Subgroups — C₁₃₁×#10919 #45696803

Order 33536.

754 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2873×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2296×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#344×C₁₃₁	C₂	1	2	yes	maximal
4	#1742×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1728×C₁₃₁	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₁₃₁	C₂²	1	2	yes	—
8	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
11	#405×C₁₃₁	—	2	4	no	—
12	C₄×M₁₆×C₁₃₁	—	2	2	no	—
13	#389×C₁₃₁	—	2	2	no	—
14	#385×C₁₃₁	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₁₃₁	—	2	2	no	—
16	#344×C₁₃₁	C₄	1	8	yes	complemented
17	#343×C₁₃₁	—	2	2	no	—
18	#336×C₁₃₁	—	2	4	no	—
19	C₂³×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₃₁	—	2	2	no	—
22	C₂×M₁₆×C₁₃₁	—	2	2	no	—
23	C₂×M₁₆×C₁₃₁	—	2	2	no	—
24	C₂×M₁₆×C₁₃₁	—	2	2	no	—
25	C₂×M₁₆×C₁₃₁	—	2	2	no	—
26	C₂×M₁₆×C₁₃₁	—	2	2	no	—
27	C₂×M₁₆×C₁₃₁	—	2	2	no	—
28	C₂×M₁₆×C₁₃₁	—	2	2	no	—
29	C₂×M₁₆×C₁₃₁	—	2	2	no	—
30	C₂×M₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₁₃₁	Q₈	1	4	yes	—
33	C₂²×C₈×C₁₃₁	—	2	2	no	—
34	C₂²×C₈×C₁₃₁	D₄	1	4	yes	—
35	#117×C₁₃₁	—	2	2	no	—
36	#117×C₁₃₁	—	2	2	no	—
37	#117×C₁₃₁	—	2	2	no	—
38	#117×C₁₃₁	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃₁	—	2	1	no	—
45	C₂×C₄²×C₁₃₁	—	2	1	no	—
46	C₂×C₄²×C₁₃₁	—	2	2	no	—
47	C₂×C₄²×C₁₃₁	—	2	1	no	—
48	C₂×C₄²×C₁₃₁	—	2	2	no	—
49	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
50	#97×C₁₃₁	—	2	4	no	—
51	C₄×C₈×C₁₃₁	—	2	4	no	—
52	#95×C₁₃₁	—	2	4	no	—
53	C₂⁴×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	—	2	2	no	—
55	C₂²×C₄×C₁₃₁	—	2	2	no	—
56	C₂²×C₄×C₁₃₁	—	2	4	no	—
57	C₂²×C₄×C₁₃₁	—	4	4	no	—
58	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
65	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
66	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
67	M₁₆×C₁₃₁	—	2	4	no	—
68	M₁₆×C₁₃₁	—	2	4	no	—
69	M₁₆×C₁₃₁	—	2	4	no	—
70	M₁₆×C₁₃₁	—	2	4	no	—
71	M₁₆×C₁₃₁	—	2	4	no	—
72	M₁₆×C₁₃₁	—	2	4	no	—
73	M₁₆×C₁₃₁	—	4	2	no	—
74	M₁₆×C₁₃₁	—	4	2	no	—
75	C₂×C₈×C₁₃₁	—	2	4	no	—
76	C₂×C₈×C₁₃₁	—	2	4	no	—
77	C₂×C₈×C₁₃₁	—	2	4	no	—
78	C₂×C₈×C₁₃₁	—	4	2	no	—
79	C₂×C₈×C₁₃₁	—	4	2	no	—
80	C₂×C₈×C₁₃₁	C₄⋊C₄	1	8	yes	complemented
81	C₄⋊C₄×C₁₃₁	—	2	4	no	—
82	C₄⋊C₄×C₁₃₁	—	2	4	no	—
83	C₄⋊C₄×C₁₃₁	—	4	8	no	—
84	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
85	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
86	C₂²⋊C₄×C₁₃₁	C₄²	1	4	yes	—
87	C₄²×C₁₃₁	—	2	4	no	—
88	C₄²×C₁₃₁	—	2	2	no	—
89	C₄²×C₁₃₁	—	2	2	no	—
90	C₄²×C₁₃₁	—	4	1	no	—
91	C₄²×C₁₃₁	—	4	2	no	—
92	C₄²×C₁₃₁	—	4	1	no	—
93	C₂³×C₁₃₁	—	4	2	no	—
94	C₂³×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
95	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₁₃₁	#117	1	1	yes	characteristic
97	C₂³×C₁₃₁	C₄×D₄	1	2	yes	—
98	C₂³×C₁₃₁	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₁₃₁	—	2	4	no	—
100	C₂×C₄×C₁₃₁	—	2	8	no	—
101	C₂×C₄×C₁₃₁	—	2	4	no	—
102	C₂×C₄×C₁₃₁	—	2	4	no	—
103	C₂×C₄×C₁₃₁	—	4	4	no	—
104	C₂×C₄×C₁₃₁	—	4	2	no	—
105	C₂×C₄×C₁₃₁	—	4	4	no	—
106	C₂×C₄×C₁₃₁	—	4	2	no	—
107	C₂×C₄×C₁₃₁	—	4	4	no	—
108	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃₁	#117	1	2	yes	—
110	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	#121	1	2	yes	—
116	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃₁	#123	1	2	yes	—
119	C₂×C₄×C₁₃₁	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₃₁	#124	1	1	yes	characteristic
121	C₂×C₄×C₁₃₁	#125	1	2	yes	—
122	C₂×C₄×C₁₃₁	#126	1	2	yes	—
123	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
124	C₈×C₁₃₁	—	2	8	no	—
125	C₈×C₁₃₁	—	4	4	no	—
126	C₂²×C₁₃₁	—	2	2	no	—
127	C₂²×C₁₃₁	—	2	4	no	—
128	C₂²×C₁₃₁	—	4	1	no	—
129	C₂²×C₁₃₁	—	4	1	no	—
130	C₂²×C₁₃₁	—	4	2	no	—
131	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₃₁	C₄×C₄⋊C₄	1	2	yes	—
133	C₂²×C₁₃₁	#386	1	1	yes	characteristic
134	C₂²×C₁₃₁	#387	1	2	yes	—
135	C₂²×C₁₃₁	#391	1	1	yes	characteristic
136	C₄×C₁₃₁	—	4	4	no	—
137	C₄×C₁₃₁	—	4	4	no	—
138	C₄×C₁₃₁	—	4	4	no	—
139	C₄×C₁₃₁	#385	1	2	yes	—
140	C₄×C₁₃₁	#387	1	2	yes	—
141	C₄×C₁₃₁	#388	1	2	yes	—
142	C₄×C₁₃₁	#389	1	2	yes	—
143	C₂×C₁₃₁	—	2	1	no	—
144	C₂×C₁₃₁	—	2	2	no	—
145	C₂×C₁₃₁	—	2	2	no	—
146	C₂×C₁₃₁	—	2	1	no	—
147	C₂×C₁₃₁	#1436	1	1	yes	characteristic
148	C₂×C₁₃₁	#1891	1	2	yes	—
149	#10919	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₁	#10919	1	1	yes	characteristic, direct product factor, Sylow
151	#2873	C₂×C₁₃₁	1	1	yes	characteristic
152	#2296	C₂×C₁₃₁	1	1	yes	characteristic
153	C₂×#344	C₂×C₁₃₁	1	2	yes	—
154	#1742	C₂×C₁₃₁	1	1	yes	characteristic
155	#1728	C₂×C₁₃₁	1	2	yes	—
156	C₂²×M₁₆	C₂²×C₁₃₁	1	2	yes	—
157	C₂²×M₁₆	C₂²×C₁₃₁	1	2	yes	—
158	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
159	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
160	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
161	#405	—	2	4	no	—
162	C₄×M₁₆	—	2	2	no	—
163	#389	—	2	2	no	—
164	#385	C₄×C₁₃₁	1	4	yes	—
165	C₄×C₄⋊C₄	—	2	2	no	—
166	#344	C₄×C₁₃₁	1	8	yes	complemented
167	#343	—	2	2	no	—
168	#336	—	2	4	no	—
169	C₂³×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
170	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
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	C₂×C₄×C₁₃₁	1	4	yes	—
181	C₂×M₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
182	C₂×M₁₆	Q₈×C₁₃₁	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	D₄×C₁₃₁	1	4	yes	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₂×C₄×C₁₃₁	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	#95	—	2	4	no	—
203	C₂⁴	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	4	yes	—
209	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
214	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
215	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
216	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	C₄⋊C₄×C₁₃₁	1	8	yes	complemented
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	8	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	C₄²×C₁₃₁	1	4	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	1	no	—
243	C₂³	—	4	2	no	—
244	C₂³	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
245	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
246	C₂³	#117×C₁₃₁	1	1	yes	characteristic
247	C₂³	C₄×D₄×C₁₃₁	1	2	yes	—
248	C₂³	C₄×Q₈×C₁₃₁	1	2	yes	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
259	C₂×C₄	#117×C₁₃₁	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
261	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
262	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
263	C₂×C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
264	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
265	C₂×C₄	#121×C₁₃₁	1	2	yes	—
266	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
267	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
268	C₂×C₄	#123×C₁₃₁	1	2	yes	—
269	C₂×C₄	#123×C₁₃₁	1	1	yes	characteristic
270	C₂×C₄	#124×C₁₃₁	1	1	yes	characteristic
271	C₂×C₄	#125×C₁₃₁	1	2	yes	—
272	C₂×C₄	#126×C₁₃₁	1	2	yes	—
273	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
274	C₈	—	2	8	no	—
275	C₈	—	4	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
282	C₂²	C₄×C₄⋊C₄×C₁₃₁	1	2	yes	—
283	C₂²	#386×C₁₃₁	1	1	yes	characteristic
284	C₂²	#387×C₁₃₁	1	2	yes	—
285	C₂²	#391×C₁₃₁	1	1	yes	characteristic
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	#385×C₁₃₁	1	2	yes	—
290	C₄	#387×C₁₃₁	1	2	yes	—
291	C₄	#388×C₁₃₁	1	2	yes	—
292	C₄	#389×C₁₃₁	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	#1436×C₁₃₁	1	1	yes	characteristic
298	C₂	#1891×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	#120
commutator	C₂³	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	#386
prime radical	C₂⁴×C₁₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
16. #344×C₁₃₁	C₄	semidirect
80. C₂×C₈×C₁₃₁	C₄⋊C₄	semidirect
149. #10919	C₁₃₁	direct (N × H)
150. C₁₃₁	#10919	direct (N × H)
166. #344	C₄×C₁₃₁	semidirect
230. C₂×C₈	C₄⋊C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2873×C₁₃₁	2
#2296×C₁₃₁	2
C₂×#344×C₁₃₁	2
#1742×C₁₃₁	2
#1728×C₁₃₁	2
#10919	131

Subgroups — C131×#10919 #45696803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#10919 #45696803