Subgroups — C₁₃₁×#33252 #45719136

Order 33536.

1026 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#492×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#487×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#486×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#2862×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#2858×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#1973×C₁₃₁	C₂	1	8	yes	maximal, complemented
8	C₄×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₁₃₁	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×#124×C₁₃₁	—	2	2	no	—
15	#500×C₁₃₁	—	2	4	no	—
16	#492×C₁₃₁	—	2	4	no	—
17	#487×C₁₃₁	—	2	4	no	—
18	#486×C₁₃₁	—	2	4	no	—
19	C₂×#107×C₁₃₁	C₂²	1	4	yes	—
20	C₂×#103×C₁₃₁	C₂²	1	4	yes	—
21	C₂×#102×C₁₃₁	C₂²	1	4	yes	—
22	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
25	#391×C₁₃₁	—	2	2	no	—
26	#391×C₁₃₁	C₂²	1	4	yes	—
27	#390×C₁₃₁	—	2	2	no	—
28	#390×C₁₃₁	C₂²	1	4	yes	—
29	#382×C₁₃₁	—	2	2	no	—
30	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃₁	—	2	1	no	—
32	C₂²×Q₈×C₁₃₁	—	2	1	no	—
33	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
34	C₂²×D₄×C₁₃₁	—	2	1	no	—
35	C₂²×D₄×C₁₃₁	—	2	1	no	—
36	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
37	C₂×SD₁₆×C₁₃₁	—	2	8	no	—
38	C₂×SD₁₆×C₁₃₁	—	2	16	no	—
39	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
42	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
43	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
44	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
45	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
46	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
47	#127×C₁₃₁	—	2	2	no	—
48	#127×C₁₃₁	—	2	2	no	—
49	#124×C₁₃₁	—	4	8	no	—
50	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₁₃₁	—	2	8	no	—
57	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
58	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
59	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
60	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
61	#107×C₁₃₁	—	2	8	no	—
62	#103×C₁₃₁	—	2	8	no	—
63	#102×C₁₃₁	—	2	8	no	—
64	#97×C₁₃₁	—	2	8	no	—
65	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
66	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
67	#95×C₁₃₁	—	2	8	no	—
68	C₂⁴×C₁₃₁	—	2	2	no	—
69	C₂×Q₈×C₁₃₁	—	2	2	no	—
70	C₂×Q₈×C₁₃₁	—	2	2	no	—
71	C₂×Q₈×C₁₃₁	—	2	8	no	—
72	C₂×Q₈×C₁₃₁	—	4	2	no	—
73	C₂×Q₈×C₁₃₁	—	4	2	no	—
74	C₂×Q₈×C₁₃₁	—	4	4	no	—
75	C₂×Q₈×C₁₃₁	—	4	4	no	—
76	C₂×D₄×C₁₃₁	—	2	2	no	—
77	C₂×D₄×C₁₃₁	—	2	2	no	—
78	C₂×D₄×C₁₃₁	—	2	8	no	—
79	C₂×D₄×C₁₃₁	—	4	2	no	—
80	C₂×D₄×C₁₃₁	—	4	2	no	—
81	C₂×D₄×C₁₃₁	—	4	4	no	—
82	C₂×D₄×C₁₃₁	—	4	4	no	—
83	C₂²×C₄×C₁₃₁	—	2	2	no	—
84	C₂²×C₄×C₁₃₁	—	2	2	no	—
85	C₂²×C₄×C₁₃₁	—	2	2	no	—
86	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
90	SD₁₆×C₁₃₁	—	4	16	no	—
91	C₂×C₈×C₁₃₁	—	2	4	no	—
92	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₁₃₁	#41	1	8	yes	—
95	C₄⋊C₄×C₁₃₁	—	2	2	no	—
96	C₄⋊C₄×C₁₃₁	—	2	2	no	—
97	C₄⋊C₄×C₁₃₁	—	2	2	no	—
98	C₄⋊C₄×C₁₃₁	—	2	2	no	—
99	C₄⋊C₄×C₁₃₁	—	4	2	no	—
100	C₄⋊C₄×C₁₃₁	—	4	2	no	—
101	C₄⋊C₄×C₁₃₁	—	4	2	no	—
102	C₄⋊C₄×C₁₃₁	—	4	2	no	—
103	C₂²⋊C₄×C₁₃₁	—	4	16	no	—
104	C₄²×C₁₃₁	—	2	4	no	—
105	C₄²×C₁₃₁	—	2	4	no	—
106	C₄²×C₁₃₁	—	2	4	no	—
107	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
108	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
109	C₂³×C₁₃₁	—	4	2	no	—
110	C₂³×C₁₃₁	—	4	4	no	—
111	C₂³×C₁₃₁	—	4	4	no	—
112	C₂³×C₁₃₁	—	4	4	no	—
113	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₁₃₁	—	4	4	no	—
115	Q₈×C₁₃₁	—	4	4	no	—
116	Q₈×C₁₃₁	—	8	4	no	—
117	Q₈×C₁₃₁	—	8	4	no	—
118	D₄×C₁₃₁	—	4	4	no	—
119	D₄×C₁₃₁	—	4	4	no	—
120	D₄×C₁₃₁	—	8	4	no	—
121	D₄×C₁₃₁	—	8	4	no	—
122	C₂×C₄×C₁₃₁	—	2	4	no	—
123	C₂×C₄×C₁₃₁	—	2	4	no	—
124	C₂×C₄×C₁₃₁	—	2	4	no	—
125	C₂×C₄×C₁₃₁	—	4	2	no	—
126	C₂×C₄×C₁₃₁	—	4	2	no	—
127	C₂×C₄×C₁₃₁	—	4	2	no	—
128	C₂×C₄×C₁₃₁	—	4	4	no	—
129	C₂×C₄×C₁₃₁	—	4	4	no	—
130	C₂×C₄×C₁₃₁	—	4	4	no	—
131	C₂×C₄×C₁₃₁	#127	1	2	yes	—
132	C₂×C₄×C₁₃₁	#127	1	2	yes	—
133	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
134	C₂×C₄×C₁₃₁	#137	1	4	yes	—
135	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
138	C₈×C₁₃₁	—	2	4	no	—
139	C₈×C₁₃₁	#124	1	8	yes	complemented
140	C₂²×C₁₃₁	—	8	2	no	—
141	C₂²×C₁₃₁	—	8	2	no	—
142	C₂²×C₁₃₁	—	8	2	no	—
143	C₂²×C₁₃₁	—	8	8	no	—
144	C₂²×C₁₃₁	C₂×#127	1	1	yes	characteristic
145	C₂²×C₁₃₁	#532	1	2	yes	—
146	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
147	C₂²×C₁₃₁	C₂×#137	1	2	yes	—
148	C₄×C₁₃₁	—	4	4	no	—
149	C₄×C₁₃₁	—	8	2	no	—
150	C₄×C₁₃₁	—	8	2	no	—
151	C₄×C₁₃₁	—	8	2	no	—
152	C₄×C₁₃₁	#496	1	4	yes	—
153	C₄×C₁₃₁	#497	1	4	yes	—
154	C₄×C₁₃₁	C₂×#124	1	2	yes	—
155	C₄×C₁₃₁	#532	1	2	yes	—
156	C₂×C₁₃₁	—	16	2	no	—
157	C₂×C₁₃₁	#2858	1	1	yes	characteristic
158	C₂×C₁₃₁	C₂×#496	1	1	yes	characteristic
159	C₂×C₁₃₁	C₂×#497	1	1	yes	characteristic
160	C₂×C₁₃₁	#3182	1	4	yes	—
161	#33252	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₁₃₁	#33252	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic, complemented
164	C₂×#492	C₂×C₁₃₁	1	1	yes	characteristic
165	C₂×#487	C₂×C₁₃₁	1	1	yes	characteristic, complemented
166	C₂×#486	C₂×C₁₃₁	1	1	yes	characteristic
167	#2862	C₂×C₁₃₁	1	1	yes	characteristic, complemented
168	#2858	C₂×C₁₃₁	1	1	yes	characteristic
169	#1973	C₂×C₁₃₁	1	8	yes	complemented
170	C₄×#97	C₂×C₁₃₁	1	1	yes	characteristic, complemented
171	C₂²×SD₁₆	C₂²×C₁₃₁	1	4	yes	—
172	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
173	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
174	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
175	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
176	C₂×#124	—	2	2	no	—
177	#500	—	2	4	no	—
178	#492	—	2	4	no	—
179	#487	—	2	4	no	—
180	#486	—	2	4	no	—
181	C₂×#107	C₂²×C₁₃₁	1	4	yes	—
182	C₂×#103	C₂²×C₁₃₁	1	4	yes	—
183	C₂×#102	C₂²×C₁₃₁	1	4	yes	—
184	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
185	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
187	#391	—	2	2	no	—
188	#391	C₂²×C₁₃₁	1	4	yes	—
189	#390	—	2	2	no	—
190	#390	C₂²×C₁₃₁	1	4	yes	—
191	#382	—	2	2	no	—
192	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
193	C₂²×Q₈	—	2	1	no	—
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	2	1	no	—
198	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
199	C₂×SD₁₆	—	2	8	no	—
200	C₂×SD₁₆	—	2	16	no	—
201	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	—	2	2	no	—
209	#127	—	2	2	no	—
210	#127	—	2	2	no	—
211	#124	—	4	8	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
218	C₂×C₂²⋊C₄	—	2	8	no	—
219	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
220	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
222	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
223	#107	—	2	8	no	—
224	#103	—	2	8	no	—
225	#102	—	2	8	no	—
226	#97	—	2	8	no	—
227	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
228	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
229	#95	—	2	8	no	—
230	C₂⁴	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	8	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	8	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
249	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
252	SD₁₆	—	4	16	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
255	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
256	C₂×C₈	#41×C₁₃₁	1	8	yes	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	16	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
270	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	4	4	no	—
274	C₂³	—	4	4	no	—
275	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
276	Q₈	—	4	4	no	—
277	Q₈	—	4	4	no	—
278	Q₈	—	8	4	no	—
279	Q₈	—	8	4	no	—
280	D₄	—	4	4	no	—
281	D₄	—	4	4	no	—
282	D₄	—	8	4	no	—
283	D₄	—	8	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	#127×C₁₃₁	1	2	yes	—
294	C₂×C₄	#127×C₁₃₁	1	2	yes	—
295	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
296	C₂×C₄	#137×C₁₃₁	1	4	yes	—
297	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
299	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
300	C₈	—	2	4	no	—
301	C₈	#124×C₁₃₁	1	8	yes	complemented
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	8	no	—
306	C₂²	C₂×#127×C₁₃₁	1	1	yes	characteristic
307	C₂²	#532×C₁₃₁	1	2	yes	—
308	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
309	C₂²	C₂×#137×C₁₃₁	1	2	yes	—
310	C₄	—	4	4	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	#496×C₁₃₁	1	4	yes	—
315	C₄	#497×C₁₃₁	1	4	yes	—
316	C₄	C₂×#124×C₁₃₁	1	2	yes	—
317	C₄	#532×C₁₃₁	1	2	yes	—
318	C₂	—	16	2	no	—
319	C₂	#2858×C₁₃₁	1	1	yes	characteristic
320	C₂	C₂×#496×C₁₃₁	1	1	yes	characteristic
321	C₂	C₂×#497×C₁₃₁	1	1	yes	characteristic
322	C₂	#3182×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₁	C₂	semidirect
3. C₂×#487×C₁₃₁	C₂	semidirect
5. #2862×C₁₃₁	C₂	semidirect
7. #1973×C₁₃₁	C₂	semidirect
8. C₄×#97×C₁₃₁	C₂	semidirect
139. C₈×C₁₃₁	#124	semidirect
161. #33252	C₁₃₁	direct (N × H)
162. C₁₃₁	#33252	direct (N × H)
163. C₂×#500	C₂×C₁₃₁	semidirect
165. C₂×#487	C₂×C₁₃₁	semidirect
167. #2862	C₂×C₁₃₁	semidirect
169. #1973	C₂×C₁₃₁	semidirect
170. C₄×#97	C₂×C₁₃₁	semidirect
301. C₈	#124×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₁	2
C₂×#492×C₁₃₁	2
C₂×#487×C₁₃₁	2
C₂×#486×C₁₃₁	2
#2862×C₁₃₁	2
#2858×C₁₃₁	2
#1973×C₁₃₁	2
C₄×#97×C₁₃₁	2
#33252	131

Subgroups — C131×#33252 #45719136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33252 #45719136