Subgroups — C₁₃₁×#29886 #45715770

Order 33536.

620 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#3079×C₁₃₁	C₂	1	2	yes	maximal
3	#2922×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2089×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#2078×C₁₃₁	C₂	1	4	yes	maximal
6	#1997×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1837×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
9	C₄×#41×C₁₃₁	C₂²	1	1	yes	characteristic
10	#500×C₁₃₁	—	2	2	no	—
11	#500×C₁₃₁	—	2	2	no	—
12	#477×C₁₃₁	—	2	2	no	—
13	#476×C₁₃₁	—	2	2	no	—
14	#475×C₁₃₁	—	2	2	no	—
15	#474×C₁₃₁	—	2	2	no	—
16	#433×C₁₃₁	—	2	2	no	—
17	#431×C₁₃₁	—	2	2	no	—
18	#428×C₁₃₁	C₂²	1	4	yes	—
19	C₂×#107×C₁₃₁	C₂²	1	4	yes	—
20	#424×C₁₃₁	C₂²	1	2	yes	—
21	#423×C₁₃₁	C₂²	1	2	yes	—
22	#417×C₁₃₁	C₂²	1	4	yes	—
23	#408×C₁₃₁	C₂²	1	2	yes	—
24	C₄×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
26	#343×C₁₃₁	C₂²	1	8	yes	—
27	#341×C₁₃₁	C₂²	1	4	yes	—
28	C₂×#41×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₂²×Q₈×C₁₃₁	—	2	2	no	—
30	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
31	C₂×M₁₆×C₁₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
33	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
34	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
35	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
36	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
37	C₄×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
39	C₄×D₄×C₁₃₁	—	2	1	no	—
40	C₄×D₄×C₁₃₁	—	2	1	no	—
41	C₄×D₄×C₁₃₁	D₄	1	1	yes	characteristic
42	C₄×D₄×C₁₃₁	D₄	1	1	yes	characteristic
43	#117×C₁₃₁	—	2	1	no	—
44	#117×C₁₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₃₁	—	2	1	no	—
48	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
49	#107×C₁₃₁	—	2	8	no	—
50	#107×C₁₃₁	—	2	4	no	—
51	#106×C₁₃₁	—	2	4	no	—
52	C₄⋊C₈×C₁₃₁	—	2	4	no	—
53	C₄⋊C₈×C₁₃₁	—	2	2	no	—
54	C₄⋊C₈×C₁₃₁	—	2	2	no	—
55	#103×C₁₃₁	—	2	4	no	—
56	#102×C₁₃₁	—	2	4	no	—
57	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
58	#97×C₁₃₁	—	2	1	no	—
59	C₄×C₈×C₁₃₁	—	2	1	no	—
60	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
61	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
62	#41×C₁₃₁	—	2	4	no	—
63	C₂×Q₈×C₁₃₁	—	4	2	no	—
64	C₂×Q₈×C₁₃₁	—	4	2	no	—
65	C₂×Q₈×C₁₃₁	—	8	4	no	—
66	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₁₃₁	—	2	1	no	—
68	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₁	—	2	1	no	—
70	C₂²×C₄×C₁₃₁	—	2	1	no	—
71	C₂²×C₄×C₁₃₁	—	2	1	no	—
72	C₂²×C₄×C₁₃₁	—	2	4	no	—
73	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₁₃₁	—	2	4	no	—
77	M₁₆×C₁₃₁	—	2	8	no	—
78	C₂×C₈×C₁₃₁	—	2	4	no	—
79	C₂×C₈×C₁₃₁	—	2	2	no	—
80	C₂×C₈×C₁₃₁	—	2	4	no	—
81	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
82	C₄⋊C₄×C₁₃₁	—	2	1	no	—
83	C₄⋊C₄×C₁₃₁	—	2	1	no	—
84	C₄⋊C₄×C₁₃₁	—	2	4	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	C₂×D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
91	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
92	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
93	C₄²×C₁₃₁	—	2	1	no	—
94	C₄²×C₁₃₁	—	2	1	no	—
95	C₄²×C₁₃₁	—	2	1	no	—
96	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂³×C₁₃₁	—	2	1	no	—
101	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
102	Q₈×C₁₃₁	—	2	2	no	—
103	Q₈×C₁₃₁	—	8	4	no	—
104	Q₈×C₁₃₁	—	8	4	no	—
105	D₄×C₁₃₁	—	2	2	no	—
106	D₄×C₁₃₁	—	4	2	no	—
107	C₂×C₄×C₁₃₁	—	2	2	no	—
108	C₂×C₄×C₁₃₁	—	2	2	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	—	4	2	no	—
113	C₂×C₄×C₁₃₁	—	4	2	no	—
114	C₂×C₄×C₁₃₁	—	4	2	no	—
115	C₂×C₄×C₁₃₁	—	4	4	no	—
116	C₂×C₄×C₁₃₁	—	8	4	no	—
117	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	4	yes	—
124	C₈×C₁₃₁	—	2	4	no	—
125	C₈×C₁₃₁	—	4	2	no	—
126	C₈×C₁₃₁	—	4	4	no	—
127	C₂²×C₁₃₁	—	2	1	no	—
128	C₂²×C₁₃₁	—	2	1	no	—
129	C₂²×C₁₃₁	—	4	1	no	—
130	C₂²×C₁₃₁	—	4	1	no	—
131	C₂²×C₁₃₁	—	4	1	no	—
132	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
133	C₂²×C₁₃₁	#541	1	2	yes	—
134	C₄×C₁₃₁	—	2	1	no	—
135	C₄×C₁₃₁	—	2	1	no	—
136	C₄×C₁₃₁	—	4	1	no	—
137	C₄×C₁₃₁	—	4	1	no	—
138	C₄×C₁₃₁	—	4	1	no	—
139	C₄×C₁₃₁	—	8	4	no	—
140	C₄×C₁₃₁	#472	1	2	yes	—
141	C₄×C₁₃₁	#473	1	2	yes	—
142	C₄×C₁₃₁	#536	1	2	yes	—
143	C₄×C₁₃₁	#536	1	2	yes	—
144	C₂×C₁₃₁	—	2	2	no	—
145	C₂×C₁₃₁	—	8	1	no	—
146	C₂×C₁₃₁	#2621	1	1	yes	characteristic
147	C₂×C₁₃₁	C₂×#472	1	1	yes	characteristic
148	C₂×C₁₃₁	C₂×#473	1	1	yes	characteristic
149	#29886	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₁	#29886	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic, complemented
152	#3079	C₂×C₁₃₁	1	2	yes	—
153	#2922	C₂×C₁₃₁	1	1	yes	characteristic
154	#2089	C₂×C₁₃₁	1	4	yes	complemented
155	#2078	C₂×C₁₃₁	1	4	yes	—
156	#1997	C₂×C₁₃₁	1	2	yes	complemented
157	#1837	C₂×C₁₃₁	1	1	yes	characteristic, complemented
158	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
159	C₄×#41	C₂²×C₁₃₁	1	1	yes	characteristic
160	#500	—	2	2	no	—
161	#500	—	2	2	no	—
162	#477	—	2	2	no	—
163	#476	—	2	2	no	—
164	#475	—	2	2	no	—
165	#474	—	2	2	no	—
166	#433	—	2	2	no	—
167	#431	—	2	2	no	—
168	#428	C₂²×C₁₃₁	1	4	yes	—
169	C₂×#107	C₂²×C₁₃₁	1	4	yes	—
170	#424	C₂²×C₁₃₁	1	2	yes	—
171	#423	C₂²×C₁₃₁	1	2	yes	—
172	#417	C₂²×C₁₃₁	1	4	yes	—
173	#408	C₂²×C₁₃₁	1	2	yes	—
174	C₄×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
176	#343	C₂²×C₁₃₁	1	8	yes	—
177	#341	C₂²×C₁₃₁	1	4	yes	—
178	C₂×#41	C₂³×C₁₃₁	1	1	yes	characteristic
179	C₂²×Q₈	—	2	2	no	—
180	C₂×M₁₆	C₂³×C₁₃₁	1	4	yes	—
181	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
182	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
188	C₄×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	D₄×C₁₃₁	1	1	yes	characteristic
192	C₄×D₄	D₄×C₁₃₁	1	1	yes	characteristic
193	#117	—	2	1	no	—
194	#117	C₂³×C₁₃₁	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
199	#107	—	2	8	no	—
200	#107	—	2	4	no	—
201	#106	—	2	4	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	2	4	no	—
206	#102	—	2	4	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	#97	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
211	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
212	#41	—	2	4	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
224	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
226	M₁₆	—	2	8	no	—
227	M₁₆	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
239	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
248	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
250	C₂³	—	2	1	no	—
251	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
252	Q₈	—	2	2	no	—
253	Q₈	—	8	4	no	—
254	Q₈	—	8	4	no	—
255	D₄	—	2	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
268	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
269	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
270	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	4	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
283	C₂²	#541×C₁₃₁	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	8	4	no	—
290	C₄	#472×C₁₃₁	1	2	yes	—
291	C₄	#473×C₁₃₁	1	2	yes	—
292	C₄	#536×C₁₃₁	1	2	yes	—
293	C₄	#536×C₁₃₁	1	2	yes	—
294	C₂	—	2	2	no	—
295	C₂	—	8	1	no	—
296	C₂	#2621×C₁₃₁	1	1	yes	characteristic
297	C₂	C₂×#472×C₁₃₁	1	1	yes	characteristic
298	C₂	C₂×#473×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#41×C₁₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₁	C₂	semidirect
4. #2089×C₁₃₁	C₂	semidirect
6. #1997×C₁₃₁	C₂	semidirect
7. #1837×C₁₃₁	C₂	semidirect
149. #29886	C₁₃₁	direct (N × H)
150. C₁₃₁	#29886	direct (N × H)
151. C₂×#500	C₂×C₁₃₁	semidirect
154. #2089	C₂×C₁₃₁	semidirect
156. #1997	C₂×C₁₃₁	semidirect
157. #1837	C₂×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
#3079×C₁₃₁	2
#2922×C₁₃₁	2
#2089×C₁₃₁	2
#2078×C₁₃₁	2
#1997×C₁₃₁	2
#1837×C₁₃₁	2
#29886	131

Subgroups — C131×#29886 #45715770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29886 #45715770