Subgroups — C₁₃₁×#7719 #45693603

Order 33536.

528 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2858×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1839×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#1831×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1270×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊Q₈×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂×#127×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×#124×C₁₃₁	—	2	1	no	—
8	C₂×#124×C₁₃₁	—	2	1	no	—
9	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
10	#390×C₁₃₁	—	2	1	no	—
11	#390×C₁₃₁	—	2	1	no	—
12	#390×C₁₃₁	C₄	1	2	yes	—
13	#390×C₁₃₁	C₂²	1	2	yes	—
14	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
15	#336×C₁₃₁	—	2	1	no	—
16	#336×C₁₃₁	—	2	1	no	—
17	#336×C₁₃₁	—	2	1	no	—
18	#336×C₁₃₁	C₂²	1	2	yes	—
19	#336×C₁₃₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃₁	—	2	1	no	—
21	C₂²×D₄×C₁₃₁	—	2	1	no	—
22	C₂²×D₄×C₁₃₁	—	2	1	no	—
23	C₂²×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃₁	—	2	2	no	—
25	C₂²×C₈×C₁₃₁	—	2	2	no	—
26	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
27	#127×C₁₃₁	—	4	2	no	—
28	#124×C₁₃₁	—	4	4	no	—
29	#124×C₁₃₁	—	4	2	no	—
30	#124×C₁₃₁	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
37	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₁₃₁	—	4	2	no	—
45	C₂⁴×C₁₃₁	—	2	2	no	—
46	C₂×Q₈×C₁₃₁	—	4	2	no	—
47	C₂×Q₈×C₁₃₁	—	4	2	no	—
48	C₂×Q₈×C₁₃₁	—	4	2	no	—
49	C₂×D₄×C₁₃₁	—	2	4	no	—
50	C₂×D₄×C₁₃₁	—	4	2	no	—
51	C₂×D₄×C₁₃₁	—	4	2	no	—
52	C₂×D₄×C₁₃₁	—	4	2	no	—
53	C₂×D₄×C₁₃₁	—	4	2	no	—
54	C₂×D₄×C₁₃₁	—	4	2	no	—
55	C₂×D₄×C₁₃₁	—	4	2	no	—
56	C₂×D₄×C₁₃₁	—	4	2	no	—
57	C₂×D₄×C₁₃₁	—	4	2	no	—
58	C₂²×C₄×C₁₃₁	—	2	2	no	—
59	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃₁	—	4	2	no	—
67	C₂×C₈×C₁₃₁	—	4	2	no	—
68	C₂×C₈×C₁₃₁	—	4	2	no	—
69	C₂×C₈×C₁₃₁	—	4	2	no	—
70	C₂×C₈×C₁₃₁	—	4	2	no	—
71	C₂×C₈×C₁₃₁	—	4	2	no	—
72	C₄⋊C₄×C₁₃₁	—	4	2	no	—
73	C₄⋊C₄×C₁₃₁	—	4	2	no	—
74	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
75	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
76	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
77	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
78	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
79	C₄²×C₁₃₁	—	2	2	no	—
80	C₄²×C₁₃₁	—	2	2	no	—
81	C₄²×C₁₃₁	—	2	2	no	—
82	C₄²×C₁₃₁	—	4	1	no	—
83	C₄²×C₁₃₁	—	4	1	no	—
84	C₄²×C₁₃₁	—	4	1	no	—
85	C₄²×C₁₃₁	D₈	1	2	yes	—
86	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
87	C₂³×C₁₃₁	—	4	2	no	—
88	C₂³×C₁₃₁	—	4	2	no	—
89	C₂³×C₁₃₁	—	4	2	no	—
90	C₂³×C₁₃₁	—	4	2	no	—
91	C₂³×C₁₃₁	—	4	2	no	—
92	C₂³×C₁₃₁	—	4	2	no	—
93	C₂³×C₁₃₁	—	4	2	no	—
94	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈×C₁₃₁	—	8	4	no	—
96	D₄×C₁₃₁	—	8	4	no	—
97	D₄×C₁₃₁	—	8	4	no	—
98	D₄×C₁₃₁	—	8	4	no	—
99	C₂×C₄×C₁₃₁	—	2	1	no	—
100	C₂×C₄×C₁₃₁	—	2	1	no	—
101	C₂×C₄×C₁₃₁	—	2	1	no	—
102	C₂×C₄×C₁₃₁	—	2	1	no	—
103	C₂×C₄×C₁₃₁	—	2	1	no	—
104	C₂×C₄×C₁₃₁	—	2	1	no	—
105	C₂×C₄×C₁₃₁	—	2	1	no	—
106	C₂×C₄×C₁₃₁	—	2	1	no	—
107	C₂×C₄×C₁₃₁	—	2	1	no	—
108	C₂×C₄×C₁₃₁	—	2	1	no	—
109	C₂×C₄×C₁₃₁	—	2	1	no	—
110	C₂×C₄×C₁₃₁	—	2	1	no	—
111	C₂×C₄×C₁₃₁	—	2	1	no	—
112	C₂×C₄×C₁₃₁	—	2	1	no	—
113	C₂×C₄×C₁₃₁	—	2	1	no	—
114	C₂×C₄×C₁₃₁	—	4	2	no	—
115	C₂×C₄×C₁₃₁	—	4	2	no	—
116	C₂×C₄×C₁₃₁	—	4	2	no	—
117	C₂×C₄×C₁₃₁	#100	1	2	yes	—
118	C₂×C₄×C₁₃₁	#102	1	2	yes	—
119	C₂×C₄×C₁₃₁	#102	1	2	yes	—
120	C₂×C₄×C₁₃₁	#131	1	2	yes	—
121	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
124	C₈×C₁₃₁	—	8	2	no	—
125	C₈×C₁₃₁	—	8	2	no	—
126	C₂²×C₁₃₁	—	8	2	no	—
127	C₂²×C₁₃₁	—	8	2	no	—
128	C₂²×C₁₃₁	—	8	2	no	—
129	C₂²×C₁₃₁	—	8	2	no	—
130	C₂²×C₁₃₁	—	8	2	no	—
131	C₂²×C₁₃₁	—	8	2	no	—
132	C₂²×C₁₃₁	—	8	2	no	—
133	C₂²×C₁₃₁	#408	1	1	yes	characteristic
134	C₂²×C₁₃₁	#410	1	1	yes	characteristic
135	C₂²×C₁₃₁	C₂×#100	1	1	yes	characteristic
136	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
137	C₂²×C₁₃₁	#418	1	1	yes	characteristic
138	C₂²×C₁₃₁	#421	1	1	yes	characteristic
139	C₂²×C₁₃₁	#421	1	1	yes	characteristic
140	C₄×C₁₃₁	—	2	2	no	—
141	C₄×C₁₃₁	—	2	2	no	—
142	C₄×C₁₃₁	—	2	2	no	—
143	C₄×C₁₃₁	—	2	2	no	—
144	C₄×C₁₃₁	—	2	1	no	—
145	C₄×C₁₃₁	—	2	1	no	—
146	C₄×C₁₃₁	—	2	2	no	—
147	C₄×C₁₃₁	—	4	1	no	—
148	C₄×C₁₃₁	—	8	2	no	—
149	C₂×C₁₃₁	—	16	2	no	—
150	C₂×C₁₃₁	#1469	1	1	yes	characteristic
151	C₂×C₁₃₁	#1480	1	1	yes	characteristic
152	C₂×C₁₃₁	#1496	1	1	yes	characteristic
153	C₂×C₁₃₁	#1514	1	1	yes	characteristic
154	C₂×C₁₃₁	#1517	1	1	yes	characteristic
155	C₂×C₁₃₁	#1533	1	1	yes	characteristic
156	C₂×C₁₃₁	#1545	1	1	yes	characteristic
157	#7719	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₁₃₁	#7719	1	1	yes	characteristic, direct product factor, Sylow
159	#2858	C₂×C₁₃₁	1	1	yes	characteristic
160	#1839	C₂×C₁₃₁	1	1	yes	characteristic, complemented
161	#1831	C₂×C₁₃₁	1	1	yes	characteristic, complemented
162	#1270	C₂×C₁₃₁	1	4	yes	complemented
163	C₂×C₄⋊Q₈	C₄×C₁₃₁	1	1	yes	characteristic
164	C₂×#127	C₄×C₁₃₁	1	1	yes	characteristic
165	C₂×#124	—	2	1	no	—
166	C₂×#124	—	2	1	no	—
167	C₂×C₄⋊C₈	—	2	1	no	—
168	#390	—	2	1	no	—
169	#390	—	2	1	no	—
170	#390	C₄×C₁₃₁	1	2	yes	—
171	#390	C₂²×C₁₃₁	1	2	yes	—
172	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
173	#336	—	2	1	no	—
174	#336	—	2	1	no	—
175	#336	—	2	1	no	—
176	#336	C₂²×C₁₃₁	1	2	yes	—
177	#336	C₂²×C₁₃₁	1	2	yes	—
178	C₂²×Q₈	—	2	1	no	—
179	C₂²×D₄	—	2	1	no	—
180	C₂²×D₄	—	2	1	no	—
181	C₂²×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₄⋊Q₈	—	4	2	no	—
185	#127	—	4	2	no	—
186	#124	—	4	4	no	—
187	#124	—	4	2	no	—
188	#124	—	4	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
202	C₄⋊C₈	—	4	2	no	—
203	C₂⁴	—	2	2	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	D₈×C₁₃₁	1	2	yes	—
244	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
253	Q₈	—	8	4	no	—
254	D₄	—	8	4	no	—
255	D₄	—	8	4	no	—
256	D₄	—	8	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#100×C₁₃₁	1	2	yes	—
276	C₂×C₄	#102×C₁₃₁	1	2	yes	—
277	C₂×C₄	#102×C₁₃₁	1	2	yes	—
278	C₂×C₄	#131×C₁₃₁	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
281	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
282	C₈	—	8	2	no	—
283	C₈	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	#408×C₁₃₁	1	1	yes	characteristic
292	C₂²	#410×C₁₃₁	1	1	yes	characteristic
293	C₂²	C₂×#100×C₁₃₁	1	1	yes	characteristic
294	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
295	C₂²	#418×C₁₃₁	1	1	yes	characteristic
296	C₂²	#421×C₁₃₁	1	1	yes	characteristic
297	C₂²	#421×C₁₃₁	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	8	2	no	—
307	C₂	—	16	2	no	—
308	C₂	#1469×C₁₃₁	1	1	yes	characteristic
309	C₂	#1480×C₁₃₁	1	1	yes	characteristic
310	C₂	#1496×C₁₃₁	1	1	yes	characteristic
311	C₂	#1514×C₁₃₁	1	1	yes	characteristic
312	C₂	#1517×C₁₃₁	1	1	yes	characteristic
313	C₂	#1533×C₁₃₁	1	1	yes	characteristic
314	C₂	#1545×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1839×C₁₃₁	C₂	semidirect
3. #1831×C₁₃₁	C₂	semidirect
4. #1270×C₁₃₁	C₂	semidirect
157. #7719	C₁₃₁	direct (N × H)
158. C₁₃₁	#7719	direct (N × H)
160. #1839	C₂×C₁₃₁	semidirect
161. #1831	C₂×C₁₃₁	semidirect
162. #1270	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2858×C₁₃₁	2
#1839×C₁₃₁	2
#1831×C₁₃₁	2
#1270×C₁₃₁	2
#7719	131

Subgroups — C131×#7719 #45693603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7719 #45693603