Subgroups — C₁₃₁×#29808 #45715692

Order 33536.

914 conjugacy classes of subgroups.

The subgroup lattice has 314 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	C₂×#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2857×C₁₃₁	C₂	1	2	yes	maximal
4	#2056×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#2001×C₁₃₁	C₂	1	4	yes	maximal
6	#1956×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1834×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₁₃₁	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	4	yes	—
11	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
12	#494×C₁₃₁	—	2	4	no	—
13	#431×C₁₃₁	—	2	4	no	—
14	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#103×C₁₃₁	C₂²	1	8	yes	—
16	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#391×C₁₃₁	C₂²	1	4	yes	—
20	#389×C₁₃₁	—	2	2	no	—
21	#389×C₁₃₁	—	2	2	no	—
22	#384×C₁₃₁	C₂²	1	2	yes	—
23	#380×C₁₃₁	—	2	2	no	—
24	#380×C₁₃₁	—	2	2	no	—
25	#340×C₁₃₁	C₂²	1	4	yes	—
26	C₂²×Q₈×C₁₃₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
30	C₂×Q₁₆×C₁₃₁	—	2	8	no	—
31	C₂×Q₁₆×C₁₃₁	—	4	8	no	—
32	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
35	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
36	C₂²⋊Q₈×C₁₃₁	—	4	8	no	—
37	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
44	C₂×C₄²×C₁₃₁	—	2	2	no	—
45	C₂×C₄²×C₁₃₁	—	2	2	no	—
46	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₁₃₁	—	2	4	no	—
51	#103×C₁₃₁	—	4	8	no	—
52	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
53	C₄×C₈×C₁₃₁	—	2	2	no	—
54	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
55	#95×C₁₃₁	—	2	4	no	—
56	#95×C₁₃₁	—	2	4	no	—
57	C₂⁴×C₁₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₁₃₁	—	2	4	no	—
59	C₂×Q₈×C₁₃₁	—	2	4	no	—
60	C₂×Q₈×C₁₃₁	—	2	8	no	—
61	C₂×Q₈×C₁₃₁	—	2	8	no	—
62	C₂²×C₄×C₁₃₁	—	2	2	no	—
63	C₂²×C₄×C₁₃₁	—	2	2	no	—
64	C₂²×C₄×C₁₃₁	—	2	2	no	—
65	C₂²×C₄×C₁₃₁	—	2	4	no	—
66	C₂²×C₄×C₁₃₁	—	2	1	no	—
67	C₂²×C₄×C₁₃₁	—	2	1	no	—
68	C₂²×C₄×C₁₃₁	—	2	4	no	—
69	C₂²×C₄×C₁₃₁	—	2	4	no	—
70	C₂²×C₄×C₁₃₁	—	2	4	no	—
71	C₂²×C₄×C₁₃₁	Q₁₆	1	4	yes	complemented
72	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₁₃₁	—	4	16	no	—
79	C₂×C₈×C₁₃₁	—	2	4	no	—
80	C₂×C₈×C₁₃₁	—	2	2	no	—
81	C₂×C₈×C₁₃₁	—	2	4	no	—
82	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
83	C₄⋊C₄×C₁₃₁	—	2	8	no	—
84	C₄⋊C₄×C₁₃₁	—	4	8	no	—
85	C₄⋊C₄×C₁₃₁	—	4	4	no	—
86	C₄⋊C₄×C₁₃₁	—	4	4	no	—
87	C₄⋊C₄×C₁₃₁	—	4	4	no	—
88	C₄⋊C₄×C₁₃₁	—	4	4	no	—
89	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
90	C₄²×C₁₃₁	—	2	2	no	—
91	C₄²×C₁₃₁	—	2	2	no	—
92	C₄²×C₁₃₁	—	2	1	no	—
93	C₄²×C₁₃₁	—	2	1	no	—
94	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
95	C₄²×C₁₃₁	#41	1	2	yes	—
96	C₂³×C₁₃₁	—	2	1	no	—
97	C₂³×C₁₃₁	—	2	2	no	—
98	C₂³×C₁₃₁	—	2	1	no	—
99	C₂³×C₁₃₁	—	2	2	no	—
100	C₂³×C₁₃₁	C₂×Q₁₆	1	2	yes	—
101	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₁₃₁	—	4	8	no	—
103	Q₈×C₁₃₁	—	4	8	no	—
104	C₂×C₄×C₁₃₁	—	2	1	no	—
105	C₂×C₄×C₁₃₁	—	2	1	no	—
106	C₂×C₄×C₁₃₁	—	2	8	no	—
107	C₂×C₄×C₁₃₁	—	2	1	no	—
108	C₂×C₄×C₁₃₁	—	2	1	no	—
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	4	2	no	—
112	C₂×C₄×C₁₃₁	—	4	2	no	—
113	C₂×C₄×C₁₃₁	—	4	4	no	—
114	C₂×C₄×C₁₃₁	—	4	4	no	—
115	C₂×C₄×C₁₃₁	—	4	4	no	—
116	C₂×C₄×C₁₃₁	—	4	4	no	—
117	C₂×C₄×C₁₃₁	—	4	4	no	—
118	C₂×C₄×C₁₃₁	—	4	4	no	—
119	C₂×C₄×C₁₃₁	—	4	4	no	—
120	C₂×C₄×C₁₃₁	#121	1	2	yes	—
121	C₂×C₄×C₁₃₁	#121	1	2	yes	—
122	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
123	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₁₃₁	#135	1	2	yes	—
125	C₂×C₄×C₁₃₁	#137	1	4	yes	—
126	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	4	yes	—
129	C₈×C₁₃₁	—	2	4	no	—
130	C₈×C₁₃₁	—	4	2	no	—
131	C₂²×C₁₃₁	—	2	4	no	—
132	C₂²×C₁₃₁	—	2	2	no	—
133	C₂²×C₁₃₁	—	4	1	no	—
134	C₂²×C₁₃₁	—	4	2	no	—
135	C₂²×C₁₃₁	—	4	1	no	—
136	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
137	C₂²×C₁₃₁	#535	1	2	yes	—
138	C₂²×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
139	C₂²×C₁₃₁	C₂×#135	1	1	yes	characteristic
140	C₂²×C₁₃₁	C₂×#137	1	2	yes	—
141	C₄×C₁₃₁	—	2	2	no	—
142	C₄×C₁₃₁	—	4	1	no	—
143	C₄×C₁₃₁	—	4	1	no	—
144	C₄×C₁₃₁	—	8	4	no	—
145	C₄×C₁₃₁	—	8	4	no	—
146	C₄×C₁₃₁	#467	1	4	yes	—
147	C₄×C₁₃₁	#470	1	4	yes	—
148	C₄×C₁₃₁	#535	1	2	yes	—
149	C₄×C₁₃₁	#540	1	2	yes	—
150	C₂×C₁₃₁	—	4	2	no	—
151	C₂×C₁₃₁	#2619	1	1	yes	characteristic
152	C₂×C₁₃₁	C₂×#467	1	1	yes	characteristic
153	C₂×C₁₃₁	C₂×#470	1	1	yes	characteristic
154	C₂×C₁₃₁	#3209	1	2	yes	—
155	C₂×C₁₃₁	#3210	1	2	yes	—
156	#29808	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₁₃₁	#29808	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×#494	C₂×C₁₃₁	1	1	yes	characteristic, complemented
159	C₂×#431	C₂×C₁₃₁	1	1	yes	characteristic
160	#2857	C₂×C₁₃₁	1	2	yes	—
161	#2056	C₂×C₁₃₁	1	4	yes	complemented
162	#2001	C₂×C₁₃₁	1	4	yes	—
163	#1956	C₂×C₁₃₁	1	2	yes	complemented
164	#1834	C₂×C₁₃₁	1	1	yes	characteristic, complemented
165	C₂²×Q₁₆	C₂²×C₁₃₁	1	4	yes	—
166	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
167	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	4	yes	—
168	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
169	#494	—	2	4	no	—
170	#431	—	2	4	no	—
171	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
172	C₂×#103	C₂²×C₁₃₁	1	8	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
176	#391	C₂²×C₁₃₁	1	4	yes	—
177	#389	—	2	2	no	—
178	#389	—	2	2	no	—
179	#384	C₂²×C₁₃₁	1	2	yes	—
180	#380	—	2	2	no	—
181	#380	—	2	2	no	—
182	#340	C₂²×C₁₃₁	1	4	yes	—
183	C₂²×Q₈	C₂³×C₁₃₁	1	4	yes	—
184	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
185	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
187	C₂×Q₁₆	—	2	8	no	—
188	C₂×Q₁₆	—	4	8	no	—
189	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	2	4	no	—
193	C₂²⋊Q₈	—	4	8	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
204	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
207	C₄⋊C₈	—	2	4	no	—
208	#103	—	4	8	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₄×C₈	—	2	2	no	—
211	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
212	#95	—	2	4	no	—
213	#95	—	2	4	no	—
214	C₂⁴	C₂×D₄×C₁₃₁	1	1	yes	characteristic
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	8	no	—
218	C₂×Q₈	—	2	8	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	Q₁₆×C₁₃₁	1	4	yes	complemented
229	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
235	Q₁₆	—	4	16	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	4	8	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	4	8	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
252	C₄²	#41×C₁₃₁	1	2	yes	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×Q₁₆×C₁₃₁	1	2	yes	—
258	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
259	Q₈	—	4	8	no	—
260	Q₈	—	4	8	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	#121×C₁₃₁	1	2	yes	—
278	C₂×C₄	#121×C₁₃₁	1	2	yes	—
279	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
280	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
281	C₂×C₄	#135×C₁₃₁	1	2	yes	—
282	C₂×C₄	#137×C₁₃₁	1	4	yes	—
283	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
285	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	4	yes	—
286	C₈	—	2	4	no	—
287	C₈	—	4	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
294	C₂²	#535×C₁₃₁	1	2	yes	—
295	C₂²	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
296	C₂²	C₂×#135×C₁₃₁	1	1	yes	characteristic
297	C₂²	C₂×#137×C₁₃₁	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	4	no	—
302	C₄	—	8	4	no	—
303	C₄	#467×C₁₃₁	1	4	yes	—
304	C₄	#470×C₁₃₁	1	4	yes	—
305	C₄	#535×C₁₃₁	1	2	yes	—
306	C₄	#540×C₁₃₁	1	2	yes	—
307	C₂	—	4	2	no	—
308	C₂	#2619×C₁₃₁	1	1	yes	characteristic
309	C₂	C₂×#467×C₁₃₁	1	1	yes	characteristic
310	C₂	C₂×#470×C₁₃₁	1	1	yes	characteristic
311	C₂	#3209×C₁₃₁	1	2	yes	—
312	C₂	#3210×C₁₃₁	1	2	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 ◁ C₂²×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. #2056×C₁₃₁	C₂	semidirect
6. #1956×C₁₃₁	C₂	semidirect
7. #1834×C₁₃₁	C₂	semidirect
71. C₂²×C₄×C₁₃₁	Q₁₆	semidirect
156. #29808	C₁₃₁	direct (N × H)
157. C₁₃₁	#29808	direct (N × H)
158. C₂×#494	C₂×C₁₃₁	semidirect
161. #2056	C₂×C₁₃₁	semidirect
163. #1956	C₂×C₁₃₁	semidirect
164. #1834	C₂×C₁₃₁	semidirect
228. C₂²×C₄	Q₁₆×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
C₂×#431×C₁₃₁	2
#2857×C₁₃₁	2
#2056×C₁₃₁	2
#2001×C₁₃₁	2
#1956×C₁₃₁	2
#1834×C₁₃₁	2
#29808	131

Subgroups — C131×#29808 #45715692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29808 #45715692