Subgroups — C₁₃₁×#33770 #45719654

Order 33536.

576 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3061×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	#2235×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2233×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2219×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#2218×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#2217×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2189×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#2186×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#2113×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#131×C₁₃₁	C₂²	1	1	yes	characteristic
12	#537×C₁₃₁	—	2	2	no	—
13	C₂×Q₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×SD₃₂×C₁₃₁	C₂²	1	2	yes	—
15	C₂×D₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
18	#468×C₁₃₁	—	2	2	no	—
19	#468×C₁₃₁	—	2	2	no	—
20	#467×C₁₃₁	C₂²	1	2	yes	—
21	#467×C₁₃₁	C₂²	1	2	yes	—
22	#466×C₁₃₁	C₂²	1	2	yes	—
23	#466×C₁₃₁	C₂²	1	2	yes	—
24	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
25	#417×C₁₃₁	—	2	2	no	—
26	#367×C₁₃₁	C₂²	1	2	yes	—
27	#367×C₁₃₁	C₂²	1	1	yes	characteristic
28	#367×C₁₃₁	C₂²	1	1	yes	characteristic
29	#358×C₁₃₁	C₂²	1	2	yes	—
30	#358×C₁₃₁	C₂²	1	2	yes	—
31	#357×C₁₃₁	C₂²	1	2	yes	—
32	#357×C₁₃₁	C₂²	1	2	yes	—
33	C₂²⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
34	C₂²⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
35	C₂×#41×C₁₃₁	—	2	2	no	—
36	C₂×#41×C₁₃₁	D₄	1	1	yes	characteristic
37	#135×C₁₃₁	—	4	2	no	—
38	#135×C₁₃₁	—	4	2	no	—
39	C₂×Q₁₆×C₁₃₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
41	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
42	#131×C₁₃₁	—	4	1	no	—
43	#131×C₁₃₁	—	4	1	no	—
44	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
45	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
48	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
49	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
50	C₂²⋊Q₈×C₁₃₁	—	4	2	no	—
51	#121×C₁₃₁	—	2	2	no	—
52	#121×C₁₃₁	—	2	2	no	—
53	#121×C₁₃₁	—	4	2	no	—
54	#121×C₁₃₁	—	4	2	no	—
55	#121×C₁₃₁	—	4	2	no	—
56	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
57	Q₃₂×C₁₃₁	—	4	1	no	—
58	SD₃₂×C₁₃₁	—	4	2	no	—
59	D₁₆×C₁₃₁	—	4	1	no	—
60	M₃₂×C₁₃₁	—	4	1	no	—
61	C₂×C₁₆×C₁₃₁	—	4	1	no	—
62	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
63	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
65	#107×C₁₃₁	—	2	2	no	—
66	#107×C₁₃₁	C₂³	1	2	yes	—
67	#107×C₁₃₁	C₂³	1	2	yes	—
68	#103×C₁₃₁	—	2	2	no	—
69	#103×C₁₃₁	—	2	2	no	—
70	#102×C₁₃₁	—	2	2	no	—
71	#102×C₁₃₁	—	2	2	no	—
72	#41×C₁₃₁	—	4	1	no	—
73	#41×C₁₃₁	—	4	1	no	—
74	#41×C₁₃₁	—	4	4	no	—
75	#41×C₁₃₁	—	4	4	no	—
76	C₂×Q₈×C₁₃₁	—	2	2	no	—
77	C₂×Q₈×C₁₃₁	D₈	1	1	yes	characteristic
78	C₂×D₄×C₁₃₁	—	2	2	no	—
79	C₂×D₄×C₁₃₁	—	4	2	no	—
80	C₂×D₄×C₁₃₁	—	4	2	no	—
81	C₂×D₄×C₁₃₁	—	4	2	no	—
82	C₂×D₄×C₁₃₁	D₈	1	1	yes	characteristic
83	C₂×D₄×C₁₃₁	D₈	1	2	yes	—
84	C₂²×C₄×C₁₃₁	—	4	2	no	—
85	C₂²×C₄×C₁₃₁	—	4	2	no	—
86	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₁₃₁	—	2	2	no	—
89	Q₁₆×C₁₃₁	—	2	2	no	—
90	SD₁₆×C₁₃₁	—	4	4	no	—
91	D₈×C₁₃₁	—	2	2	no	—
92	D₈×C₁₃₁	—	2	2	no	—
93	M₁₆×C₁₃₁	—	4	2	no	—
94	M₁₆×C₁₃₁	—	4	1	no	—
95	C₂×C₈×C₁₃₁	—	2	1	no	—
96	C₂×C₈×C₁₃₁	—	2	1	no	—
97	C₂×C₈×C₁₃₁	—	2	2	no	—
98	C₂×C₈×C₁₃₁	—	2	2	no	—
99	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₁	—	2	2	no	—
103	C₄⋊C₄×C₁₃₁	—	2	2	no	—
104	C₄⋊C₄×C₁₃₁	—	4	2	no	—
105	C₄⋊C₄×C₁₃₁	—	4	2	no	—
106	C₄⋊C₄×C₁₃₁	—	4	2	no	—
107	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
108	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
111	C₁₆×C₁₃₁	—	2	2	no	—
112	C₁₆×C₁₃₁	—	2	1	no	—
113	C₁₆×C₁₃₁	—	2	1	no	—
114	C₂³×C₁₃₁	—	4	2	no	—
115	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
116	C₂³×C₁₃₁	C₂×D₈	1	2	yes	—
117	Q₈×C₁₃₁	—	4	1	no	—
118	Q₈×C₁₃₁	—	4	2	no	—
119	Q₈×C₁₃₁	—	4	2	no	—
120	D₄×C₁₃₁	—	4	2	no	—
121	D₄×C₁₃₁	—	4	1	no	—
122	D₄×C₁₃₁	—	4	2	no	—
123	D₄×C₁₃₁	—	4	2	no	—
124	D₄×C₁₃₁	—	8	4	no	—
125	D₄×C₁₃₁	—	16	2	no	—
126	D₄×C₁₃₁	—	16	2	no	—
127	C₂×C₄×C₁₃₁	—	2	1	no	—
128	C₂×C₄×C₁₃₁	—	2	2	no	—
129	C₂×C₄×C₁₃₁	—	2	1	no	—
130	C₂×C₄×C₁₃₁	—	2	2	no	—
131	C₂×C₄×C₁₃₁	—	4	2	no	—
132	C₂×C₄×C₁₃₁	—	4	2	no	—
133	C₂×C₄×C₁₃₁	—	4	2	no	—
134	C₂×C₄×C₁₃₁	—	8	4	no	—
135	C₂×C₄×C₁₃₁	—	16	2	no	—
136	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₁₃₁	—	2	1	no	—
140	C₈×C₁₃₁	—	2	2	no	—
141	C₈×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
142	C₈×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂²×C₁₃₁	—	2	2	no	—
144	C₂²×C₁₃₁	—	2	2	no	—
145	C₂²×C₁₃₁	—	2	2	no	—
146	C₂²×C₁₃₁	—	2	2	no	—
147	C₂²×C₁₃₁	—	2	1	no	—
148	C₂²×C₁₃₁	—	8	2	no	—
149	C₂²×C₁₃₁	—	8	2	no	—
150	C₂²×C₁₃₁	—	8	2	no	—
151	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₁₃₁	—	2	2	no	—
153	C₄×C₁₃₁	—	2	1	no	—
154	C₄×C₁₃₁	—	8	2	no	—
155	C₄×C₁₃₁	—	8	2	no	—
156	C₄×C₁₃₁	—	8	2	no	—
157	C₄×C₁₃₁	#534	1	1	yes	characteristic
158	C₄×C₁₃₁	#575	1	1	yes	characteristic
159	C₂×C₁₃₁	—	4	2	no	—
160	C₂×C₁₃₁	—	4	1	no	—
161	C₂×C₁₃₁	—	16	2	no	—
162	C₂×C₁₃₁	#3186	1	1	yes	characteristic
163	C₂×C₁₃₁	#3416	1	2	yes	—
164	#33770	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₁₃₁	#33770	1	1	yes	characteristic, direct product factor, Sylow
166	#3061	C₂×C₁₃₁	1	2	yes	complemented
167	#2235	C₂×C₁₃₁	1	2	yes	complemented
168	#2233	C₂×C₁₃₁	1	2	yes	complemented
169	#2219	C₂×C₁₃₁	1	1	yes	characteristic, complemented
170	#2218	C₂×C₁₃₁	1	1	yes	characteristic, complemented
171	#2217	C₂×C₁₃₁	1	2	yes	complemented
172	#2189	C₂×C₁₃₁	1	2	yes	complemented
173	#2186	C₂×C₁₃₁	1	2	yes	complemented
174	#2113	C₂×C₁₃₁	1	1	yes	characteristic, complemented
175	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
176	C₂×#131	C₂²×C₁₃₁	1	1	yes	characteristic
177	#537	—	2	2	no	—
178	C₂×Q₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
179	C₂×SD₃₂	C₂²×C₁₃₁	1	2	yes	—
180	C₂×D₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
181	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
182	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
183	#468	—	2	2	no	—
184	#468	—	2	2	no	—
185	#467	C₂²×C₁₃₁	1	2	yes	—
186	#467	C₂²×C₁₃₁	1	2	yes	—
187	#466	C₂²×C₁₃₁	1	2	yes	—
188	#466	C₂²×C₁₃₁	1	2	yes	—
189	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
190	#417	—	2	2	no	—
191	#367	C₂²×C₁₃₁	1	2	yes	—
192	#367	C₂²×C₁₃₁	1	1	yes	characteristic
193	#367	C₂²×C₁₃₁	1	1	yes	characteristic
194	#358	C₂²×C₁₃₁	1	2	yes	—
195	#358	C₂²×C₁₃₁	1	2	yes	—
196	#357	C₂²×C₁₃₁	1	2	yes	—
197	#357	C₂²×C₁₃₁	1	2	yes	—
198	C₂²⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
199	C₂²⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
200	C₂×#41	—	2	2	no	—
201	C₂×#41	D₄×C₁₃₁	1	1	yes	characteristic
202	#135	—	4	2	no	—
203	#135	—	4	2	no	—
204	C₂×Q₁₆	C₂³×C₁₃₁	1	2	yes	—
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
207	#131	—	4	1	no	—
208	#131	—	4	1	no	—
209	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
210	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
212	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	4	2	no	—
216	#121	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	4	2	no	—
219	#121	—	4	2	no	—
220	#121	—	4	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	Q₃₂	—	4	1	no	—
223	SD₃₂	—	4	2	no	—
224	D₁₆	—	4	1	no	—
225	M₃₂	—	4	1	no	—
226	C₂×C₁₆	—	4	1	no	—
227	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
228	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
229	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
230	#107	—	2	2	no	—
231	#107	C₂³×C₁₃₁	1	2	yes	—
232	#107	C₂³×C₁₃₁	1	2	yes	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	#102	—	2	2	no	—
236	#102	—	2	2	no	—
237	#41	—	4	1	no	—
238	#41	—	4	1	no	—
239	#41	—	4	4	no	—
240	#41	—	4	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	D₈×C₁₃₁	1	1	yes	characteristic
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	D₈×C₁₃₁	1	1	yes	characteristic
248	C₂×D₄	D₈×C₁₃₁	1	2	yes	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	SD₁₆	—	4	4	no	—
256	D₈	—	2	2	no	—
257	D₈	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
266	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₁₆	—	2	2	no	—
277	C₁₆	—	2	1	no	—
278	C₁₆	—	2	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
281	C₂³	C₂×D₈×C₁₃₁	1	2	yes	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	16	2	no	—
291	D₄	—	16	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	16	2	no	—
301	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
303	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
307	C₈	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	#534×C₁₃₁	1	1	yes	characteristic
323	C₄	#575×C₁₃₁	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	4	1	no	—
326	C₂	—	16	2	no	—
327	C₂	#3186×C₁₃₁	1	1	yes	characteristic
328	C₂	#3416×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×#41×C₁₃₁ ◁ C₂×#131×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3061×C₁₃₁	C₂	semidirect
2. #2235×C₁₃₁	C₂	semidirect
3. #2233×C₁₃₁	C₂	semidirect
4. #2219×C₁₃₁	C₂	semidirect
5. #2218×C₁₃₁	C₂	semidirect
6. #2217×C₁₃₁	C₂	semidirect
7. #2189×C₁₃₁	C₂	semidirect
8. #2186×C₁₃₁	C₂	semidirect
9. #2113×C₁₃₁	C₂	semidirect
164. #33770	C₁₃₁	direct (N × H)
165. C₁₃₁	#33770	direct (N × H)
166. #3061	C₂×C₁₃₁	semidirect
167. #2235	C₂×C₁₃₁	semidirect
168. #2233	C₂×C₁₃₁	semidirect
169. #2219	C₂×C₁₃₁	semidirect
170. #2218	C₂×C₁₃₁	semidirect
171. #2217	C₂×C₁₃₁	semidirect
172. #2189	C₂×C₁₃₁	semidirect
173. #2186	C₂×C₁₃₁	semidirect
174. #2113	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3061×C₁₃₁	2
#2235×C₁₃₁	2
#2233×C₁₃₁	2
#2219×C₁₃₁	2
#2218×C₁₃₁	2
#2217×C₁₃₁	2
#2189×C₁₃₁	2
#2186×C₁₃₁	2
#2113×C₁₃₁	2
#33770	131

Subgroups — C131×#33770 #45719654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33770 #45719654