Subgroups — C₁₃₁×#8613 #45694497

Order 33536.

460 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#435×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1810×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1465×C₁₃₁	C₂	1	2	yes	maximal
4	#1293×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
8	#435×C₁₃₁	—	2	2	no	—
9	#435×C₁₃₁	—	2	2	no	—
10	C₂×#108×C₁₃₁	—	2	2	no	—
11	C₂×C₄⋊C₈×C₁₃₁	C₄	1	1	yes	characteristic
12	#407×C₁₃₁	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#97×C₁₃₁	C₄	1	1	yes	characteristic
16	#341×C₁₃₁	—	2	2	no	—
17	#324×C₁₃₁	—	2	2	no	—
18	#323×C₁₃₁	—	2	2	no	—
19	C₂²×D₄×C₁₃₁	Q₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₃₁	—	2	1	no	—
23	C₂×M₁₆×C₁₃₁	—	2	1	no	—
24	C₂×M₁₆×C₁₃₁	—	2	4	no	—
25	C₂×M₁₆×C₁₃₁	—	4	2	no	—
26	C₂²×C₈×C₁₃₁	—	4	2	no	—
27	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
33	C₄×D₄×C₁₃₁	—	2	2	no	—
34	C₄×D₄×C₁₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₃₁	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
39	#108×C₁₃₁	—	4	4	no	—
40	C₄⋊C₈×C₁₃₁	—	2	2	no	—
41	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
42	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
43	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
44	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
45	#97×C₁₃₁	—	2	1	no	—
46	#97×C₁₃₁	—	2	1	no	—
47	C₂⁴×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×D₄×C₁₃₁	—	2	2	no	—
50	C₂×D₄×C₁₃₁	—	4	2	no	—
51	C₂²×C₄×C₁₃₁	—	2	1	no	—
52	C₂²×C₄×C₁₃₁	—	2	1	no	—
53	C₂²×C₄×C₁₃₁	—	4	1	no	—
54	C₂²×C₄×C₁₃₁	—	4	1	no	—
55	C₂²×C₄×C₁₃₁	—	4	1	no	—
56	C₂²×C₄×C₁₃₁	—	4	1	no	—
57	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
64	M₁₆×C₁₃₁	—	4	2	no	—
65	M₁₆×C₁₃₁	—	4	2	no	—
66	M₁₆×C₁₃₁	—	4	4	no	—
67	M₁₆×C₁₃₁	—	4	4	no	—
68	C₂×C₈×C₁₃₁	—	2	4	no	—
69	C₂×C₈×C₁₃₁	—	2	2	no	—
70	C₂×C₈×C₁₃₁	—	2	1	no	—
71	C₂×C₈×C₁₃₁	—	2	1	no	—
72	C₂×C₈×C₁₃₁	—	4	1	no	—
73	C₂×C₈×C₁₃₁	—	4	2	no	—
74	C₂×C₈×C₁₃₁	—	4	1	no	—
75	C₂×C₈×C₁₃₁	—	4	1	no	—
76	C₂×C₈×C₁₃₁	—	4	4	no	—
77	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₃₁	—	2	2	no	—
81	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
82	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
83	C₄²×C₁₃₁	—	2	1	no	—
84	C₄²×C₁₃₁	—	2	1	no	—
85	C₂³×C₁₃₁	—	2	2	no	—
86	C₂³×C₁₃₁	—	2	1	no	—
87	C₂³×C₁₃₁	—	2	2	no	—
88	C₂³×C₁₃₁	—	4	1	no	—
89	C₂³×C₁₃₁	—	4	1	no	—
90	C₂³×C₁₃₁	—	4	1	no	—
91	C₂³×C₁₃₁	—	4	1	no	—
92	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
94	D₄×C₁₃₁	—	4	4	no	—
95	C₂×C₄×C₁₃₁	—	2	1	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	2	1	no	—
98	C₂×C₄×C₁₃₁	—	2	2	no	—
99	C₂×C₄×C₁₃₁	—	2	1	no	—
100	C₂×C₄×C₁₃₁	—	2	1	no	—
101	C₂×C₄×C₁₃₁	—	2	1	no	—
102	C₂×C₄×C₁₃₁	—	2	2	no	—
103	C₂×C₄×C₁₃₁	—	4	1	no	—
104	C₂×C₄×C₁₃₁	—	4	2	no	—
105	C₂×C₄×C₁₃₁	—	4	2	no	—
106	C₂×C₄×C₁₃₁	—	4	1	no	—
107	C₂×C₄×C₁₃₁	—	4	1	no	—
108	C₂×C₄×C₁₃₁	—	4	1	no	—
109	C₂×C₄×C₁₃₁	—	4	1	no	—
110	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃₁	#121	1	2	yes	—
113	C₈×C₁₃₁	—	2	2	no	—
114	C₈×C₁₃₁	—	2	1	no	—
115	C₈×C₁₃₁	—	4	1	no	—
116	C₈×C₁₃₁	—	4	1	no	—
117	C₈×C₁₃₁	—	8	4	no	—
118	C₈×C₁₃₁	#99	1	2	yes	—
119	C₂²×C₁₃₁	—	2	1	no	—
120	C₂²×C₁₃₁	—	2	1	no	—
121	C₂²×C₁₃₁	—	2	2	no	—
122	C₂²×C₁₃₁	—	2	2	no	—
123	C₂²×C₁₃₁	—	2	2	no	—
124	C₂²×C₁₃₁	—	4	4	no	—
125	C₂²×C₁₃₁	—	4	1	no	—
126	C₂²×C₁₃₁	—	4	1	no	—
127	C₂²×C₁₃₁	—	4	1	no	—
128	C₂²×C₁₃₁	—	8	2	no	—
129	C₂²×C₁₃₁	#380	1	1	yes	characteristic
130	C₂²×C₁₃₁	#430	1	2	yes	—
131	C₄×C₁₃₁	—	2	2	no	—
132	C₄×C₁₃₁	—	2	1	no	—
133	C₄×C₁₃₁	—	4	1	no	—
134	C₄×C₁₃₁	—	4	1	no	—
135	C₄×C₁₃₁	—	4	1	no	—
136	C₄×C₁₃₁	C₂×#99	1	1	yes	characteristic
137	C₄×C₁₃₁	#410	1	1	yes	characteristic
138	C₂×C₁₃₁	—	2	2	no	—
139	C₂×C₁₃₁	—	4	2	no	—
140	C₂×C₁₃₁	—	8	1	no	—
141	C₂×C₁₃₁	#1799	1	1	yes	characteristic
142	C₂×C₁₃₁	#1811	1	1	yes	characteristic
143	C₂×C₁₃₁	#1813	1	1	yes	characteristic
144	#8613	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₃₁	#8613	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#435	C₂×C₁₃₁	1	1	yes	characteristic
147	#1810	C₂×C₁₃₁	1	2	yes	complemented
148	#1465	C₂×C₁₃₁	1	2	yes	—
149	#1293	C₂×C₁₃₁	1	2	yes	complemented
150	C₂²×M₁₆	C₄×C₁₃₁	1	1	yes	characteristic
151	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
152	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
153	#435	—	2	2	no	—
154	#435	—	2	2	no	—
155	C₂×#108	—	2	2	no	—
156	C₂×C₄⋊C₈	C₄×C₁₃₁	1	1	yes	characteristic
157	#407	C₂²×C₁₃₁	1	4	yes	—
158	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	1	yes	characteristic
159	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
160	C₂×#97	C₄×C₁₃₁	1	1	yes	characteristic
161	#341	—	2	2	no	—
162	#324	—	2	2	no	—
163	#323	—	2	2	no	—
164	C₂²×D₄	Q₈×C₁₃₁	1	1	yes	characteristic
165	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	4	2	no	—
171	C₂²×C₈	—	4	2	no	—
172	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
173	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
177	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	Q₈×C₁₃₁	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
184	#108	—	4	4	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	4	no	—
189	C₂²⋊C₈	—	2	4	no	—
190	#97	—	2	1	no	—
191	#97	—	2	1	no	—
192	C₂⁴	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
193	C₂⁴	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
208	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	4	no	—
212	M₁₆	—	4	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
224	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	4	1	no	—
234	C₂³	—	4	1	no	—
235	C₂³	—	4	1	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
238	C₂³	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
239	D₄	—	4	4	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
256	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
257	C₂×C₄	#121×C₁₃₁	1	2	yes	—
258	C₈	—	2	1	no	—
259	C₈	—	2	2	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	8	4	no	—
263	C₈	#99×C₁₃₁	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	8	2	no	—
274	C₂²	#380×C₁₃₁	1	1	yes	characteristic
275	C₂²	#430×C₁₃₁	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	C₂×#99×C₁₃₁	1	1	yes	characteristic
282	C₄	#410×C₁₃₁	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	4	2	no	—
285	C₂	—	8	1	no	—
286	C₂	#1799×C₁₃₁	1	1	yes	characteristic
287	C₂	#1811×C₁₃₁	1	1	yes	characteristic
288	C₂	#1813×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1810×C₁₃₁	C₂	semidirect
4. #1293×C₁₃₁	C₂	semidirect
144. #8613	C₁₃₁	direct (N × H)
145. C₁₃₁	#8613	direct (N × H)
147. #1810	C₂×C₁₃₁	semidirect
149. #1293	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×#435×C₁₃₁	2
#1810×C₁₃₁	2
#1465×C₁₃₁	2
#1293×C₁₃₁	2
#8613	131

Subgroups — C131×#8613 #45694497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8613 #45694497