Subgroups — C₁₃₁×#29862 #45715746

Order 33536.

660 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#491×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3080×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#3039×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2923×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#2064×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2052×C₁₃₁	C₂	1	4	yes	maximal
7	#1998×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#1966×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#1849×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₁₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#125×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄×#41×C₁₃₁	C₂²	1	1	yes	characteristic
14	#491×C₁₃₁	—	2	2	no	—
15	#491×C₁₃₁	—	2	2	no	—
16	#479×C₁₃₁	—	2	2	no	—
17	#478×C₁₃₁	—	2	2	no	—
18	#462×C₁₃₁	—	2	2	no	—
19	#461×C₁₃₁	—	2	2	no	—
20	#433×C₁₃₁	—	2	2	no	—
21	#432×C₁₃₁	—	2	2	no	—
22	C₂×#108×C₁₃₁	C₂²	1	4	yes	—
23	#428×C₁₃₁	C₂²	1	2	yes	—
24	#424×C₁₃₁	C₂²	1	2	yes	—
25	#423×C₁₃₁	C₂²	1	1	yes	characteristic
26	#423×C₁₃₁	C₂²	1	1	yes	characteristic
27	#419×C₁₃₁	C₂²	1	2	yes	—
28	#417×C₁₃₁	C₂²	1	2	yes	—
29	C₂×#103×C₁₃₁	C₂²	1	4	yes	—
30	C₂×#101×C₁₃₁	C₂²	1	4	yes	—
31	#408×C₁₃₁	C₂²	1	2	yes	—
32	C₄×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
33	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
34	#343×C₁₃₁	C₂²	1	4	yes	—
35	C₂×#41×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
37	#137×C₁₃₁	—	4	4	no	—
38	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
39	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
40	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
41	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
45	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
46	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
47	#125×C₁₃₁	—	2	2	no	—
48	#125×C₁₃₁	—	2	1	no	—
49	#125×C₁₃₁	—	2	1	no	—
50	C₄×Q₈×C₁₃₁	D₄	1	2	yes	—
51	C₄×D₄×C₁₃₁	—	2	2	no	—
52	C₄×D₄×C₁₃₁	D₄	1	2	yes	—
53	#117×C₁₃₁	—	2	1	no	—
54	#117×C₁₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₁₃₁	—	2	1	no	—
59	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
60	#108×C₁₃₁	—	4	4	no	—
61	#107×C₁₃₁	—	2	2	no	—
62	#106×C₁₃₁	—	2	2	no	—
63	C₄⋊C₈×C₁₃₁	—	2	4	no	—
64	C₄⋊C₈×C₁₃₁	—	2	2	no	—
65	C₄⋊C₈×C₁₃₁	—	2	2	no	—
66	#103×C₁₃₁	—	2	2	no	—
67	#103×C₁₃₁	—	2	8	no	—
68	#103×C₁₃₁	—	2	4	no	—
69	#102×C₁₃₁	—	2	2	no	—
70	#101×C₁₃₁	—	4	4	no	—
71	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
72	#97×C₁₃₁	—	2	1	no	—
73	#97×C₁₃₁	D₄	1	2	yes	—
74	#97×C₁₃₁	D₄	1	2	yes	—
75	C₄×C₈×C₁₃₁	—	2	1	no	—
76	#41×C₁₃₁	—	2	2	no	—
77	#41×C₁₃₁	—	2	2	no	—
78	C₂×Q₈×C₁₃₁	—	2	2	no	—
79	C₂×Q₈×C₁₃₁	—	2	2	no	—
80	C₂×Q₈×C₁₃₁	—	2	4	no	—
81	C₂×Q₈×C₁₃₁	—	2	4	no	—
82	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×D₄×C₁₃₁	—	2	1	no	—
84	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₁	—	2	1	no	—
86	C₂²×C₄×C₁₃₁	—	2	2	no	—
87	C₂²×C₄×C₁₃₁	—	2	2	no	—
88	C₂²×C₄×C₁₃₁	—	2	2	no	—
89	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
90	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
91	Q₁₆×C₁₃₁	—	4	4	no	—
92	SD₁₆×C₁₃₁	—	4	4	no	—
93	M₁₆×C₁₃₁	—	2	2	no	—
94	M₁₆×C₁₃₁	—	2	4	no	—
95	M₁₆×C₁₃₁	—	2	4	no	—
96	M₁₆×C₁₃₁	—	2	2	no	—
97	C₂×C₈×C₁₃₁	—	2	2	no	—
98	C₂×C₈×C₁₃₁	—	2	2	no	—
99	C₂×C₈×C₁₃₁	—	2	2	no	—
100	C₂×C₈×C₁₃₁	—	2	2	no	—
101	C₂×C₈×C₁₃₁	—	2	2	no	—
102	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
103	C₄⋊C₄×C₁₃₁	—	2	2	no	—
104	C₄⋊C₄×C₁₃₁	—	2	2	no	—
105	C₄⋊C₄×C₁₃₁	—	2	2	no	—
106	C₄⋊C₄×C₁₃₁	—	4	2	no	—
107	C₄⋊C₄×C₁₃₁	—	4	2	no	—
108	C₄⋊C₄×C₁₃₁	—	4	2	no	—
109	C₄⋊C₄×C₁₃₁	—	4	2	no	—
110	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
112	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
113	C₄²×C₁₃₁	—	2	2	no	—
114	C₄²×C₁₃₁	—	2	1	no	—
115	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
118	C₂³×C₁₃₁	—	2	1	no	—
119	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
120	Q₈×C₁₃₁	—	2	1	no	—
121	Q₈×C₁₃₁	—	2	1	no	—
122	Q₈×C₁₃₁	—	4	2	no	—
123	Q₈×C₁₃₁	—	4	2	no	—
124	Q₈×C₁₃₁	—	4	4	no	—
125	D₄×C₁₃₁	—	2	1	no	—
126	D₄×C₁₃₁	—	2	1	no	—
127	D₄×C₁₃₁	—	4	2	no	—
128	C₂×C₄×C₁₃₁	—	2	2	no	—
129	C₂×C₄×C₁₃₁	—	2	1	no	—
130	C₂×C₄×C₁₃₁	—	4	2	no	—
131	C₂×C₄×C₁₃₁	—	4	2	no	—
132	C₂×C₄×C₁₃₁	—	4	2	no	—
133	C₂×C₄×C₁₃₁	—	4	2	no	—
134	C₂×C₄×C₁₃₁	—	4	4	no	—
135	C₂×C₄×C₁₃₁	—	8	2	no	—
136	C₂×C₄×C₁₃₁	—	8	2	no	—
137	C₂×C₄×C₁₃₁	#121	1	2	yes	—
138	C₂×C₄×C₁₃₁	#121	1	2	yes	—
139	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
142	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
143	C₈×C₁₃₁	—	2	4	no	—
144	C₈×C₁₃₁	—	4	2	no	—
145	C₈×C₁₃₁	—	4	2	no	—
146	C₈×C₁₃₁	—	4	2	no	—
147	C₂²×C₁₃₁	—	2	2	no	—
148	C₂²×C₁₃₁	—	4	1	no	—
149	C₂²×C₁₃₁	—	4	2	no	—
150	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
151	C₂²×C₁₃₁	#542	1	2	yes	—
152	C₄×C₁₃₁	—	2	2	no	—
153	C₄×C₁₃₁	—	2	2	no	—
154	C₄×C₁₃₁	—	4	1	no	—
155	C₄×C₁₃₁	—	4	2	no	—
156	C₄×C₁₃₁	—	8	2	no	—
157	C₄×C₁₃₁	—	8	2	no	—
158	C₄×C₁₃₁	#535	1	1	yes	characteristic
159	C₄×C₁₃₁	#536	1	1	yes	characteristic
160	C₄×C₁₃₁	#537	1	2	yes	—
161	C₂×C₁₃₁	—	2	2	no	—
162	C₂×C₁₃₁	—	8	1	no	—
163	C₂×C₁₃₁	#2620	1	1	yes	characteristic
164	C₂×C₁₃₁	#3069	1	2	yes	—
165	#29862	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₁	#29862	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#491	C₂×C₁₃₁	1	1	yes	characteristic, complemented
168	#3080	C₂×C₁₃₁	1	1	yes	characteristic
169	#3039	C₂×C₁₃₁	1	1	yes	characteristic
170	#2923	C₂×C₁₃₁	1	1	yes	characteristic
171	#2064	C₂×C₁₃₁	1	4	yes	complemented
172	#2052	C₂×C₁₃₁	1	4	yes	—
173	#1998	C₂×C₁₃₁	1	1	yes	characteristic, complemented
174	#1966	C₂×C₁₃₁	1	1	yes	characteristic, complemented
175	#1849	C₂×C₁₃₁	1	1	yes	characteristic, complemented
176	C₂×#137	C₂²×C₁₃₁	1	2	yes	—
177	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
178	C₂×#125	C₂²×C₁₃₁	1	1	yes	characteristic
179	C₄×#41	C₂²×C₁₃₁	1	1	yes	characteristic
180	#491	—	2	2	no	—
181	#491	—	2	2	no	—
182	#479	—	2	2	no	—
183	#478	—	2	2	no	—
184	#462	—	2	2	no	—
185	#461	—	2	2	no	—
186	#433	—	2	2	no	—
187	#432	—	2	2	no	—
188	C₂×#108	C₂²×C₁₃₁	1	4	yes	—
189	#428	C₂²×C₁₃₁	1	2	yes	—
190	#424	C₂²×C₁₃₁	1	2	yes	—
191	#423	C₂²×C₁₃₁	1	1	yes	characteristic
192	#423	C₂²×C₁₃₁	1	1	yes	characteristic
193	#419	C₂²×C₁₃₁	1	2	yes	—
194	#417	C₂²×C₁₃₁	1	2	yes	—
195	C₂×#103	C₂²×C₁₃₁	1	4	yes	—
196	C₂×#101	C₂²×C₁₃₁	1	4	yes	—
197	#408	C₂²×C₁₃₁	1	2	yes	—
198	C₄×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
199	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
200	#343	C₂²×C₁₃₁	1	4	yes	—
201	C₂×#41	C₂³×C₁₃₁	1	1	yes	characteristic
202	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
203	#137	—	4	4	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
207	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
208	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
209	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	#125	—	2	2	no	—
214	#125	—	2	1	no	—
215	#125	—	2	1	no	—
216	C₄×Q₈	D₄×C₁₃₁	1	2	yes	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	D₄×C₁₃₁	1	2	yes	—
219	#117	—	2	1	no	—
220	#117	C₂³×C₁₃₁	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
226	#108	—	4	4	no	—
227	#107	—	2	2	no	—
228	#106	—	2	2	no	—
229	C₄⋊C₈	—	2	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	#103	—	2	2	no	—
233	#103	—	2	8	no	—
234	#103	—	2	4	no	—
235	#102	—	2	2	no	—
236	#101	—	4	4	no	—
237	C₂²⋊C₈	—	2	4	no	—
238	#97	—	2	1	no	—
239	#97	D₄×C₁₃₁	1	2	yes	—
240	#97	D₄×C₁₃₁	1	2	yes	—
241	C₄×C₈	—	2	1	no	—
242	#41	—	2	2	no	—
243	#41	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
257	Q₁₆	—	4	4	no	—
258	SD₁₆	—	4	4	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
284	C₂³	—	2	1	no	—
285	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	4	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	4	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	#121×C₁₃₁	1	2	yes	—
304	C₂×C₄	#121×C₁₃₁	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
307	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
308	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
309	C₈	—	2	4	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
317	C₂²	#542×C₁₃₁	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	#535×C₁₃₁	1	1	yes	characteristic
325	C₄	#536×C₁₃₁	1	1	yes	characteristic
326	C₄	#537×C₁₃₁	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	8	1	no	—
329	C₂	#2620×C₁₃₁	1	1	yes	characteristic
330	C₂	#3069×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#41×C₁₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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×#491×C₁₃₁	C₂	semidirect
5. #2064×C₁₃₁	C₂	semidirect
7. #1998×C₁₃₁	C₂	semidirect
8. #1966×C₁₃₁	C₂	semidirect
9. #1849×C₁₃₁	C₂	semidirect
165. #29862	C₁₃₁	direct (N × H)
166. C₁₃₁	#29862	direct (N × H)
167. C₂×#491	C₂×C₁₃₁	semidirect
171. #2064	C₂×C₁₃₁	semidirect
173. #1998	C₂×C₁₃₁	semidirect
174. #1966	C₂×C₁₃₁	semidirect
175. #1849	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×#491×C₁₃₁	2
#3080×C₁₃₁	2
#3039×C₁₃₁	2
#2923×C₁₃₁	2
#2064×C₁₃₁	2
#2052×C₁₃₁	2
#1998×C₁₃₁	2
#1966×C₁₃₁	2
#1849×C₁₃₁	2
#29862	131

Subgroups — C131×#29862 #45715746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29862 #45715746