Subgroups — C₁₃₁×#33246 #45719130

Order 33536.

922 conjugacy classes of subgroups.

The subgroup lattice has 308 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, complemented
2	C₂×#492×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#491×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#490×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#2642×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#2641×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#1973×C₁₃₁	C₂	1	4	yes	maximal, complemented
8	#1968×C₁₃₁	C₂	1	4	yes	maximal, complemented
9	#1726×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₁₃₁	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×#125×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#125×C₁₃₁	C₂²	1	1	yes	characteristic
16	#499×C₁₃₁	—	2	4	no	—
17	#492×C₁₃₁	—	2	4	no	—
18	#491×C₁₃₁	—	2	4	no	—
19	#490×C₁₃₁	—	2	4	no	—
20	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
21	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
22	C₂×#103×C₁₃₁	C₂²	1	4	yes	—
23	C₂×#102×C₁₃₁	C₂²	1	4	yes	—
24	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
25	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
28	#400×C₁₃₁	—	2	4	no	—
29	#394×C₁₃₁	—	2	4	no	—
30	#391×C₁₃₁	C₂²	1	4	yes	—
31	#390×C₁₃₁	C₂²	1	4	yes	—
32	#376×C₁₃₁	C₂²	1	1	yes	characteristic
33	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
34	C₂²×D₄×C₁₃₁	—	2	2	no	—
35	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
37	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
38	C₂×SD₁₆×C₁₃₁	—	2	8	no	—
39	C₂×SD₁₆×C₁₃₁	—	2	8	no	—
40	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
44	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
45	#127×C₁₃₁	—	2	2	no	—
46	#127×C₁₃₁	—	2	2	no	—
47	#125×C₁₃₁	—	2	2	no	—
48	#125×C₁₃₁	—	2	2	no	—
49	#125×C₁₃₁	—	2	2	no	—
50	#125×C₁₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₁₃₁	—	2	8	no	—
57	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
59	#107×C₁₃₁	—	2	4	no	—
60	#106×C₁₃₁	—	2	4	no	—
61	#103×C₁₃₁	—	2	8	no	—
62	#102×C₁₃₁	—	2	8	no	—
63	#97×C₁₃₁	—	2	4	no	—
64	#97×C₁₃₁	D₄	1	4	yes	—
65	C₄×C₈×C₁₃₁	—	2	4	no	—
66	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
67	#95×C₁₃₁	—	2	8	no	—
68	#95×C₁₃₁	D₄	1	4	yes	—
69	C₂⁴×C₁₃₁	—	2	2	no	—
70	C₂×Q₈×C₁₃₁	—	2	2	no	—
71	C₂×Q₈×C₁₃₁	—	2	2	no	—
72	C₂×Q₈×C₁₃₁	—	2	4	no	—
73	C₂×Q₈×C₁₃₁	—	2	4	no	—
74	C₂×D₄×C₁₃₁	—	2	2	no	—
75	C₂×D₄×C₁₃₁	—	2	2	no	—
76	C₂×D₄×C₁₃₁	—	2	4	no	—
77	C₂×D₄×C₁₃₁	—	2	4	no	—
78	C₂×D₄×C₁₃₁	—	4	4	no	—
79	C₂×D₄×C₁₃₁	—	8	4	no	—
80	C₂²×C₄×C₁₃₁	—	2	2	no	—
81	C₂²×C₄×C₁₃₁	—	2	2	no	—
82	C₂²×C₄×C₁₃₁	—	2	2	no	—
83	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
86	SD₁₆×C₁₃₁	—	4	8	no	—
87	SD₁₆×C₁₃₁	—	4	8	no	—
88	C₂×C₈×C₁₃₁	—	2	2	no	—
89	C₂×C₈×C₁₃₁	—	2	2	no	—
90	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃₁	#41	1	4	yes	—
94	C₂×C₈×C₁₃₁	#41	1	4	yes	—
95	C₄⋊C₄×C₁₃₁	—	2	2	no	—
96	C₄⋊C₄×C₁₃₁	—	2	2	no	—
97	C₄⋊C₄×C₁₃₁	—	2	2	no	—
98	C₄⋊C₄×C₁₃₁	—	2	2	no	—
99	C₄⋊C₄×C₁₃₁	—	4	4	no	—
100	C₄⋊C₄×C₁₃₁	—	4	4	no	—
101	C₄⋊C₄×C₁₃₁	—	4	4	no	—
102	C₂²⋊C₄×C₁₃₁	—	8	8	no	—
103	C₄²×C₁₃₁	—	2	2	no	—
104	C₄²×C₁₃₁	—	2	2	no	—
105	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
106	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
107	C₂³×C₁₃₁	—	4	2	no	—
108	C₂³×C₁₃₁	—	4	4	no	—
109	C₂³×C₁₃₁	—	4	8	no	—
110	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₁₃₁	—	4	2	no	—
112	Q₈×C₁₃₁	—	4	2	no	—
113	Q₈×C₁₃₁	—	4	4	no	—
114	D₄×C₁₃₁	—	4	2	no	—
115	D₄×C₁₃₁	—	4	2	no	—
116	D₄×C₁₃₁	—	4	4	no	—
117	D₄×C₁₃₁	—	8	8	no	—
118	C₂×C₄×C₁₃₁	—	2	4	no	—
119	C₂×C₄×C₁₃₁	—	2	4	no	—
120	C₂×C₄×C₁₃₁	—	2	8	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	4	no	—
125	C₂×C₄×C₁₃₁	—	4	4	no	—
126	C₂×C₄×C₁₃₁	—	4	4	no	—
127	C₂×C₄×C₁₃₁	#127	1	4	yes	—
128	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
132	C₈×C₁₃₁	—	2	4	no	—
133	C₈×C₁₃₁	#124	1	4	yes	—
134	C₈×C₁₃₁	#124	1	4	yes	—
135	C₂²×C₁₃₁	—	8	2	no	—
136	C₂²×C₁₃₁	—	8	4	no	—
137	C₂²×C₁₃₁	—	8	8	no	—
138	C₂²×C₁₃₁	C₂×#127	1	1	yes	characteristic
139	C₂²×C₁₃₁	#532	1	2	yes	—
140	C₂²×C₁₃₁	#577	1	4	yes	—
141	C₄×C₁₃₁	—	2	4	no	—
142	C₄×C₁₃₁	—	4	4	no	—
143	C₄×C₁₃₁	—	8	2	no	—
144	C₄×C₁₃₁	—	8	2	no	—
145	C₄×C₁₃₁	—	8	2	no	—
146	C₄×C₁₃₁	C₂×#124	1	1	yes	characteristic
147	C₄×C₁₃₁	C₂×#124	1	1	yes	characteristic
148	C₄×C₁₃₁	#532	1	2	yes	—
149	C₂×C₁₃₁	—	16	2	no	—
150	C₂×C₁₃₁	#2858	1	1	yes	characteristic
151	C₂×C₁₃₁	#3154	1	2	yes	—
152	C₂×C₁₃₁	#3174	1	4	yes	—
153	#33246	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₃₁	#33246	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#499	C₂×C₁₃₁	1	1	yes	characteristic, complemented
156	C₂×#492	C₂×C₁₃₁	1	1	yes	characteristic
157	C₂×#491	C₂×C₁₃₁	1	1	yes	characteristic, complemented
158	C₂×#490	C₂×C₁₃₁	1	1	yes	characteristic
159	#2642	C₂×C₁₃₁	1	1	yes	characteristic, complemented
160	#2641	C₂×C₁₃₁	1	1	yes	characteristic
161	#1973	C₂×C₁₃₁	1	4	yes	complemented
162	#1968	C₂×C₁₃₁	1	4	yes	complemented
163	#1726	C₂×C₁₃₁	1	1	yes	characteristic, complemented
164	C₂²×SD₁₆	C₂²×C₁₃₁	1	2	yes	—
165	C₂²×SD₁₆	C₂²×C₁₃₁	1	2	yes	—
166	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
167	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
168	C₂×#125	C₂²×C₁₃₁	1	1	yes	characteristic
169	C₂×#125	C₂²×C₁₃₁	1	1	yes	characteristic
170	#499	—	2	4	no	—
171	#492	—	2	4	no	—
172	#491	—	2	4	no	—
173	#490	—	2	4	no	—
174	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
175	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
176	C₂×#103	C₂²×C₁₃₁	1	4	yes	—
177	C₂×#102	C₂²×C₁₃₁	1	4	yes	—
178	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
179	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
182	#400	—	2	4	no	—
183	#394	—	2	4	no	—
184	#391	C₂²×C₁₃₁	1	4	yes	—
185	#390	C₂²×C₁₃₁	1	4	yes	—
186	#376	C₂²×C₁₃₁	1	1	yes	characteristic
187	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
190	C₂×SD₁₆	—	2	4	no	—
191	C₂×SD₁₆	—	2	4	no	—
192	C₂×SD₁₆	—	2	8	no	—
193	C₂×SD₁₆	—	2	8	no	—
194	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	#127	—	2	2	no	—
200	#127	—	2	2	no	—
201	#125	—	2	2	no	—
202	#125	—	2	2	no	—
203	#125	—	2	2	no	—
204	#125	—	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	C₂³×C₁₃₁	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	8	no	—
211	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
212	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
213	#107	—	2	4	no	—
214	#106	—	2	4	no	—
215	#103	—	2	8	no	—
216	#102	—	2	8	no	—
217	#97	—	2	4	no	—
218	#97	D₄×C₁₃₁	1	4	yes	—
219	C₄×C₈	—	2	4	no	—
220	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
221	#95	—	2	8	no	—
222	#95	D₄×C₁₃₁	1	4	yes	—
223	C₂⁴	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	8	4	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	4	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
240	SD₁₆	—	4	8	no	—
241	SD₁₆	—	4	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
247	C₂×C₈	#41×C₁₃₁	1	4	yes	—
248	C₂×C₈	#41×C₁₃₁	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	8	8	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
260	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	8	no	—
264	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	Q₈	—	4	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	8	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	#127×C₁₃₁	1	4	yes	—
282	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
285	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
286	C₈	—	2	4	no	—
287	C₈	#124×C₁₃₁	1	4	yes	—
288	C₈	#124×C₁₃₁	1	4	yes	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	4	no	—
291	C₂²	—	8	8	no	—
292	C₂²	C₂×#127×C₁₃₁	1	1	yes	characteristic
293	C₂²	#532×C₁₃₁	1	2	yes	—
294	C₂²	#577×C₁₃₁	1	4	yes	—
295	C₄	—	2	4	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	C₂×#124×C₁₃₁	1	1	yes	characteristic
301	C₄	C₂×#124×C₁₃₁	1	1	yes	characteristic
302	C₄	#532×C₁₃₁	1	2	yes	—
303	C₂	—	16	2	no	—
304	C₂	#2858×C₁₃₁	1	1	yes	characteristic
305	C₂	#3154×C₁₃₁	1	2	yes	—
306	C₂	#3174×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₁₃₁	C₂	semidirect
3. C₂×#491×C₁₃₁	C₂	semidirect
5. #2642×C₁₃₁	C₂	semidirect
7. #1973×C₁₃₁	C₂	semidirect
8. #1968×C₁₃₁	C₂	semidirect
9. #1726×C₁₃₁	C₂	semidirect
153. #33246	C₁₃₁	direct (N × H)
154. C₁₃₁	#33246	direct (N × H)
155. C₂×#499	C₂×C₁₃₁	semidirect
157. C₂×#491	C₂×C₁₃₁	semidirect
159. #2642	C₂×C₁₃₁	semidirect
161. #1973	C₂×C₁₃₁	semidirect
162. #1968	C₂×C₁₃₁	semidirect
163. #1726	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×#499×C₁₃₁	2
C₂×#492×C₁₃₁	2
C₂×#491×C₁₃₁	2
C₂×#490×C₁₃₁	2
#2642×C₁₃₁	2
#2641×C₁₃₁	2
#1973×C₁₃₁	2
#1968×C₁₃₁	2
#1726×C₁₃₁	2
#33246	131

Subgroups — C131×#33246 #45719130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33246 #45719130