Subgroups — C₁₃₁×#13260 #45699144

Order 33536.

482 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	C₂×#495×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#362×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#361×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#2141×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#2140×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#1772×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₁₃₁	—	2	1	no	—
10	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂×#124×C₁₃₁	—	2	1	no	—
12	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
14	#495×C₁₃₁	—	2	2	no	—
15	#495×C₁₃₁	C₄	1	4	yes	complemented
16	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
17	#427×C₁₃₁	C₂²	1	1	yes	characteristic
18	#405×C₁₃₁	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	#364×C₁₃₁	—	2	2	no	—
21	#362×C₁₃₁	—	2	2	no	—
22	#361×C₁₃₁	—	2	2	no	—
23	#358×C₁₃₁	—	2	2	no	—
24	#357×C₁₃₁	—	2	2	no	—
25	#339×C₁₃₁	—	2	1	no	—
26	C₂²×Q₈×C₁₃₁	—	2	1	no	—
27	C₂²×D₄×C₁₃₁	—	2	1	no	—
28	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
29	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
30	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
31	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
32	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
33	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
34	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
35	C₂×SD₁₆×C₁₃₁	—	4	1	no	—
36	C₂×SD₁₆×C₁₃₁	—	4	1	no	—
37	C₂×SD₁₆×C₁₃₁	—	4	2	no	—
38	C₂×D₈×C₁₃₁	—	2	1	no	—
39	C₂×D₈×C₁₃₁	—	2	1	no	—
40	C₂×D₈×C₁₃₁	—	2	1	no	—
41	C₂×D₈×C₁₃₁	—	2	1	no	—
42	C₂×D₈×C₁₃₁	—	2	1	no	—
43	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₁₃₁	—	2	1	no	—
45	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
47	#124×C₁₃₁	—	2	4	no	—
48	#124×C₁₃₁	—	4	2	no	—
49	#117×C₁₃₁	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₁₃₁	—	4	2	no	—
51	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
52	M₃₂×C₁₃₁	—	2	1	no	—
53	M₃₂×C₁₃₁	—	2	1	no	—
54	M₃₂×C₁₃₁	—	2	1	no	—
55	M₃₂×C₁₃₁	—	2	1	no	—
56	C₂×C₁₆×C₁₃₁	—	2	1	no	—
57	C₂×C₁₆×C₁₃₁	—	2	1	no	—
58	#108×C₁₃₁	—	2	1	no	—
59	#108×C₁₃₁	D₄	1	2	yes	—
60	#107×C₁₃₁	D₄	1	2	yes	—
61	#106×C₁₃₁	—	2	1	no	—
62	#97×C₁₃₁	—	4	1	no	—
63	C₄×C₈×C₁₃₁	—	4	1	no	—
64	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
66	C₂⁴×C₁₃₁	—	4	1	no	—
67	C₂×Q₈×C₁₃₁	—	2	2	no	—
68	C₂×Q₈×C₁₃₁	—	4	1	no	—
69	C₂×Q₈×C₁₃₁	—	4	1	no	—
70	C₂×Q₈×C₁₃₁	—	4	1	no	—
71	C₂×Q₈×C₁₃₁	—	4	1	no	—
72	C₂×Q₈×C₁₃₁	—	4	1	no	—
73	C₂×D₄×C₁₃₁	—	2	2	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	C₂×D₄×C₁₃₁	—	4	1	no	—
79	C₂²×C₄×C₁₃₁	—	2	1	no	—
80	C₂²×C₄×C₁₃₁	—	4	1	no	—
81	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₁₃₁	—	2	2	no	—
83	Q₁₆×C₁₃₁	—	2	2	no	—
84	Q₁₆×C₁₃₁	—	4	1	no	—
85	Q₁₆×C₁₃₁	—	4	1	no	—
86	SD₁₆×C₁₃₁	—	4	4	no	—
87	SD₁₆×C₁₃₁	—	4	4	no	—
88	D₈×C₁₃₁	—	2	2	no	—
89	D₈×C₁₃₁	—	2	2	no	—
90	D₈×C₁₃₁	—	4	1	no	—
91	D₈×C₁₃₁	—	4	1	no	—
92	M₁₆×C₁₃₁	—	2	2	no	—
93	M₁₆×C₁₃₁	—	4	1	no	—
94	C₂×C₈×C₁₃₁	—	4	1	no	—
95	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
97	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
98	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
99	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
104	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₁	—	2	2	no	—
106	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
107	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
108	C₄²×C₁₃₁	—	2	1	no	—
109	C₄²×C₁₃₁	—	4	1	no	—
110	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
111	C₁₆×C₁₃₁	—	4	1	no	—
112	C₁₆×C₁₃₁	—	4	1	no	—
113	C₂³×C₁₃₁	—	4	2	no	—
114	C₂³×C₁₃₁	—	8	4	no	—
115	C₂³×C₁₃₁	—	8	1	no	—
116	C₂³×C₁₃₁	—	8	1	no	—
117	C₂³×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	Q₈×C₁₃₁	—	4	2	no	—
119	Q₈×C₁₃₁	—	4	2	no	—
120	Q₈×C₁₃₁	—	8	1	no	—
121	Q₈×C₁₃₁	—	8	1	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	—	4	2	no	—
129	C₂×C₄×C₁₃₁	—	4	2	no	—
130	C₂×C₄×C₁₃₁	—	8	1	no	—
131	C₂×C₄×C₁₃₁	—	8	1	no	—
132	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	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	1	yes	characteristic
136	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₈×C₁₃₁	—	4	2	no	—
139	C₈×C₁₃₁	#102	1	2	yes	—
140	C₈×C₁₃₁	#102	1	2	yes	—
141	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
142	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
143	C₈×C₁₃₁	#124	1	1	yes	characteristic
144	C₈×C₁₃₁	#127	1	1	yes	characteristic
145	C₂²×C₁₃₁	—	2	2	no	—
146	C₂²×C₁₃₁	—	8	4	no	—
147	C₂²×C₁₃₁	—	8	2	no	—
148	C₂²×C₁₃₁	—	16	2	no	—
149	C₂²×C₁₃₁	—	16	2	no	—
150	C₂²×C₁₃₁	#418	1	1	yes	characteristic
151	C₂²×C₁₃₁	#459	1	1	yes	characteristic
152	C₂²×C₁₃₁	#460	1	1	yes	characteristic
153	C₄×C₁₃₁	—	4	2	no	—
154	C₄×C₁₃₁	—	8	2	no	—
155	C₄×C₁₃₁	—	8	2	no	—
156	C₄×C₁₃₁	#390	1	1	yes	characteristic
157	C₄×C₁₃₁	C₂×#102	1	1	yes	characteristic
158	C₄×C₁₃₁	#459	1	1	yes	characteristic
159	C₄×C₁₃₁	#460	1	1	yes	characteristic
160	C₂×C₁₃₁	—	2	1	no	—
161	C₂×C₁₃₁	—	2	1	no	—
162	C₂×C₁₃₁	—	16	2	no	—
163	C₂×C₁₃₁	#1982	1	1	yes	characteristic
164	C₂×C₁₃₁	#2212	1	2	yes	—
165	#13260	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₁	#13260	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#495	C₂×C₁₃₁	1	1	yes	characteristic
168	C₂×#364	C₂×C₁₃₁	1	1	yes	characteristic, complemented
169	C₂×#362	C₂×C₁₃₁	1	1	yes	characteristic, complemented
170	C₂×#361	C₂×C₁₃₁	1	1	yes	characteristic
171	#2141	C₂×C₁₃₁	1	1	yes	characteristic, complemented
172	#2140	C₂×C₁₃₁	1	1	yes	characteristic
173	#1772	C₂×C₁₃₁	1	1	yes	characteristic, complemented
174	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
175	C₂²×SD₁₆	—	2	1	no	—
176	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
177	C₂×#124	—	2	1	no	—
178	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
179	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
180	#495	—	2	2	no	—
181	#495	C₄×C₁₃₁	1	4	yes	complemented
182	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
183	#427	C₂²×C₁₃₁	1	1	yes	characteristic
184	#405	—	2	1	no	—
185	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
186	#364	—	2	2	no	—
187	#362	—	2	2	no	—
188	#361	—	2	2	no	—
189	#358	—	2	2	no	—
190	#357	—	2	2	no	—
191	#339	—	2	1	no	—
192	C₂²×Q₈	—	2	1	no	—
193	C₂²×D₄	—	2	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
200	C₂×SD₁₆	—	2	4	no	—
201	C₂×SD₁₆	—	4	1	no	—
202	C₂×SD₁₆	—	4	1	no	—
203	C₂×SD₁₆	—	4	2	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
213	#124	—	2	4	no	—
214	#124	—	4	2	no	—
215	#117	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	4	2	no	—
217	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	#108	—	2	1	no	—
225	#108	D₄×C₁₃₁	1	2	yes	—
226	#107	D₄×C₁₃₁	1	2	yes	—
227	#106	—	2	1	no	—
228	#97	—	4	1	no	—
229	C₄×C₈	—	4	1	no	—
230	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
231	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
232	C₂⁴	—	4	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×D₄	—	2	2	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	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	4	1	no	—
251	Q₁₆	—	4	1	no	—
252	SD₁₆	—	4	4	no	—
253	SD₁₆	—	4	4	no	—
254	D₈	—	2	2	no	—
255	D₈	—	2	2	no	—
256	D₈	—	4	1	no	—
257	D₈	—	4	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
262	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
263	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
264	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
265	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
269	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
270	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	8	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
277	C₁₆	—	4	1	no	—
278	C₁₆	—	4	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	1	no	—
287	Q₈	—	8	1	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	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
299	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
303	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	#102×C₁₃₁	1	2	yes	—
306	C₈	#102×C₁₃₁	1	2	yes	—
307	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
308	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
309	C₈	#124×C₁₃₁	1	1	yes	characteristic
310	C₈	#127×C₁₃₁	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	16	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	#418×C₁₃₁	1	1	yes	characteristic
317	C₂²	#459×C₁₃₁	1	1	yes	characteristic
318	C₂²	#460×C₁₃₁	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	#390×C₁₃₁	1	1	yes	characteristic
323	C₄	C₂×#102×C₁₃₁	1	1	yes	characteristic
324	C₄	#459×C₁₃₁	1	1	yes	characteristic
325	C₄	#460×C₁₃₁	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	16	2	no	—
329	C₂	#1982×C₁₃₁	1	1	yes	characteristic
330	C₂	#2212×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₃₁	C₂	semidirect
3. C₂×#362×C₁₃₁	C₂	semidirect
5. #2141×C₁₃₁	C₂	semidirect
7. #1772×C₁₃₁	C₂	semidirect
15. #495×C₁₃₁	C₄	semidirect
165. #13260	C₁₃₁	direct (N × H)
166. C₁₃₁	#13260	direct (N × H)
168. C₂×#364	C₂×C₁₃₁	semidirect
169. C₂×#362	C₂×C₁₃₁	semidirect
171. #2141	C₂×C₁₃₁	semidirect
173. #1772	C₂×C₁₃₁	semidirect
181. #495	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#495×C₁₃₁	2
C₂×#364×C₁₃₁	2
C₂×#362×C₁₃₁	2
C₂×#361×C₁₃₁	2
#2141×C₁₃₁	2
#2140×C₁₃₁	2
#1772×C₁₃₁	2
#13260	131

Subgroups — C131×#13260 #45699144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13260 #45699144