Subgroups — C₁₃₁×#29880 #45715764

Order 33536.

688 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×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#433×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2628×C₁₃₁	C₂	1	2	yes	maximal
4	#2087×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2086×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2076×C₁₃₁	C₂	1	2	yes	maximal
7	#2075×C₁₃₁	C₂	1	2	yes	maximal
8	#1993×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#1835×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
12	#500×C₁₃₁	—	2	4	no	—
13	#433×C₁₃₁	—	2	4	no	—
14	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	#400×C₁₃₁	—	2	2	no	—
23	#399×C₁₃₁	—	2	2	no	—
24	#395×C₁₃₁	—	2	2	no	—
25	#393×C₁₃₁	—	2	2	no	—
26	#389×C₁₃₁	C₂²	1	4	yes	—
27	#382×C₁₃₁	C₂²	1	2	yes	—
28	#380×C₁₃₁	C₂²	1	4	yes	—
29	#340×C₁₃₁	C₂²	1	4	yes	—
30	#340×C₁₃₁	C₂²	1	4	yes	—
31	#340×C₁₃₁	C₂²	1	2	yes	—
32	#340×C₁₃₁	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₃₁	—	2	2	no	—
34	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
40	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
41	#117×C₁₃₁	—	2	2	no	—
42	#117×C₁₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
54	#107×C₁₃₁	—	2	4	no	—
55	#107×C₁₃₁	—	2	4	no	—
56	C₄⋊C₈×C₁₃₁	—	2	2	no	—
57	C₄⋊C₈×C₁₃₁	—	2	2	no	—
58	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
59	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
60	#97×C₁₃₁	—	2	2	no	—
61	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
62	#95×C₁₃₁	—	2	4	no	—
63	#95×C₁₃₁	—	2	2	no	—
64	#95×C₁₃₁	—	2	4	no	—
65	#95×C₁₃₁	—	2	2	no	—
66	C₂⁴×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₁₃₁	—	4	4	no	—
68	C₂×Q₈×C₁₃₁	—	8	4	no	—
69	C₂²×C₄×C₁₃₁	—	2	1	no	—
70	C₂²×C₄×C₁₃₁	—	2	1	no	—
71	C₂²×C₄×C₁₃₁	—	2	2	no	—
72	C₂²×C₄×C₁₃₁	—	2	2	no	—
73	C₂²×C₄×C₁₃₁	—	2	4	no	—
74	C₂²×C₄×C₁₃₁	—	2	4	no	—
75	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₁₃₁	—	2	2	no	—
82	C₂×C₈×C₁₃₁	—	2	2	no	—
83	C₂×C₈×C₁₃₁	—	2	1	no	—
84	C₂×C₈×C₁₃₁	—	2	1	no	—
85	C₂×C₈×C₁₃₁	—	2	2	no	—
86	C₂×C₈×C₁₃₁	—	2	2	no	—
87	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₁₃₁	—	2	2	no	—
90	C₄⋊C₄×C₁₃₁	—	2	2	no	—
91	C₄⋊C₄×C₁₃₁	—	2	4	no	—
92	C₄⋊C₄×C₁₃₁	—	2	4	no	—
93	C₄⋊C₄×C₁₃₁	—	2	4	no	—
94	C₄⋊C₄×C₁₃₁	—	2	4	no	—
95	C₄⋊C₄×C₁₃₁	—	4	4	no	—
96	C₄⋊C₄×C₁₃₁	—	8	4	no	—
97	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
98	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
99	C₂²⋊C₄×C₁₃₁	—	8	4	no	—
100	C₄²×C₁₃₁	—	2	1	no	—
101	C₄²×C₁₃₁	—	2	1	no	—
102	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
103	C₄²×C₁₃₁	#41	1	2	yes	—
104	C₂³×C₁₃₁	—	2	1	no	—
105	C₂³×C₁₃₁	—	2	2	no	—
106	C₂³×C₁₃₁	—	2	2	no	—
107	C₂³×C₁₃₁	—	2	2	no	—
108	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₁₃₁	—	8	8	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	2	2	no	—
112	C₂×C₄×C₁₃₁	—	2	1	no	—
113	C₂×C₄×C₁₃₁	—	2	1	no	—
114	C₂×C₄×C₁₃₁	—	2	1	no	—
115	C₂×C₄×C₁₃₁	—	2	2	no	—
116	C₂×C₄×C₁₃₁	—	2	2	no	—
117	C₂×C₄×C₁₃₁	—	2	2	no	—
118	C₂×C₄×C₁₃₁	—	4	1	no	—
119	C₂×C₄×C₁₃₁	—	4	1	no	—
120	C₂×C₄×C₁₃₁	—	4	2	no	—
121	C₂×C₄×C₁₃₁	—	4	4	no	—
122	C₂×C₄×C₁₃₁	—	4	4	no	—
123	C₂×C₄×C₁₃₁	—	4	4	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	#121	1	2	yes	—
128	C₂×C₄×C₁₃₁	#121	1	2	yes	—
129	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
132	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
133	C₈×C₁₃₁	—	2	2	no	—
134	C₈×C₁₃₁	—	2	2	no	—
135	C₈×C₁₃₁	—	4	1	no	—
136	C₈×C₁₃₁	—	4	1	no	—
137	C₂²×C₁₃₁	—	4	1	no	—
138	C₂²×C₁₃₁	—	4	4	no	—
139	C₂²×C₁₃₁	—	4	1	no	—
140	C₂²×C₁₃₁	—	4	1	no	—
141	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
142	C₂²×C₁₃₁	#536	1	2	yes	—
143	C₂²×C₁₃₁	#576	1	2	yes	—
144	C₂²×C₁₃₁	#578	1	2	yes	—
145	C₄×C₁₃₁	—	2	2	no	—
146	C₄×C₁₃₁	—	2	2	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	—	8	4	no	—
152	C₄×C₁₃₁	#536	1	1	yes	characteristic
153	C₄×C₁₃₁	#536	1	1	yes	characteristic
154	C₄×C₁₃₁	#541	1	2	yes	—
155	C₂×C₁₃₁	—	8	1	no	—
156	C₂×C₁₃₁	#2621	1	1	yes	characteristic
157	C₂×C₁₃₁	#3058	1	1	yes	characteristic
158	C₂×C₁₃₁	#3059	1	1	yes	characteristic
159	C₂×C₁₃₁	#3243	1	4	yes	—
160	#29880	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₁₃₁	#29880	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic, complemented
163	C₂×#433	C₂×C₁₃₁	1	1	yes	characteristic
164	#2628	C₂×C₁₃₁	1	2	yes	—
165	#2087	C₂×C₁₃₁	1	2	yes	complemented
166	#2086	C₂×C₁₃₁	1	2	yes	complemented
167	#2076	C₂×C₁₃₁	1	2	yes	—
168	#2075	C₂×C₁₃₁	1	2	yes	—
169	#1993	C₂×C₁₃₁	1	2	yes	complemented
170	#1835	C₂×C₁₃₁	1	1	yes	characteristic, complemented
171	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
172	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
173	#500	—	2	4	no	—
174	#433	—	2	4	no	—
175	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
176	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
177	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
179	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
180	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
181	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
183	#400	—	2	2	no	—
184	#399	—	2	2	no	—
185	#395	—	2	2	no	—
186	#393	—	2	2	no	—
187	#389	C₂²×C₁₃₁	1	4	yes	—
188	#382	C₂²×C₁₃₁	1	2	yes	—
189	#380	C₂²×C₁₃₁	1	4	yes	—
190	#340	C₂²×C₁₃₁	1	2	yes	—
191	#340	C₂²×C₁₃₁	1	4	yes	—
192	#340	C₂²×C₁₃₁	1	4	yes	—
193	#340	C₂²×C₁₃₁	1	2	yes	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
207	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
215	#107	—	2	4	no	—
216	#107	—	2	4	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	C₂²⋊C₈	—	2	2	no	—
221	#97	—	2	2	no	—
222	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
223	#95	—	2	4	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	4	no	—
227	C₂⁴	C₂×D₄×C₁₃₁	1	1	yes	characteristic
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	8	4	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
241	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	4	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	8	4	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
264	C₄²	#41×C₁₃₁	1	2	yes	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
270	Q₈	—	8	8	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	#121×C₁₃₁	1	2	yes	—
289	C₂×C₄	#121×C₁₃₁	1	2	yes	—
290	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
292	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
293	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	4	1	no	—
297	C₈	—	4	1	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
303	C₂²	#536×C₁₃₁	1	2	yes	—
304	C₂²	#576×C₁₃₁	1	2	yes	—
305	C₂²	#578×C₁₃₁	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	8	4	no	—
312	C₄	—	8	4	no	—
313	C₄	#536×C₁₃₁	1	1	yes	characteristic
314	C₄	#536×C₁₃₁	1	1	yes	characteristic
315	C₄	#541×C₁₃₁	1	2	yes	—
316	C₂	—	8	1	no	—
317	C₂	#2621×C₁₃₁	1	1	yes	characteristic
318	C₂	#3058×C₁₃₁	1	1	yes	characteristic
319	C₂	#3059×C₁₃₁	1	1	yes	characteristic
320	C₂	#3243×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂⁴×C₁₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×#117×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₁	C₂	semidirect
4. #2087×C₁₃₁	C₂	semidirect
5. #2086×C₁₃₁	C₂	semidirect
8. #1993×C₁₃₁	C₂	semidirect
9. #1835×C₁₃₁	C₂	semidirect
160. #29880	C₁₃₁	direct (N × H)
161. C₁₃₁	#29880	direct (N × H)
162. C₂×#500	C₂×C₁₃₁	semidirect
165. #2087	C₂×C₁₃₁	semidirect
166. #2086	C₂×C₁₃₁	semidirect
169. #1993	C₂×C₁₃₁	semidirect
170. #1835	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×#500×C₁₃₁	2
C₂×#433×C₁₃₁	2
#2628×C₁₃₁	2
#2087×C₁₃₁	2
#2086×C₁₃₁	2
#2076×C₁₃₁	2
#2075×C₁₃₁	2
#1993×C₁₃₁	2
#1835×C₁₃₁	2
#29880	131

Subgroups — C131×#29880 #45715764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29880 #45715764