Subgroups — C₁₃₁×#11365 #45697249

Order 33536.

634 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#343×C₁₃₁	C₂	1	2	yes	maximal
3	C₂×#341×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1736×C₁₃₁	C₂	1	2	yes	maximal
5	#1729×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₃₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#107×C₁₃₁	—	2	2	no	—
12	C₂×#107×C₁₃₁	—	2	2	no	—
13	C₂×#107×C₁₃₁	C₄	1	4	yes	complemented
14	#343×C₁₃₁	—	2	4	no	—
15	#341×C₁₃₁	—	2	2	no	—
16	#340×C₁₃₁	—	2	4	no	—
17	#336×C₁₃₁	—	2	2	no	—
18	C₂³×C₄×C₁₃₁	—	2	2	no	—
19	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₃₁	—	2	1	no	—
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	D₄	1	2	yes	—
29	C₂×M₁₆×C₁₃₁	Q₈	1	4	yes	—
30	C₂²×C₈×C₁₃₁	—	2	2	no	—
31	C₂²×C₈×C₁₃₁	—	2	2	no	—
32	C₂²×C₈×C₁₃₁	—	2	1	no	—
33	C₂²×C₈×C₁₃₁	—	2	2	no	—
34	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
37	#117×C₁₃₁	—	2	2	no	—
38	#117×C₁₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₁	—	2	2	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	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
47	C₂×C₄²×C₁₃₁	—	2	1	no	—
48	#107×C₁₃₁	—	2	8	no	—
49	C₂⁴×C₁₃₁	#41	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	—	2	2	no	—
51	C₂²×C₄×C₁₃₁	—	2	4	no	—
52	C₂²×C₄×C₁₃₁	—	4	2	no	—
53	C₂²×C₄×C₁₃₁	—	4	2	no	—
54	C₂²×C₄×C₁₃₁	—	8	4	no	—
55	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
62	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
63	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
65	M₁₆×C₁₃₁	—	2	4	no	—
66	M₁₆×C₁₃₁	—	2	4	no	—
67	M₁₆×C₁₃₁	—	2	4	no	—
68	M₁₆×C₁₃₁	—	4	2	no	—
69	M₁₆×C₁₃₁	—	4	2	no	—
70	M₁₆×C₁₃₁	—	4	2	no	—
71	C₂×C₈×C₁₃₁	—	2	4	no	—
72	C₂×C₈×C₁₃₁	—	2	2	no	—
73	C₂×C₈×C₁₃₁	—	2	2	no	—
74	C₂×C₈×C₁₃₁	—	2	4	no	—
75	C₂×C₈×C₁₃₁	—	2	2	no	—
76	C₂×C₈×C₁₃₁	—	2	4	no	—
77	C₂×C₈×C₁₃₁	—	4	1	no	—
78	C₂×C₈×C₁₃₁	—	4	2	no	—
79	C₂×C₈×C₁₃₁	—	4	1	no	—
80	C₂×C₈×C₁₃₁	—	4	2	no	—
81	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
82	C₄⋊C₄×C₁₃₁	—	2	4	no	—
83	C₄⋊C₄×C₁₃₁	—	2	4	no	—
84	C₄⋊C₄×C₁₃₁	—	2	4	no	—
85	C₄⋊C₄×C₁₃₁	—	4	2	no	—
86	C₄⋊C₄×C₁₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
88	C₄²×C₁₃₁	—	4	1	no	—
89	C₄²×C₁₃₁	—	4	1	no	—
90	C₂³×C₁₃₁	—	2	2	no	—
91	C₂³×C₁₃₁	—	2	2	no	—
92	C₂³×C₁₃₁	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₁₃₁	#121	1	1	yes	characteristic
94	C₂³×C₁₃₁	#124	1	1	yes	characteristic
95	C₂³×C₁₃₁	#125	1	2	yes	—
96	C₂³×C₁₃₁	#126	1	2	yes	—
97	C₂×C₄×C₁₃₁	—	4	4	no	—
98	C₂×C₄×C₁₃₁	—	4	2	no	—
99	C₂×C₄×C₁₃₁	—	4	2	no	—
100	C₂×C₄×C₁₃₁	—	4	2	no	—
101	C₂×C₄×C₁₃₁	—	8	4	no	—
102	C₂×C₄×C₁₃₁	—	8	4	no	—
103	C₂×C₄×C₁₃₁	—	8	4	no	—
104	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	#117	1	2	yes	—
106	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₃₁	#123	1	1	yes	characteristic
113	C₂×C₄×C₁₃₁	#123	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	#123	1	2	yes	—
115	C₂×C₄×C₁₃₁	#123	1	2	yes	—
116	C₂×C₄×C₁₃₁	#123	1	2	yes	—
117	C₂×C₄×C₁₃₁	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	#125	1	2	yes	—
119	C₂×C₄×C₁₃₁	#126	1	2	yes	—
120	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
121	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
122	C₈×C₁₃₁	—	2	4	no	—
123	C₈×C₁₃₁	—	4	2	no	—
124	C₈×C₁₃₁	—	4	4	no	—
125	C₂²×C₁₃₁	—	2	4	no	—
126	C₂²×C₁₃₁	—	2	4	no	—
127	C₂²×C₁₃₁	—	2	1	no	—
128	C₂²×C₁₃₁	—	2	1	no	—
129	C₂²×C₁₃₁	—	2	1	no	—
130	C₂²×C₁₃₁	—	2	1	no	—
131	C₂²×C₁₃₁	#387	1	2	yes	—
132	C₂²×C₁₃₁	#390	1	1	yes	characteristic
133	C₂²×C₁₃₁	#397	1	1	yes	characteristic
134	C₂²×C₁₃₁	#398	1	1	yes	characteristic
135	C₂²×C₁₃₁	#400	1	2	yes	—
136	C₂²×C₁₃₁	#472	1	2	yes	—
137	C₂²×C₁₃₁	#473	1	2	yes	—
138	C₄×C₁₃₁	—	8	2	no	—
139	C₄×C₁₃₁	—	8	4	no	—
140	C₄×C₁₃₁	#381	1	1	yes	characteristic
141	C₄×C₁₃₁	#387	1	2	yes	—
142	C₄×C₁₃₁	#391	1	1	yes	characteristic
143	C₄×C₁₃₁	#397	1	1	yes	characteristic
144	C₄×C₁₃₁	#398	1	1	yes	characteristic
145	C₄×C₁₃₁	#399	1	2	yes	—
146	C₂×C₁₃₁	—	2	2	no	—
147	C₂×C₁₃₁	—	2	2	no	—
148	C₂×C₁₃₁	#1445	1	1	yes	characteristic
149	C₂×C₁₃₁	#1978	1	1	yes	characteristic
150	C₂×C₁₃₁	#1979	1	1	yes	characteristic
151	C₂×C₁₃₁	#2078	1	2	yes	—
152	C₂×C₁₃₁	#2089	1	2	yes	—
153	#11365	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₃₁	#11365	1	1	yes	characteristic, direct product factor, Sylow
155	C₂²×#107	C₂×C₁₃₁	1	1	yes	characteristic
156	C₂×#343	C₂×C₁₃₁	1	2	yes	—
157	C₂×#341	C₂×C₁₃₁	1	1	yes	characteristic
158	#1736	C₂×C₁₃₁	1	2	yes	—
159	#1729	C₂×C₁₃₁	1	1	yes	characteristic
160	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
161	C₂²×M₁₆	C₂²×C₁₃₁	1	2	yes	—
162	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
163	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
165	C₂×#107	—	2	2	no	—
166	C₂×#107	—	2	2	no	—
167	C₂×#107	C₄×C₁₃₁	1	4	yes	complemented
168	#343	—	2	4	no	—
169	#341	—	2	2	no	—
170	#340	—	2	4	no	—
171	#336	—	2	2	no	—
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
174	C₂×M₁₆	—	2	1	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	D₄×C₁₃₁	1	2	yes	—
183	C₂×M₁₆	Q₈×C₁₃₁	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
189	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
190	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
199	C₂×C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	#107	—	2	8	no	—
203	C₂⁴	#41×C₁₃₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	8	4	no	—
209	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
215	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
216	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
217	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
218	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₄×D₄×C₁₃₁	1	1	yes	characteristic
247	C₂³	#121×C₁₃₁	1	1	yes	characteristic
248	C₂³	#124×C₁₃₁	1	1	yes	characteristic
249	C₂³	#125×C₁₃₁	1	2	yes	—
250	C₂³	#126×C₁₃₁	1	2	yes	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
259	C₂×C₄	#117×C₁₃₁	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
261	C₂×C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
262	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
264	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
265	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
266	C₂×C₄	#123×C₁₃₁	1	2	yes	—
267	C₂×C₄	#123×C₁₃₁	1	1	yes	characteristic
268	C₂×C₄	#123×C₁₃₁	1	1	yes	characteristic
269	C₂×C₄	#123×C₁₃₁	1	2	yes	—
270	C₂×C₄	#123×C₁₃₁	1	2	yes	—
271	C₂×C₄	#124×C₁₃₁	1	1	yes	characteristic
272	C₂×C₄	#125×C₁₃₁	1	2	yes	—
273	C₂×C₄	#126×C₁₃₁	1	2	yes	—
274	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
275	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	4	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	#387×C₁₃₁	1	2	yes	—
286	C₂²	#390×C₁₃₁	1	1	yes	characteristic
287	C₂²	#397×C₁₃₁	1	1	yes	characteristic
288	C₂²	#398×C₁₃₁	1	1	yes	characteristic
289	C₂²	#400×C₁₃₁	1	2	yes	—
290	C₂²	#472×C₁₃₁	1	2	yes	—
291	C₂²	#473×C₁₃₁	1	2	yes	—
292	C₄	—	8	4	no	—
293	C₄	—	8	2	no	—
294	C₄	#381×C₁₃₁	1	1	yes	characteristic
295	C₄	#387×C₁₃₁	1	2	yes	—
296	C₄	#391×C₁₃₁	1	1	yes	characteristic
297	C₄	#397×C₁₃₁	1	1	yes	characteristic
298	C₄	#398×C₁₃₁	1	1	yes	characteristic
299	C₄	#399×C₁₃₁	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	#1445×C₁₃₁	1	1	yes	characteristic
303	C₂	#1978×C₁₃₁	1	1	yes	characteristic
304	C₂	#1979×C₁₃₁	1	1	yes	characteristic
305	C₂	#2078×C₁₃₁	1	2	yes	—
306	C₂	#2089×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₁
prime radical	C₂⁴×C₁₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#107×C₁₃₁	C₄	semidirect
153. #11365	C₁₃₁	direct (N × H)
154. C₁₃₁	#11365	direct (N × H)
167. C₂×#107	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#107×C₁₃₁	2
C₂×#343×C₁₃₁	2
C₂×#341×C₁₃₁	2
#1736×C₁₃₁	2
#1729×C₁₃₁	2
#11365	131

Subgroups — C131×#11365 #45697249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#11365 #45697249