Subgroups — C₁₃₁×#13010 #45698894

Order 33536.

548 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×#469×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#430×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	C₂×#349×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1383×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₃₁	—	2	1	no	—
8	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#121×C₁₃₁	—	2	1	no	—
10	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
11	#469×C₁₃₁	—	2	2	no	—
12	#469×C₁₃₁	C₄	1	4	yes	—
13	#430×C₁₃₁	—	2	4	no	—
14	C₂×#108×C₁₃₁	—	2	1	no	—
15	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
16	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#102×C₁₃₁	—	2	1	no	—
18	#361×C₁₃₁	—	2	4	no	—
19	#349×C₁₃₁	—	2	2	no	—
20	C₂²×D₄×C₁₃₁	—	2	1	no	—
21	C₂²×D₄×C₁₃₁	—	4	1	no	—
22	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
23	C₂×D₈×C₁₃₁	—	2	2	no	—
24	C₂×D₈×C₁₃₁	—	2	1	no	—
25	C₂×D₈×C₁₃₁	—	2	2	no	—
26	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₁₃₁	—	2	2	no	—
28	C₂×M₁₆×C₁₃₁	—	2	2	no	—
29	C₂×M₁₆×C₁₃₁	—	2	4	no	—
30	C₂×M₁₆×C₁₃₁	—	2	2	no	—
31	C₂×M₁₆×C₁₃₁	—	4	1	no	—
32	C₂×M₁₆×C₁₃₁	—	4	2	no	—
33	C₂×M₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
35	C₂²×C₈×C₁₃₁	—	4	1	no	—
36	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
38	#121×C₁₃₁	—	2	4	no	—
39	#121×C₁₃₁	—	4	2	no	—
40	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
42	M₃₂×C₁₃₁	—	2	2	no	—
43	M₃₂×C₁₃₁	—	2	2	no	—
44	C₂×C₁₆×C₁₃₁	—	2	2	no	—
45	#108×C₁₃₁	—	2	4	no	—
46	#108×C₁₃₁	—	2	2	no	—
47	#108×C₁₃₁	D₄	1	4	yes	complemented
48	#106×C₁₃₁	—	2	1	no	—
49	#106×C₁₃₁	C₂×C₄	1	2	yes	—
50	#102×C₁₃₁	—	2	4	no	—
51	C₂⁴×C₁₃₁	—	4	1	no	—
52	C₂⁴×C₁₃₁	SD₁₆	1	1	yes	characteristic
53	C₂×D₄×C₁₃₁	—	2	2	no	—
54	C₂×D₄×C₁₃₁	—	4	4	no	—
55	C₂×D₄×C₁₃₁	—	4	1	no	—
56	C₂×D₄×C₁₃₁	—	4	1	no	—
57	C₂×D₄×C₁₃₁	—	4	1	no	—
58	C₂×D₄×C₁₃₁	—	4	1	no	—
59	C₂×D₄×C₁₃₁	—	4	1	no	—
60	C₂×D₄×C₁₃₁	—	8	4	no	—
61	C₂²×C₄×C₁₃₁	—	2	1	no	—
62	C₂²×C₄×C₁₃₁	—	2	1	no	—
63	C₂²×C₄×C₁₃₁	—	2	1	no	—
64	C₂²×C₄×C₁₃₁	—	2	1	no	—
65	C₂²×C₄×C₁₃₁	—	4	1	no	—
66	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
68	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
70	D₈×C₁₃₁	—	2	4	no	—
71	D₈×C₁₃₁	—	4	2	no	—
72	M₁₆×C₁₃₁	—	2	4	no	—
73	M₁₆×C₁₃₁	—	2	4	no	—
74	M₁₆×C₁₃₁	—	2	4	no	—
75	M₁₆×C₁₃₁	—	4	4	no	—
76	M₁₆×C₁₃₁	—	4	2	no	—
77	C₂×C₈×C₁₃₁	—	2	1	no	—
78	C₂×C₈×C₁₃₁	—	2	1	no	—
79	C₂×C₈×C₁₃₁	—	4	4	no	—
80	C₂×C₈×C₁₃₁	—	4	2	no	—
81	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃₁	#41	1	2	yes	—
86	C₄⋊C₄×C₁₃₁	—	2	2	no	—
87	C₄⋊C₄×C₁₃₁	—	4	1	no	—
88	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
89	C₁₆×C₁₃₁	—	4	2	no	—
90	C₂³×C₁₃₁	—	2	1	no	—
91	C₂³×C₁₃₁	—	2	1	no	—
92	C₂³×C₁₃₁	—	4	2	no	—
93	C₂³×C₁₃₁	—	4	2	no	—
94	C₂³×C₁₃₁	—	8	2	no	—
95	C₂³×C₁₃₁	—	8	2	no	—
96	C₂³×C₁₃₁	—	8	1	no	—
97	C₂³×C₁₃₁	—	8	1	no	—
98	C₂³×C₁₃₁	#102	1	2	yes	—
99	C₂³×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
100	D₄×C₁₃₁	—	4	2	no	—
101	D₄×C₁₃₁	—	4	2	no	—
102	D₄×C₁₃₁	—	8	8	no	—
103	D₄×C₁₃₁	—	8	1	no	—
104	D₄×C₁₃₁	—	8	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	—	2	2	no	—
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	4	2	no	—
112	C₂×C₄×C₁₃₁	—	8	2	no	—
113	C₂×C₄×C₁₃₁	#102	1	2	yes	—
114	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	#123	1	1	yes	characteristic
117	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
120	C₈×C₁₃₁	—	2	2	no	—
121	C₈×C₁₃₁	—	4	4	no	—
122	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
123	C₈×C₁₃₁	#121	1	2	yes	—
124	C₂²×C₁₃₁	—	2	2	no	—
125	C₂²×C₁₃₁	—	2	2	no	—
126	C₂²×C₁₃₁	—	4	2	no	—
127	C₂²×C₁₃₁	—	4	2	no	—
128	C₂²×C₁₃₁	—	8	1	no	—
129	C₂²×C₁₃₁	—	8	2	no	—
130	C₂²×C₁₃₁	—	8	2	no	—
131	C₂²×C₁₃₁	—	8	2	no	—
132	C₂²×C₁₃₁	—	16	2	no	—
133	C₂²×C₁₃₁	—	16	2	no	—
134	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
135	C₂²×C₁₃₁	#448	1	1	yes	characteristic
136	C₂²×C₁₃₁	#481	1	1	yes	characteristic
137	C₄×C₁₃₁	—	2	2	no	—
138	C₄×C₁₃₁	—	8	2	no	—
139	C₄×C₁₃₁	#386	1	1	yes	characteristic
140	C₄×C₁₃₁	#417	1	1	yes	characteristic
141	C₄×C₁₃₁	#447	1	1	yes	characteristic
142	C₄×C₁₃₁	#480	1	1	yes	characteristic
143	C₂×C₁₃₁	—	2	1	no	—
144	C₂×C₁₃₁	—	2	1	no	—
145	C₂×C₁₃₁	—	4	2	no	—
146	C₂×C₁₃₁	—	16	2	no	—
147	C₂×C₁₃₁	#1874	1	1	yes	characteristic
148	C₂×C₁₃₁	#2238	1	2	yes	—
149	#13010	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₁	#13010	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#469	C₂×C₁₃₁	1	1	yes	characteristic
152	C₂×#430	C₂×C₁₃₁	1	1	yes	characteristic, complemented
153	C₂×#361	C₂×C₁₃₁	1	2	yes	complemented
154	C₂×#349	C₂×C₁₃₁	1	1	yes	characteristic, complemented
155	#1383	C₂×C₁₃₁	1	2	yes	complemented
156	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
157	C₂²×M₁₆	—	2	1	no	—
158	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
159	C₂×#121	—	2	1	no	—
160	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
161	#469	—	2	2	no	—
162	#469	C₄×C₁₃₁	1	4	yes	—
163	#430	—	2	4	no	—
164	C₂×#108	—	2	1	no	—
165	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
166	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
167	C₂×#102	—	2	1	no	—
168	#361	—	2	4	no	—
169	#349	—	2	2	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	4	1	no	—
182	C₂×M₁₆	—	4	2	no	—
183	C₂×M₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
184	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
185	C₂²×C₈	—	4	1	no	—
186	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
188	#121	—	2	4	no	—
189	#121	—	4	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₂²⋊C₄	—	4	1	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	2	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	#108	—	2	4	no	—
196	#108	—	2	2	no	—
197	#108	D₄×C₁₃₁	1	4	yes	complemented
198	#106	—	2	1	no	—
199	#106	C₂×C₄×C₁₃₁	1	2	yes	—
200	#102	—	2	4	no	—
201	C₂⁴	—	4	1	no	—
202	C₂⁴	SD₁₆×C₁₃₁	1	1	yes	characteristic
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	8	4	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
217	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
218	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
220	D₈	—	2	4	no	—
221	D₈	—	4	2	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	4	no	—
226	M₁₆	—	4	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
233	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
235	C₂×C₈	#41×C₁₃₁	1	2	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₁₆	—	4	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	1	no	—
247	C₂³	—	8	1	no	—
248	C₂³	#102×C₁₃₁	1	2	yes	—
249	C₂³	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	8	no	—
253	D₄	—	8	1	no	—
254	D₄	—	8	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	8	2	no	—
263	C₂×C₄	#102×C₁₃₁	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
265	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
266	C₂×C₄	#123×C₁₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
268	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
269	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	4	4	no	—
272	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
273	C₈	#121×C₁₃₁	1	2	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	8	1	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	16	2	no	—
283	C₂²	—	16	2	no	—
284	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
285	C₂²	#448×C₁₃₁	1	1	yes	characteristic
286	C₂²	#481×C₁₃₁	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	8	2	no	—
289	C₄	#386×C₁₃₁	1	1	yes	characteristic
290	C₄	#417×C₁₃₁	1	1	yes	characteristic
291	C₄	#447×C₁₃₁	1	1	yes	characteristic
292	C₄	#480×C₁₃₁	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	4	2	no	—
296	C₂	—	16	2	no	—
297	C₂	#1874×C₁₃₁	1	1	yes	characteristic
298	C₂	#2238×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#481
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂×#469×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×M₁₆×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#430×C₁₃₁	C₂	semidirect
3. C₂×#361×C₁₃₁	C₂	semidirect
4. C₂×#349×C₁₃₁	C₂	semidirect
5. #1383×C₁₃₁	C₂	semidirect
47. #108×C₁₃₁	D₄	semidirect
149. #13010	C₁₃₁	direct (N × H)
150. C₁₃₁	#13010	direct (N × H)
152. C₂×#430	C₂×C₁₃₁	semidirect
153. C₂×#361	C₂×C₁₃₁	semidirect
154. C₂×#349	C₂×C₁₃₁	semidirect
155. #1383	C₂×C₁₃₁	semidirect
197. #108	D₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#469×C₁₃₁	2
C₂×#430×C₁₃₁	2
C₂×#361×C₁₃₁	2
C₂×#349×C₁₃₁	2
#1383×C₁₃₁	2
#13010	131

Subgroups — C131×#13010 #45698894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13010 #45698894