Subgroups — C₁₃₁×#33794 #45719678

Order 33536.

504 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3169×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#3160×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2254×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2252×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#2240×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#2240×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#2234×C₁₃₁	C₂	1	4	yes	maximal, complemented
8	#2228×C₁₃₁	C₂	1	4	yes	maximal, complemented
9	#2162×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#533×C₁₃₁	—	2	1	no	—
11	#532×C₁₃₁	—	2	1	no	—
12	#500×C₁₃₁	C₂²	1	1	yes	characteristic
13	#500×C₁₃₁	C₂²	1	1	yes	characteristic
14	#500×C₁₃₁	C₂²	1	1	yes	characteristic
15	#499×C₁₃₁	—	2	1	no	—
16	#499×C₁₃₁	C₂²	1	2	yes	—
17	#498×C₁₃₁	—	2	1	no	—
18	#493×C₁₃₁	C₂²	1	1	yes	characteristic
19	#486×C₁₃₁	—	2	1	no	—
20	#466×C₁₃₁	—	2	1	no	—
21	#466×C₁₃₁	C₂²	1	2	yes	—
22	C₄×D₈×C₁₃₁	—	2	1	no	—
23	C₄×D₈×C₁₃₁	C₂²	1	2	yes	—
24	#427×C₁₃₁	—	2	1	no	—
25	#427×C₁₃₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
27	#367×C₁₃₁	C₂²	1	4	yes	—
28	#366×C₁₃₁	C₂²	1	4	yes	—
29	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
30	#357×C₁₃₁	C₂²	1	4	yes	—
31	#357×C₁₃₁	C₂²	1	4	yes	—
32	C₂²⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
33	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₂×#41×C₁₃₁	—	4	1	no	—
36	C₂×D₈×C₁₃₁	—	2	1	no	—
37	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
41	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
42	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
43	#127×C₁₃₁	—	2	1	no	—
44	#125×C₁₃₁	—	2	2	no	—
45	#124×C₁₃₁	—	4	1	no	—
46	C₂²⋊Q₈×C₁₃₁	—	4	1	no	—
47	C₂²⋊Q₈×C₁₃₁	—	4	1	no	—
48	#121×C₁₃₁	—	2	2	no	—
49	#121×C₁₃₁	—	4	1	no	—
50	C₄×Q₈×C₁₃₁	—	4	1	no	—
51	C₄×Q₈×C₁₃₁	—	4	1	no	—
52	C₄×D₄×C₁₃₁	—	2	2	no	—
53	C₄×D₄×C₁₃₁	—	4	1	no	—
54	#117×C₁₃₁	—	2	2	no	—
55	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
58	#107×C₁₃₁	—	2	1	no	—
59	#107×C₁₃₁	—	2	1	no	—
60	#107×C₁₃₁	—	2	1	no	—
61	#107×C₁₃₁	C₂³	1	2	yes	—
62	#107×C₁₃₁	C₂³	1	2	yes	—
63	#107×C₁₃₁	C₂³	1	2	yes	—
64	#106×C₁₃₁	—	2	2	no	—
65	#106×C₁₃₁	—	2	2	no	—
66	#102×C₁₃₁	—	2	2	no	—
67	#102×C₁₃₁	—	2	2	no	—
68	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
72	#41×C₁₃₁	—	8	4	no	—
73	C₂×Q₈×C₁₃₁	—	4	1	no	—
74	C₂×Q₈×C₁₃₁	—	4	1	no	—
75	C₂×Q₈×C₁₃₁	—	4	1	no	—
76	C₂×D₄×C₁₃₁	—	2	2	no	—
77	C₂×D₄×C₁₃₁	—	4	2	no	—
78	C₂×D₄×C₁₃₁	—	4	1	no	—
79	C₂×D₄×C₁₃₁	—	4	1	no	—
80	C₂²×C₄×C₁₃₁	—	4	2	no	—
81	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
84	D₈×C₁₃₁	—	2	4	no	—
85	D₈×C₁₃₁	—	4	2	no	—
86	C₂×C₈×C₁₃₁	—	2	2	no	—
87	C₂×C₈×C₁₃₁	—	2	2	no	—
88	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
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	—	4	1	no	—
96	C₄⋊C₄×C₁₃₁	—	4	2	no	—
97	C₄⋊C₄×C₁₃₁	—	4	2	no	—
98	C₄⋊C₄×C₁₃₁	—	4	1	no	—
99	C₄⋊C₄×C₁₃₁	—	4	1	no	—
100	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
103	C₄²×C₁₃₁	—	4	2	no	—
104	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
105	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
106	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
108	C₁₆×C₁₃₁	—	2	2	no	—
109	C₁₆×C₁₃₁	—	2	2	no	—
110	C₂³×C₁₃₁	—	4	2	no	—
111	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
112	Q₈×C₁₃₁	—	8	2	no	—
113	Q₈×C₁₃₁	—	8	2	no	—
114	Q₈×C₁₃₁	—	8	2	no	—
115	D₄×C₁₃₁	—	4	2	no	—
116	D₄×C₁₃₁	—	4	2	no	—
117	D₄×C₁₃₁	—	8	4	no	—
118	D₄×C₁₃₁	—	8	2	no	—
119	D₄×C₁₃₁	—	8	2	no	—
120	C₂×C₄×C₁₃₁	—	2	1	no	—
121	C₂×C₄×C₁₃₁	—	2	1	no	—
122	C₂×C₄×C₁₃₁	—	2	2	no	—
123	C₂×C₄×C₁₃₁	—	2	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	—	8	4	no	—
128	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₁₃₁	—	2	1	no	—
135	C₈×C₁₃₁	—	2	1	no	—
136	C₈×C₁₃₁	#137	1	2	yes	—
137	C₈×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
138	C₂²×C₁₃₁	—	2	1	no	—
139	C₂²×C₁₃₁	—	2	1	no	—
140	C₂²×C₁₃₁	—	2	1	no	—
141	C₂²×C₁₃₁	—	8	2	no	—
142	C₂²×C₁₃₁	—	8	2	no	—
143	C₂²×C₁₃₁	—	8	2	no	—
144	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₁₃₁	—	2	1	no	—
146	C₄×C₁₃₁	—	2	1	no	—
147	C₄×C₁₃₁	—	2	1	no	—
148	C₄×C₁₃₁	—	8	2	no	—
149	C₄×C₁₃₁	—	8	2	no	—
150	C₄×C₁₃₁	—	8	2	no	—
151	C₄×C₁₃₁	#508	1	2	yes	—
152	C₄×C₁₃₁	#509	1	2	yes	—
153	C₄×C₁₃₁	#536	1	1	yes	characteristic
154	C₄×C₁₃₁	C₂×#137	1	1	yes	characteristic
155	C₂×C₁₃₁	—	4	1	no	—
156	C₂×C₁₃₁	—	16	2	no	—
157	C₂×C₁₃₁	#3226	1	1	yes	characteristic
158	C₂×C₁₃₁	C₂×#508	1	1	yes	characteristic
159	C₂×C₁₃₁	C₂×#509	1	1	yes	characteristic
160	#33794	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₁₃₁	#33794	1	1	yes	characteristic, direct product factor, Sylow
162	#3169	C₂×C₁₃₁	1	1	yes	characteristic
163	#3160	C₂×C₁₃₁	1	1	yes	characteristic, complemented
164	#2254	C₂×C₁₃₁	1	1	yes	characteristic, complemented
165	#2252	C₂×C₁₃₁	1	1	yes	characteristic, complemented
166	#2240	C₂×C₁₃₁	1	1	yes	characteristic, complemented
167	#2240	C₂×C₁₃₁	1	1	yes	characteristic, complemented
168	#2234	C₂×C₁₃₁	1	4	yes	complemented
169	#2228	C₂×C₁₃₁	1	4	yes	complemented
170	#2162	C₂×C₁₃₁	1	1	yes	characteristic, complemented
171	#533	—	2	1	no	—
172	#532	—	2	1	no	—
173	#500	C₂²×C₁₃₁	1	1	yes	characteristic
174	#500	C₂²×C₁₃₁	1	1	yes	characteristic
175	#500	C₂²×C₁₃₁	1	1	yes	characteristic
176	#499	—	2	1	no	—
177	#499	C₂²×C₁₃₁	1	2	yes	—
178	#498	—	2	1	no	—
179	#493	C₂²×C₁₃₁	1	1	yes	characteristic
180	#486	—	2	1	no	—
181	#466	—	2	1	no	—
182	#466	C₂²×C₁₃₁	1	2	yes	—
183	C₄×D₈	—	2	1	no	—
184	C₄×D₈	C₂²×C₁₃₁	1	2	yes	—
185	#427	—	2	1	no	—
186	#427	C₂²×C₁₃₁	1	2	yes	—
187	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
188	#367	C₂²×C₁₃₁	1	4	yes	—
189	#366	C₂²×C₁₃₁	1	4	yes	—
190	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
191	#357	C₂²×C₁₃₁	1	4	yes	—
192	#357	C₂²×C₁₃₁	1	4	yes	—
193	C₂²⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
194	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
195	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
196	C₂×#41	—	4	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
199	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
201	C₄⋊Q₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	#127	—	2	1	no	—
205	#125	—	2	2	no	—
206	#124	—	4	1	no	—
207	C₂²⋊Q₈	—	4	1	no	—
208	C₂²⋊Q₈	—	4	1	no	—
209	#121	—	2	2	no	—
210	#121	—	4	1	no	—
211	C₄×Q₈	—	4	1	no	—
212	C₄×Q₈	—	4	1	no	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	—	4	1	no	—
215	#117	—	2	2	no	—
216	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
217	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
218	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	2	1	no	—
222	#107	C₂³×C₁₃₁	1	2	yes	—
223	#107	C₂³×C₁₃₁	1	2	yes	—
224	#107	C₂³×C₁₃₁	1	2	yes	—
225	#106	—	2	2	no	—
226	#106	—	2	2	no	—
227	#102	—	2	2	no	—
228	#102	—	2	2	no	—
229	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
230	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
233	#41	—	8	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
243	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
245	D₈	—	2	4	no	—
246	D₈	—	4	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₄²	—	4	2	no	—
265	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
266	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
267	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
273	Q₈	—	8	2	no	—
274	Q₈	—	8	2	no	—
275	Q₈	—	8	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	2	no	—
280	D₄	—	8	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#137×C₁₃₁	1	2	yes	—
298	C₈	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	#508×C₁₃₁	1	2	yes	—
313	C₄	#509×C₁₃₁	1	2	yes	—
314	C₄	#536×C₁₃₁	1	1	yes	characteristic
315	C₄	C₂×#137×C₁₃₁	1	1	yes	characteristic
316	C₂	—	4	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#3226×C₁₃₁	1	1	yes	characteristic
319	C₂	C₂×#508×C₁₃₁	1	1	yes	characteristic
320	C₂	C₂×#509×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	#3169×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #3160×C₁₃₁	C₂	semidirect
3. #2254×C₁₃₁	C₂	semidirect
4. #2252×C₁₃₁	C₂	semidirect
5. #2240×C₁₃₁	C₂	semidirect
6. #2240×C₁₃₁	C₂	semidirect
7. #2234×C₁₃₁	C₂	semidirect
8. #2228×C₁₃₁	C₂	semidirect
9. #2162×C₁₃₁	C₂	semidirect
160. #33794	C₁₃₁	direct (N × H)
161. C₁₃₁	#33794	direct (N × H)
163. #3160	C₂×C₁₃₁	semidirect
164. #2254	C₂×C₁₃₁	semidirect
165. #2252	C₂×C₁₃₁	semidirect
166. #2240	C₂×C₁₃₁	semidirect
167. #2240	C₂×C₁₃₁	semidirect
168. #2234	C₂×C₁₃₁	semidirect
169. #2228	C₂×C₁₃₁	semidirect
170. #2162	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3169×C₁₃₁	2
#3160×C₁₃₁	2
#2254×C₁₃₁	2
#2252×C₁₃₁	2
#2240×C₁₃₁	2
#2234×C₁₃₁	2
#2228×C₁₃₁	2
#2162×C₁₃₁	2
#33794	131

Subgroups — C131×#33794 #45719678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33794 #45719678