×C₁₅₁	—	4	1	no	—
78	C₂×C₈×C₁₅₁	—	2	4	no	—
79	C₂×C₈×C₁₅₁	—	2	1	no	—
80	C₂×C₈×C₁₅₁	—	2	1	no	—
81	C₂×C₈×C₁₅₁	—	2	4	no	—
82	C₂×C₈×C₁₅₁	—	2	1	no	—
83	C₂×C₈×C₁₅₁	—	2	1	no	—
84	C₂×C₈×C₁₅₁	—	2	2	no	—
85	C₂×C₈×C₁₅₁	—	2	4	no	—
86	C₂×C₈×C₁₅₁	—	4	2	no	—
87	C₂×C₈×C₁₅₁	—	4	2	no	—
88	C₂×C₈×C₁₅₁	—	4	2	no	—
89	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	2	yes	—
91	C₄⋊C₄×C₁₅₁	—	2	2	no	—
92	C₄⋊C₄×C₁₅₁	—	2	4	no	—
93	C₄⋊C₄×C₁₅₁	—	4	2	no	—
94	C₄⋊C₄×C₁₅₁	C₄⋊C₄	1	4	yes	—
95	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
96	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
97	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
98	C₂²⋊C₄×C₁₅₁	—	4	2	no	—
99	C₄²×C₁₅₁	—	2	2	no	—
100	C₄²×C₁₅₁	—	2	2	no	—
101	C₄²×C₁₅₁	—	2	4	no	—
102	C₄²×C₁₅₁	—	4	2	no	—
103	C₂³×C₁₅₁	—	4	2	no	—
104	C₂³×C₁₅₁	#117	1	1	yes	characteristic
105	C₂³×C₁₅₁	#121	1	1	yes	characteristic
106	C₂³×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₁₅₁	#123	1	2	yes	—
108	C₂³×C₁₅₁	#125	1	2	yes	—
109	C₂×C₄×C₁₅₁	—	2	4	no	—
110	C₂×C₄×C₁₅₁	—	4	4	no	—
111	C₂×C₄×C₁₅₁	—	4	4	no	—
112	C₂×C₄×C₁₅₁	—	8	4	no	—
113	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₅₁	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₁₅₁	#120	1	1	yes	characteristic
118	C₂×C₄×C₁₅₁	#121	1	2	yes	—
119	C₂×C₄×C₁₅₁	#121	1	2	yes	—
120	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₅₁	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₅₁	#124	1	1	yes	characteristic
127	C₂×C₄×C₁₅₁	#125	1	2	yes	—
128	C₂×C₄×C₁₅₁	#126	1	2	yes	—
129	C₂×C₄×C₁₅₁	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
131	C₈×C₁₅₁	—	2	2	no	—
132	C₈×C₁₅₁	—	2	2	no	—
133	C₈×C₁₅₁	—	4	2	no	—
134	C₈×C₁₅₁	—	4	4	no	—
135	C₂²×C₁₅₁	—	2	2	no	—
136	C₂²×C₁₅₁	—	2	2	no	—
137	C₂²×C₁₅₁	—	2	2	no	—
138	C₂²×C₁₅₁	—	4	2	no	—
139	C₂²×C₁₅₁	—	4	2	no	—
140	C₂²×C₁₅₁	#380	1	1	yes	characteristic
141	C₂²×C₁₅₁	#387	1	2	yes	—
142	C₂²×C₁₅₁	#394	1	1	yes	characteristic
143	C₂²×C₁₅₁	#398	1	2	yes	—
144	C₂²×C₁₅₁	#398	1	1	yes	characteristic
145	C₄×C₁₅₁	—	4	4	no	—
146	C₄×C₁₅₁	—	8	4	no	—
147	C₄×C₁₅₁	#386	1	1	yes	characteristic
148	C₄×C₁₅₁	#389	1	2	yes	—
149	C₄×C₁₅₁	#391	1	1	yes	characteristic
150	C₄×C₁₅₁	#395	1	1	yes	characteristic
151	C₄×C₁₅₁	#396	1	2	yes	—
152	C₄×C₁₅₁	#400	1	1	yes	characteristic
153	C₂×C₁₅₁	—	2	1	no	—
154	C₂×C₁₅₁	—	2	2	no	—
155	C₂×C₁₅₁	—	2	2	no	—
156	C₂×C₁₅₁	—	2	1	no	—
157	C₂×C₁₅₁	#1441	1	1	yes	characteristic
158	C₂×C₁₅₁	#1814	1	2	yes	—
159	#11241	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₁₅₁	#11241	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×#427	C₂×C₁₅₁	1	1	yes	characteristic
162	C₂×#424	C₂×C₁₅₁	1	1	yes	characteristic
163	C₂×#344	C₂×C₁₅₁	1	2	yes	—
164	C₂×#341	C₂×C₁₅₁	1	1	yes	characteristic
165	#1733	C₂×C₁₅₁	1	2	yes	—
166	C₂²×M₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₁₅₁	1	2	yes	—
168	C₂³×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
169	C₂×#117	C₂²×C₁₅₁	1	2	yes	—
170	C₂×#117	C₂²×C₁₅₁	1	1	yes	characteristic
171	#427	—	2	4	no	—
172	C₂×#107	—	2	1	no	—
173	C₂×#106	—	2	1	no	—
174	#424	—	2	1	no	—
175	#424	—	2	1	no	—
176	#424	C₄×C₁₅₁	1	4	yes	complemented
177	C₂×C₄⋊C₈	—	2	2	no	—
178	#344	—	2	4	no	—
179	#341	—	2	2	no	—
180	#339	—	2	4	no	—
181	C₂³×C₄	C₂³×C₁₅₁	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	C₂×C₄×C₁₅₁	1	2	yes	—
191	C₂×M₁₆	Q₈×C₁₅₁	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
197	C₂²×C₈	D₄×C₁₅₁	1	2	yes	—
198	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
199	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
200	#117	—	2	1	no	—
201	#117	—	2	1	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	2	2	no	—
205	#117	C₂×C₄×C₁₅₁	1	2	yes	—
206	#117	D₄×C₁₅₁	1	4	yes	—
207	#117	D₄×C₁₅₁	1	2	yes	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	Q₈×C₁₅₁	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	2	no	—
214	#107	—	2	4	no	—
215	#106	—	2	4	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₂⁴	#41×C₁₅₁	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₅₁	1	2	yes	—
226	C₂²×C₄	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×Q₈×C₁₅₁	1	2	yes	—
228	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
229	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
230	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	—	4	1	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	2	yes	—
250	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	2	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	C₄⋊C₄×C₁₅₁	1	4	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	#117×C₁₅₁	1	1	yes	characteristic
265	C₂³	#121×C₁₅₁	1	1	yes	characteristic
266	C₂³	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
267	C₂³	#123×C₁₅₁	1	2	yes	—
268	C₂³	#125×C₁₅₁	1	2	yes	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	4	no	—
273	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
275	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
276	C₂×C₄	C₄×Q₈×C₁₅₁	1	2	yes	—
277	C₂×C₄	#120×C₁₅₁	1	1	yes	characteristic
278	C₂×C₄	#121×C₁₅₁	1	2	yes	—
279	C₂×C₄	#121×C₁₅₁	1	2	yes	—
280	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	2	yes	—
282	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	2	yes	—
283	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	2	yes	—
284	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
285	C₂×C₄	#123×C₁₅₁	1	1	yes	characteristic
286	C₂×C₄	#124×C₁₅₁	1	1	yes	characteristic
287	C₂×C₄	#125×C₁₅₁	1	2	yes	—
288	C₂×C₄	#126×C₁₅₁	1	2	yes	—
289	C₂×C₄	C₄⋊Q₈×C₁₅₁	1	2	yes	—
290	C₂×C₄	C₄⋊Q₈×C₁₅₁	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	#380×C₁₅₁	1	1	yes	characteristic
301	C₂²	#387×C₁₅₁	1	2	yes	—
302	C₂²	#394×C₁₅₁	1	1	yes	characteristic
303	C₂²	#398×C₁₅₁	1	2	yes	—
304	C₂²	#398×C₁₅₁	1	1	yes	characteristic
305	C₄	—	4	4	no	—
306	C₄	—	8	4	no	—
307	C₄	#386×C₁₅₁	1	1	yes	characteristic
308	C₄	#389×C₁₅₁	1	2	yes	—
309	C₄	#391×C₁₅₁	1	1	yes	characteristic
310	C₄	#395×C₁₅₁	1	1	yes	characteristic
311	C₄	#396×C₁₅₁	1	2	yes	—
312	C₄	#400×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	#1441×C₁₅₁	1	1	yes	characteristic
318	C₂	#1814×C₁₅₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₅₁	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₅₁
socle	C₂²×C₁₅₁	#380
prime radical	C₂⁴×C₁₅₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂³×C₄×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
16. #424×C₁₅₁	C₄	semidirect
159. #11241	C₁₅₁	direct (N × H)
160. C₁₅₁	#11241	direct (N × H)
176. #424	C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#427×C₁₅₁	2
C₂×#424×C₁₅₁	2
C₂×#344×C₁₅₁	2
C₂×#341×C₁₅₁	2
#1733×C₁₅₁	2
#11241	151

Subgroups — C151×#11241 #50316166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#11241 #50316166