Subgroups — C₁₃₁×#8684 #45694568

Order 33536.

430 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2931×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1848×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#335×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1547×C₁₃₁	C₂	1	2	yes	maximal
5	#1292×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₁₃₁	—	2	1	no	—
7	#532×C₁₃₁	C₂²	1	1	yes	characteristic
8	#436×C₁₃₁	—	2	1	no	—
9	#436×C₁₃₁	C₄	1	2	yes	—
10	C₈×D₄×C₁₃₁	—	2	1	no	—
11	C₈×D₄×C₁₃₁	C₄	1	2	yes	—
12	C₂×#108×C₁₃₁	—	2	1	no	—
13	C₂×#108×C₁₃₁	—	2	1	no	—
14	#423×C₁₃₁	C₂²	1	2	yes	—
15	#423×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
17	#408×C₁₃₁	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#335×C₁₃₁	—	2	1	no	—
20	#335×C₁₃₁	—	2	1	no	—
21	#332×C₁₃₁	—	2	2	no	—
22	#331×C₁₃₁	—	2	2	no	—
23	C₂×#41×C₁₃₁	—	2	1	no	—
24	#131×C₁₃₁	—	4	2	no	—
25	#131×C₁₃₁	—	4	2	no	—
26	C₂×M₁₆×C₁₃₁	—	2	1	no	—
27	C₂×M₁₆×C₁₃₁	—	2	2	no	—
28	C₂×M₁₆×C₁₃₁	—	2	2	no	—
29	C₂²×C₈×C₁₃₁	—	2	2	no	—
30	C₂²×C₈×C₁₃₁	—	2	2	no	—
31	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₃₁	D₄	1	1	yes	characteristic
34	#127×C₁₃₁	D₄	1	1	yes	characteristic
35	#124×C₁₃₁	—	2	1	no	—
36	#121×C₁₃₁	—	2	1	no	—
37	#121×C₁₃₁	Q₈	1	2	yes	—
38	C₄×D₄×C₁₃₁	—	2	1	no	—
39	C₄×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
41	#108×C₁₃₁	—	4	2	no	—
42	#108×C₁₃₁	—	4	2	no	—
43	C₄⋊C₈×C₁₃₁	—	2	1	no	—
44	C₄⋊C₈×C₁₃₁	—	2	2	no	—
45	C₄⋊C₈×C₁₃₁	—	2	2	no	—
46	C₄⋊C₈×C₁₃₁	—	2	2	no	—
47	C₄⋊C₈×C₁₃₁	—	2	2	no	—
48	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
49	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
50	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
51	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
55	#41×C₁₃₁	—	4	1	no	—
56	#41×C₁₃₁	—	4	1	no	—
57	#41×C₁₃₁	—	4	1	no	—
58	#41×C₁₃₁	—	4	1	no	—
59	C₂×Q₈×C₁₃₁	—	2	1	no	—
60	C₂×D₄×C₁₃₁	—	2	2	no	—
61	C₂×D₄×C₁₃₁	—	2	1	no	—
62	C₂×D₄×C₁₃₁	—	2	1	no	—
63	C₂×D₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃₁	—	2	2	no	—
65	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
68	M₁₆×C₁₃₁	—	4	1	no	—
69	M₁₆×C₁₃₁	—	4	1	no	—
70	M₁₆×C₁₃₁	—	4	4	no	—
71	M₁₆×C₁₃₁	—	4	2	no	—
72	M₁₆×C₁₃₁	—	4	2	no	—
73	C₂×C₈×C₁₃₁	—	2	1	no	—
74	C₂×C₈×C₁₃₁	—	2	1	no	—
75	C₂×C₈×C₁₃₁	—	2	2	no	—
76	C₂×C₈×C₁₃₁	—	4	4	no	—
77	C₂×C₈×C₁₃₁	—	4	2	no	—
78	C₂×C₈×C₁₃₁	—	4	2	no	—
79	C₂×C₈×C₁₃₁	—	8	2	no	—
80	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃₁	D₈	1	2	yes	—
85	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
86	C₄⋊C₄×C₁₃₁	—	2	1	no	—
87	C₄⋊C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
88	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
89	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
90	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃₁	#41	1	1	yes	characteristic
94	C₂³×C₁₃₁	—	2	2	no	—
95	C₂³×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
96	Q₈×C₁₃₁	—	4	1	no	—
97	Q₈×C₁₃₁	—	4	1	no	—
98	D₄×C₁₃₁	—	4	2	no	—
99	D₄×C₁₃₁	—	4	2	no	—
100	D₄×C₁₃₁	—	4	2	no	—
101	D₄×C₁₃₁	—	4	1	no	—
102	D₄×C₁₃₁	—	4	1	no	—
103	D₄×C₁₃₁	—	4	1	no	—
104	D₄×C₁₃₁	—	4	1	no	—
105	C₂×C₄×C₁₃₁	—	2	1	no	—
106	C₂×C₄×C₁₃₁	—	2	1	no	—
107	C₂×C₄×C₁₃₁	—	2	2	no	—
108	C₂×C₄×C₁₃₁	—	4	1	no	—
109	C₂×C₄×C₁₃₁	—	4	2	no	—
110	C₂×C₄×C₁₃₁	—	4	2	no	—
111	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
121	C₈×C₁₃₁	—	2	1	no	—
122	C₈×C₁₃₁	—	2	1	no	—
123	C₈×C₁₃₁	—	4	2	no	—
124	C₈×C₁₃₁	—	8	2	no	—
125	C₈×C₁₃₁	—	8	2	no	—
126	C₈×C₁₃₁	#102	1	2	yes	—
127	C₈×C₁₃₁	#102	1	2	yes	—
128	C₂²×C₁₃₁	—	2	1	no	—
129	C₂²×C₁₃₁	—	2	1	no	—
130	C₂²×C₁₃₁	—	4	2	no	—
131	C₂²×C₁₃₁	—	4	2	no	—
132	C₂²×C₁₃₁	—	4	2	no	—
133	C₂²×C₁₃₁	#380	1	1	yes	characteristic
134	C₂²×C₁₃₁	#474	1	1	yes	characteristic
135	C₂²×C₁₃₁	#476	1	1	yes	characteristic
136	C₄×C₁₃₁	—	2	1	no	—
137	C₄×C₁₃₁	—	2	1	no	—
138	C₄×C₁₃₁	—	2	1	no	—
139	C₄×C₁₃₁	—	4	2	no	—
140	C₄×C₁₃₁	C₂×#102	1	1	yes	characteristic
141	C₄×C₁₃₁	#417	1	1	yes	characteristic
142	C₄×C₁₃₁	#459	1	1	yes	characteristic
143	C₄×C₁₃₁	#461	1	1	yes	characteristic
144	C₄×C₁₃₁	#472	1	1	yes	characteristic
145	C₄×C₁₃₁	#473	1	1	yes	characteristic
146	C₂×C₁₃₁	—	2	1	no	—
147	C₂×C₁₃₁	—	2	1	no	—
148	C₂×C₁₃₁	—	8	2	no	—
149	C₂×C₁₃₁	#1802	1	1	yes	characteristic
150	C₂×C₁₃₁	#1813	1	1	yes	characteristic
151	C₂×C₁₃₁	#1816	1	1	yes	characteristic
152	#8684	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₁₃₁	#8684	1	1	yes	characteristic, direct product factor, Sylow
154	#2931	C₂×C₁₃₁	1	1	yes	characteristic
155	#1848	C₂×C₁₃₁	1	1	yes	characteristic, complemented
156	C₂×#335	C₂×C₁₃₁	1	1	yes	characteristic, complemented
157	#1547	C₂×C₁₃₁	1	2	yes	—
158	#1292	C₂×C₁₃₁	1	2	yes	complemented
159	C₂×#131	—	2	1	no	—
160	#532	C₂²×C₁₃₁	1	1	yes	characteristic
161	#436	—	2	1	no	—
162	#436	C₄×C₁₃₁	1	2	yes	—
163	C₈×D₄	—	2	1	no	—
164	C₈×D₄	C₄×C₁₃₁	1	2	yes	—
165	C₂×#108	—	2	1	no	—
166	C₂×#108	—	2	1	no	—
167	#423	C₂²×C₁₃₁	1	2	yes	—
168	#423	C₂²×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
170	#408	—	2	1	no	—
171	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
172	#335	—	2	1	no	—
173	#335	—	2	1	no	—
174	#332	—	2	2	no	—
175	#331	—	2	2	no	—
176	C₂×#41	—	2	1	no	—
177	#131	—	4	2	no	—
178	#131	—	4	2	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
185	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
186	C₄⋊Q₈	D₄×C₁₃₁	1	1	yes	characteristic
187	#127	D₄×C₁₃₁	1	1	yes	characteristic
188	#124	—	2	1	no	—
189	#121	—	2	1	no	—
190	#121	Q₈×C₁₃₁	1	2	yes	—
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
193	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
194	#108	—	4	2	no	—
195	#108	—	4	2	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
205	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
206	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
207	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
208	#41	—	4	1	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
220	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	4	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	8	2	no	—
233	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
237	C₂×C₈	D₈×C₁₃₁	1	2	yes	—
238	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
246	C₄²	#41×C₁₃₁	1	1	yes	characteristic
247	C₂³	—	2	2	no	—
248	C₂³	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	Q₈	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	1	no	—
257	D₄	—	4	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
266	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
267	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
268	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
269	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
272	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
273	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	4	2	no	—
277	C₈	—	8	2	no	—
278	C₈	—	8	2	no	—
279	C₈	#102×C₁₃₁	1	2	yes	—
280	C₈	#102×C₁₃₁	1	2	yes	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#380×C₁₃₁	1	1	yes	characteristic
287	C₂²	#474×C₁₃₁	1	1	yes	characteristic
288	C₂²	#476×C₁₃₁	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	2	no	—
293	C₄	C₂×#102×C₁₃₁	1	1	yes	characteristic
294	C₄	#417×C₁₃₁	1	1	yes	characteristic
295	C₄	#459×C₁₃₁	1	1	yes	characteristic
296	C₄	#461×C₁₃₁	1	1	yes	characteristic
297	C₄	#472×C₁₃₁	1	1	yes	characteristic
298	C₄	#473×C₁₃₁	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	#1802×C₁₃₁	1	1	yes	characteristic
303	C₂	#1813×C₁₃₁	1	1	yes	characteristic
304	C₂	#1816×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#380
commutator	C₄²	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	#532×C₁₃₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ 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
2. #1848×C₁₃₁	C₂	semidirect
3. C₂×#335×C₁₃₁	C₂	semidirect
5. #1292×C₁₃₁	C₂	semidirect
152. #8684	C₁₃₁	direct (N × H)
153. C₁₃₁	#8684	direct (N × H)
155. #1848	C₂×C₁₃₁	semidirect
156. C₂×#335	C₂×C₁₃₁	semidirect
158. #1292	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2931×C₁₃₁	2
#1848×C₁₃₁	2
C₂×#335×C₁₃₁	2
#1547×C₁₃₁	2
#1292×C₁₃₁	2
#8684	131

Subgroups — C131×#8684 #45694568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8684 #45694568