Subgroups — C₁₃₁×#19293 #45705177

Order 33536.

680 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×#494×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3079×C₁₃₁	C₂	1	2	yes	maximal
3	#2965×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1690×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#1689×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#1566×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1549×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆×C₁₃₁	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
10	C₄×#41×C₁₃₁	C₂²	1	1	yes	characteristic
11	#494×C₁₃₁	—	2	1	no	—
12	#494×C₁₃₁	—	2	1	no	—
13	#494×C₁₃₁	C₂²	1	4	yes	—
14	#477×C₁₃₁	—	2	2	no	—
15	#476×C₁₃₁	—	2	2	no	—
16	#475×C₁₃₁	C₂²	1	4	yes	—
17	#474×C₁₃₁	C₂²	1	4	yes	—
18	#446×C₁₃₁	C₂²	1	2	yes	—
19	#436×C₁₃₁	C₂²	1	2	yes	—
20	#435×C₁₃₁	—	2	2	no	—
21	#431×C₁₃₁	—	2	1	no	—
22	#431×C₁₃₁	C₂²	1	1	yes	characteristic
23	#428×C₁₃₁	—	2	2	no	—
24	#423×C₁₃₁	C₂²	1	2	yes	—
25	#417×C₁₃₁	—	2	2	no	—
26	C₄×M₁₆×C₁₃₁	—	2	1	no	—
27	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
28	#334×C₁₃₁	C₂²	1	4	yes	—
29	#332×C₁₃₁	C₂²	1	8	yes	—
30	C₂×#41×C₁₃₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
33	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
34	C₂×Q₁₆×C₁₃₁	—	4	8	no	—
35	C₂×M₁₆×C₁₃₁	—	2	2	no	—
36	C₂×M₁₆×C₁₃₁	—	2	2	no	—
37	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
38	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
39	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
40	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
41	C₄⋊Q₈×C₁₃₁	C₂³	1	4	yes	—
42	C₄×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₁₃₁	—	2	1	no	—
45	C₄×D₄×C₁₃₁	—	2	1	no	—
46	C₄×D₄×C₁₃₁	D₄	1	1	yes	characteristic
47	C₄×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
48	#117×C₁₃₁	—	2	1	no	—
49	#117×C₁₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
52	C₂×C₄²×C₁₃₁	—	2	1	no	—
53	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
54	#107×C₁₃₁	—	2	4	no	—
55	#106×C₁₃₁	—	2	4	no	—
56	C₄⋊C₈×C₁₃₁	—	2	2	no	—
57	C₄⋊C₈×C₁₃₁	—	2	2	no	—
58	C₄⋊C₈×C₁₃₁	C₂³	1	4	yes	—
59	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
60	#103×C₁₃₁	—	2	4	no	—
61	#102×C₁₃₁	—	2	4	no	—
62	C₂²⋊C₈×C₁₃₁	—	2	1	no	—
63	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
64	#97×C₁₃₁	—	2	2	no	—
65	C₄×C₈×C₁₃₁	—	2	1	no	—
66	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
69	#41×C₁₃₁	—	2	4	no	—
70	C₂×Q₈×C₁₃₁	—	2	4	no	—
71	C₂×Q₈×C₁₃₁	—	4	8	no	—
72	C₂×Q₈×C₁₃₁	—	4	2	no	—
73	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₁₃₁	—	2	1	no	—
75	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₁	—	2	1	no	—
77	C₂²×C₄×C₁₃₁	—	2	1	no	—
78	C₂²×C₄×C₁₃₁	—	2	1	no	—
79	C₂²×C₄×C₁₃₁	—	4	2	no	—
80	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
83	Q₁₆×C₁₃₁	—	4	8	no	—
84	Q₁₆×C₁₃₁	—	4	8	no	—
85	M₁₆×C₁₃₁	—	4	4	no	—
86	M₁₆×C₁₃₁	—	4	4	no	—
87	C₂×C₈×C₁₃₁	—	2	4	no	—
88	C₂×C₈×C₁₃₁	—	2	2	no	—
89	C₂×C₈×C₁₃₁	—	2	4	no	—
90	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₁₃₁	—	2	1	no	—
93	C₄⋊C₄×C₁₃₁	—	2	1	no	—
94	C₄⋊C₄×C₁₃₁	—	2	4	no	—
95	C₄⋊C₄×C₁₃₁	—	2	4	no	—
96	C₄⋊C₄×C₁₃₁	—	4	2	no	—
97	C₄⋊C₄×C₁₃₁	—	4	2	no	—
98	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
101	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
102	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
103	C₄²×C₁₃₁	—	2	1	no	—
104	C₄²×C₁₃₁	—	2	1	no	—
105	C₄²×C₁₃₁	—	2	1	no	—
106	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃₁	#41	1	1	yes	characteristic
109	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
110	C₂³×C₁₃₁	—	2	1	no	—
111	C₂³×C₁₃₁	#121	1	1	yes	characteristic
112	Q₈×C₁₃₁	—	2	2	no	—
113	Q₈×C₁₃₁	—	4	8	no	—
114	Q₈×C₁₃₁	—	8	4	no	—
115	D₄×C₁₃₁	—	2	2	no	—
116	D₄×C₁₃₁	—	4	2	no	—
117	C₂×C₄×C₁₃₁	—	2	2	no	—
118	C₂×C₄×C₁₃₁	—	2	2	no	—
119	C₂×C₄×C₁₃₁	—	2	1	no	—
120	C₂×C₄×C₁₃₁	—	2	1	no	—
121	C₂×C₄×C₁₃₁	—	2	1	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	4	no	—
126	C₂×C₄×C₁₃₁	—	8	4	no	—
127	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	#137	1	4	yes	—
131	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₃₁	—	2	2	no	—
135	C₈×C₁₃₁	—	4	2	no	—
136	C₈×C₁₃₁	—	4	4	no	—
137	C₈×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
138	C₂²×C₁₃₁	—	2	1	no	—
139	C₂²×C₁₃₁	—	2	1	no	—
140	C₂²×C₁₃₁	—	4	1	no	—
141	C₂²×C₁₃₁	—	4	1	no	—
142	C₂²×C₁₃₁	—	4	1	no	—
143	C₂²×C₁₃₁	#470	1	2	yes	—
144	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
145	C₄×C₁₃₁	—	2	1	no	—
146	C₄×C₁₃₁	—	2	1	no	—
147	C₄×C₁₃₁	—	4	1	no	—
148	C₄×C₁₃₁	—	4	1	no	—
149	C₄×C₁₃₁	—	4	1	no	—
150	C₄×C₁₃₁	—	8	4	no	—
151	C₄×C₁₃₁	#470	1	2	yes	—
152	C₄×C₁₃₁	#473	1	2	yes	—
153	C₄×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
154	C₄×C₁₃₁	C₂×#137	1	2	yes	—
155	C₂×C₁₃₁	—	2	2	no	—
156	C₂×C₁₃₁	—	8	1	no	—
157	C₂×C₁₃₁	#3038	1	1	yes	characteristic
158	C₂×C₁₃₁	C₂×#470	1	1	yes	characteristic
159	C₂×C₁₃₁	C₂×#473	1	1	yes	characteristic
160	#19293	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₁₃₁	#19293	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#494	C₂×C₁₃₁	1	1	yes	characteristic, complemented
163	#3079	C₂×C₁₃₁	1	2	yes	—
164	#2965	C₂×C₁₃₁	1	1	yes	characteristic
165	#1690	C₂×C₁₃₁	1	4	yes	complemented
166	#1689	C₂×C₁₃₁	1	4	yes	complemented
167	#1566	C₂×C₁₃₁	1	1	yes	characteristic, complemented
168	#1549	C₂×C₁₃₁	1	2	yes	complemented
169	C₂²×Q₁₆	—	2	2	no	—
170	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
171	C₄×#41	C₂²×C₁₃₁	1	1	yes	characteristic
172	#494	—	2	1	no	—
173	#494	—	2	1	no	—
174	#494	C₂²×C₁₃₁	1	4	yes	—
175	#477	—	2	2	no	—
176	#476	—	2	2	no	—
177	#475	C₂²×C₁₃₁	1	4	yes	—
178	#474	C₂²×C₁₃₁	1	4	yes	—
179	#446	C₂²×C₁₃₁	1	2	yes	—
180	#436	C₂²×C₁₃₁	1	2	yes	—
181	#435	—	2	2	no	—
182	#431	—	2	1	no	—
183	#431	C₂²×C₁₃₁	1	1	yes	characteristic
184	#428	—	2	2	no	—
185	#423	C₂²×C₁₃₁	1	2	yes	—
186	#417	—	2	2	no	—
187	C₄×M₁₆	—	2	1	no	—
188	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
189	#334	C₂²×C₁₃₁	1	4	yes	—
190	#332	C₂²×C₁₃₁	1	8	yes	—
191	C₂×#41	D₄×C₁₃₁	1	1	yes	characteristic
192	C₂²×Q₈	—	2	2	no	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	—	2	4	no	—
195	C₂×Q₁₆	—	4	8	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	C₂³×C₁₃₁	1	4	yes	—
203	C₄×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
204	C₄×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	D₄×C₁₃₁	1	1	yes	characteristic
208	C₄×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
209	#117	—	2	1	no	—
210	#117	D₄×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
215	#107	—	2	4	no	—
216	#106	—	2	4	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
220	C₄⋊C₈	C₂³×C₁₃₁	1	4	yes	—
221	#103	—	2	4	no	—
222	#102	—	2	4	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	#97	—	2	2	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
228	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
230	#41	—	2	4	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
243	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
244	Q₁₆	—	4	8	no	—
245	Q₁₆	—	4	8	no	—
246	M₁₆	—	4	4	no	—
247	M₁₆	—	4	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
269	C₄²	#41×C₁₃₁	1	1	yes	characteristic
270	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
271	C₂³	—	2	1	no	—
272	C₂³	#121×C₁₃₁	1	1	yes	characteristic
273	Q₈	—	2	2	no	—
274	Q₈	—	4	8	no	—
275	Q₈	—	8	4	no	—
276	D₄	—	2	2	no	—
277	D₄	—	4	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
289	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
290	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
291	C₂×C₄	#137×C₁₃₁	1	4	yes	—
292	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	2	no	—
297	C₈	—	4	4	no	—
298	C₈	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	#470×C₁₃₁	1	2	yes	—
305	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	8	4	no	—
312	C₄	#470×C₁₃₁	1	2	yes	—
313	C₄	#473×C₁₃₁	1	2	yes	—
314	C₄	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
315	C₄	C₂×#137×C₁₃₁	1	2	yes	—
316	C₂	—	2	2	no	—
317	C₂	—	8	1	no	—
318	C₂	#3038×C₁₃₁	1	1	yes	characteristic
319	C₂	C₂×#470×C₁₃₁	1	1	yes	characteristic
320	C₂	C₂×#473×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
prime radical	C₂×#41×C₁₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#494×C₁₃₁	C₂	semidirect
4. #1690×C₁₃₁	C₂	semidirect
5. #1689×C₁₃₁	C₂	semidirect
6. #1566×C₁₃₁	C₂	semidirect
7. #1549×C₁₃₁	C₂	semidirect
160. #19293	C₁₃₁	direct (N × H)
161. C₁₃₁	#19293	direct (N × H)
162. C₂×#494	C₂×C₁₃₁	semidirect
165. #1690	C₂×C₁₃₁	semidirect
166. #1689	C₂×C₁₃₁	semidirect
167. #1566	C₂×C₁₃₁	semidirect
168. #1549	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×#494×C₁₃₁	2
#3079×C₁₃₁	2
#2965×C₁₃₁	2
#1690×C₁₃₁	2
#1689×C₁₃₁	2
#1566×C₁₃₁	2
#1549×C₁₃₁	2
#19293	131

Subgroups — C131×#19293 #45705177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#19293 #45705177