Subgroups — C₁₃₁×#23159 #45709043

Order 33536.

718 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2370×C₁₃₁	C₂	1	2	yes	maximal
3	#1989×C₁₃₁	C₂	1	2	yes	maximal
4	#1941×C₁₃₁	C₂	1	2	yes	maximal
5	#1921×C₁₃₁	C₂	1	2	yes	maximal
6	#1918×C₁₃₁	C₂	1	2	yes	maximal
7	C₄×#106×C₁₃₁	C₂	1	2	yes	maximal
8	C₄×#106×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₈×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
11	#499×C₁₃₁	C₄	1	8	yes	complemented
12	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
16	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	#389×C₁₃₁	—	2	2	no	—
21	#389×C₁₃₁	C₂²	1	2	yes	—
22	#389×C₁₃₁	C₂²	1	2	yes	—
23	#387×C₁₃₁	—	2	2	no	—
24	#387×C₁₃₁	—	2	2	no	—
25	C₄×C₄⋊C₄×C₁₃₁	—	2	2	no	—
26	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
30	#340×C₁₃₁	C₂²	1	2	yes	—
31	#340×C₁₃₁	C₂²	1	2	yes	—
32	#340×C₁₃₁	C₂²	1	2	yes	—
33	#340×C₁₃₁	C₂²	1	2	yes	—
34	#340×C₁₃₁	C₂²	1	2	yes	—
35	#336×C₁₃₁	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
39	#125×C₁₃₁	—	2	4	no	—
40	#125×C₁₃₁	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃₁	—	2	2	no	—
51	C₂×C₄²×C₁₃₁	—	2	2	no	—
52	C₂×C₄²×C₁₃₁	—	2	2	no	—
53	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
55	#107×C₁₃₁	C₂×C₄	1	4	yes	—
56	#107×C₁₃₁	C₂×C₄	1	4	yes	—
57	#106×C₁₃₁	—	2	2	no	—
58	#106×C₁₃₁	C₂×C₄	1	8	yes	—
59	#106×C₁₃₁	Q₈	1	4	yes	—
60	C₄⋊C₈×C₁₃₁	—	2	2	no	—
61	C₄×C₈×C₁₃₁	—	2	4	no	—
62	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
63	#95×C₁₃₁	—	2	2	no	—
64	#95×C₁₃₁	—	2	2	no	—
65	#95×C₁₃₁	—	2	2	no	—
66	#95×C₁₃₁	—	2	2	no	—
67	#95×C₁₃₁	—	2	2	no	—
68	#95×C₁₃₁	D₄	1	1	yes	characteristic
69	#95×C₁₃₁	D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₁	—	2	2	no	—
71	C₂²×C₄×C₁₃₁	—	2	2	no	—
72	C₂²×C₄×C₁₃₁	—	2	2	no	—
73	C₂²×C₄×C₁₃₁	—	2	2	no	—
74	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃₁	—	2	1	no	—
80	C₂×C₈×C₁₃₁	—	2	1	no	—
81	C₂×C₈×C₁₃₁	—	2	2	no	—
82	C₂×C₈×C₁₃₁	—	2	2	no	—
83	C₂×C₈×C₁₃₁	—	2	2	no	—
84	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₁₃₁	#41	1	2	yes	—
88	C₄⋊C₄×C₁₃₁	—	2	8	no	—
89	C₄⋊C₄×C₁₃₁	—	2	8	no	—
90	C₄⋊C₄×C₁₃₁	—	2	4	no	—
91	C₄⋊C₄×C₁₃₁	—	2	2	no	—
92	C₄⋊C₄×C₁₃₁	—	2	2	no	—
93	C₄⋊C₄×C₁₃₁	—	2	2	no	—
94	C₄⋊C₄×C₁₃₁	—	2	2	no	—
95	C₄⋊C₄×C₁₃₁	—	2	2	no	—
96	C₄⋊C₄×C₁₃₁	—	4	4	no	—
97	C₄⋊C₄×C₁₃₁	—	8	2	no	—
98	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
99	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
100	C₄⋊C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
101	C₄²×C₁₃₁	—	2	2	no	—
102	C₄²×C₁₃₁	—	2	2	no	—
103	C₄²×C₁₃₁	—	4	4	no	—
104	C₄²×C₁₃₁	—	8	2	no	—
105	C₄²×C₁₃₁	—	8	2	no	—
106	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
107	C₄²×C₁₃₁	#41	1	2	yes	—
108	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	2	2	no	—
112	C₂×C₄×C₁₃₁	—	2	2	no	—
113	C₂×C₄×C₁₃₁	—	2	4	no	—
114	C₂×C₄×C₁₃₁	—	2	4	no	—
115	C₂×C₄×C₁₃₁	—	2	2	no	—
116	C₂×C₄×C₁₃₁	—	2	4	no	—
117	C₂×C₄×C₁₃₁	—	2	2	no	—
118	C₂×C₄×C₁₃₁	—	2	2	no	—
119	C₂×C₄×C₁₃₁	—	4	2	no	—
120	C₂×C₄×C₁₃₁	—	4	2	no	—
121	C₂×C₄×C₁₃₁	—	4	2	no	—
122	C₂×C₄×C₁₃₁	—	4	2	no	—
123	C₂×C₄×C₁₃₁	—	4	2	no	—
124	C₂×C₄×C₁₃₁	—	4	2	no	—
125	C₂×C₄×C₁₃₁	—	4	2	no	—
126	C₂×C₄×C₁₃₁	—	4	2	no	—
127	C₂×C₄×C₁₃₁	—	4	2	no	—
128	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃₁	#135	1	4	yes	—
131	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
137	C₈×C₁₃₁	—	2	2	no	—
138	C₈×C₁₃₁	—	4	2	no	—
139	C₈×C₁₃₁	C₄×Q₈	1	4	yes	—
140	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₁₃₁	#547	1	1	yes	characteristic
142	C₂²×C₁₃₁	D₄×Q₈	1	1	yes	characteristic
143	C₂²×C₁₃₁	C₂×#135	1	2	yes	—
144	C₂²×C₁₃₁	#577	1	2	yes	—
145	C₄×C₁₃₁	—	2	4	no	—
146	C₄×C₁₃₁	—	2	2	no	—
147	C₄×C₁₃₁	—	2	2	no	—
148	C₄×C₁₃₁	—	4	2	no	—
149	C₄×C₁₃₁	—	4	4	no	—
150	C₄×C₁₃₁	—	4	4	no	—
151	C₄×C₁₃₁	—	4	4	no	—
152	C₄×C₁₃₁	—	8	2	no	—
153	C₄×C₁₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₄×C₁₃₁	#519	1	1	yes	characteristic
155	C₄×C₁₃₁	#556	1	1	yes	characteristic
156	C₄×C₁₃₁	#557	1	1	yes	characteristic
157	C₂×C₁₃₁	#2365	1	1	yes	characteristic
158	C₂×C₁₃₁	C₄×#135	1	1	yes	characteristic
159	C₂×C₁₃₁	#2952	1	1	yes	characteristic
160	C₂×C₁₃₁	#3337	1	2	yes	—
161	C₂×C₁₃₁	#3381	1	2	yes	—
162	#23159	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₁₃₁	#23159	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×#499	C₂×C₁₃₁	1	1	yes	characteristic
165	#2370	C₂×C₁₃₁	1	2	yes	—
166	#1989	C₂×C₁₃₁	1	2	yes	—
167	#1941	C₂×C₁₃₁	1	2	yes	—
168	#1921	C₂×C₁₃₁	1	2	yes	—
169	#1918	C₂×C₁₃₁	1	2	yes	—
170	C₄×#106	C₂×C₁₃₁	1	1	yes	characteristic
171	C₄×#106	C₂×C₁₃₁	1	2	yes	—
172	C₈×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic
173	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
174	#499	C₄×C₁₃₁	1	8	yes	complemented
175	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
176	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
177	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
178	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
179	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
180	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
182	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
183	#389	—	2	2	no	—
184	#389	C₂²×C₁₃₁	1	2	yes	—
185	#389	C₂²×C₁₃₁	1	2	yes	—
186	#387	—	2	2	no	—
187	#387	—	2	2	no	—
188	C₄×C₄⋊C₄	—	2	2	no	—
189	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
190	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
191	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
192	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
193	#340	C₂²×C₁₃₁	1	2	yes	—
194	#340	C₂²×C₁₃₁	1	2	yes	—
195	#340	C₂²×C₁₃₁	1	2	yes	—
196	#340	C₂²×C₁₃₁	1	2	yes	—
197	#340	C₂²×C₁₃₁	1	2	yes	—
198	#336	C₂²×C₁₃₁	1	2	yes	—
199	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
200	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
202	#125	—	2	4	no	—
203	#125	C₂×C₄×C₁₃₁	1	8	yes	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
217	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
218	#107	C₂×C₄×C₁₃₁	1	4	yes	—
219	#107	C₂×C₄×C₁₃₁	1	4	yes	—
220	#106	—	2	2	no	—
221	#106	C₂×C₄×C₁₃₁	1	8	yes	—
222	#106	Q₈×C₁₃₁	1	4	yes	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄×C₈	—	2	4	no	—
225	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	D₄×C₁₃₁	1	1	yes	characteristic
232	#95	D₄×C₁₃₁	1	1	yes	characteristic
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
248	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
249	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
250	C₂×C₈	#41×C₁₃₁	1	2	yes	—
251	C₄⋊C₄	—	2	8	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	8	2	no	—
261	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
262	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
263	C₄⋊C₄	C₂×Q₈×C₁₃₁	1	4	yes	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	4	no	—
267	C₄²	—	8	2	no	—
268	C₄²	—	8	2	no	—
269	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
270	C₄²	#41×C₁₃₁	1	2	yes	—
271	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
292	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
293	C₂×C₄	#135×C₁₃₁	1	4	yes	—
294	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
297	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
298	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
299	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
300	C₈	—	2	2	no	—
301	C₈	—	4	2	no	—
302	C₈	C₄×Q₈×C₁₃₁	1	4	yes	—
303	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
304	C₂²	#547×C₁₃₁	1	1	yes	characteristic
305	C₂²	D₄×Q₈×C₁₃₁	1	1	yes	characteristic
306	C₂²	C₂×#135×C₁₃₁	1	2	yes	—
307	C₂²	#577×C₁₃₁	1	2	yes	—
308	C₄	—	2	4	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	4	no	—
313	C₄	—	4	4	no	—
314	C₄	—	4	4	no	—
315	C₄	—	8	2	no	—
316	C₄	C₂×C₄×Q₈×C₁₃₁	1	1	yes	characteristic
317	C₄	#519×C₁₃₁	1	1	yes	characteristic
318	C₄	#556×C₁₃₁	1	1	yes	characteristic
319	C₄	#557×C₁₃₁	1	1	yes	characteristic
320	C₂	#2365×C₁₃₁	1	1	yes	characteristic
321	C₂	C₄×#135×C₁₃₁	1	1	yes	characteristic
322	C₂	#2952×C₁₃₁	1	1	yes	characteristic
323	C₂	#3337×C₁₃₁	1	2	yes	—
324	C₂	#3381×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄×C₄⋊C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. #499×C₁₃₁	C₄	semidirect
162. #23159	C₁₃₁	direct (N × H)
163. C₁₃₁	#23159	direct (N × H)
174. #499	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×#499×C₁₃₁	2
#2370×C₁₃₁	2
#1989×C₁₃₁	2
#1941×C₁₃₁	2
#1921×C₁₃₁	2
#1918×C₁₃₁	2
C₄×#106×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
#23159	131

Subgroups — C131×#23159 #45709043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23159 #45709043