Subgroups — C₁₃₁×#19271 #45705155

Order 33536.

702 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3039×C₁₃₁	C₂	1	2	yes	maximal
3	C₈×#41×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1674×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#1673×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#1567×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1546×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×#137×C₁₃₁	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
10	C₄×#41×C₁₃₁	C₂²	1	1	yes	characteristic
11	#500×C₁₃₁	—	2	1	no	—
12	#500×C₁₃₁	—	2	1	no	—
13	#500×C₁₃₁	C₂²	1	4	yes	—
14	#462×C₁₃₁	—	2	2	no	—
15	#462×C₁₃₁	C₂²	1	4	yes	—
16	#461×C₁₃₁	—	2	2	no	—
17	#461×C₁₃₁	C₂²	1	4	yes	—
18	C₈×Q₈×C₁₃₁	C₂²	1	2	yes	—
19	C₈×D₄×C₁₃₁	—	2	2	no	—
20	C₈×D₄×C₁₃₁	C₂²	1	2	yes	—
21	#431×C₁₃₁	—	2	1	no	—
22	#431×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×#107×C₁₃₁	—	2	2	no	—
24	#423×C₁₃₁	C₂²	1	2	yes	—
25	#419×C₁₃₁	—	2	2	no	—
26	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
27	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
28	#335×C₁₃₁	C₂²	1	4	yes	—
29	#332×C₁₃₁	C₂²	1	8	yes	—
30	C₂×#41×C₁₃₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃₁	—	2	2	no	—
32	#137×C₁₃₁	—	4	8	no	—
33	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
35	C₂×M₁₆×C₁₃₁	—	2	2	no	—
36	C₂²×C₈×C₁₃₁	—	2	2	no	—
37	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
38	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
39	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
40	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
41	C₄⋊Q₈×C₁₃₁	C₂³	1	4	yes	—
42	C₄×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₁₃₁	—	2	1	no	—
45	C₄×D₄×C₁₃₁	—	2	1	no	—
46	C₄×D₄×C₁₃₁	D₄	1	1	yes	characteristic
47	C₄×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
48	#117×C₁₃₁	—	2	1	no	—
49	#117×C₁₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
52	C₂×C₄²×C₁₃₁	—	2	1	no	—
53	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
54	#107×C₁₃₁	—	2	4	no	—
55	#107×C₁₃₁	—	2	4	no	—
56	C₄⋊C₈×C₁₃₁	—	2	2	no	—
57	C₄⋊C₈×C₁₃₁	—	2	2	no	—
58	C₄⋊C₈×C₁₃₁	C₂³	1	4	yes	—
59	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
60	#103×C₁₃₁	—	2	4	no	—
61	#103×C₁₃₁	—	2	4	no	—
62	C₂²⋊C₈×C₁₃₁	—	2	1	no	—
63	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
64	C₄×C₈×C₁₃₁	—	2	2	no	—
65	C₄×C₈×C₁₃₁	—	2	1	no	—
66	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
69	#41×C₁₃₁	—	2	2	no	—
70	#41×C₁₃₁	D₈	1	4	yes	—
71	C₂×Q₈×C₁₃₁	—	2	4	no	—
72	C₂×Q₈×C₁₃₁	—	4	2	no	—
73	C₂×Q₈×C₁₃₁	—	4	4	no	—
74	C₂×Q₈×C₁₃₁	—	4	4	no	—
75	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×D₄×C₁₃₁	—	2	1	no	—
77	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₁	—	2	1	no	—
79	C₂²×C₄×C₁₃₁	—	2	1	no	—
80	C₂²×C₄×C₁₃₁	—	2	1	no	—
81	C₂²×C₄×C₁₃₁	—	4	2	no	—
82	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
85	Q₁₆×C₁₃₁	—	4	8	no	—
86	SD₁₆×C₁₃₁	—	4	8	no	—
87	M₁₆×C₁₃₁	—	4	4	no	—
88	C₂×C₈×C₁₃₁	—	2	4	no	—
89	C₂×C₈×C₁₃₁	—	2	2	no	—
90	C₂×C₈×C₁₃₁	—	2	4	no	—
91	C₂×C₈×C₁₃₁	—	4	4	no	—
92	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₁₃₁	—	2	1	no	—
95	C₄⋊C₄×C₁₃₁	—	2	1	no	—
96	C₄⋊C₄×C₁₃₁	—	2	4	no	—
97	C₄⋊C₄×C₁₃₁	—	2	4	no	—
98	C₄⋊C₄×C₁₃₁	—	4	2	no	—
99	C₄⋊C₄×C₁₃₁	—	4	2	no	—
100	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
104	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
105	C₄²×C₁₃₁	—	2	1	no	—
106	C₄²×C₁₃₁	—	2	1	no	—
107	C₄²×C₁₃₁	—	2	1	no	—
108	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃₁	#41	1	1	yes	characteristic
111	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
112	C₂³×C₁₃₁	—	2	1	no	—
113	C₂³×C₁₃₁	#121	1	1	yes	characteristic
114	Q₈×C₁₃₁	—	2	1	no	—
115	Q₈×C₁₃₁	—	4	4	no	—
116	Q₈×C₁₃₁	—	4	4	no	—
117	Q₈×C₁₃₁	—	8	4	no	—
118	Q₈×C₁₃₁	C₂×D₈	1	2	yes	—
119	D₄×C₁₃₁	—	2	4	no	—
120	D₄×C₁₃₁	—	2	1	no	—
121	D₄×C₁₃₁	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₁₃₁	—	2	1	no	—
123	C₂×C₄×C₁₃₁	—	2	2	no	—
124	C₂×C₄×C₁₃₁	—	2	4	no	—
125	C₂×C₄×C₁₃₁	—	2	2	no	—
126	C₂×C₄×C₁₃₁	—	2	1	no	—
127	C₂×C₄×C₁₃₁	—	2	1	no	—
128	C₂×C₄×C₁₃₁	—	4	2	no	—
129	C₂×C₄×C₁₃₁	—	4	2	no	—
130	C₂×C₄×C₁₃₁	—	4	4	no	—
131	C₂×C₄×C₁₃₁	—	8	4	no	—
132	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃₁	#135	1	2	yes	—
137	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
140	C₈×C₁₃₁	—	2	2	no	—
141	C₈×C₁₃₁	—	4	4	no	—
142	C₈×C₁₃₁	—	4	2	no	—
143	C₈×C₁₃₁	#137	1	4	yes	complemented
144	C₂²×C₁₃₁	—	2	1	no	—
145	C₂²×C₁₃₁	—	2	1	no	—
146	C₂²×C₁₃₁	—	2	2	no	—
147	C₂²×C₁₃₁	—	4	1	no	—
148	C₂²×C₁₃₁	—	4	1	no	—
149	C₂²×C₁₃₁	#466	1	2	yes	—
150	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
151	C₄×C₁₃₁	—	2	1	no	—
152	C₄×C₁₃₁	—	2	2	no	—
153	C₄×C₁₃₁	—	2	1	no	—
154	C₄×C₁₃₁	—	4	1	no	—
155	C₄×C₁₃₁	—	4	1	no	—
156	C₄×C₁₃₁	—	8	4	no	—
157	C₄×C₁₃₁	#466	1	2	yes	—
158	C₄×C₁₃₁	#473	1	2	yes	—
159	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₁₃₁	C₂×#135	1	1	yes	characteristic
161	C₄×C₁₃₁	C₂×#137	1	2	yes	—
162	C₂×C₁₃₁	—	2	2	no	—
163	C₂×C₁₃₁	—	4	2	no	—
164	C₂×C₁₃₁	#3039	1	1	yes	characteristic
165	C₂×C₁₃₁	C₂×#466	1	1	yes	characteristic
166	C₂×C₁₃₁	C₂×#473	1	1	yes	characteristic
167	#19271	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₁₃₁	#19271	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic, complemented
170	#3039	C₂×C₁₃₁	1	2	yes	—
171	C₈×#41	C₂×C₁₃₁	1	1	yes	characteristic
172	#1674	C₂×C₁₃₁	1	4	yes	complemented
173	#1673	C₂×C₁₃₁	1	4	yes	complemented
174	#1567	C₂×C₁₃₁	1	1	yes	characteristic, complemented
175	#1546	C₂×C₁₃₁	1	2	yes	complemented
176	C₂×#137	—	2	2	no	—
177	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
178	C₄×#41	C₂²×C₁₃₁	1	1	yes	characteristic
179	#500	—	2	1	no	—
180	#500	—	2	1	no	—
181	#500	C₂²×C₁₃₁	1	4	yes	—
182	#462	—	2	2	no	—
183	#462	C₂²×C₁₃₁	1	4	yes	—
184	#461	—	2	2	no	—
185	#461	C₂²×C₁₃₁	1	4	yes	—
186	C₈×Q₈	C₂²×C₁₃₁	1	2	yes	—
187	C₈×D₄	—	2	2	no	—
188	C₈×D₄	C₂²×C₁₃₁	1	2	yes	—
189	#431	—	2	1	no	—
190	#431	C₂²×C₁₃₁	1	1	yes	characteristic
191	C₂×#107	—	2	2	no	—
192	#423	C₂²×C₁₃₁	1	2	yes	—
193	#419	—	2	2	no	—
194	C₂×C₄×C₈	—	2	1	no	—
195	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
196	#335	C₂²×C₁₃₁	1	4	yes	—
197	#332	C₂²×C₁₃₁	1	8	yes	—
198	C₂×#41	D₄×C₁₃₁	1	1	yes	characteristic
199	C₂²×Q₈	—	2	2	no	—
200	#137	—	4	8	no	—
201	C₂×Q₁₆	—	2	4	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	C₂³×C₁₃₁	1	4	yes	—
210	C₄×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
211	C₄×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	D₄×C₁₃₁	1	1	yes	characteristic
215	C₄×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
216	#117	—	2	1	no	—
217	#117	D₄×C₁₃₁	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
222	#107	—	2	4	no	—
223	#107	—	2	4	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
227	C₄⋊C₈	C₂³×C₁₃₁	1	4	yes	—
228	#103	—	2	4	no	—
229	#103	—	2	4	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	C₂²⋊C₈	—	2	2	no	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
235	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
237	#41	—	2	2	no	—
238	#41	D₈×C₁₃₁	1	4	yes	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	4	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
252	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
253	Q₁₆	—	4	8	no	—
254	SD₁₆	—	4	8	no	—
255	M₁₆	—	4	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
269	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
278	C₄²	#41×C₁₃₁	1	1	yes	characteristic
279	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	#121×C₁₃₁	1	1	yes	characteristic
282	Q₈	—	2	1	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	8	4	no	—
286	Q₈	C₂×D₈×C₁₃₁	1	2	yes	—
287	D₄	—	2	4	no	—
288	D₄	—	2	1	no	—
289	D₄	C₂×D₈×C₁₃₁	1	2	yes	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
301	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
302	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
304	C₂×C₄	#135×C₁₃₁	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	4	no	—
310	C₈	—	4	2	no	—
311	C₈	#137×C₁₃₁	1	4	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	#466×C₁₃₁	1	2	yes	—
318	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	4	no	—
325	C₄	#466×C₁₃₁	1	2	yes	—
326	C₄	#473×C₁₃₁	1	2	yes	—
327	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
328	C₄	C₂×#135×C₁₃₁	1	1	yes	characteristic
329	C₄	C₂×#137×C₁₃₁	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	4	2	no	—
332	C₂	#3039×C₁₃₁	1	1	yes	characteristic
333	C₂	C₂×#466×C₁₃₁	1	1	yes	characteristic
334	C₂	C₂×#473×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
prime radical	C₂×#41×C₁₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₁	C₂	semidirect
4. #1674×C₁₃₁	C₂	semidirect
5. #1673×C₁₃₁	C₂	semidirect
6. #1567×C₁₃₁	C₂	semidirect
7. #1546×C₁₃₁	C₂	semidirect
143. C₈×C₁₃₁	#137	semidirect
167. #19271	C₁₃₁	direct (N × H)
168. C₁₃₁	#19271	direct (N × H)
169. C₂×#500	C₂×C₁₃₁	semidirect
172. #1674	C₂×C₁₃₁	semidirect
173. #1673	C₂×C₁₃₁	semidirect
174. #1567	C₂×C₁₃₁	semidirect
175. #1546	C₂×C₁₃₁	semidirect
311. C₈	#137×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₁	2
#3039×C₁₃₁	2
C₈×#41×C₁₃₁	2
#1674×C₁₃₁	2
#1673×C₁₃₁	2
#1567×C₁₃₁	2
#1546×C₁₃₁	2
#19271	131

Subgroups — C131×#19271 #45705155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#19271 #45705155