Subgroups — C₁₃₁×#29792 #45715676

Order 33536.

686 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3095×C₁₃₁	C₂	1	2	yes	maximal
2	C₂×#470×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2905×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2090×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2064×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2047×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1901×C₁₃₁	C₂	1	2	yes	maximal
8	#1811×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
14	#485×C₁₃₁	—	2	4	no	—
15	#484×C₁₃₁	—	2	2	no	—
16	#483×C₁₃₁	—	2	2	no	—
17	#470×C₁₃₁	—	2	2	no	—
18	#470×C₁₃₁	—	2	2	no	—
19	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
20	#428×C₁₃₁	C₂²	1	4	yes	—
21	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
22	#424×C₁₃₁	—	2	2	no	—
23	#424×C₁₃₁	C₂²	1	2	yes	—
24	#424×C₁₃₁	C₂²	1	2	yes	—
25	#423×C₁₃₁	—	2	2	no	—
26	#419×C₁₃₁	C₂²	1	4	yes	—
27	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
28	C₂×#101×C₁₃₁	C₂²	1	4	yes	—
29	#407×C₁₃₁	C₂²	1	2	yes	—
30	#343×C₁₃₁	C₂²	1	4	yes	—
31	#341×C₁₃₁	C₂²	1	2	yes	—
32	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
35	C₂×Q₁₆×C₁₃₁	—	4	2	no	—
36	C₂×M₁₆×C₁₃₁	—	2	1	no	—
37	C₂×M₁₆×C₁₃₁	—	4	2	no	—
38	C₂×M₁₆×C₁₃₁	—	4	2	no	—
39	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
40	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
41	C₂×M₁₆×C₁₃₁	C₂³	1	4	yes	—
42	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃₁	—	2	1	no	—
44	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
47	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
48	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
49	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
50	#117×C₁₃₁	—	2	1	no	—
51	#117×C₁₃₁	—	2	1	no	—
52	#117×C₁₃₁	D₄	1	2	yes	—
53	#117×C₁₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
59	C₂×C₄²×C₁₃₁	—	2	1	no	—
60	#108×C₁₃₁	—	4	2	no	—
61	#107×C₁₃₁	—	2	4	no	—
62	#106×C₁₃₁	—	2	2	no	—
63	#106×C₁₃₁	—	2	4	no	—
64	C₄⋊C₈×C₁₃₁	—	2	4	no	—
65	C₄⋊C₈×C₁₃₁	—	2	4	no	—
66	#103×C₁₃₁	—	2	4	no	—
67	#103×C₁₃₁	—	2	4	no	—
68	#103×C₁₃₁	—	2	4	no	—
69	#101×C₁₃₁	—	4	4	no	—
70	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
71	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
72	C₂⁴×C₁₃₁	#41	1	1	yes	characteristic
73	C₂×Q₈×C₁₃₁	—	2	2	no	—
74	C₂×Q₈×C₁₃₁	—	2	2	no	—
75	C₂×Q₈×C₁₃₁	—	2	4	no	—
76	C₂×Q₈×C₁₃₁	—	2	4	no	—
77	C₂²×C₄×C₁₃₁	—	2	2	no	—
78	C₂²×C₄×C₁₃₁	—	2	2	no	—
79	C₂²×C₄×C₁₃₁	—	2	2	no	—
80	C₂²×C₄×C₁₃₁	—	2	2	no	—
81	C₂²×C₄×C₁₃₁	—	2	2	no	—
82	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
87	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
88	Q₁₆×C₁₃₁	—	4	4	no	—
89	M₁₆×C₁₃₁	—	2	4	no	—
90	M₁₆×C₁₃₁	—	2	4	no	—
91	M₁₆×C₁₃₁	—	2	4	no	—
92	M₁₆×C₁₃₁	—	4	2	no	—
93	M₁₆×C₁₃₁	—	4	2	no	—
94	M₁₆×C₁₃₁	—	4	2	no	—
95	C₂×C₈×C₁₃₁	—	2	2	no	—
96	C₂×C₈×C₁₃₁	—	2	2	no	—
97	C₂×C₈×C₁₃₁	—	2	4	no	—
98	C₂×C₈×C₁₃₁	—	2	2	no	—
99	C₂×C₈×C₁₃₁	—	4	2	no	—
100	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₁₃₁	—	2	2	no	—
103	C₄⋊C₄×C₁₃₁	—	2	2	no	—
104	C₄⋊C₄×C₁₃₁	—	2	2	no	—
105	C₄⋊C₄×C₁₃₁	—	4	2	no	—
106	C₄⋊C₄×C₁₃₁	—	4	2	no	—
107	C₄⋊C₄×C₁₃₁	C₂×D₄	1	4	yes	—
108	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
109	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
110	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
111	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
112	C₄²×C₁₃₁	—	2	2	no	—
113	C₄²×C₁₃₁	—	2	2	no	—
114	C₂³×C₁₃₁	—	2	2	no	—
115	C₂³×C₁₃₁	—	4	2	no	—
116	C₂³×C₁₃₁	#121	1	1	yes	characteristic
117	C₂³×C₁₃₁	#121	1	1	yes	characteristic
118	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
119	Q₈×C₁₃₁	—	4	2	no	—
120	Q₈×C₁₃₁	—	4	2	no	—
121	Q₈×C₁₃₁	—	4	4	no	—
122	C₂×C₄×C₁₃₁	—	2	2	no	—
123	C₂×C₄×C₁₃₁	—	2	1	no	—
124	C₂×C₄×C₁₃₁	—	2	1	no	—
125	C₂×C₄×C₁₃₁	—	2	1	no	—
126	C₂×C₄×C₁₃₁	—	2	1	no	—
127	C₂×C₄×C₁₃₁	—	2	2	no	—
128	C₂×C₄×C₁₃₁	—	2	2	no	—
129	C₂×C₄×C₁₃₁	—	2	4	no	—
130	C₂×C₄×C₁₃₁	—	4	2	no	—
131	C₂×C₄×C₁₃₁	—	4	2	no	—
132	C₂×C₄×C₁₃₁	—	4	4	no	—
133	C₂×C₄×C₁₃₁	—	8	2	no	—
134	C₂×C₄×C₁₃₁	—	8	2	no	—
135	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
140	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
141	C₈×C₁₃₁	—	2	2	no	—
142	C₈×C₁₃₁	—	2	2	no	—
143	C₈×C₁₃₁	—	4	2	no	—
144	C₈×C₁₃₁	—	4	4	no	—
145	C₂²×C₁₃₁	—	2	2	no	—
146	C₂²×C₁₃₁	—	2	1	no	—
147	C₂²×C₁₃₁	—	2	1	no	—
148	C₂²×C₁₃₁	—	2	2	no	—
149	C₂²×C₁₃₁	—	2	2	no	—
150	C₂²×C₁₃₁	—	8	2	no	—
151	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
152	C₂²×C₁₃₁	#539	1	2	yes	—
153	C₄×C₁₃₁	—	2	1	no	—
154	C₄×C₁₃₁	—	2	1	no	—
155	C₄×C₁₃₁	—	4	4	no	—
156	C₄×C₁₃₁	—	8	2	no	—
157	C₄×C₁₃₁	—	8	2	no	—
158	C₄×C₁₃₁	#534	1	1	yes	characteristic
159	C₄×C₁₃₁	#536	1	1	yes	characteristic
160	C₄×C₁₃₁	#537	1	2	yes	—
161	C₂×C₁₃₁	—	2	2	no	—
162	C₂×C₁₃₁	—	4	1	no	—
163	C₂×C₁₃₁	—	4	1	no	—
164	C₂×C₁₃₁	#2618	1	1	yes	characteristic
165	C₂×C₁₃₁	#3069	1	2	yes	—
166	#29792	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₁₃₁	#29792	1	1	yes	characteristic, direct product factor, Sylow
168	#3095	C₂×C₁₃₁	1	2	yes	—
169	C₂×#470	C₂×C₁₃₁	1	1	yes	characteristic
170	#2905	C₂×C₁₃₁	1	1	yes	characteristic
171	#2090	C₂×C₁₃₁	1	2	yes	complemented
172	#2064	C₂×C₁₃₁	1	4	yes	complemented
173	#2047	C₂×C₁₃₁	1	2	yes	complemented
174	#1901	C₂×C₁₃₁	1	2	yes	—
175	#1811	C₂×C₁₃₁	1	1	yes	characteristic
176	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
177	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
178	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
179	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
180	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
181	#485	—	2	4	no	—
182	#484	—	2	2	no	—
183	#483	—	2	2	no	—
184	#470	—	2	2	no	—
185	#470	—	2	2	no	—
186	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
187	#428	C₂²×C₁₃₁	1	4	yes	—
188	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
189	#424	—	2	2	no	—
190	#424	C₂²×C₁₃₁	1	2	yes	—
191	#424	C₂²×C₁₃₁	1	2	yes	—
192	#423	—	2	2	no	—
193	#419	C₂²×C₁₃₁	1	4	yes	—
194	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
195	C₂×#101	C₂²×C₁₃₁	1	4	yes	—
196	#407	C₂²×C₁₃₁	1	2	yes	—
197	#343	C₂²×C₁₃₁	1	4	yes	—
198	#341	C₂²×C₁₃₁	1	2	yes	—
199	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
200	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	4	2	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	4	2	no	—
205	C₂×M₁₆	—	4	2	no	—
206	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
207	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
208	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
209	C₂×M₁₆	C₂³×C₁₃₁	1	4	yes	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	#117	—	2	1	no	—
218	#117	—	2	1	no	—
219	#117	D₄×C₁₃₁	1	2	yes	—
220	#117	C₂³×C₁₃₁	1	2	yes	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₄²	—	2	1	no	—
227	#108	—	4	2	no	—
228	#107	—	2	4	no	—
229	#106	—	2	2	no	—
230	#106	—	2	4	no	—
231	C₄⋊C₈	—	2	4	no	—
232	C₄⋊C₈	—	2	4	no	—
233	#103	—	2	4	no	—
234	#103	—	2	4	no	—
235	#103	—	2	4	no	—
236	#101	—	4	4	no	—
237	C₂²⋊C₈	—	2	2	no	—
238	C₂²⋊C₈	—	2	2	no	—
239	C₂⁴	#41×C₁₃₁	1	1	yes	characteristic
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
253	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
254	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
255	Q₁₆	—	4	4	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	2	4	no	—
258	M₁₆	—	2	4	no	—
259	M₁₆	—	4	2	no	—
260	M₁₆	—	4	2	no	—
261	M₁₆	—	4	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	C₂×D₄×C₁₃₁	1	4	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	#121×C₁₃₁	1	1	yes	characteristic
284	C₂³	#121×C₁₃₁	1	1	yes	characteristic
285	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
303	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
306	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
307	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
319	C₂²	#539×C₁₃₁	1	2	yes	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	#534×C₁₃₁	1	1	yes	characteristic
326	C₄	#536×C₁₃₁	1	1	yes	characteristic
327	C₄	#537×C₁₃₁	1	2	yes	—
328	C₂	—	2	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	4	1	no	—
331	C₂	#2618×C₁₃₁	1	1	yes	characteristic
332	C₂	#3069×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂⁴×C₁₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. #2090×C₁₃₁	C₂	semidirect
5. #2064×C₁₃₁	C₂	semidirect
6. #2047×C₁₃₁	C₂	semidirect
166. #29792	C₁₃₁	direct (N × H)
167. C₁₃₁	#29792	direct (N × H)
171. #2090	C₂×C₁₃₁	semidirect
172. #2064	C₂×C₁₃₁	semidirect
173. #2047	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3095×C₁₃₁	2
C₂×#470×C₁₃₁	2
#2905×C₁₃₁	2
#2090×C₁₃₁	2
#2064×C₁₃₁	2
#2047×C₁₃₁	2
#1901×C₁₃₁	2
#1811×C₁₃₁	2
#29792	131

Subgroups — C131×#29792 #45715676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29792 #45715676