Subgroups — C₁₃₁×#33793 #45719677

Order 33536.

538 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3169×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2254×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2240×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2233×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2226×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#2162×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#532×C₁₃₁	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₁₃₁	—	2	1	no	—
10	#500×C₁₃₁	—	2	1	no	—
11	#500×C₁₃₁	—	2	1	no	—
12	#500×C₁₃₁	C₂²	1	2	yes	—
13	#500×C₁₃₁	C₂²	1	2	yes	—
14	#500×C₁₃₁	C₂²	1	1	yes	characteristic
15	#493×C₁₃₁	C₂²	1	1	yes	characteristic
16	#486×C₁₃₁	—	2	1	no	—
17	#466×C₁₃₁	—	2	1	no	—
18	#466×C₁₃₁	C₂²	1	2	yes	—
19	C₄×D₈×C₁₃₁	—	2	1	no	—
20	C₄×D₈×C₁₃₁	C₂²	1	2	yes	—
21	C₂×#107×C₁₃₁	—	2	1	no	—
22	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	#367×C₁₃₁	C₂²	1	8	yes	—
25	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
26	#357×C₁₃₁	C₂²	1	8	yes	—
27	C₂²⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
28	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	—
29	C₂×#41×C₁₃₁	—	4	1	no	—
30	C₂²×Q₈×C₁₃₁	—	4	1	no	—
31	C₂×D₈×C₁₃₁	—	2	1	no	—
32	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
36	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
37	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
38	C₄⋊Q₈×C₁₃₁	—	4	1	no	—
39	C₄⋊Q₈×C₁₃₁	—	4	1	no	—
40	#127×C₁₃₁	—	2	1	no	—
41	#124×C₁₃₁	—	4	1	no	—
42	#121×C₁₃₁	—	2	2	no	—
43	#121×C₁₃₁	—	4	1	no	—
44	C₄×D₄×C₁₃₁	—	2	2	no	—
45	C₄×D₄×C₁₃₁	—	4	1	no	—
46	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
48	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
49	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
50	#107×C₁₃₁	—	2	2	no	—
51	#107×C₁₃₁	—	2	2	no	—
52	#107×C₁₃₁	—	2	2	no	—
53	#107×C₁₃₁	—	2	1	no	—
54	#107×C₁₃₁	C₂³	1	4	yes	—
55	#107×C₁₃₁	C₂³	1	2	yes	—
56	#102×C₁₃₁	—	2	2	no	—
57	#102×C₁₃₁	—	2	2	no	—
58	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
62	#41×C₁₃₁	—	8	4	no	—
63	C₂×Q₈×C₁₃₁	—	4	1	no	—
64	C₂×Q₈×C₁₃₁	—	4	2	no	—
65	C₂×Q₈×C₁₃₁	—	4	2	no	—
66	C₂×Q₈×C₁₃₁	—	8	4	no	—
67	C₂×D₄×C₁₃₁	—	2	2	no	—
68	C₂×D₄×C₁₃₁	—	4	2	no	—
69	C₂×D₄×C₁₃₁	—	4	1	no	—
70	C₂×D₄×C₁₃₁	—	4	1	no	—
71	C₂²×C₄×C₁₃₁	—	4	2	no	—
72	C₂²×C₄×C₁₃₁	—	4	2	no	—
73	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	D₈×C₁₃₁	—	2	4	no	—
77	D₈×C₁₃₁	—	4	2	no	—
78	C₂×C₈×C₁₃₁	—	2	2	no	—
79	C₂×C₈×C₁₃₁	—	2	2	no	—
80	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₁₃₁	—	2	2	no	—
85	C₄⋊C₄×C₁₃₁	—	2	4	no	—
86	C₄⋊C₄×C₁₃₁	—	4	2	no	—
87	C₄⋊C₄×C₁₃₁	—	4	2	no	—
88	C₄⋊C₄×C₁₃₁	—	4	2	no	—
89	C₄⋊C₄×C₁₃₁	—	4	1	no	—
90	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
91	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
92	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
93	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
94	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
96	C₁₆×C₁₃₁	—	2	4	no	—
97	C₂³×C₁₃₁	—	4	2	no	—
98	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₁₃₁	—	8	4	no	—
100	Q₈×C₁₃₁	—	8	2	no	—
101	Q₈×C₁₃₁	—	8	4	no	—
102	D₄×C₁₃₁	—	4	2	no	—
103	D₄×C₁₃₁	—	4	2	no	—
104	D₄×C₁₃₁	—	8	4	no	—
105	D₄×C₁₃₁	—	8	2	no	—
106	D₄×C₁₃₁	—	8	2	no	—
107	C₂×C₄×C₁₃₁	—	2	1	no	—
108	C₂×C₄×C₁₃₁	—	2	1	no	—
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	4	2	no	—
111	C₂×C₄×C₁₃₁	—	4	4	no	—
112	C₂×C₄×C₁₃₁	—	8	4	no	—
113	C₂×C₄×C₁₃₁	—	8	4	no	—
114	C₂×C₄×C₁₃₁	SD₃₂	1	4	yes	—
115	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₁₃₁	—	2	1	no	—
122	C₈×C₁₃₁	—	2	1	no	—
123	C₈×C₁₃₁	#137	1	2	yes	—
124	C₈×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
125	C₂²×C₁₃₁	—	2	1	no	—
126	C₂²×C₁₃₁	—	2	1	no	—
127	C₂²×C₁₃₁	—	8	2	no	—
128	C₂²×C₁₃₁	—	8	2	no	—
129	C₂²×C₁₃₁	—	8	2	no	—
130	C₂²×C₁₃₁	C₂×SD₃₂	1	2	yes	—
131	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃₁	—	2	1	no	—
133	C₄×C₁₃₁	—	2	1	no	—
134	C₄×C₁₃₁	—	8	2	no	—
135	C₄×C₁₃₁	—	8	4	no	—
136	C₄×C₁₃₁	C₂×SD₃₂	1	2	yes	—
137	C₄×C₁₃₁	C₂×SD₃₂	1	2	yes	—
138	C₄×C₁₃₁	#509	1	2	yes	—
139	C₄×C₁₃₁	#536	1	1	yes	characteristic
140	C₄×C₁₃₁	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₃₁	—	2	2	no	—
142	C₂×C₁₃₁	—	16	2	no	—
143	C₂×C₁₃₁	#3226	1	1	yes	characteristic
144	C₂×C₁₃₁	C₂²×SD₃₂	1	1	yes	characteristic
145	C₂×C₁₃₁	C₂×#509	1	1	yes	characteristic
146	#33793	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₁₃₁	#33793	1	1	yes	characteristic, direct product factor, Sylow
148	#3169	C₂×C₁₃₁	1	1	yes	characteristic
149	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic, complemented
150	#2254	C₂×C₁₃₁	1	2	yes	complemented
151	#2240	C₂×C₁₃₁	1	2	yes	complemented
152	#2233	C₂×C₁₃₁	1	4	yes	complemented
153	#2226	C₂×C₁₃₁	1	4	yes	complemented
154	#2162	C₂×C₁₃₁	1	1	yes	characteristic, complemented
155	#532	—	2	1	no	—
156	C₂×C₄⋊Q₈	—	2	1	no	—
157	#500	—	2	1	no	—
158	#500	—	2	1	no	—
159	#500	C₂²×C₁₃₁	1	2	yes	—
160	#500	C₂²×C₁₃₁	1	2	yes	—
161	#500	C₂²×C₁₃₁	1	1	yes	characteristic
162	#493	C₂²×C₁₃₁	1	1	yes	characteristic
163	#486	—	2	1	no	—
164	#466	—	2	1	no	—
165	#466	C₂²×C₁₃₁	1	2	yes	—
166	C₄×D₈	—	2	1	no	—
167	C₄×D₈	C₂²×C₁₃₁	1	2	yes	—
168	C₂×#107	—	2	1	no	—
169	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
171	#367	C₂²×C₁₃₁	1	8	yes	—
172	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
173	#357	C₂²×C₁₃₁	1	8	yes	—
174	C₂²⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
175	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	—
176	C₂×#41	—	4	1	no	—
177	C₂²×Q₈	—	4	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
180	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	4	1	no	—
186	C₄⋊Q₈	—	4	1	no	—
187	#127	—	2	1	no	—
188	#124	—	4	1	no	—
189	#121	—	2	2	no	—
190	#121	—	4	1	no	—
191	C₄×D₄	—	2	2	no	—
192	C₄×D₄	—	4	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	4	1	no	—
195	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
196	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
197	#107	—	2	2	no	—
198	#107	—	2	2	no	—
199	#107	—	2	2	no	—
200	#107	—	2	1	no	—
201	#107	C₂³×C₁₃₁	1	4	yes	—
202	#107	C₂³×C₁₃₁	1	2	yes	—
203	#102	—	2	2	no	—
204	#102	—	2	2	no	—
205	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
206	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
209	#41	—	8	4	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	8	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
221	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
223	D₈	—	2	4	no	—
224	D₈	—	4	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
240	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
243	C₁₆	—	2	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
246	Q₈	—	8	4	no	—
247	Q₈	—	8	2	no	—
248	Q₈	—	8	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	2	no	—
253	D₄	—	8	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	SD₃₂×C₁₃₁	1	4	yes	—
262	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	#137×C₁₃₁	1	2	yes	—
271	C₈	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	C₂×SD₃₂×C₁₃₁	1	2	yes	—
278	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₂×SD₃₂×C₁₃₁	1	2	yes	—
284	C₄	C₂×SD₃₂×C₁₃₁	1	2	yes	—
285	C₄	#509×C₁₃₁	1	2	yes	—
286	C₄	#536×C₁₃₁	1	1	yes	characteristic
287	C₄	C₂×#137×C₁₃₁	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	16	2	no	—
290	C₂	#3226×C₁₃₁	1	1	yes	characteristic
291	C₂	C₂²×SD₃₂×C₁₃₁	1	1	yes	characteristic
292	C₂	C₂×#509×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
2. C₂×#500×C₁₃₁	C₂	semidirect
3. #2254×C₁₃₁	C₂	semidirect
4. #2240×C₁₃₁	C₂	semidirect
5. #2233×C₁₃₁	C₂	semidirect
6. #2226×C₁₃₁	C₂	semidirect
7. #2162×C₁₃₁	C₂	semidirect
146. #33793	C₁₃₁	direct (N × H)
147. C₁₃₁	#33793	direct (N × H)
149. C₂×#500	C₂×C₁₃₁	semidirect
150. #2254	C₂×C₁₃₁	semidirect
151. #2240	C₂×C₁₃₁	semidirect
152. #2233	C₂×C₁₃₁	semidirect
153. #2226	C₂×C₁₃₁	semidirect
154. #2162	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3169×C₁₃₁	2
C₂×#500×C₁₃₁	2
#2254×C₁₃₁	2
#2240×C₁₃₁	2
#2233×C₁₃₁	2
#2226×C₁₃₁	2
#2162×C₁₃₁	2
#33793	131

Subgroups — C131×#33793 #45719677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33793 #45719677