Subgroups — C₁₃₁×#10161 #45696045

Order 33536.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3208×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#3165×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#486×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1965×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1548×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1493×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1278×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#533×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
11	#501×C₁₃₁	—	2	2	no	—
12	#500×C₁₃₁	—	2	1	no	—
13	#498×C₁₃₁	—	2	1	no	—
14	#492×C₁₃₁	—	2	1	no	—
15	#486×C₁₃₁	—	2	2	no	—
16	#486×C₁₃₁	—	2	1	no	—
17	#486×C₁₃₁	—	2	2	no	—
18	#460×C₁₃₁	—	2	2	no	—
19	#450×C₁₃₁	—	2	2	no	—
20	#431×C₁₃₁	C₂²	1	1	yes	characteristic
21	#428×C₁₃₁	—	2	2	no	—
22	#423×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×#102×C₁₃₁	—	2	2	no	—
24	C₂×#100×C₁₃₁	—	2	2	no	—
25	C₄×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
27	#333×C₁₃₁	—	2	2	no	—
28	Q₈⋊C₈×C₁₃₁	—	2	2	no	—
29	C₂²×Q₈×C₁₃₁	—	2	1	no	—
30	C₂²×D₄×C₁₃₁	—	2	1	no	—
31	C₂²×D₄×C₁₃₁	—	2	1	no	—
32	C₂²×D₄×C₁₃₁	D₄	1	2	yes	—
33	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
34	C₂×M₁₆×C₁₃₁	—	2	1	no	—
35	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃₁	—	2	1	no	—
37	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
38	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
39	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
40	C₄⋊Q₈×C₁₃₁	D₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₁₃₁	D₄	1	2	yes	—
42	C₄⋊Q₈×C₁₃₁	D₄	1	1	yes	characteristic
43	#127×C₁₃₁	—	2	1	no	—
44	#127×C₁₃₁	—	2	1	no	—
45	#127×C₁₃₁	—	2	1	no	—
46	#127×C₁₃₁	—	2	1	no	—
47	#125×C₁₃₁	—	2	1	no	—
48	C₂²⋊Q₈×C₁₃₁	—	2	1	no	—
49	C₂²⋊Q₈×C₁₃₁	—	2	1	no	—
50	C₄×Q₈×C₁₃₁	—	2	1	no	—
51	C₄×Q₈×C₁₃₁	—	2	1	no	—
52	#117×C₁₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
55	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
56	#107×C₁₃₁	—	2	4	no	—
57	#106×C₁₃₁	—	2	4	no	—
58	C₄⋊C₈×C₁₃₁	—	2	1	no	—
59	C₄⋊C₈×C₁₃₁	—	2	1	no	—
60	C₄⋊C₈×C₁₃₁	D₄	1	2	yes	—
61	#102×C₁₃₁	—	2	8	no	—
62	#102×C₁₃₁	—	2	4	no	—
63	#100×C₁₃₁	—	4	4	no	—
64	C₂²⋊C₈×C₁₃₁	—	2	1	no	—
65	#97×C₁₃₁	—	2	1	no	—
66	C₄×C₈×C₁₃₁	—	2	1	no	—
67	C₄×C₈×C₁₃₁	—	2	1	no	—
68	C₄×C₈×C₁₃₁	—	2	1	no	—
69	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
70	C₂⁴×C₁₃₁	—	2	2	no	—
71	C₂×Q₈×C₁₃₁	—	2	1	no	—
72	C₂×Q₈×C₁₃₁	—	2	1	no	—
73	C₂×Q₈×C₁₃₁	—	2	2	no	—
74	C₂×Q₈×C₁₃₁	—	4	1	no	—
75	C₂×Q₈×C₁₃₁	—	8	2	no	—
76	C₂×D₄×C₁₃₁	—	2	2	no	—
77	C₂×D₄×C₁₃₁	—	2	2	no	—
78	C₂×D₄×C₁₃₁	—	4	4	no	—
79	C₂×D₄×C₁₃₁	—	4	1	no	—
80	C₂×D₄×C₁₃₁	—	4	1	no	—
81	C₂×D₄×C₁₃₁	—	4	1	no	—
82	C₂×D₄×C₁₃₁	—	4	1	no	—
83	C₂×D₄×C₁₃₁	—	4	2	no	—
84	C₂×D₄×C₁₃₁	—	4	2	no	—
85	C₂×D₄×C₁₃₁	—	8	2	no	—
86	C₂²×C₄×C₁₃₁	—	4	1	no	—
87	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
90	SD₁₆×C₁₃₁	—	8	4	no	—
91	M₁₆×C₁₃₁	—	2	2	no	—
92	M₁₆×C₁₃₁	—	4	2	no	—
93	M₁₆×C₁₃₁	SD₁₆	1	4	yes	complemented
94	C₂×C₈×C₁₃₁	—	2	2	no	—
95	C₂×C₈×C₁₃₁	—	2	2	no	—
96	C₂×C₈×C₁₃₁	—	2	2	no	—
97	C₂×C₈×C₁₃₁	—	4	2	no	—
98	C₂×C₈×C₁₃₁	—	4	1	no	—
99	C₄⋊C₄×C₁₃₁	—	2	2	no	—
100	C₄⋊C₄×C₁₃₁	—	2	1	no	—
101	C₄⋊C₄×C₁₃₁	—	2	1	no	—
102	C₄⋊C₄×C₁₃₁	—	2	2	no	—
103	C₄⋊C₄×C₁₃₁	—	2	1	no	—
104	C₄⋊C₄×C₁₃₁	—	2	1	no	—
105	C₄⋊C₄×C₁₃₁	—	4	1	no	—
106	C₄⋊C₄×C₁₃₁	—	4	1	no	—
107	C₄⋊C₄×C₁₃₁	—	4	1	no	—
108	C₄⋊C₄×C₁₃₁	—	4	1	no	—
109	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
110	C₄²×C₁₃₁	—	4	1	no	—
111	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₁₃₁	—	4	2	no	—
116	C₂³×C₁₃₁	—	4	4	no	—
117	C₂³×C₁₃₁	—	8	2	no	—
118	C₂³×C₁₃₁	—	8	2	no	—
119	C₂³×C₁₃₁	#121	1	1	yes	characteristic
120	Q₈×C₁₃₁	—	4	2	no	—
121	Q₈×C₁₃₁	—	4	2	no	—
122	Q₈×C₁₃₁	—	8	2	no	—
123	Q₈×C₁₃₁	—	8	2	no	—
124	D₄×C₁₃₁	—	4	4	no	—
125	D₄×C₁₃₁	—	4	4	no	—
126	D₄×C₁₃₁	—	8	2	no	—
127	D₄×C₁₃₁	—	8	2	no	—
128	D₄×C₁₃₁	—	8	2	no	—
129	D₄×C₁₃₁	—	8	1	no	—
130	D₄×C₁₃₁	—	8	1	no	—
131	C₂×C₄×C₁₃₁	—	2	2	no	—
132	C₂×C₄×C₁₃₁	—	2	2	no	—
133	C₂×C₄×C₁₃₁	—	4	1	no	—
134	C₂×C₄×C₁₃₁	—	4	1	no	—
135	C₂×C₄×C₁₃₁	—	4	2	no	—
136	C₂×C₄×C₁₃₁	—	8	2	no	—
137	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
143	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
144	C₂×C₄×C₁₃₁	#137	1	2	yes	—
145	C₈×C₁₃₁	—	2	4	no	—
146	C₈×C₁₃₁	—	4	2	no	—
147	C₈×C₁₃₁	—	4	2	no	—
148	C₈×C₁₃₁	—	8	1	no	—
149	C₂²×C₁₃₁	—	2	1	no	—
150	C₂²×C₁₃₁	—	2	1	no	—
151	C₂²×C₁₃₁	—	8	2	no	—
152	C₂²×C₁₃₁	—	8	2	no	—
153	C₂²×C₁₃₁	—	8	2	no	—
154	C₂²×C₁₃₁	—	16	4	no	—
155	C₂²×C₁₃₁	#392	1	1	yes	characteristic
156	C₂²×C₁₃₁	#461	1	2	yes	—
157	C₄×C₁₃₁	—	2	1	no	—
158	C₄×C₁₃₁	—	2	1	no	—
159	C₄×C₁₃₁	—	8	1	no	—
160	C₄×C₁₃₁	—	8	2	no	—
161	C₄×C₁₃₁	—	8	1	no	—
162	C₄×C₁₃₁	#448	1	2	yes	—
163	C₄×C₁₃₁	#455	1	2	yes	—
164	C₄×C₁₃₁	#472	1	2	yes	—
165	C₄×C₁₃₁	#492	1	1	yes	characteristic
166	C₄×C₁₃₁	#497	1	1	yes	characteristic
167	C₂×C₁₃₁	—	2	2	no	—
168	C₂×C₁₃₁	—	16	2	no	—
169	C₂×C₁₃₁	#2004	1	1	yes	characteristic
170	C₂×C₁₃₁	#2013	1	1	yes	characteristic
171	C₂×C₁₃₁	#2018	1	1	yes	characteristic
172	#10161	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₁₃₁	#10161	1	1	yes	characteristic, direct product factor, Sylow
174	#3208	C₂×C₁₃₁	1	1	yes	characteristic
175	#3165	C₂×C₁₃₁	1	1	yes	characteristic, complemented
176	C₂×#486	C₂×C₁₃₁	1	1	yes	characteristic
177	#1965	C₂×C₁₃₁	1	1	yes	characteristic
178	#1548	C₂×C₁₃₁	1	1	yes	characteristic, complemented
179	#1493	C₂×C₁₃₁	1	1	yes	characteristic, complemented
180	#1278	C₂×C₁₃₁	1	1	yes	characteristic, complemented
181	#533	C₂²×C₁₃₁	1	1	yes	characteristic
182	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
183	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
184	#501	—	2	2	no	—
185	#500	—	2	1	no	—
186	#498	—	2	1	no	—
187	#492	—	2	1	no	—
188	#486	—	2	2	no	—
189	#486	—	2	1	no	—
190	#486	—	2	2	no	—
191	#460	—	2	2	no	—
192	#450	—	2	2	no	—
193	#431	C₂²×C₁₃₁	1	1	yes	characteristic
194	#428	—	2	2	no	—
195	#423	C₂²×C₁₃₁	1	1	yes	characteristic
196	C₂×#102	—	2	2	no	—
197	C₂×#100	—	2	2	no	—
198	C₄×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
200	#333	—	2	2	no	—
201	Q₈⋊C₈	—	2	2	no	—
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	D₄×C₁₃₁	1	2	yes	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
209	C₂²×C₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	D₄×C₁₃₁	1	2	yes	—
214	C₄⋊Q₈	D₄×C₁₃₁	1	1	yes	characteristic
215	C₄⋊Q₈	D₄×C₁₃₁	1	1	yes	characteristic
216	#127	—	2	1	no	—
217	#127	—	2	1	no	—
218	#127	—	2	1	no	—
219	#127	—	2	1	no	—
220	#125	—	2	1	no	—
221	C₂²⋊Q₈	—	2	1	no	—
222	C₂²⋊Q₈	—	2	1	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	2	1	no	—
225	#117	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
229	#107	—	2	4	no	—
230	#106	—	2	4	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	D₄×C₁₃₁	1	2	yes	—
234	#102	—	2	8	no	—
235	#102	—	2	4	no	—
236	#100	—	4	4	no	—
237	C₂²⋊C₈	—	2	1	no	—
238	#97	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
243	C₂⁴	—	2	2	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	8	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	8	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
262	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
263	SD₁₆	—	8	4	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	2	no	—
266	M₁₆	SD₁₆×C₁₃₁	1	4	yes	complemented
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	4	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
288	C₂³	—	4	2	no	—
289	C₂³	—	4	4	no	—
290	C₂³	—	8	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	#121×C₁₃₁	1	1	yes	characteristic
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	Q₈	—	8	2	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	1	no	—
303	D₄	—	8	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
311	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
312	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
313	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
314	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
315	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
316	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	2	yes	—
317	C₂×C₄	#137×C₁₃₁	1	2	yes	—
318	C₈	—	2	4	no	—
319	C₈	—	4	2	no	—
320	C₈	—	4	2	no	—
321	C₈	—	8	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	#392×C₁₃₁	1	1	yes	characteristic
329	C₂²	#461×C₁₃₁	1	2	yes	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	1	no	—
335	C₄	#448×C₁₃₁	1	2	yes	—
336	C₄	#455×C₁₃₁	1	2	yes	—
337	C₄	#472×C₁₃₁	1	2	yes	—
338	C₄	#492×C₁₃₁	1	1	yes	characteristic
339	C₄	#497×C₁₃₁	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	16	2	no	—
342	C₂	#2004×C₁₃₁	1	1	yes	characteristic
343	C₂	#2013×C₁₃₁	1	1	yes	characteristic
344	C₂	#2018×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₂²
LCS	G ▷ C₂×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
2. #3165×C₁₃₁	C₂	semidirect
5. #1548×C₁₃₁	C₂	semidirect
6. #1493×C₁₃₁	C₂	semidirect
7. #1278×C₁₃₁	C₂	semidirect
93. M₁₆×C₁₃₁	SD₁₆	semidirect
172. #10161	C₁₃₁	direct (N × H)
173. C₁₃₁	#10161	direct (N × H)
175. #3165	C₂×C₁₃₁	semidirect
178. #1548	C₂×C₁₃₁	semidirect
179. #1493	C₂×C₁₃₁	semidirect
180. #1278	C₂×C₁₃₁	semidirect
266. M₁₆	SD₁₆×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3208×C₁₃₁	2
#3165×C₁₃₁	2
C₂×#486×C₁₃₁	2
#1965×C₁₃₁	2
#1548×C₁₃₁	2
#1493×C₁₃₁	2
#1278×C₁₃₁	2
#10161	131

Subgroups — C131×#10161 #45696045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#10161 #45696045