Subgroups — C₁₃₁×#33820 #45719704

Order 33536.

528 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	#3170×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3169×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2227×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2224×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2220×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2208×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2207×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#2150×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#533×C₁₃₁	—	2	1	no	—
10	#532×C₁₃₁	—	2	1	no	—
11	C₂×SD₃₂×C₁₃₁	C₂²	1	4	yes	—
12	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
13	#500×C₁₃₁	C₂²	1	1	yes	characteristic
14	#500×C₁₃₁	C₂²	1	1	yes	characteristic
15	#494×C₁₃₁	C₂²	1	1	yes	characteristic
16	#493×C₁₃₁	C₂²	1	1	yes	characteristic
17	#487×C₁₃₁	—	2	1	no	—
18	#486×C₁₃₁	—	2	1	no	—
19	#467×C₁₃₁	—	2	1	no	—
20	#467×C₁₃₁	C₂²	1	2	yes	—
21	#466×C₁₃₁	—	2	1	no	—
22	#466×C₁₃₁	C₂²	1	2	yes	—
23	C₄×Q₁₆×C₁₃₁	—	2	1	no	—
24	C₄×Q₁₆×C₁₃₁	C₂²	1	2	yes	—
25	C₄×D₈×C₁₃₁	—	2	1	no	—
26	C₄×D₈×C₁₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
28	#366×C₁₃₁	C₂²	1	4	yes	—
29	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
30	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
31	#358×C₁₃₁	C₂²	1	4	yes	—
32	#357×C₁₃₁	C₂²	1	4	yes	—
33	C₂×#41×C₁₃₁	—	4	1	no	—
34	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
35	C₂×Q₁₆×C₁₃₁	C₂³	1	2	yes	—
36	C₂×D₈×C₁₃₁	—	2	1	no	—
37	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
41	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
42	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
43	#127×C₁₃₁	—	2	1	no	—
44	#125×C₁₃₁	—	4	1	no	—
45	#124×C₁₃₁	—	4	1	no	—
46	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
47	C₂²⋊Q₈×C₁₃₁	—	4	1	no	—
48	#121×C₁₃₁	—	2	2	no	—
49	#121×C₁₃₁	—	4	1	no	—
50	C₄×Q₈×C₁₃₁	—	2	2	no	—
51	C₄×Q₈×C₁₃₁	—	4	1	no	—
52	C₄×D₄×C₁₃₁	—	2	2	no	—
53	C₄×D₄×C₁₃₁	—	4	1	no	—
54	#117×C₁₃₁	—	4	1	no	—
55	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
56	SD₃₂×C₁₃₁	—	4	4	no	—
57	M₃₂×C₁₃₁	—	4	2	no	—
58	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
59	#107×C₁₃₁	—	2	1	no	—
60	#107×C₁₃₁	—	2	1	no	—
61	#107×C₁₃₁	C₂³	1	2	yes	—
62	#107×C₁₃₁	C₂³	1	2	yes	—
63	#103×C₁₃₁	—	2	2	no	—
64	#103×C₁₃₁	—	2	2	no	—
65	#102×C₁₃₁	—	2	2	no	—
66	#102×C₁₃₁	—	2	2	no	—
67	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
71	#41×C₁₃₁	—	8	4	no	—
72	C₂×Q₈×C₁₃₁	—	2	2	no	—
73	C₂×Q₈×C₁₃₁	—	4	1	no	—
74	C₂×Q₈×C₁₃₁	—	4	1	no	—
75	C₂×D₄×C₁₃₁	—	2	2	no	—
76	C₂×D₄×C₁₃₁	—	4	2	no	—
77	C₂×D₄×C₁₃₁	—	4	1	no	—
78	C₂×D₄×C₁₃₁	—	4	1	no	—
79	C₂²×C₄×C₁₃₁	—	4	2	no	—
80	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₁₃₁	—	2	2	no	—
84	Q₁₆×C₁₃₁	—	2	2	no	—
85	Q₁₆×C₁₃₁	—	4	2	no	—
86	D₈×C₁₃₁	—	2	2	no	—
87	D₈×C₁₃₁	—	2	2	no	—
88	D₈×C₁₃₁	—	4	2	no	—
89	C₂×C₈×C₁₃₁	—	2	1	no	—
90	C₂×C₈×C₁₃₁	—	2	1	no	—
91	C₂×C₈×C₁₃₁	—	4	1	no	—
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	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₁₃₁	—	2	2	no	—
97	C₄⋊C₄×C₁₃₁	—	2	2	no	—
98	C₄⋊C₄×C₁₃₁	—	4	1	no	—
99	C₄⋊C₄×C₁₃₁	—	4	1	no	—
100	C₄⋊C₄×C₁₃₁	—	4	1	no	—
101	C₄⋊C₄×C₁₃₁	—	4	2	no	—
102	C₄⋊C₄×C₁₃₁	—	4	2	no	—
103	C₄⋊C₄×C₁₃₁	—	4	1	no	—
104	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
106	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
107	C₄²×C₁₃₁	—	4	2	no	—
108	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
109	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
110	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₁₃₁	—	2	4	no	—
113	C₂³×C₁₃₁	—	4	2	no	—
114	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₁₃₁	—	4	2	no	—
116	Q₈×C₁₃₁	—	4	2	no	—
117	Q₈×C₁₃₁	—	8	2	no	—
118	Q₈×C₁₃₁	—	8	2	no	—
119	D₄×C₁₃₁	—	4	2	no	—
120	D₄×C₁₃₁	—	4	2	no	—
121	D₄×C₁₃₁	—	8	2	no	—
122	D₄×C₁₃₁	—	8	4	no	—
123	D₄×C₁₃₁	—	8	2	no	—
124	C₂×C₄×C₁₃₁	—	2	1	no	—
125	C₂×C₄×C₁₃₁	—	2	1	no	—
126	C₂×C₄×C₁₃₁	—	2	2	no	—
127	C₂×C₄×C₁₃₁	—	2	2	no	—
128	C₂×C₄×C₁₃₁	—	4	2	no	—
129	C₂×C₄×C₁₃₁	—	4	2	no	—
130	C₂×C₄×C₁₃₁	—	4	2	no	—
131	C₂×C₄×C₁₃₁	—	8	4	no	—
132	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₁₃₁	—	2	1	no	—
139	C₈×C₁₃₁	—	2	1	no	—
140	C₈×C₁₃₁	—	2	1	no	—
141	C₈×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
142	C₈×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²×C₁₃₁	—	2	1	no	—
144	C₂²×C₁₃₁	—	2	1	no	—
145	C₂²×C₁₃₁	—	2	1	no	—
146	C₂²×C₁₃₁	—	8	2	no	—
147	C₂²×C₁₃₁	—	8	2	no	—
148	C₂²×C₁₃₁	—	8	2	no	—
149	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₁₃₁	—	2	1	no	—
151	C₄×C₁₃₁	—	2	1	no	—
152	C₄×C₁₃₁	—	2	1	no	—
153	C₄×C₁₃₁	—	8	2	no	—
154	C₄×C₁₃₁	—	8	2	no	—
155	C₄×C₁₃₁	—	8	2	no	—
156	C₄×C₁₃₁	#509	1	2	yes	—
157	C₄×C₁₃₁	#510	1	2	yes	—
158	C₄×C₁₃₁	#537	1	1	yes	characteristic
159	C₄×C₁₃₁	#575	1	1	yes	characteristic
160	C₂×C₁₃₁	—	4	1	no	—
161	C₂×C₁₃₁	—	16	2	no	—
162	C₂×C₁₃₁	#3245	1	1	yes	characteristic
163	C₂×C₁₃₁	C₂×#509	1	1	yes	characteristic
164	C₂×C₁₃₁	C₂×#510	1	1	yes	characteristic
165	#33820	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₁	#33820	1	1	yes	characteristic, direct product factor, Sylow
167	#3170	C₂×C₁₃₁	1	1	yes	characteristic, complemented
168	#3169	C₂×C₁₃₁	1	1	yes	characteristic
169	#2227	C₂×C₁₃₁	1	2	yes	complemented
170	#2224	C₂×C₁₃₁	1	2	yes	complemented
171	#2220	C₂×C₁₃₁	1	4	yes	complemented
172	#2208	C₂×C₁₃₁	1	2	yes	complemented
173	#2207	C₂×C₁₃₁	1	2	yes	complemented
174	#2150	C₂×C₁₃₁	1	1	yes	characteristic, complemented
175	#533	—	2	1	no	—
176	#532	—	2	1	no	—
177	C₂×SD₃₂	C₂²×C₁₃₁	1	4	yes	—
178	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
179	#500	C₂²×C₁₃₁	1	1	yes	characteristic
180	#500	C₂²×C₁₃₁	1	1	yes	characteristic
181	#494	C₂²×C₁₃₁	1	1	yes	characteristic
182	#493	C₂²×C₁₃₁	1	1	yes	characteristic
183	#487	—	2	1	no	—
184	#486	—	2	1	no	—
185	#467	—	2	1	no	—
186	#467	C₂²×C₁₃₁	1	2	yes	—
187	#466	—	2	1	no	—
188	#466	C₂²×C₁₃₁	1	2	yes	—
189	C₄×Q₁₆	—	2	1	no	—
190	C₄×Q₁₆	C₂²×C₁₃₁	1	2	yes	—
191	C₄×D₈	—	2	1	no	—
192	C₄×D₈	C₂²×C₁₃₁	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
194	#366	C₂²×C₁₃₁	1	4	yes	—
195	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
196	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
197	#358	C₂²×C₁₃₁	1	4	yes	—
198	#357	C₂²×C₁₃₁	1	4	yes	—
199	C₂×#41	—	4	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	C₂³×C₁₃₁	1	2	yes	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
204	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	#127	—	2	1	no	—
210	#125	—	4	1	no	—
211	#124	—	4	1	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	4	1	no	—
214	#121	—	2	2	no	—
215	#121	—	4	1	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	4	1	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	4	1	no	—
220	#117	—	4	1	no	—
221	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
222	SD₃₂	—	4	4	no	—
223	M₃₂	—	4	2	no	—
224	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#107	C₂³×C₁₃₁	1	2	yes	—
228	#107	C₂³×C₁₃₁	1	2	yes	—
229	#103	—	2	2	no	—
230	#103	—	2	2	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
234	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
237	#41	—	8	4	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
247	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	4	2	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	4	1	no	—
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	C₂⁴×C₁₃₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
275	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
278	C₁₆	—	2	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	8	2	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
299	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
308	C₈	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	#509×C₁₃₁	1	2	yes	—
323	C₄	#510×C₁₃₁	1	2	yes	—
324	C₄	#537×C₁₃₁	1	1	yes	characteristic
325	C₄	#575×C₁₃₁	1	1	yes	characteristic
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	#3245×C₁₃₁	1	1	yes	characteristic
329	C₂	C₂×#509×C₁₃₁	1	1	yes	characteristic
330	C₂	C₂×#510×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	#3169×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
1. #3170×C₁₃₁	C₂	semidirect
3. #2227×C₁₃₁	C₂	semidirect
4. #2224×C₁₃₁	C₂	semidirect
5. #2220×C₁₃₁	C₂	semidirect
6. #2208×C₁₃₁	C₂	semidirect
7. #2207×C₁₃₁	C₂	semidirect
8. #2150×C₁₃₁	C₂	semidirect
165. #33820	C₁₃₁	direct (N × H)
166. C₁₃₁	#33820	direct (N × H)
167. #3170	C₂×C₁₃₁	semidirect
169. #2227	C₂×C₁₃₁	semidirect
170. #2224	C₂×C₁₃₁	semidirect
171. #2220	C₂×C₁₃₁	semidirect
172. #2208	C₂×C₁₃₁	semidirect
173. #2207	C₂×C₁₃₁	semidirect
174. #2150	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3170×C₁₃₁	2
#3169×C₁₃₁	2
#2227×C₁₃₁	2
#2224×C₁₃₁	2
#2220×C₁₃₁	2
#2208×C₁₃₁	2
#2207×C₁₃₁	2
#2150×C₁₃₁	2
#33820	131

Subgroups — C131×#33820 #45719704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33820 #45719704