Subgroups — C₁₃₁×#13237 #45699121

Order 33536.

474 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×SD₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#367×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#358×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#357×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1970×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#1969×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×SD₃₂×C₁₃₁	—	2	2	no	—
11	C₂×SD₃₂×C₁₃₁	—	2	2	no	—
12	C₂×SD₃₂×C₁₃₁	C₄	1	4	yes	complemented
13	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#103×C₁₃₁	—	2	1	no	—
18	C₂×#102×C₁₃₁	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	#391×C₁₃₁	—	2	1	no	—
21	#390×C₁₃₁	—	2	1	no	—
22	#367×C₁₃₁	—	2	2	no	—
23	#358×C₁₃₁	—	2	2	no	—
24	#357×C₁₃₁	—	2	2	no	—
25	C₄×C₁₆×C₁₃₁	—	2	2	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×D₈×C₁₃₁	—	2	1	no	—
35	C₂×D₈×C₁₃₁	—	2	1	no	—
36	C₂×D₈×C₁₃₁	—	2	1	no	—
37	C₂×D₈×C₁₃₁	—	2	1	no	—
38	C₂×D₈×C₁₃₁	—	2	1	no	—
39	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₁₃₁	—	4	2	no	—
45	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
46	SD₃₂×C₁₃₁	—	2	8	no	—
47	C₂×C₁₆×C₁₃₁	—	2	1	no	—
48	C₂×C₁₆×C₁₃₁	—	2	1	no	—
49	C₂×C₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
51	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
52	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
53	#107×C₁₃₁	—	2	1	no	—
54	#107×C₁₃₁	—	2	1	no	—
55	#107×C₁₃₁	—	2	1	no	—
56	#107×C₁₃₁	—	2	1	no	—
57	#103×C₁₃₁	—	4	2	no	—
58	#102×C₁₃₁	—	4	2	no	—
59	C₄×C₈×C₁₃₁	—	2	1	no	—
60	C₄×C₈×C₁₃₁	—	2	1	no	—
61	#95×C₁₃₁	—	4	2	no	—
62	C₂⁴×C₁₃₁	—	4	1	no	—
63	C₂×Q₈×C₁₃₁	—	2	2	no	—
64	C₂×Q₈×C₁₃₁	—	4	1	no	—
65	C₂×Q₈×C₁₃₁	—	4	1	no	—
66	C₂×Q₈×C₁₃₁	—	4	1	no	—
67	C₂×Q₈×C₁₃₁	—	4	1	no	—
68	C₂×Q₈×C₁₃₁	—	4	1	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	C₂²×C₄×C₁₃₁	—	4	1	no	—
76	C₂²×C₄×C₁₃₁	—	4	1	no	—
77	C₂²×C₄×C₁₃₁	—	4	1	no	—
78	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
79	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₁₃₁	—	2	2	no	—
81	Q₁₆×C₁₃₁	—	2	2	no	—
82	Q₁₆×C₁₃₁	—	4	1	no	—
83	Q₁₆×C₁₃₁	—	4	1	no	—
84	D₈×C₁₃₁	—	2	2	no	—
85	D₈×C₁₃₁	—	2	2	no	—
86	D₈×C₁₃₁	—	4	1	no	—
87	D₈×C₁₃₁	—	4	1	no	—
88	C₂×C₈×C₁₃₁	—	2	1	no	—
89	C₂×C₈×C₁₃₁	—	2	1	no	—
90	C₂×C₈×C₁₃₁	D₈	1	2	yes	—
91	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
96	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
97	C₄⋊C₄×C₁₃₁	—	4	1	no	—
98	C₄⋊C₄×C₁₃₁	—	4	1	no	—
99	C₄⋊C₄×C₁₃₁	—	4	1	no	—
100	C₄⋊C₄×C₁₃₁	—	4	1	no	—
101	C₂²⋊C₄×C₁₃₁	—	8	4	no	—
102	C₄²×C₁₃₁	—	2	1	no	—
103	C₄²×C₁₃₁	—	2	1	no	—
104	C₁₆×C₁₃₁	—	2	2	no	—
105	C₁₆×C₁₃₁	C₂²⋊C₄	1	4	yes	complemented
106	C₂³×C₁₃₁	—	4	2	no	—
107	C₂³×C₁₃₁	—	8	1	no	—
108	C₂³×C₁₃₁	—	8	1	no	—
109	C₂³×C₁₃₁	—	8	2	no	—
110	C₂³×C₁₃₁	—	8	2	no	—
111	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
112	Q₈×C₁₃₁	—	4	2	no	—
113	Q₈×C₁₃₁	—	4	2	no	—
114	Q₈×C₁₃₁	—	8	1	no	—
115	Q₈×C₁₃₁	—	8	1	no	—
116	D₄×C₁₃₁	—	4	2	no	—
117	D₄×C₁₃₁	—	4	2	no	—
118	D₄×C₁₃₁	—	8	1	no	—
119	D₄×C₁₃₁	—	8	1	no	—
120	C₂×C₄×C₁₃₁	—	2	2	no	—
121	C₂×C₄×C₁₃₁	—	2	2	no	—
122	C₂×C₄×C₁₃₁	—	4	2	no	—
123	C₂×C₄×C₁₃₁	—	8	1	no	—
124	C₂×C₄×C₁₃₁	—	8	1	no	—
125	C₂×C₄×C₁₃₁	—	8	1	no	—
126	C₂×C₄×C₁₃₁	—	8	1	no	—
127	C₂×C₄×C₁₃₁	—	8	2	no	—
128	C₂×C₄×C₁₃₁	—	8	2	no	—
129	C₂×C₄×C₁₃₁	SD₃₂	1	4	yes	complemented
130	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
136	C₈×C₁₃₁	—	2	2	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	#124	1	1	yes	characteristic
141	C₂²×C₁₃₁	—	8	4	no	—
142	C₂²×C₁₃₁	—	8	2	no	—
143	C₂²×C₁₃₁	—	16	2	no	—
144	C₂²×C₁₃₁	—	16	1	no	—
145	C₂²×C₁₃₁	—	16	1	no	—
146	C₂²×C₁₃₁	C₄×D₈	1	1	yes	characteristic
147	C₂²×C₁₃₁	#466	1	1	yes	characteristic
148	C₂²×C₁₃₁	#493	1	1	yes	characteristic
149	C₂²×C₁₃₁	C₂×SD₃₂	1	2	yes	—
150	C₂²×C₁₃₁	#508	1	2	yes	—
151	C₄×C₁₃₁	—	2	4	no	—
152	C₄×C₁₃₁	—	8	2	no	—
153	C₄×C₁₃₁	—	16	1	no	—
154	C₄×C₁₃₁	—	16	1	no	—
155	C₄×C₁₃₁	#390	1	1	yes	characteristic
156	C₄×C₁₃₁	C₄×D₈	1	1	yes	characteristic
157	C₄×C₁₃₁	#466	1	1	yes	characteristic
158	C₄×C₁₃₁	#495	1	1	yes	characteristic
159	C₂×C₁₃₁	—	16	2	no	—
160	C₂×C₁₃₁	#1970	1	1	yes	characteristic
161	C₂×C₁₃₁	C₄×SD₃₂	1	2	yes	—
162	C₂×C₁₃₁	#2217	1	2	yes	—
163	C₂×C₁₃₁	#2248	1	1	yes	characteristic
164	C₂×C₁₃₁	#2249	1	1	yes	characteristic
165	#13237	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₁	#13237	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×SD₃₂	C₂×C₁₃₁	1	1	yes	characteristic
168	C₂×#367	C₂×C₁₃₁	1	1	yes	characteristic, complemented
169	C₂×#358	C₂×C₁₃₁	1	1	yes	characteristic, complemented
170	C₂×#357	C₂×C₁₃₁	1	1	yes	characteristic
171	C₂×C₄×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
172	#1970	C₂×C₁₃₁	1	1	yes	characteristic
173	#1969	C₂×C₁₃₁	1	1	yes	characteristic, complemented
174	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
175	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
176	C₂×SD₃₂	—	2	2	no	—
177	C₂×SD₃₂	—	2	2	no	—
178	C₂×SD₃₂	C₄×C₁₃₁	1	4	yes	complemented
179	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
180	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
181	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
182	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
183	C₂×#103	—	2	1	no	—
184	C₂×#102	—	2	1	no	—
185	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
186	#391	—	2	1	no	—
187	#390	—	2	1	no	—
188	#367	—	2	2	no	—
189	#358	—	2	2	no	—
190	#357	—	2	2	no	—
191	C₄×C₁₆	—	2	2	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×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
206	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	4	2	no	—
211	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
212	SD₃₂	—	2	8	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
216	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
217	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
218	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#103	—	4	2	no	—
224	#102	—	4	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	4	2	no	—
228	C₂⁴	—	4	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	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	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
246	Q₁₆	—	2	2	no	—
247	Q₁₆	—	2	2	no	—
248	Q₁₆	—	4	1	no	—
249	Q₁₆	—	4	1	no	—
250	D₈	—	2	2	no	—
251	D₈	—	2	2	no	—
252	D₈	—	4	1	no	—
253	D₈	—	4	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	D₈×C₁₃₁	1	2	yes	—
257	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
261	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
262	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	8	4	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	C₂²⋊C₄×C₁₃₁	1	4	yes	complemented
272	C₂³	—	4	2	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	8	1	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	1	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	SD₃₂×C₁₃₁	1	4	yes	complemented
296	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
297	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
298	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
299	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
301	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
302	C₈	—	2	2	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	#124×C₁₃₁	1	1	yes	characteristic
307	C₂²	—	8	2	no	—
308	C₂²	—	8	4	no	—
309	C₂²	—	16	2	no	—
310	C₂²	—	16	1	no	—
311	C₂²	—	16	1	no	—
312	C₂²	C₄×D₈×C₁₃₁	1	1	yes	characteristic
313	C₂²	#466×C₁₃₁	1	1	yes	characteristic
314	C₂²	#493×C₁₃₁	1	1	yes	characteristic
315	C₂²	C₂×SD₃₂×C₁₃₁	1	2	yes	—
316	C₂²	#508×C₁₃₁	1	2	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	16	1	no	—
320	C₄	—	16	1	no	—
321	C₄	#390×C₁₃₁	1	1	yes	characteristic
322	C₄	C₄×D₈×C₁₃₁	1	1	yes	characteristic
323	C₄	#466×C₁₃₁	1	1	yes	characteristic
324	C₄	#495×C₁₃₁	1	1	yes	characteristic
325	C₂	—	16	2	no	—
326	C₂	#1970×C₁₃₁	1	1	yes	characteristic
327	C₂	C₄×SD₃₂×C₁₃₁	1	2	yes	—
328	C₂	#2217×C₁₃₁	1	2	yes	—
329	C₂	#2248×C₁₃₁	1	1	yes	characteristic
330	C₂	#2249×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×D₈
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 ◁ C₂³×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. C₂×#367×C₁₃₁	C₂	semidirect
3. C₂×#358×C₁₃₁	C₂	semidirect
5. C₂×C₄×C₁₆×C₁₃₁	C₂	semidirect
7. #1969×C₁₃₁	C₂	semidirect
12. C₂×SD₃₂×C₁₃₁	C₄	semidirect
105. C₁₆×C₁₃₁	C₂²⋊C₄	semidirect
129. C₂×C₄×C₁₃₁	SD₃₂	semidirect
165. #13237	C₁₃₁	direct (N × H)
166. C₁₃₁	#13237	direct (N × H)
168. C₂×#367	C₂×C₁₃₁	semidirect
169. C₂×#358	C₂×C₁₃₁	semidirect
171. C₂×C₄×C₁₆	C₂×C₁₃₁	semidirect
173. #1969	C₂×C₁₃₁	semidirect
178. C₂×SD₃₂	C₄×C₁₃₁	semidirect
271. C₁₆	C₂²⋊C₄×C₁₃₁	semidirect
295. C₂×C₄	SD₃₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂×C₁₃₁	2
C₂×#367×C₁₃₁	2
C₂×#358×C₁₃₁	2
C₂×#357×C₁₃₁	2
C₂×C₄×C₁₆×C₁₃₁	2
#1970×C₁₃₁	2
#1969×C₁₃₁	2
#13237	131

Subgroups — C131×#13237 #45699121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13237 #45699121