Subgroups — C₁₃₁×#13017 #45698901

Order 33536.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2948×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2140×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1978×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1382×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1379×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	#518×C₁₃₁	—	2	1	no	—
11	C₂×D₁₆×C₁₃₁	—	2	2	no	—
12	C₂×D₁₆×C₁₃₁	—	2	2	no	—
13	C₂×D₁₆×C₁₃₁	C₄	1	4	yes	complemented
14	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
16	#442×C₁₃₁	—	2	2	no	—
17	C₄×D₈×C₁₃₁	—	2	2	no	—
18	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
19	#427×C₁₃₁	C₂²	1	1	yes	characteristic
20	#418×C₁₃₁	—	2	1	no	—
21	#418×C₁₃₁	—	2	1	no	—
22	C₂×#100×C₁₃₁	—	2	1	no	—
23	#405×C₁₃₁	C₂²	1	1	yes	characteristic
24	#361×C₁₃₁	—	2	2	no	—
25	#357×C₁₃₁	—	2	2	no	—
26	C₂²×D₄×C₁₃₁	—	2	1	no	—
27	C₂²×D₄×C₁₃₁	—	2	1	no	—
28	C₂×D₈×C₁₃₁	—	2	1	no	—
29	C₂×D₈×C₁₃₁	—	2	1	no	—
30	C₂×D₈×C₁₃₁	—	2	1	no	—
31	C₂×D₈×C₁₃₁	—	2	1	no	—
32	C₂×D₈×C₁₃₁	—	2	1	no	—
33	C₂×D₈×C₁₃₁	—	2	1	no	—
34	C₂×D₈×C₁₃₁	—	2	1	no	—
35	C₂×D₈×C₁₃₁	—	2	1	no	—
36	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
37	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	complemented
38	C₂×D₈×C₁₃₁	D₄	1	2	yes	—
39	C₂×D₈×C₁₃₁	D₄	1	2	yes	—
40	C₂×M₁₆×C₁₃₁	—	2	1	no	—
41	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₁₃₁	—	2	4	no	—
44	C₄×D₄×C₁₃₁	—	4	2	no	—
45	#117×C₁₃₁	—	2	1	no	—
46	#117×C₁₃₁	D₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₃₁	—	4	2	no	—
48	D₁₆×C₁₃₁	—	2	8	no	—
49	M₃₂×C₁₃₁	—	2	1	no	—
50	M₃₂×C₁₃₁	—	2	1	no	—
51	C₂×C₁₆×C₁₃₁	—	2	1	no	—
52	C₂×C₁₆×C₁₃₁	—	2	1	no	—
53	C₂×C₁₆×C₁₃₁	—	2	1	no	—
54	C₂×C₁₆×C₁₃₁	C₂×C₄	1	2	yes	complemented
55	#108×C₁₃₁	—	2	1	no	—
56	#108×C₁₃₁	—	2	1	no	—
57	#107×C₁₃₁	D₄	1	2	yes	—
58	#106×C₁₃₁	—	2	1	no	—
59	#102×C₁₃₁	—	2	4	no	—
60	#102×C₁₃₁	—	4	2	no	—
61	#100×C₁₃₁	—	4	2	no	—
62	#97×C₁₃₁	—	2	1	no	—
63	C₄×C₈×C₁₃₁	—	2	1	no	—
64	C₂⁴×C₁₃₁	—	4	1	no	—
65	C₂⁴×C₁₃₁	—	4	1	no	—
66	C₂×D₄×C₁₃₁	—	2	2	no	—
67	C₂×D₄×C₁₃₁	—	2	2	no	—
68	C₂×D₄×C₁₃₁	—	2	2	no	—
69	C₂×D₄×C₁₃₁	—	2	2	no	—
70	C₂×D₄×C₁₃₁	—	4	1	no	—
71	C₂×D₄×C₁₃₁	—	4	1	no	—
72	C₂×D₄×C₁₃₁	—	4	1	no	—
73	C₂×D₄×C₁₃₁	—	4	1	no	—
74	C₂×D₄×C₁₃₁	—	4	1	no	—
75	C₂×D₄×C₁₃₁	—	4	1	no	—
76	C₂×D₄×C₁₃₁	—	4	1	no	—
77	C₂×D₄×C₁₃₁	—	4	1	no	—
78	C₂²×C₄×C₁₃₁	—	4	4	no	—
79	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
80	D₈×C₁₃₁	—	2	2	no	—
81	D₈×C₁₃₁	—	2	2	no	—
82	D₈×C₁₃₁	—	2	2	no	—
83	D₈×C₁₃₁	—	2	2	no	—
84	D₈×C₁₃₁	—	2	2	no	—
85	D₈×C₁₃₁	—	4	1	no	—
86	D₈×C₁₃₁	—	4	1	no	—
87	D₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
88	M₁₆×C₁₃₁	—	2	1	no	—
89	M₁₆×C₁₃₁	—	2	1	no	—
90	M₁₆×C₁₃₁	—	4	1	no	—
91	M₁₆×C₁₃₁	—	4	1	no	—
92	C₂×C₈×C₁₃₁	—	2	1	no	—
93	C₂×C₈×C₁₃₁	—	4	1	no	—
94	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₁	—	2	2	no	—
101	C₄⋊C₄×C₁₃₁	D₈	1	2	yes	—
102	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
104	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
105	C₄²×C₁₃₁	—	2	1	no	—
106	C₄²×C₁₃₁	—	4	1	no	—
107	C₁₆×C₁₃₁	—	2	2	no	—
108	C₁₆×C₁₃₁	—	4	1	no	—
109	C₂³×C₁₃₁	—	4	2	no	—
110	C₂³×C₁₃₁	—	4	2	no	—
111	C₂³×C₁₃₁	—	4	2	no	—
112	C₂³×C₁₃₁	—	4	2	no	—
113	C₂³×C₁₃₁	—	8	1	no	—
114	C₂³×C₁₃₁	—	8	1	no	—
115	C₂³×C₁₃₁	—	16	2	no	—
116	C₂³×C₁₃₁	—	16	2	no	—
117	C₂³×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	D₄×C₁₃₁	—	2	4	no	—
119	D₄×C₁₃₁	—	4	2	no	—
120	D₄×C₁₃₁	—	4	2	no	—
121	D₄×C₁₃₁	—	4	2	no	—
122	D₄×C₁₃₁	—	4	2	no	—
123	D₄×C₁₃₁	—	4	2	no	—
124	D₄×C₁₃₁	—	8	1	no	—
125	D₄×C₁₃₁	—	8	1	no	—
126	C₂×C₄×C₁₃₁	—	2	2	no	—
127	C₂×C₄×C₁₃₁	—	4	2	no	—
128	C₂×C₄×C₁₃₁	—	16	4	no	—
129	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
134	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₈×C₁₃₁	—	4	1	no	—
136	C₈×C₁₃₁	—	8	1	no	—
137	C₈×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
139	C₈×C₁₃₁	#121	1	1	yes	characteristic
140	C₈×C₁₃₁	#123	1	1	yes	characteristic
141	C₂²×C₁₃₁	—	2	2	no	—
142	C₂²×C₁₃₁	—	4	4	no	—
143	C₂²×C₁₃₁	—	8	4	no	—
144	C₂²×C₁₃₁	—	8	2	no	—
145	C₂²×C₁₃₁	—	16	2	no	—
146	C₂²×C₁₃₁	—	16	2	no	—
147	C₂²×C₁₃₁	—	16	2	no	—
148	C₂²×C₁₃₁	—	16	2	no	—
149	C₂²×C₁₃₁	—	16	2	no	—
150	C₂²×C₁₃₁	#442	1	1	yes	characteristic
151	C₂²×C₁₃₁	#447	1	1	yes	characteristic
152	C₂²×C₁₃₁	#463	1	1	yes	characteristic
153	C₄×C₁₃₁	—	4	2	no	—
154	C₄×C₁₃₁	—	8	2	no	—
155	C₄×C₁₃₁	#386	1	1	yes	characteristic
156	C₄×C₁₃₁	C₄×D₈	1	1	yes	characteristic
157	C₄×C₁₃₁	#449	1	1	yes	characteristic
158	C₄×C₁₃₁	#459	1	1	yes	characteristic
159	C₂×C₁₃₁	—	2	1	no	—
160	C₂×C₁₃₁	—	2	1	no	—
161	C₂×C₁₃₁	—	8	4	no	—
162	C₂×C₁₃₁	—	16	2	no	—
163	C₂×C₁₃₁	#1879	1	1	yes	characteristic
164	C₂×C₁₃₁	#2192	1	2	yes	—
165	#13017	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₁	#13017	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×D₁₆	C₂×C₁₃₁	1	1	yes	characteristic
168	#2948	C₂×C₁₃₁	1	1	yes	characteristic, complemented
169	C₂×#361	C₂×C₁₃₁	1	1	yes	characteristic, complemented
170	#2140	C₂×C₁₃₁	1	1	yes	characteristic, complemented
171	#1978	C₂×C₁₃₁	1	1	yes	characteristic, complemented
172	#1382	C₂×C₁₃₁	1	1	yes	characteristic, complemented
173	#1379	C₂×C₁₃₁	1	1	yes	characteristic, complemented
174	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
175	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
176	#518	—	2	1	no	—
177	C₂×D₁₆	—	2	2	no	—
178	C₂×D₁₆	—	2	2	no	—
179	C₂×D₁₆	C₄×C₁₃₁	1	4	yes	complemented
180	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
181	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
182	#442	—	2	2	no	—
183	C₄×D₈	—	2	2	no	—
184	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
185	#427	C₂²×C₁₃₁	1	1	yes	characteristic
186	#418	—	2	1	no	—
187	#418	—	2	1	no	—
188	C₂×#100	—	2	1	no	—
189	#405	C₂²×C₁₃₁	1	1	yes	characteristic
190	#361	—	2	2	no	—
191	#357	—	2	2	no	—
192	C₂²×D₄	—	2	1	no	—
193	C₂²×D₄	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
203	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	complemented
204	C₂×D₈	D₄×C₁₃₁	1	2	yes	—
205	C₂×D₈	D₄×C₁₃₁	1	2	yes	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
209	C₄×D₄	—	2	4	no	—
210	C₄×D₄	—	4	2	no	—
211	#117	—	2	1	no	—
212	#117	D₄×C₁₃₁	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	4	2	no	—
214	D₁₆	—	2	8	no	—
215	M₃₂	—	2	1	no	—
216	M₃₂	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	—	2	1	no	—
220	C₂×C₁₆	C₂×C₄×C₁₃₁	1	2	yes	complemented
221	#108	—	2	1	no	—
222	#108	—	2	1	no	—
223	#107	D₄×C₁₃₁	1	2	yes	—
224	#106	—	2	1	no	—
225	#102	—	2	4	no	—
226	#102	—	4	2	no	—
227	#100	—	4	2	no	—
228	#97	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₂⁴	—	4	1	no	—
231	C₂⁴	—	4	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	4	1	no	—
252	D₈	—	4	1	no	—
253	D₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
264	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
265	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	D₈×C₁₃₁	1	2	yes	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	4	1	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	—	4	1	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	8	1	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	16	2	no	—
282	C₂³	—	16	2	no	—
283	C₂³	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
284	D₄	—	2	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	1	no	—
291	D₄	—	8	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	4	no	—
295	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
296	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
297	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
299	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
301	C₈	—	4	1	no	—
302	C₈	—	8	1	no	—
303	C₈	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
304	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
305	C₈	#121×C₁₃₁	1	1	yes	characteristic
306	C₈	#123×C₁₃₁	1	1	yes	characteristic
307	C₂²	—	2	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	4	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	16	2	no	—
312	C₂²	—	16	2	no	—
313	C₂²	—	16	2	no	—
314	C₂²	—	16	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	#442×C₁₃₁	1	1	yes	characteristic
317	C₂²	#447×C₁₃₁	1	1	yes	characteristic
318	C₂²	#463×C₁₃₁	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	#386×C₁₃₁	1	1	yes	characteristic
322	C₄	C₄×D₈×C₁₃₁	1	1	yes	characteristic
323	C₄	#449×C₁₃₁	1	1	yes	characteristic
324	C₄	#459×C₁₃₁	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	8	4	no	—
328	C₂	—	16	2	no	—
329	C₂	#1879×C₁₃₁	1	1	yes	characteristic
330	C₂	#2192×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#447
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂²×D₁₆×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ #405×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2948×C₁₃₁	C₂	semidirect
3. C₂×#361×C₁₃₁	C₂	semidirect
4. #2140×C₁₃₁	C₂	semidirect
5. #1978×C₁₃₁	C₂	semidirect
6. #1382×C₁₃₁	C₂	semidirect
7. #1379×C₁₃₁	C₂	semidirect
13. C₂×D₁₆×C₁₃₁	C₄	semidirect
37. C₂×D₈×C₁₃₁	C₂×C₄	semidirect
54. C₂×C₁₆×C₁₃₁	C₂×C₄	semidirect
165. #13017	C₁₃₁	direct (N × H)
166. C₁₃₁	#13017	direct (N × H)
168. #2948	C₂×C₁₃₁	semidirect
169. C₂×#361	C₂×C₁₃₁	semidirect
170. #2140	C₂×C₁₃₁	semidirect
171. #1978	C₂×C₁₃₁	semidirect
172. #1382	C₂×C₁₃₁	semidirect
173. #1379	C₂×C₁₃₁	semidirect
179. C₂×D₁₆	C₄×C₁₃₁	semidirect
203. C₂×D₈	C₂×C₄×C₁₃₁	semidirect
220. C₂×C₁₆	C₂×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×D₁₆×C₁₃₁	2
#2948×C₁₃₁	2
C₂×#361×C₁₃₁	2
#2140×C₁₃₁	2
#1978×C₁₃₁	2
#1382×C₁₃₁	2
#1379×C₁₃₁	2
#13017	131

Subgroups — C131×#13017 #45698901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13017 #45698901