Subgroups — C₁₃₁×#33774 #45719658

Order 33536.

686 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3170×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2248×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2240×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2207×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2187×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#2162×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₁	—	2	1	no	—
9	C₂²×D₈×C₁₃₁	C₂²	1	2	yes	—
10	#533×C₁₃₁	—	2	1	no	—
11	C₂×#127×C₁₃₁	—	2	1	no	—
12	C₂×SD₃₂×C₁₃₁	C₂²	1	8	yes	complemented
13	#500×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
14	#494×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
15	#493×C₁₃₁	—	2	1	no	—
16	#493×C₁₃₁	—	2	1	no	—
17	#493×C₁₃₁	C₂²	1	2	yes	—
18	#493×C₁₃₁	C₂²	1	2	yes	—
19	#487×C₁₃₁	—	2	1	no	—
20	#467×C₁₃₁	—	2	1	no	—
21	#467×C₁₃₁	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₁₃₁	—	2	1	no	—
23	C₄×Q₁₆×C₁₃₁	C₂²	1	2	yes	complemented
24	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
26	#357×C₁₃₁	C₂²	1	8	yes	complemented
27	C₂²⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
28	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
29	C₂²×D₄×C₁₃₁	—	2	2	no	—
30	C₂²×D₄×C₁₃₁	—	4	1	no	—
31	C₂²×D₄×C₁₃₁	—	4	1	no	—
32	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
33	C₂×Q₁₆×C₁₃₁	C₂³	1	2	yes	—
34	C₂×D₈×C₁₃₁	—	2	2	no	—
35	C₂×D₈×C₁₃₁	—	2	2	no	—
36	C₂×D₈×C₁₃₁	—	2	2	no	—
37	C₂×D₈×C₁₃₁	—	4	4	no	—
38	C₂×D₈×C₁₃₁	—	4	4	no	—
39	C₂×D₈×C₁₃₁	C₂³	1	4	yes	—
40	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
43	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
44	#127×C₁₃₁	—	2	2	no	—
45	#127×C₁₃₁	—	2	2	no	—
46	#127×C₁₃₁	—	4	1	no	—
47	#127×C₁₃₁	—	4	1	no	—
48	#125×C₁₃₁	—	4	1	no	—
49	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
50	C₂²⋊Q₈×C₁₃₁	—	4	1	no	—
51	C₄×Q₈×C₁₃₁	—	2	2	no	—
52	C₄×Q₈×C₁₃₁	—	4	1	no	—
53	#117×C₁₃₁	—	4	1	no	—
54	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
55	SD₃₂×C₁₃₁	—	4	8	no	—
56	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
57	#107×C₁₃₁	—	2	1	no	—
58	#107×C₁₃₁	C₂³	1	2	yes	—
59	#103×C₁₃₁	—	2	2	no	—
60	#103×C₁₃₁	—	2	2	no	—
61	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
65	C₂⁴×C₁₃₁	—	4	2	no	—
66	C₂×Q₈×C₁₃₁	—	2	2	no	—
67	C₂×Q₈×C₁₃₁	—	4	1	no	—
68	C₂×D₄×C₁₃₁	—	2	4	no	—
69	C₂×D₄×C₁₃₁	—	4	4	no	—
70	C₂×D₄×C₁₃₁	—	4	4	no	—
71	C₂×D₄×C₁₃₁	—	4	2	no	—
72	C₂×D₄×C₁₃₁	—	4	2	no	—
73	C₂×D₄×C₁₃₁	—	4	2	no	—
74	C₂×D₄×C₁₃₁	—	4	2	no	—
75	C₂×D₄×C₁₃₁	—	4	2	no	—
76	C₂×D₄×C₁₃₁	—	8	4	no	—
77	C₂×D₄×C₁₃₁	—	8	4	no	—
78	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₁₃₁	—	2	4	no	—
82	Q₁₆×C₁₃₁	—	4	2	no	—
83	D₈×C₁₃₁	—	2	8	no	—
84	D₈×C₁₃₁	—	4	4	no	—
85	D₈×C₁₃₁	—	4	4	no	—
86	D₈×C₁₃₁	—	4	4	no	—
87	C₂×C₈×C₁₃₁	—	2	2	no	—
88	C₂×C₈×C₁₃₁	—	2	2	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	—	4	1	no	—
95	C₄⋊C₄×C₁₃₁	—	4	1	no	—
96	C₄⋊C₄×C₁₃₁	—	4	2	no	—
97	C₄⋊C₄×C₁₃₁	—	4	2	no	—
98	C₄⋊C₄×C₁₃₁	—	4	1	no	—
99	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
100	C₄²×C₁₃₁	—	4	2	no	—
101	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
102	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
103	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
105	C₁₆×C₁₃₁	—	2	4	no	—
106	C₂³×C₁₃₁	—	4	4	no	—
107	C₂³×C₁₃₁	—	8	4	no	—
108	C₂³×C₁₃₁	—	8	4	no	—
109	C₂³×C₁₃₁	—	8	4	no	—
110	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
111	Q₈×C₁₃₁	—	4	2	no	—
112	Q₈×C₁₃₁	—	4	2	no	—
113	Q₈×C₁₃₁	—	8	2	no	—
114	D₄×C₁₃₁	—	4	4	no	—
115	D₄×C₁₃₁	—	4	4	no	—
116	D₄×C₁₃₁	—	8	4	no	—
117	D₄×C₁₃₁	—	8	4	no	—
118	D₄×C₁₃₁	—	8	4	no	—
119	D₄×C₁₃₁	—	8	4	no	—
120	D₄×C₁₃₁	—	8	4	no	—
121	C₂×C₄×C₁₃₁	—	2	1	no	—
122	C₂×C₄×C₁₃₁	—	2	1	no	—
123	C₂×C₄×C₁₃₁	—	2	2	no	—
124	C₂×C₄×C₁₃₁	—	4	2	no	—
125	C₂×C₄×C₁₃₁	—	4	2	no	—
126	C₂×C₄×C₁₃₁	SD₃₂	1	4	yes	complemented
127	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₁₃₁	—	2	1	no	—
134	C₈×C₁₃₁	—	2	1	no	—
135	C₈×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
136	C₈×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
137	C₂²×C₁₃₁	—	2	1	no	—
138	C₂²×C₁₃₁	—	2	1	no	—
139	C₂²×C₁₃₁	—	8	4	no	—
140	C₂²×C₁₃₁	—	8	4	no	—
141	C₂²×C₁₃₁	—	8	4	no	—
142	C₂²×C₁₃₁	—	16	8	no	—
143	C₂²×C₁₃₁	C₂×SD₃₂	1	2	yes	—
144	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₁₃₁	—	2	1	no	—
146	C₄×C₁₃₁	—	2	1	no	—
147	C₄×C₁₃₁	—	8	2	no	—
148	C₄×C₁₃₁	—	8	2	no	—
149	C₄×C₁₃₁	C₂×SD₃₂	1	2	yes	—
150	C₄×C₁₃₁	C₂×SD₃₂	1	2	yes	—
151	C₄×C₁₃₁	#509	1	2	yes	—
152	C₄×C₁₃₁	#535	1	1	yes	characteristic
153	C₄×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₁₃₁	—	2	2	no	—
155	C₂×C₁₃₁	—	16	4	no	—
156	C₂×C₁₃₁	#3205	1	1	yes	characteristic
157	C₂×C₁₃₁	C₂²×SD₃₂	1	1	yes	characteristic
158	C₂×C₁₃₁	C₂×#509	1	1	yes	characteristic
159	#33774	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₁₃₁	#33774	1	1	yes	characteristic, direct product factor, Sylow
161	#3170	C₂×C₁₃₁	1	1	yes	characteristic, complemented
162	C₂×#493	C₂×C₁₃₁	1	1	yes	characteristic
163	#2248	C₂×C₁₃₁	1	2	yes	complemented
164	#2240	C₂×C₁₃₁	1	2	yes	complemented
165	#2207	C₂×C₁₃₁	1	4	yes	complemented
166	#2187	C₂×C₁₃₁	1	4	yes	complemented
167	#2162	C₂×C₁₃₁	1	1	yes	characteristic, complemented
168	C₂²×D₈	—	2	1	no	—
169	C₂²×D₈	C₂²×C₁₃₁	1	2	yes	—
170	#533	—	2	1	no	—
171	C₂×#127	—	2	1	no	—
172	C₂×SD₃₂	C₂²×C₁₃₁	1	8	yes	complemented
173	#500	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
174	#494	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
175	#493	—	2	1	no	—
176	#493	—	2	1	no	—
177	#493	C₂²×C₁₃₁	1	2	yes	—
178	#493	C₂²×C₁₃₁	1	2	yes	—
179	#487	—	2	1	no	—
180	#467	—	2	1	no	—
181	#467	C₂²×C₁₃₁	1	2	yes	—
182	C₄×Q₁₆	—	2	1	no	—
183	C₄×Q₁₆	C₂²×C₁₃₁	1	2	yes	complemented
184	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
185	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
186	#357	C₂²×C₁₃₁	1	8	yes	complemented
187	C₂²⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
188	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	4	1	no	—
191	C₂²×D₄	—	4	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	C₂³×C₁₃₁	1	2	yes	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	4	4	no	—
198	C₂×D₈	—	4	4	no	—
199	C₂×D₈	C₂³×C₁₃₁	1	4	yes	—
200	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	#127	—	2	2	no	—
205	#127	—	2	2	no	—
206	#127	—	4	1	no	—
207	#127	—	4	1	no	—
208	#125	—	4	1	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	4	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	4	1	no	—
213	#117	—	4	1	no	—
214	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
215	SD₃₂	—	4	8	no	—
216	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
217	#107	—	2	1	no	—
218	#107	C₂³×C₁₃₁	1	2	yes	—
219	#103	—	2	2	no	—
220	#103	—	2	2	no	—
221	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
222	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
225	C₂⁴	—	4	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	8	4	no	—
237	C₂×D₄	—	8	4	no	—
238	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
239	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	4	2	no	—
243	D₈	—	2	8	no	—
244	D₈	—	4	4	no	—
245	D₈	—	4	4	no	—
246	D₈	—	4	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
262	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
265	C₁₆	—	2	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	—	8	4	no	—
270	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	8	2	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	8	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	SD₃₂×C₁₃₁	1	4	yes	complemented
287	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
296	C₈	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	16	8	no	—
303	C₂²	C₂×SD₃₂×C₁₃₁	1	2	yes	—
304	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₂×SD₃₂×C₁₃₁	1	2	yes	—
310	C₄	C₂×SD₃₂×C₁₃₁	1	2	yes	—
311	C₄	#509×C₁₃₁	1	2	yes	—
312	C₄	#535×C₁₃₁	1	1	yes	characteristic
313	C₄	C₂×AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	16	4	no	—
316	C₂	#3205×C₁₃₁	1	1	yes	characteristic
317	C₂	C₂²×SD₃₂×C₁₃₁	1	1	yes	characteristic
318	C₂	C₂×#509×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	C₂×#493×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. #3170×C₁₃₁	C₂	semidirect
3. #2248×C₁₃₁	C₂	semidirect
4. #2240×C₁₃₁	C₂	semidirect
5. #2207×C₁₃₁	C₂	semidirect
6. #2187×C₁₃₁	C₂	semidirect
7. #2162×C₁₃₁	C₂	semidirect
12. C₂×SD₃₂×C₁₃₁	C₂²	semidirect
13. #500×C₁₃₁	C₂²	semidirect
14. #494×C₁₃₁	C₂²	semidirect
23. C₄×Q₁₆×C₁₃₁	C₂²	semidirect
25. C₄⋊C₁₆×C₁₃₁	C₂²	semidirect
26. #357×C₁₃₁	C₂²	semidirect
28. C₄×C₁₆×C₁₃₁	C₂²	semidirect
126. C₂×C₄×C₁₃₁	SD₃₂	semidirect
159. #33774	C₁₃₁	direct (N × H)
160. C₁₃₁	#33774	direct (N × H)
161. #3170	C₂×C₁₃₁	semidirect
163. #2248	C₂×C₁₃₁	semidirect
164. #2240	C₂×C₁₃₁	semidirect
165. #2207	C₂×C₁₃₁	semidirect
166. #2187	C₂×C₁₃₁	semidirect
167. #2162	C₂×C₁₃₁	semidirect
172. C₂×SD₃₂	C₂²×C₁₃₁	semidirect
173. #500	C₂²×C₁₃₁	semidirect
174. #494	C₂²×C₁₃₁	semidirect
183. C₄×Q₁₆	C₂²×C₁₃₁	semidirect
185. C₄⋊C₁₆	C₂²×C₁₃₁	semidirect
186. #357	C₂²×C₁₃₁	semidirect
188. C₄×C₁₆	C₂²×C₁₃₁	semidirect
286. C₂×C₄	SD₃₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3170×C₁₃₁	2
C₂×#493×C₁₃₁	2
#2248×C₁₃₁	2
#2240×C₁₃₁	2
#2207×C₁₃₁	2
#2187×C₁₃₁	2
#2162×C₁₃₁	2
#33774	131

Subgroups — C131×#33774 #45719658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33774 #45719658