Subgroups — C₁₃₁×#33750 #45719634

Order 33536.

516 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	#3168×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	#2253×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2249×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2242×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2220×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2219×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#2218×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#2177×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#2176×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#2175×C₁₃₁	C₂	1	2	yes	maximal, complemented
11	C₄×M₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#529×C₁₃₁	—	2	2	no	—
13	C₂×Q₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×SD₃₂×C₁₃₁	C₂²	1	2	yes	—
15	C₂×D₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
18	#499×C₁₃₁	C₂²	1	2	yes	—
19	#495×C₁₃₁	C₂²	1	2	yes	—
20	#488×C₁₃₁	—	2	2	no	—
21	#467×C₁₃₁	C₂²	1	2	yes	—
22	#466×C₁₃₁	C₂²	1	2	yes	—
23	#465×C₁₃₁	—	2	2	no	—
24	C₄×Q₁₆×C₁₃₁	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₁₃₁	—	2	2	no	—
26	C₄×D₈×C₁₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
28	#367×C₁₃₁	C₂²	1	1	yes	characteristic
29	#367×C₁₃₁	C₂²	1	1	yes	characteristic
30	#366×C₁₃₁	C₂²	1	2	yes	—
31	#358×C₁₃₁	C₂²	1	2	yes	—
32	#358×C₁₃₁	C₂²	1	2	yes	—
33	#357×C₁₃₁	C₂²	1	2	yes	—
34	#357×C₁₃₁	C₂²	1	2	yes	—
35	#346×C₁₃₁	C₂²	1	1	yes	characteristic
36	#346×C₁₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₁₃₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
41	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
44	#126×C₁₃₁	—	4	2	no	—
45	#125×C₁₃₁	—	2	2	no	—
46	#124×C₁₃₁	—	2	2	no	—
47	#123×C₁₃₁	—	4	2	no	—
48	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
49	#121×C₁₃₁	—	2	2	no	—
50	C₄×Q₈×C₁₃₁	—	2	2	no	—
51	C₄×D₄×C₁₃₁	—	2	2	no	—
52	C₄×D₄×C₁₃₁	—	4	2	no	—
53	#117×C₁₃₁	—	4	2	no	—
54	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
55	Q₃₂×C₁₃₁	—	2	2	no	—
56	SD₃₂×C₁₃₁	—	2	4	no	—
57	D₁₆×C₁₃₁	—	2	2	no	—
58	M₃₂×C₁₃₁	—	2	2	no	—
59	M₃₂×C₁₃₁	—	2	2	no	—
60	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
64	#107×C₁₃₁	C₂³	1	2	yes	—
65	#107×C₁₃₁	C₂³	1	2	yes	—
66	#106×C₁₃₁	—	2	2	no	—
67	#103×C₁₃₁	—	2	2	no	—
68	#103×C₁₃₁	—	2	2	no	—
69	#102×C₁₃₁	—	2	2	no	—
70	#102×C₁₃₁	—	2	2	no	—
71	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₁₃₁	—	2	2	no	—
76	C₂×D₄×C₁₃₁	—	2	2	no	—
77	C₂×D₄×C₁₃₁	—	4	2	no	—
78	C₂²×C₄×C₁₃₁	—	4	2	no	—
79	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
80	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₁₃₁	—	2	2	no	—
82	Q₁₆×C₁₃₁	—	2	2	no	—
83	SD₁₆×C₁₃₁	—	4	4	no	—
84	D₈×C₁₃₁	—	2	2	no	—
85	D₈×C₁₃₁	—	2	2	no	—
86	C₂×C₈×C₁₃₁	—	2	2	no	—
87	C₂×C₈×C₁₃₁	—	2	1	no	—
88	C₂×C₈×C₁₃₁	—	2	1	no	—
89	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₁₃₁	—	2	2	no	—
94	C₄⋊C₄×C₁₃₁	—	2	2	no	—
95	C₄⋊C₄×C₁₃₁	—	4	2	no	—
96	C₄⋊C₄×C₁₃₁	—	4	2	no	—
97	C₄⋊C₄×C₁₃₁	—	4	2	no	—
98	C₄⋊C₄×C₁₃₁	—	4	2	no	—
99	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
100	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
103	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
104	C₄²×C₁₃₁	—	4	2	no	—
105	C₄²×C₁₃₁	D₈	1	2	yes	—
106	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
108	C₁₆×C₁₃₁	—	2	1	no	—
109	C₁₆×C₁₃₁	—	2	1	no	—
110	C₁₆×C₁₃₁	#41	1	2	yes	—
111	C₁₆×C₁₃₁	#41	1	2	yes	—
112	C₂³×C₁₃₁	—	4	2	no	—
113	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₁₃₁	—	4	2	no	—
115	Q₈×C₁₃₁	—	4	2	no	—
116	D₄×C₁₃₁	—	4	2	no	—
117	D₄×C₁₃₁	—	4	2	no	—
118	D₄×C₁₃₁	—	8	4	no	—
119	C₂×C₄×C₁₃₁	—	2	1	no	—
120	C₂×C₄×C₁₃₁	—	2	1	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	—	8	4	no	—
126	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₁₃₁	—	2	1	no	—
131	C₈×C₁₃₁	—	2	1	no	—
132	C₈×C₁₃₁	#135	1	2	yes	—
133	C₈×C₁₃₁	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₃₁	C₂×#41	1	1	yes	characteristic
135	C₂²×C₁₃₁	—	2	2	no	—
136	C₂²×C₁₃₁	—	2	1	no	—
137	C₂²×C₁₃₁	—	8	2	no	—
138	C₂²×C₁₃₁	—	8	2	no	—
139	C₂²×C₁₃₁	—	8	2	no	—
140	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₁₃₁	—	2	2	no	—
142	C₄×C₁₃₁	—	2	2	no	—
143	C₄×C₁₃₁	—	2	1	no	—
144	C₄×C₁₃₁	—	8	2	no	—
145	C₄×C₁₃₁	—	8	2	no	—
146	C₄×C₁₃₁	—	8	2	no	—
147	C₄×C₁₃₁	#532	1	1	yes	characteristic
148	C₄×C₁₃₁	C₂×#135	1	1	yes	characteristic
149	C₂×C₁₃₁	—	4	1	no	—
150	C₂×C₁₃₁	—	16	2	no	—
151	C₂×C₁₃₁	#3169	1	1	yes	characteristic
152	C₂×C₁₃₁	#3414	1	2	yes	—
153	#33750	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₃₁	#33750	1	1	yes	characteristic, direct product factor, Sylow
155	#3168	C₂×C₁₃₁	1	2	yes	complemented
156	#2253	C₂×C₁₃₁	1	1	yes	characteristic, complemented
157	#2249	C₂×C₁₃₁	1	1	yes	characteristic, complemented
158	#2242	C₂×C₁₃₁	1	2	yes	complemented
159	#2220	C₂×C₁₃₁	1	2	yes	complemented
160	#2219	C₂×C₁₃₁	1	1	yes	characteristic, complemented
161	#2218	C₂×C₁₃₁	1	1	yes	characteristic
162	#2177	C₂×C₁₃₁	1	1	yes	characteristic, complemented
163	#2176	C₂×C₁₃₁	1	1	yes	characteristic, complemented
164	#2175	C₂×C₁₃₁	1	2	yes	complemented
165	C₄×M₃₂	C₂×C₁₃₁	1	1	yes	characteristic, complemented
166	#529	—	2	2	no	—
167	C₂×Q₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
168	C₂×SD₃₂	C₂²×C₁₃₁	1	2	yes	—
169	C₂×D₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
170	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
171	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
172	#499	C₂²×C₁₃₁	1	2	yes	—
173	#495	C₂²×C₁₃₁	1	2	yes	—
174	#488	—	2	2	no	—
175	#467	C₂²×C₁₃₁	1	2	yes	—
176	#466	C₂²×C₁₃₁	1	2	yes	—
177	#465	—	2	2	no	—
178	C₄×Q₁₆	C₂²×C₁₃₁	1	2	yes	—
179	C₄×SD₁₆	—	2	2	no	—
180	C₄×D₈	C₂²×C₁₃₁	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
182	#367	C₂²×C₁₃₁	1	1	yes	characteristic
183	#367	C₂²×C₁₃₁	1	1	yes	characteristic
184	#366	C₂²×C₁₃₁	1	2	yes	—
185	#358	C₂²×C₁₃₁	1	2	yes	—
186	#358	C₂²×C₁₃₁	1	2	yes	—
187	#357	C₂²×C₁₃₁	1	2	yes	—
188	#357	C₂²×C₁₃₁	1	2	yes	—
189	#346	C₂²×C₁₃₁	1	1	yes	characteristic
190	#346	C₂²×C₁₃₁	1	1	yes	characteristic
191	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
192	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
193	C₂×Q₁₆	C₂³×C₁₃₁	1	2	yes	—
194	C₂×SD₁₆	—	2	2	no	—
195	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
196	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
198	#126	—	4	2	no	—
199	#125	—	2	2	no	—
200	#124	—	2	2	no	—
201	#123	—	4	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	#121	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×D₄	—	2	2	no	—
206	C₄×D₄	—	4	2	no	—
207	#117	—	4	2	no	—
208	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
209	Q₃₂	—	2	2	no	—
210	SD₃₂	—	2	4	no	—
211	D₁₆	—	2	2	no	—
212	M₃₂	—	2	2	no	—
213	M₃₂	—	2	2	no	—
214	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
215	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
216	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
217	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
218	#107	C₂³×C₁₃₁	1	2	yes	—
219	#107	C₂³×C₁₃₁	1	2	yes	—
220	#106	—	2	2	no	—
221	#103	—	2	2	no	—
222	#103	—	2	2	no	—
223	#102	—	2	2	no	—
224	#102	—	2	2	no	—
225	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
226	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
229	C₂×Q₈	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	2	2	no	—
239	D₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	D₈×C₁₃₁	1	2	yes	—
260	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
262	C₁₆	—	2	1	no	—
263	C₁₆	—	2	1	no	—
264	C₁₆	#41×C₁₃₁	1	2	yes	—
265	C₁₆	#41×C₁₃₁	1	2	yes	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	8	4	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
281	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	#135×C₁₃₁	1	2	yes	—
287	C₈	C₂×#41×C₁₃₁	1	1	yes	characteristic
288	C₈	C₂×#41×C₁₃₁	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	#532×C₁₃₁	1	1	yes	characteristic
302	C₄	C₂×#135×C₁₃₁	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	#3169×C₁₃₁	1	1	yes	characteristic
306	C₂	#3414×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	#2218×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3168×C₁₃₁	C₂	semidirect
2. #2253×C₁₃₁	C₂	semidirect
3. #2249×C₁₃₁	C₂	semidirect
4. #2242×C₁₃₁	C₂	semidirect
5. #2220×C₁₃₁	C₂	semidirect
6. #2219×C₁₃₁	C₂	semidirect
8. #2177×C₁₃₁	C₂	semidirect
9. #2176×C₁₃₁	C₂	semidirect
10. #2175×C₁₃₁	C₂	semidirect
11. C₄×M₃₂×C₁₃₁	C₂	semidirect
153. #33750	C₁₃₁	direct (N × H)
154. C₁₃₁	#33750	direct (N × H)
155. #3168	C₂×C₁₃₁	semidirect
156. #2253	C₂×C₁₃₁	semidirect
157. #2249	C₂×C₁₃₁	semidirect
158. #2242	C₂×C₁₃₁	semidirect
159. #2220	C₂×C₁₃₁	semidirect
160. #2219	C₂×C₁₃₁	semidirect
162. #2177	C₂×C₁₃₁	semidirect
163. #2176	C₂×C₁₃₁	semidirect
164. #2175	C₂×C₁₃₁	semidirect
165. C₄×M₃₂	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3168×C₁₃₁	2
#2253×C₁₃₁	2
#2249×C₁₃₁	2
#2242×C₁₃₁	2
#2220×C₁₃₁	2
#2219×C₁₃₁	2
#2218×C₁₃₁	2
#2177×C₁₃₁	2
#2176×C₁₃₁	2
#2175×C₁₃₁	2
C₄×M₃₂×C₁₃₁	2
#33750	131

Subgroups — C131×#33750 #45719634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33750 #45719634