Subgroups — C₁₃₁×#29138 #45715022

Order 33536.

642 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2647×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2327×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2101×C₁₃₁	C₂	1	2	yes	maximal
4	#2043×C₁₃₁	C₂	1	2	yes	maximal
5	#1935×C₁₃₁	C₂	1	2	yes	maximal
6	#1933×C₁₃₁	C₂	1	2	yes	maximal
7	#1805×C₁₃₁	C₂	1	2	yes	maximal
8	#1803×C₁₃₁	C₂	1	2	yes	maximal
9	#1768×C₁₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
14	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
16	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#400×C₁₃₁	—	2	2	no	—
20	#398×C₁₃₁	—	2	2	no	—
21	#391×C₁₃₁	C₂²	1	2	yes	—
22	#389×C₁₃₁	C₂²	1	2	yes	—
23	#389×C₁₃₁	C₂²	1	2	yes	—
24	#387×C₁₃₁	—	2	2	no	—
25	#387×C₁₃₁	—	2	2	no	—
26	#384×C₁₃₁	C₂²	1	1	yes	characteristic
27	#382×C₁₃₁	C₂²	1	1	yes	characteristic
28	#382×C₁₃₁	C₂²	1	1	yes	characteristic
29	#376×C₁₃₁	C₂²	1	1	yes	characteristic
30	#340×C₁₃₁	C₂²	1	2	yes	—
31	#340×C₁₃₁	C₂²	1	2	yes	—
32	#340×C₁₃₁	C₂²	1	2	yes	—
33	#340×C₁₃₁	C₂²	1	2	yes	—
34	#336×C₁₃₁	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
39	C₄×Q₈×C₁₃₁	—	2	4	no	—
40	C₄×Q₈×C₁₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₁	—	2	2	no	—
50	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₁₃₁	—	2	4	no	—
54	#103×C₁₃₁	—	2	4	no	—
55	#103×C₁₃₁	—	2	4	no	—
56	#103×C₁₃₁	—	4	2	no	—
57	#103×C₁₃₁	—	4	2	no	—
58	#97×C₁₃₁	—	4	1	no	—
59	C₄×C₈×C₁₃₁	—	4	1	no	—
60	#95×C₁₃₁	—	2	2	no	—
61	#95×C₁₃₁	—	2	2	no	—
62	#95×C₁₃₁	—	2	2	no	—
63	#95×C₁₃₁	—	2	2	no	—
64	#95×C₁₃₁	—	2	2	no	—
65	#95×C₁₃₁	—	2	2	no	—
66	#95×C₁₃₁	D₄	1	2	yes	—
67	#95×C₁₃₁	D₄	1	2	yes	—
68	C₂×Q₈×C₁₃₁	—	2	2	no	—
69	C₂×Q₈×C₁₃₁	—	2	2	no	—
70	C₂×Q₈×C₁₃₁	—	2	2	no	—
71	C₂×Q₈×C₁₃₁	—	2	2	no	—
72	C₂×Q₈×C₁₃₁	—	2	2	no	—
73	C₂×Q₈×C₁₃₁	#41	1	4	yes	—
74	C₂²×C₄×C₁₃₁	—	2	2	no	—
75	C₂²×C₄×C₁₃₁	—	2	2	no	—
76	C₂²×C₄×C₁₃₁	—	2	2	no	—
77	C₂²×C₄×C₁₃₁	—	2	2	no	—
78	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₁₃₁	—	2	2	no	—
84	C₂×C₈×C₁₃₁	—	2	2	no	—
85	C₂×C₈×C₁₃₁	—	2	2	no	—
86	C₂×C₈×C₁₃₁	—	2	2	no	—
87	C₂×C₈×C₁₃₁	—	2	2	no	—
88	C₂×C₈×C₁₃₁	—	2	2	no	—
89	C₄⋊C₄×C₁₃₁	—	2	8	no	—
90	C₄⋊C₄×C₁₃₁	—	2	2	no	—
91	C₄⋊C₄×C₁₃₁	—	2	2	no	—
92	C₄⋊C₄×C₁₃₁	—	2	2	no	—
93	C₄⋊C₄×C₁₃₁	—	2	2	no	—
94	C₄⋊C₄×C₁₃₁	—	2	2	no	—
95	C₄⋊C₄×C₁₃₁	—	2	2	no	—
96	C₄⋊C₄×C₁₃₁	—	4	4	no	—
97	C₄⋊C₄×C₁₃₁	—	8	2	no	—
98	C₄⋊C₄×C₁₃₁	—	8	2	no	—
99	C₄⋊C₄×C₁₃₁	—	8	2	no	—
100	C₄²×C₁₃₁	—	2	2	no	—
101	C₄²×C₁₃₁	—	2	1	no	—
102	C₄²×C₁₃₁	—	2	1	no	—
103	C₄²×C₁₃₁	—	4	4	no	—
104	C₄²×C₁₃₁	#41	1	4	yes	—
105	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₁₃₁	—	2	4	no	—
107	Q₈×C₁₃₁	—	2	4	no	—
108	Q₈×C₁₃₁	—	4	2	no	—
109	Q₈×C₁₃₁	—	4	2	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	2	2	no	—
112	C₂×C₄×C₁₃₁	—	2	2	no	—
113	C₂×C₄×C₁₃₁	—	2	4	no	—
114	C₂×C₄×C₁₃₁	—	2	1	no	—
115	C₂×C₄×C₁₃₁	—	2	1	no	—
116	C₂×C₄×C₁₃₁	—	2	2	no	—
117	C₂×C₄×C₁₃₁	—	2	2	no	—
118	C₂×C₄×C₁₃₁	—	2	2	no	—
119	C₂×C₄×C₁₃₁	—	2	2	no	—
120	C₂×C₄×C₁₃₁	—	4	2	no	—
121	C₂×C₄×C₁₃₁	—	4	2	no	—
122	C₂×C₄×C₁₃₁	—	4	2	no	—
123	C₂×C₄×C₁₃₁	—	4	2	no	—
124	C₂×C₄×C₁₃₁	—	4	2	no	—
125	C₂×C₄×C₁₃₁	—	4	2	no	—
126	C₂×C₄×C₁₃₁	—	4	2	no	—
127	C₂×C₄×C₁₃₁	—	4	2	no	—
128	C₂×C₄×C₁₃₁	—	4	2	no	—
129	C₂×C₄×C₁₃₁	—	4	2	no	—
130	C₂×C₄×C₁₃₁	—	4	2	no	—
131	C₂×C₄×C₁₃₁	#124	1	4	yes	—
132	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
134	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
136	C₈×C₁₃₁	—	4	2	no	—
137	C₈×C₁₃₁	—	4	2	no	—
138	C₂²×C₁₃₁	#525	1	1	yes	characteristic
139	C₂²×C₁₃₁	C₂×#124	1	1	yes	characteristic
140	C₂²×C₁₃₁	#537	1	1	yes	characteristic
141	C₂²×C₁₃₁	#577	1	2	yes	—
142	C₂²×C₁₃₁	#578	1	2	yes	—
143	C₄×C₁₃₁	—	2	4	no	—
144	C₄×C₁₃₁	—	4	4	no	—
145	C₄×C₁₃₁	—	4	2	no	—
146	C₄×C₁₃₁	—	4	1	no	—
147	C₄×C₁₃₁	—	4	1	no	—
148	C₄×C₁₃₁	—	8	2	no	—
149	C₄×C₁₃₁	—	8	2	no	—
150	C₄×C₁₃₁	—	8	2	no	—
151	C₄×C₁₃₁	#552	1	2	yes	—
152	C₄×C₁₃₁	#553	1	2	yes	—
153	C₂×C₁₃₁	#2562	1	1	yes	characteristic
154	C₂×C₁₃₁	#3121	1	2	yes	—
155	C₂×C₁₃₁	#3137	1	2	yes	—
156	C₂×C₁₃₁	#3260	1	1	yes	characteristic
157	C₂×C₁₃₁	#3261	1	1	yes	characteristic
158	#29138	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₁₃₁	#29138	1	1	yes	characteristic, direct product factor, Sylow
160	#2647	C₂×C₁₃₁	1	1	yes	characteristic
161	#2327	C₂×C₁₃₁	1	1	yes	characteristic
162	#2101	C₂×C₁₃₁	1	2	yes	—
163	#2043	C₂×C₁₃₁	1	2	yes	—
164	#1935	C₂×C₁₃₁	1	2	yes	—
165	#1933	C₂×C₁₃₁	1	2	yes	—
166	#1805	C₂×C₁₃₁	1	2	yes	—
167	#1803	C₂×C₁₃₁	1	2	yes	—
168	#1768	C₂×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
170	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
172	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
173	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
174	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
175	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
176	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
178	#400	—	2	2	no	—
179	#398	—	2	2	no	—
180	#391	C₂²×C₁₃₁	1	2	yes	—
181	#389	C₂²×C₁₃₁	1	2	yes	—
182	#389	C₂²×C₁₃₁	1	2	yes	—
183	#387	—	2	2	no	—
184	#387	—	2	2	no	—
185	#384	C₂²×C₁₃₁	1	1	yes	characteristic
186	#382	C₂²×C₁₃₁	1	1	yes	characteristic
187	#382	C₂²×C₁₃₁	1	1	yes	characteristic
188	#376	C₂²×C₁₃₁	1	1	yes	characteristic
189	#340	C₂²×C₁₃₁	1	2	yes	—
190	#340	C₂²×C₁₃₁	1	2	yes	—
191	#340	C₂²×C₁₃₁	1	2	yes	—
192	#340	C₂²×C₁₃₁	1	2	yes	—
193	#336	C₂²×C₁₃₁	1	2	yes	—
194	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
197	C₄⋊Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	4	no	—
199	C₄×Q₈	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
212	C₄⋊C₈	—	2	4	no	—
213	#103	—	2	4	no	—
214	#103	—	2	4	no	—
215	#103	—	4	2	no	—
216	#103	—	4	2	no	—
217	#97	—	4	1	no	—
218	C₄×C₈	—	4	1	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	D₄×C₁₃₁	1	2	yes	—
226	#95	D₄×C₁₃₁	1	2	yes	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	#41×C₁₃₁	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	8	2	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	—	8	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	4	4	no	—
263	C₄²	#41×C₁₃₁	1	4	yes	—
264	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
265	Q₈	—	2	4	no	—
266	Q₈	—	2	4	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	#124×C₁₃₁	1	4	yes	—
291	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
292	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
293	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
294	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
295	C₈	—	4	2	no	—
296	C₈	—	4	2	no	—
297	C₂²	#525×C₁₃₁	1	1	yes	characteristic
298	C₂²	C₂×#124×C₁₃₁	1	1	yes	characteristic
299	C₂²	#537×C₁₃₁	1	1	yes	characteristic
300	C₂²	#577×C₁₃₁	1	2	yes	—
301	C₂²	#578×C₁₃₁	1	2	yes	—
302	C₄	—	2	4	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	#552×C₁₃₁	1	2	yes	—
311	C₄	#553×C₁₃₁	1	2	yes	—
312	C₂	#2562×C₁₃₁	1	1	yes	characteristic
313	C₂	#3121×C₁₃₁	1	2	yes	—
314	C₂	#3137×C₁₃₁	1	2	yes	—
315	C₂	#3260×C₁₃₁	1	1	yes	characteristic
316	C₂	#3261×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
158. #29138	C₁₃₁	direct (N × H)
159. C₁₃₁	#29138	direct (N × H)

Maximal subgroups

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

subgroup	index
#2647×C₁₃₁	2
#2327×C₁₃₁	2
#2101×C₁₃₁	2
#2043×C₁₃₁	2
#1935×C₁₃₁	2
#1933×C₁₃₁	2
#1805×C₁₃₁	2
#1803×C₁₃₁	2
#1768×C₁₃₁	2
#29138	131

Subgroups — C131×#29138 #45715022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29138 #45715022