Subgroups — C₁₅₁×#10812 #50315737

Order 38656.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2873×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#344×C₁₅₁	C₂	1	2	yes	maximal
3	#1738×C₁₅₁	C₂	1	2	yes	maximal
4	C₂×#338×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#338×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₅₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₁₅₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₅₁	C₂²	1	1	yes	characteristic
11	#405×C₁₅₁	—	2	2	no	—
12	#405×C₁₅₁	C₄	1	4	yes	—
13	C₄×M₁₆×C₁₅₁	—	2	2	no	—
14	#344×C₁₅₁	—	2	4	no	—
15	#341×C₁₅₁	C₄	1	8	yes	complemented
16	#338×C₁₅₁	—	2	2	no	—
17	#338×C₁₅₁	—	2	2	no	—
18	C₂³×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₁₅₁	—	2	1	no	—
20	C₂×M₁₆×C₁₅₁	—	2	2	no	—
21	C₂×M₁₆×C₁₅₁	—	2	2	no	—
22	C₂×M₁₆×C₁₅₁	—	2	2	no	—
23	C₂×M₁₆×C₁₅₁	—	2	2	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	—	2	2	no	—
31	C₂×M₁₆×C₁₅₁	—	2	2	no	—
32	C₂×M₁₆×C₁₅₁	—	2	2	no	—
33	C₂×M₁₆×C₁₅₁	—	2	2	no	—
34	C₂×M₁₆×C₁₅₁	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₁₅₁	C₂×C₄	1	4	yes	—
36	C₂×M₁₆×C₁₅₁	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₁₅₁	D₄	1	2	yes	—
38	C₂×M₁₆×C₁₅₁	D₄	1	2	yes	—
39	C₂²×C₈×C₁₅₁	—	2	1	no	—
40	C₂²×C₈×C₁₅₁	—	2	2	no	—
41	C₂²×C₈×C₁₅₁	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
43	#117×C₁₅₁	—	2	1	no	—
44	#117×C₁₅₁	—	2	1	no	—
45	#117×C₁₅₁	C₂×C₄	1	2	yes	—
46	#117×C₁₅₁	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₁₅₁	Q₈	1	2	yes	—
49	C₂×C₄²×C₁₅₁	—	2	1	no	—
50	#97×C₁₅₁	—	2	4	no	—
51	C₄×C₈×C₁₅₁	—	2	4	no	—
52	C₂⁴×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₅₁	—	2	2	no	—
54	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
62	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
63	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
64	M₁₆×C₁₅₁	—	2	2	no	—
65	M₁₆×C₁₅₁	—	2	4	no	—
66	M₁₆×C₁₅₁	—	2	4	no	—
67	M₁₆×C₁₅₁	—	2	4	no	—
68	M₁₆×C₁₅₁	—	2	4	no	—
69	M₁₆×C₁₅₁	—	2	4	no	—
70	M₁₆×C₁₅₁	—	4	2	no	—
71	M₁₆×C₁₅₁	—	4	2	no	—
72	M₁₆×C₁₅₁	—	4	2	no	—
73	M₁₆×C₁₅₁	—	4	2	no	—
74	M₁₆×C₁₅₁	—	4	2	no	—
75	M₁₆×C₁₅₁	—	4	2	no	—
76	M₁₆×C₁₅₁	C₄²	1	4	yes	—
77	C₂×C₈×C₁₅₁	—	2	2	no	—
78	C₂×C₈×C₁₅₁	—	2	2	no	—
79	C₂×C₈×C₁₅₁	—	2	2	no	—
80	C₂×C₈×C₁₅₁	—	2	4	no	—
81	C₂×C₈×C₁₅₁	—	2	4	no	—
82	C₂×C₈×C₁₅₁	—	2	4	no	—
83	C₂×C₈×C₁₅₁	—	2	4	no	—
84	C₂×C₈×C₁₅₁	—	2	4	no	—
85	C₂×C₈×C₁₅₁	—	4	2	no	—
86	C₂×C₈×C₁₅₁	—	4	2	no	—
87	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₁₅₁	—	2	2	no	—
89	C₄⋊C₄×C₁₅₁	—	2	2	no	—
90	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
91	C₂²⋊C₄×C₁₅₁	C₄⋊C₄	1	4	yes	—
92	C₄²×C₁₅₁	—	2	2	no	—
93	C₄²×C₁₅₁	—	2	2	no	—
94	C₂³×C₁₅₁	—	8	1	no	—
95	C₂³×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₁₅₁	C₄×Q₈	1	2	yes	—
97	C₂³×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₁₅₁	#125	1	2	yes	—
100	C₂×C₄×C₁₅₁	—	2	4	no	—
101	C₂×C₄×C₁₅₁	—	8	2	no	—
102	C₂×C₄×C₁₅₁	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₅₁	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₅₁	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₅₁	#117	1	2	yes	—
107	C₂×C₄×C₁₅₁	#117	1	2	yes	—
108	C₂×C₄×C₁₅₁	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₅₁	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₅₁	#123	1	2	yes	—
116	C₂×C₄×C₁₅₁	#123	1	2	yes	—
117	C₂×C₄×C₁₅₁	#124	1	2	yes	—
118	C₂×C₄×C₁₅₁	#126	1	2	yes	—
119	C₂×C₄×C₁₅₁	#126	1	2	yes	—
120	C₈×C₁₅₁	—	2	4	no	—
121	C₈×C₁₅₁	—	2	4	no	—
122	C₈×C₁₅₁	—	4	4	no	—
123	C₈×C₁₅₁	—	4	4	no	—
124	C₂²×C₁₅₁	—	4	1	no	—
125	C₂²×C₁₅₁	—	4	1	no	—
126	C₂²×C₁₅₁	—	4	2	no	—
127	C₂²×C₁₅₁	—	4	1	no	—
128	C₂²×C₁₅₁	—	4	2	no	—
129	C₂²×C₁₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₁₅₁	#380	1	1	yes	characteristic
131	C₂²×C₁₅₁	#382	1	1	yes	characteristic
132	C₂²×C₁₅₁	#387	1	2	yes	—
133	C₂²×C₁₅₁	#387	1	2	yes	—
134	C₄×C₁₅₁	—	4	4	no	—
135	C₄×C₁₅₁	#376	1	1	yes	characteristic
136	C₄×C₁₅₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₁₅₁	#381	1	1	yes	characteristic
138	C₄×C₁₅₁	#383	1	1	yes	characteristic
139	C₄×C₁₅₁	#388	1	2	yes	—
140	C₄×C₁₅₁	#388	1	2	yes	—
141	C₂×C₁₅₁	—	2	1	no	—
142	C₂×C₁₅₁	—	2	1	no	—
143	C₂×C₁₅₁	—	2	1	no	—
144	C₂×C₁₅₁	—	2	2	no	—
145	C₂×C₁₅₁	—	2	2	no	—
146	C₂×C₁₅₁	#1434	1	1	yes	characteristic
147	#10812	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₅₁	#10812	1	1	yes	characteristic, direct product factor, Sylow
149	#2873	C₂×C₁₅₁	1	1	yes	characteristic
150	C₂×#344	C₂×C₁₅₁	1	2	yes	—
151	#1738	C₂×C₁₅₁	1	2	yes	—
152	C₂×#338	C₂×C₁₅₁	1	1	yes	characteristic
153	C₂×#338	C₂×C₁₅₁	1	1	yes	characteristic
154	C₂²×M₁₆	C₂²×C₁₅₁	1	2	yes	—
155	C₂²×M₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
156	C₂²×M₁₆	C₂²×C₁₅₁	1	2	yes	—
157	C₂²×M₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
158	C₂×#117	C₂²×C₁₅₁	1	1	yes	characteristic
159	#405	—	2	2	no	—
160	#405	C₄×C₁₅₁	1	4	yes	—
161	C₄×M₁₆	—	2	2	no	—
162	#344	—	2	4	no	—
163	#341	C₄×C₁₅₁	1	8	yes	complemented
164	#338	—	2	2	no	—
165	#338	—	2	2	no	—
166	C₂³×C₄	C₂³×C₁₅₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	C₂×C₄×C₁₅₁	1	4	yes	—
183	C₂×M₁₆	C₂×C₄×C₁₅₁	1	4	yes	—
184	C₂×M₁₆	C₂×C₄×C₁₅₁	1	2	yes	—
185	C₂×M₁₆	D₄×C₁₅₁	1	2	yes	—
186	C₂×M₁₆	D₄×C₁₅₁	1	2	yes	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂×C₄×C₁₅₁	1	4	yes	—
190	C₂²×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
191	#117	—	2	1	no	—
192	#117	—	2	1	no	—
193	#117	C₂×C₄×C₁₅₁	1	2	yes	—
194	#117	D₄×C₁₅₁	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	Q₈×C₁₅₁	1	2	yes	—
197	C₂×C₄²	—	2	1	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	4	no	—
200	C₂⁴	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₁₅₁	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
207	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
208	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
209	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
210	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
211	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	C₄²×C₁₅₁	1	4	yes	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	C₄⋊C₄×C₁₅₁	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
244	C₂³	C₄×Q₈×C₁₅₁	1	2	yes	—
245	C₂³	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
246	C₂³	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
247	C₂³	#125×C₁₅₁	1	2	yes	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₂×C₄²×C₁₅₁	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
252	C₂×C₄	#117×C₁₅₁	1	1	yes	characteristic
253	C₂×C₄	#117×C₁₅₁	1	1	yes	characteristic
254	C₂×C₄	#117×C₁₅₁	1	2	yes	—
255	C₂×C₄	#117×C₁₅₁	1	2	yes	—
256	C₂×C₄	#117×C₁₅₁	1	1	yes	characteristic
257	C₂×C₄	#117×C₁₅₁	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
261	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
262	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
263	C₂×C₄	#123×C₁₅₁	1	2	yes	—
264	C₂×C₄	#123×C₁₅₁	1	2	yes	—
265	C₂×C₄	#124×C₁₅₁	1	2	yes	—
266	C₂×C₄	#126×C₁₅₁	1	2	yes	—
267	C₂×C₄	#126×C₁₅₁	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	2	4	no	—
270	C₈	—	4	4	no	—
271	C₈	—	4	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₄×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
278	C₂²	#380×C₁₅₁	1	1	yes	characteristic
279	C₂²	#382×C₁₅₁	1	1	yes	characteristic
280	C₂²	#387×C₁₅₁	1	2	yes	—
281	C₂²	#387×C₁₅₁	1	2	yes	—
282	C₄	—	4	4	no	—
283	C₄	#376×C₁₅₁	1	1	yes	characteristic
284	C₄	C₄×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
285	C₄	#381×C₁₅₁	1	1	yes	characteristic
286	C₄	#383×C₁₅₁	1	1	yes	characteristic
287	C₄	#388×C₁₅₁	1	2	yes	—
288	C₄	#388×C₁₅₁	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	#1434×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₅₁	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₅₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₅₁
socle	C₂×C₁₅₁	#1434
prime radical	C₂⁴×C₁₅₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₅₁ ◁ C₂²×M₁₆×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
15. #341×C₁₅₁	C₄	semidirect
147. #10812	C₁₅₁	direct (N × H)
148. C₁₅₁	#10812	direct (N × H)
163. #341	C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#2873×C₁₅₁	2
C₂×#344×C₁₅₁	2
#1738×C₁₅₁	2
C₂×#338×C₁₅₁	2
#10812	151

Subgroups — C151×#10812 #50315737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#10812 #50315737