Subgroups — C₁₃₁×#10805 #45696689

Order 33536.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#344×C₁₃₁	C₂	1	2	yes	maximal
3	C₂×#339×C₁₃₁	C₂	1	2	yes	maximal
4	C₂×#338×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×H(3,ℤ/4)×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₁₃₁	—	2	2	no	—
12	C₄×M₁₆×C₁₃₁	—	2	2	no	—
13	C₂×#97×C₁₃₁	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₁₃₁	C₄	1	2	yes	complemented
15	#344×C₁₃₁	C₄	1	8	yes	complemented
16	#339×C₁₃₁	—	2	4	no	—
17	#338×C₁₃₁	—	2	2	no	—
18	H(3,ℤ/4)×C₁₃₁	—	2	2	no	—
19	C₂³×C₄×C₁₃₁	—	2	1	no	—
20	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₃₁	—	2	1	no	—
22	C₂×M₁₆×C₁₃₁	—	2	1	no	—
23	C₂×M₁₆×C₁₃₁	—	2	2	no	—
24	C₂×M₁₆×C₁₃₁	—	2	2	no	—
25	C₂×M₁₆×C₁₃₁	—	2	2	no	—
26	C₂×M₁₆×C₁₃₁	—	2	2	no	—
27	C₂×M₁₆×C₁₃₁	—	2	2	no	—
28	C₂×M₁₆×C₁₃₁	—	2	2	no	—
29	C₂×M₁₆×C₁₃₁	—	2	2	no	—
30	C₂×M₁₆×C₁₃₁	—	2	2	no	—
31	C₂×M₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
34	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
35	C₂²×C₈×C₁₃₁	—	2	2	no	—
36	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
38	#117×C₁₃₁	—	2	2	no	—
39	#117×C₁₃₁	—	2	2	no	—
40	#117×C₁₃₁	—	2	2	no	—
41	#117×C₁₃₁	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
44	C₂×C₄²×C₁₃₁	—	2	1	no	—
45	C₂×C₄²×C₁₃₁	—	2	1	no	—
46	C₂×C₄²×C₁₃₁	—	2	1	no	—
47	C₂×C₄²×C₁₃₁	—	2	2	no	—
48	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃₁	Q₈	1	2	yes	—
52	#97×C₁₃₁	—	2	4	no	—
53	C₄×C₈×C₁₃₁	—	2	4	no	—
54	C₂⁴×C₁₃₁	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	—	2	1	no	—
56	C₂²×C₄×C₁₃₁	—	2	1	no	—
57	C₂²×C₄×C₁₃₁	—	2	2	no	—
58	C₂²×C₄×C₁₃₁	—	2	2	no	—
59	C₂²×C₄×C₁₃₁	—	4	4	no	—
60	C₂²×C₄×C₁₃₁	—	4	2	no	—
61	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
69	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
70	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
71	M₁₆×C₁₃₁	—	2	4	no	—
72	M₁₆×C₁₃₁	—	2	4	no	—
73	M₁₆×C₁₃₁	—	2	4	no	—
74	M₁₆×C₁₃₁	—	2	4	no	—
75	M₁₆×C₁₃₁	—	2	4	no	—
76	M₁₆×C₁₃₁	—	2	4	no	—
77	M₁₆×C₁₃₁	—	4	2	no	—
78	M₁₆×C₁₃₁	—	4	2	no	—
79	C₂×C₈×C₁₃₁	—	2	2	no	—
80	C₂×C₈×C₁₃₁	—	2	2	no	—
81	C₂×C₈×C₁₃₁	—	2	2	no	—
82	C₂×C₈×C₁₃₁	—	2	2	no	—
83	C₂×C₈×C₁₃₁	—	2	4	no	—
84	C₂×C₈×C₁₃₁	—	4	2	no	—
85	C₂×C₈×C₁₃₁	—	4	2	no	—
86	C₂×C₈×C₁₃₁	C₄²	1	4	yes	—
87	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₁₃₁	—	2	4	no	—
89	C₄⋊C₄×C₁₃₁	—	4	2	no	—
90	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
91	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
92	C₄²×C₁₃₁	—	2	1	no	—
93	C₄²×C₁₃₁	—	2	1	no	—
94	C₄²×C₁₃₁	—	2	2	no	—
95	C₄²×C₁₃₁	—	2	2	no	—
96	C₄²×C₁₃₁	—	2	4	no	—
97	C₄²×C₁₃₁	—	4	2	no	—
98	C₄²×C₁₃₁	C₄⋊C₄	1	2	yes	—
99	C₄²×C₁₃₁	C₄⋊C₄	1	2	yes	—
100	C₂³×C₁₃₁	—	4	2	no	—
101	C₂³×C₁₃₁	#117	1	1	yes	characteristic
102	C₂³×C₁₃₁	#117	1	2	yes	—
103	C₂³×C₁₃₁	#121	1	1	yes	characteristic
104	C₂³×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₁₃₁	#126	1	2	yes	—
106	C₂×C₄×C₁₃₁	—	2	4	no	—
107	C₂×C₄×C₁₃₁	—	2	2	no	—
108	C₂×C₄×C₁₃₁	—	2	2	no	—
109	C₂×C₄×C₁₃₁	—	4	4	no	—
110	C₂×C₄×C₁₃₁	—	4	4	no	—
111	C₂×C₄×C₁₃₁	—	4	2	no	—
112	C₂×C₄×C₁₃₁	—	4	4	no	—
113	C₂×C₄×C₁₃₁	—	8	4	no	—
114	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₃₁	#117	1	2	yes	—
120	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃₁	#123	1	2	yes	—
128	C₂×C₄×C₁₃₁	#123	1	2	yes	—
129	C₂×C₄×C₁₃₁	#124	1	2	yes	—
130	C₂×C₄×C₁₃₁	#125	1	2	yes	—
131	C₂×C₄×C₁₃₁	#126	1	2	yes	—
132	C₈×C₁₃₁	—	2	4	no	—
133	C₈×C₁₃₁	—	2	4	no	—
134	C₈×C₁₃₁	—	4	4	no	—
135	C₂²×C₁₃₁	—	2	2	no	—
136	C₂²×C₁₃₁	—	2	4	no	—
137	C₂²×C₁₃₁	—	4	1	no	—
138	C₂²×C₁₃₁	—	4	1	no	—
139	C₂²×C₁₃₁	—	4	2	no	—
140	C₂²×C₁₃₁	#376	1	1	yes	characteristic
141	C₂²×C₁₃₁	#380	1	1	yes	characteristic
142	C₂²×C₁₃₁	#383	1	1	yes	characteristic
143	C₂²×C₁₃₁	#387	1	2	yes	—
144	C₂²×C₁₃₁	#388	1	2	yes	—
145	C₄×C₁₃₁	—	2	4	no	—
146	C₄×C₁₃₁	—	4	2	no	—
147	C₄×C₁₃₁	—	8	4	no	—
148	C₄×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₁₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₁₃₁	#381	1	1	yes	characteristic
151	C₄×C₁₃₁	#382	1	1	yes	characteristic
152	C₄×C₁₃₁	#387	1	2	yes	—
153	C₄×C₁₃₁	#388	1	2	yes	—
154	C₂×C₁₃₁	—	2	1	no	—
155	C₂×C₁₃₁	—	2	2	no	—
156	C₂×C₁₃₁	—	2	1	no	—
157	C₂×C₁₃₁	—	2	2	no	—
158	C₂×C₁₃₁	#1434	1	1	yes	characteristic
159	C₂×C₁₃₁	#1841	1	2	yes	—
160	#10805	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₁₃₁	#10805	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×C₄×M₁₆	C₂×C₁₃₁	1	1	yes	characteristic
163	C₂×#344	C₂×C₁₃₁	1	2	yes	—
164	C₂×#339	C₂×C₁₃₁	1	2	yes	—
165	C₂×#338	C₂×C₁₃₁	1	1	yes	characteristic
166	C₂×H(3,ℤ/4)	C₂×C₁₃₁	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₁₃₁	1	2	yes	—
168	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
169	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
170	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
171	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
172	C₄×M₁₆	—	2	2	no	—
173	C₄×M₁₆	—	2	2	no	—
174	C₂×#97	C₄×C₁₃₁	1	2	yes	complemented
175	C₂×C₄×C₈	C₄×C₁₃₁	1	2	yes	complemented
176	#344	C₄×C₁₃₁	1	8	yes	complemented
177	#339	—	2	4	no	—
178	#338	—	2	2	no	—
179	H(3,ℤ/4)	—	2	2	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
193	C₂×M₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
194	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
195	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
198	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	C₂×C₄×C₁₃₁	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
210	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
212	C₂×C₄²	Q₈×C₁₃₁	1	2	yes	—
213	#97	—	2	4	no	—
214	C₄×C₈	—	2	4	no	—
215	C₂⁴	#41×C₁₃₁	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
228	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
229	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
230	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
231	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	C₄²×C₁₃₁	1	4	yes	—
248	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	4	2	no	—
259	C₄²	C₄⋊C₄×C₁₃₁	1	2	yes	—
260	C₄²	C₄⋊C₄×C₁₃₁	1	2	yes	—
261	C₂³	—	4	2	no	—
262	C₂³	#117×C₁₃₁	1	1	yes	characteristic
263	C₂³	#117×C₁₃₁	1	2	yes	—
264	C₂³	#121×C₁₃₁	1	1	yes	characteristic
265	C₂³	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
266	C₂³	#126×C₁₃₁	1	2	yes	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
278	C₂×C₄	#117×C₁₃₁	1	2	yes	—
279	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
280	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
281	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
283	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
284	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
285	C₂×C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
286	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
288	C₂×C₄	#123×C₁₃₁	1	2	yes	—
289	C₂×C₄	#123×C₁₃₁	1	2	yes	—
290	C₂×C₄	#124×C₁₃₁	1	2	yes	—
291	C₂×C₄	#125×C₁₃₁	1	2	yes	—
292	C₂×C₄	#126×C₁₃₁	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	2	4	no	—
295	C₈	—	4	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	#376×C₁₃₁	1	1	yes	characteristic
302	C₂²	#380×C₁₃₁	1	1	yes	characteristic
303	C₂²	#383×C₁₃₁	1	1	yes	characteristic
304	C₂²	#387×C₁₃₁	1	2	yes	—
305	C₂²	#388×C₁₃₁	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	4	no	—
309	C₄	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
310	C₄	C₄×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
311	C₄	#381×C₁₃₁	1	1	yes	characteristic
312	C₄	#382×C₁₃₁	1	1	yes	characteristic
313	C₄	#387×C₁₃₁	1	2	yes	—
314	C₄	#388×C₁₃₁	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	#1434×C₁₃₁	1	1	yes	characteristic
320	C₂	#1841×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	#117
commutator	C₂×C₄	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	#383
prime radical	C₂⁴×C₁₃₁	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂²×M₁₆×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#97×C₁₃₁	C₄	semidirect
14. C₂×C₄×C₈×C₁₃₁	C₄	semidirect
15. #344×C₁₃₁	C₄	semidirect
160. #10805	C₁₃₁	direct (N × H)
161. C₁₃₁	#10805	direct (N × H)
174. C₂×#97	C₄×C₁₃₁	semidirect
175. C₂×C₄×C₈	C₄×C₁₃₁	semidirect
176. #344	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₁₃₁	2
C₂×#344×C₁₃₁	2
C₂×#339×C₁₃₁	2
C₂×#338×C₁₃₁	2
C₂×H(3,ℤ/4)×C₁₃₁	2
#10805	131

Subgroups — C131×#10805 #45696689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#10805 #45696689