Subgroups — C₁₃₁×#22275 #45708159

Order 33536.

938 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2328×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2327×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1961×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1960×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1880×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#1877×C₁₃₁	C₂	1	4	yes	maximal, complemented
8	#1835×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#1746×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₁₃₁	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
15	#492×C₁₃₁	C₄	1	8	yes	complemented
16	C₂×#103×C₁₃₁	C₂²	1	4	yes	—
17	C₂×#102×C₁₃₁	C₂²	1	4	yes	—
18	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	#391×C₁₃₁	C₂²	1	2	yes	—
21	#390×C₁₃₁	C₂²	1	2	yes	—
22	#387×C₁₃₁	—	2	4	no	—
23	#386×C₁₃₁	—	2	4	no	—
24	#382×C₁₃₁	C₂²	1	1	yes	characteristic
25	#382×C₁₃₁	C₂²	1	1	yes	characteristic
26	#376×C₁₃₁	C₂²	1	1	yes	characteristic
27	#340×C₁₃₁	C₂²	1	4	yes	—
28	#336×C₁₃₁	C₂²	1	2	yes	—
29	#336×C₁₃₁	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
31	C₂²×D₄×C₁₃₁	—	2	2	no	—
32	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₃₁	—	2	2	no	—
34	C₂×SD₁₆×C₁₃₁	—	4	4	no	—
35	C₂×SD₁₆×C₁₃₁	—	4	4	no	—
36	C₂×SD₁₆×C₁₃₁	C₂×C₄	1	16	yes	complemented
37	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
40	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
41	#127×C₁₃₁	—	2	2	no	—
42	#127×C₁₃₁	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₁₃₁	—	4	4	no	—
44	C₄×D₄×C₁₃₁	—	4	4	no	—
45	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
51	C₂×C₄²×C₁₃₁	—	2	2	no	—
52	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
55	#103×C₁₃₁	—	4	4	no	—
56	#102×C₁₃₁	—	4	4	no	—
57	#97×C₁₃₁	—	2	2	no	—
58	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
59	#95×C₁₃₁	—	2	4	no	—
60	#95×C₁₃₁	—	2	2	no	—
61	#95×C₁₃₁	—	2	2	no	—
62	#95×C₁₃₁	D₄	1	4	yes	—
63	C₂⁴×C₁₃₁	—	2	2	no	—
64	C₂×Q₈×C₁₃₁	—	2	2	no	—
65	C₂×Q₈×C₁₃₁	—	2	4	no	—
66	C₂×Q₈×C₁₃₁	—	2	4	no	—
67	C₂×Q₈×C₁₃₁	C₂²×C₄	1	4	yes	—
68	C₂×D₄×C₁₃₁	—	2	8	no	—
69	C₂×D₄×C₁₃₁	—	2	2	no	—
70	C₂×D₄×C₁₃₁	—	2	4	no	—
71	C₂×D₄×C₁₃₁	—	2	4	no	—
72	C₂×D₄×C₁₃₁	—	8	4	no	—
73	C₂×D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₁₃₁	—	2	2	no	—
75	C₂²×C₄×C₁₃₁	—	2	2	no	—
76	C₂²×C₄×C₁₃₁	—	2	2	no	—
77	C₂²×C₄×C₁₃₁	—	4	4	no	—
78	C₂²×C₄×C₁₃₁	—	8	4	no	—
79	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
83	SD₁₆×C₁₃₁	—	4	16	no	—
84	C₂×C₈×C₁₃₁	—	2	2	no	—
85	C₂×C₈×C₁₃₁	—	2	2	no	—
86	C₂×C₈×C₁₃₁	—	2	2	no	—
87	C₂×C₈×C₁₃₁	—	2	2	no	—
88	C₂×C₈×C₁₃₁	—	2	2	no	—
89	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
90	C₄⋊C₄×C₁₃₁	—	2	8	no	—
91	C₄⋊C₄×C₁₃₁	—	2	4	no	—
92	C₄⋊C₄×C₁₃₁	—	2	4	no	—
93	C₄⋊C₄×C₁₃₁	—	4	4	no	—
94	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
95	C₂²⋊C₄×C₁₃₁	—	8	4	no	—
96	C₄²×C₁₃₁	—	2	2	no	—
97	C₄²×C₁₃₁	—	2	2	no	—
98	C₄²×C₁₃₁	—	4	4	no	—
99	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
100	C₄²×C₁₃₁	#41	1	2	yes	—
101	C₂³×C₁₃₁	—	2	4	no	—
102	C₂³×C₁₃₁	—	4	2	no	—
103	C₂³×C₁₃₁	—	4	2	no	—
104	C₂³×C₁₃₁	—	4	4	no	—
105	C₂³×C₁₃₁	—	4	4	no	—
106	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₁₃₁	—	2	8	no	—
108	Q₈×C₁₃₁	—	4	4	no	—
109	D₄×C₁₃₁	—	2	8	no	—
110	D₄×C₁₃₁	—	4	4	no	—
111	D₄×C₁₃₁	—	8	8	no	—
112	C₂×C₄×C₁₃₁	—	2	4	no	—
113	C₂×C₄×C₁₃₁	—	2	4	no	—
114	C₂×C₄×C₁₃₁	—	2	2	no	—
115	C₂×C₄×C₁₃₁	—	2	2	no	—
116	C₂×C₄×C₁₃₁	—	2	4	no	—
117	C₂×C₄×C₁₃₁	—	2	4	no	—
118	C₂×C₄×C₁₃₁	—	4	2	no	—
119	C₂×C₄×C₁₃₁	—	4	2	no	—
120	C₂×C₄×C₁₃₁	—	4	2	no	—
121	C₂×C₄×C₁₃₁	—	4	2	no	—
122	C₂×C₄×C₁₃₁	—	4	4	no	—
123	C₂×C₄×C₁₃₁	—	4	4	no	—
124	C₂×C₄×C₁₃₁	—	8	8	no	—
125	C₂×C₄×C₁₃₁	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
129	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
130	C₈×C₁₃₁	—	2	4	no	—
131	C₈×C₁₃₁	—	4	2	no	—
132	C₂²×C₁₃₁	—	4	4	no	—
133	C₂²×C₁₃₁	—	4	8	no	—
134	C₂²×C₁₃₁	—	8	4	no	—
135	C₂²×C₁₃₁	—	8	2	no	—
136	C₂²×C₁₃₁	—	8	2	no	—
137	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₁₃₁	#535	1	1	yes	characteristic
139	C₂²×C₁₃₁	#535	1	1	yes	characteristic
140	C₂²×C₁₃₁	#577	1	4	yes	—
141	C₄×C₁₃₁	—	2	4	no	—
142	C₄×C₁₃₁	—	4	4	no	—
143	C₄×C₁₃₁	—	4	2	no	—
144	C₄×C₁₃₁	—	4	2	no	—
145	C₄×C₁₃₁	—	8	2	no	—
146	C₄×C₁₃₁	—	8	2	no	—
147	C₄×C₁₃₁	#518	1	2	yes	—
148	C₄×C₁₃₁	#540	1	2	yes	—
149	C₂×C₁₃₁	—	8	4	no	—
150	C₂×C₁₃₁	#2328	1	1	yes	characteristic
151	C₂×C₁₃₁	#2949	1	2	yes	—
152	C₂×C₁₃₁	#3221	1	2	yes	—
153	C₂×C₁₃₁	#3222	1	2	yes	—
154	#22275	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₁₃₁	#22275	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×#492	C₂×C₁₃₁	1	1	yes	characteristic
157	#2328	C₂×C₁₃₁	1	1	yes	characteristic
158	#2327	C₂×C₁₃₁	1	1	yes	characteristic, complemented
159	#1961	C₂×C₁₃₁	1	1	yes	characteristic
160	#1960	C₂×C₁₃₁	1	1	yes	characteristic, complemented
161	#1880	C₂×C₁₃₁	1	4	yes	complemented
162	#1877	C₂×C₁₃₁	1	4	yes	complemented
163	#1835	C₂×C₁₃₁	1	1	yes	characteristic, complemented
164	#1746	C₂×C₁₃₁	1	1	yes	characteristic, complemented
165	C₂²×SD₁₆	C₂²×C₁₃₁	1	4	yes	—
166	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
167	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
169	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
170	#492	C₄×C₁₃₁	1	8	yes	complemented
171	C₂×#103	C₂²×C₁₃₁	1	4	yes	—
172	C₂×#102	C₂²×C₁₃₁	1	4	yes	—
173	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
175	#391	C₂²×C₁₃₁	1	2	yes	—
176	#390	C₂²×C₁₃₁	1	2	yes	—
177	#387	—	2	4	no	—
178	#386	—	2	4	no	—
179	#382	C₂²×C₁₃₁	1	1	yes	characteristic
180	#382	C₂²×C₁₃₁	1	1	yes	characteristic
181	#376	C₂²×C₁₃₁	1	1	yes	characteristic
182	#340	C₂²×C₁₃₁	1	4	yes	—
183	#336	C₂²×C₁₃₁	1	2	yes	—
184	#336	C₂²×C₁₃₁	1	2	yes	—
185	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
186	C₂²×D₄	—	2	2	no	—
187	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
188	C₂³×C₄	—	2	2	no	—
189	C₂×SD₁₆	—	4	4	no	—
190	C₂×SD₁₆	—	4	4	no	—
191	C₂×SD₁₆	C₂×C₄×C₁₃₁	1	16	yes	complemented
192	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
193	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
196	#127	—	2	2	no	—
197	#127	C₂×C₄×C₁₃₁	1	4	yes	—
198	C₄×Q₈	—	4	4	no	—
199	C₄×D₄	—	4	4	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
210	#103	—	4	4	no	—
211	#102	—	4	4	no	—
212	#97	—	2	2	no	—
213	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	4	no	—
217	#95	D₄×C₁₃₁	1	4	yes	—
218	C₂⁴	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	C₂²×C₄×C₁₃₁	1	4	yes	—
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	8	4	no	—
228	C₂×D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	8	4	no	—
234	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
238	SD₁₆	—	4	16	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	4	4	no	—
250	C₂²⋊C₄	—	8	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
255	C₄²	#41×C₁₃₁	1	2	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
262	Q₈	—	2	8	no	—
263	Q₈	—	4	4	no	—
264	D₄	—	2	8	no	—
265	D₄	—	4	4	no	—
266	D₄	—	8	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	C₄×D₄×C₁₃₁	1	4	yes	—
281	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
282	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
283	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
284	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
285	C₈	—	2	4	no	—
286	C₈	—	4	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	8	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
293	C₂²	#535×C₁₃₁	1	1	yes	characteristic
294	C₂²	#535×C₁₃₁	1	1	yes	characteristic
295	C₂²	#577×C₁₃₁	1	4	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	4	4	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	#518×C₁₃₁	1	2	yes	—
303	C₄	#540×C₁₃₁	1	2	yes	—
304	C₂	—	8	4	no	—
305	C₂	#2328×C₁₃₁	1	1	yes	characteristic
306	C₂	#2949×C₁₃₁	1	2	yes	—
307	C₂	#3221×C₁₃₁	1	2	yes	—
308	C₂	#3222×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2327×C₁₃₁	C₂	semidirect
5. #1960×C₁₃₁	C₂	semidirect
6. #1880×C₁₃₁	C₂	semidirect
7. #1877×C₁₃₁	C₂	semidirect
8. #1835×C₁₃₁	C₂	semidirect
9. #1746×C₁₃₁	C₂	semidirect
15. #492×C₁₃₁	C₄	semidirect
36. C₂×SD₁₆×C₁₃₁	C₂×C₄	semidirect
40. C₄⋊Q₈×C₁₃₁	C₂×C₄	semidirect
58. C₄×C₈×C₁₃₁	C₂×C₄	semidirect
154. #22275	C₁₃₁	direct (N × H)
155. C₁₃₁	#22275	direct (N × H)
158. #2327	C₂×C₁₃₁	semidirect
160. #1960	C₂×C₁₃₁	semidirect
161. #1880	C₂×C₁₃₁	semidirect
162. #1877	C₂×C₁₃₁	semidirect
163. #1835	C₂×C₁₃₁	semidirect
164. #1746	C₂×C₁₃₁	semidirect
170. #492	C₄×C₁₃₁	semidirect
191. C₂×SD₁₆	C₂×C₄×C₁₃₁	semidirect
195. C₄⋊Q₈	C₂×C₄×C₁₃₁	semidirect
213. C₄×C₈	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
C₂×#492×C₁₃₁	2
#2328×C₁₃₁	2
#2327×C₁₃₁	2
#1961×C₁₃₁	2
#1960×C₁₃₁	2
#1880×C₁₃₁	2
#1877×C₁₃₁	2
#1835×C₁₃₁	2
#1746×C₁₃₁	2
#22275	131

Subgroups — C131×#22275 #45708159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#22275 #45708159