Subgroups — C₁₃₁×#29861 #45715745

Order 33536.

712 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#490×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3080×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#3038×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2923×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#2063×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2051×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#1998×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#1965×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#1849×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#127×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	#490×C₁₃₁	—	2	2	no	—
15	#490×C₁₃₁	—	2	2	no	—
16	#479×C₁₃₁	—	2	2	no	—
17	#478×C₁₃₁	—	2	2	no	—
18	#460×C₁₃₁	—	2	2	no	—
19	#459×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	#418×C₁₃₁	C₂²	1	2	yes	—
28	#417×C₁₃₁	C₂²	1	2	yes	—
29	C₂×#102×C₁₃₁	C₂²	1	4	yes	—
30	C₂×#100×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×D₄×C₁₃₁	—	2	2	no	—
37	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
38	AGL(1,ℤ/8)×C₁₃₁	—	4	4	no	—
39	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
40	C₂×D₈×C₁₃₁	—	2	2	no	—
41	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
43	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
45	#127×C₁₃₁	—	2	2	no	—
46	#127×C₁₃₁	—	2	1	no	—
47	#127×C₁₃₁	—	2	1	no	—
48	#125×C₁₃₁	—	2	2	no	—
49	#125×C₁₃₁	—	2	1	no	—
50	#125×C₁₃₁	—	2	1	no	—
51	C₄×Q₈×C₁₃₁	D₄	1	2	yes	—
52	C₄×D₄×C₁₃₁	—	2	2	no	—
53	C₄×D₄×C₁₃₁	D₄	1	2	yes	—
54	#117×C₁₃₁	—	2	1	no	—
55	#117×C₁₃₁	C₂³	1	1	yes	characteristic
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	#102×C₁₃₁	—	2	2	no	—
68	#102×C₁₃₁	—	2	8	no	—
69	#102×C₁₃₁	—	2	4	no	—
70	#100×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	C₂⁴×C₁₃₁	—	2	2	no	—
77	#41×C₁₃₁	—	2	2	no	—
78	#41×C₁₃₁	—	2	2	no	—
79	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×D₄×C₁₃₁	—	2	1	no	—
81	C₂×D₄×C₁₃₁	—	2	2	no	—
82	C₂×D₄×C₁₃₁	—	2	2	no	—
83	C₂×D₄×C₁₃₁	—	2	4	no	—
84	C₂×D₄×C₁₃₁	—	2	4	no	—
85	C₂×D₄×C₁₃₁	—	4	2	no	—
86	C₂×D₄×C₁₃₁	—	4	2	no	—
87	C₂×D₄×C₁₃₁	—	8	4	no	—
88	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₁	—	2	1	no	—
90	C₂²×C₄×C₁₃₁	—	2	2	no	—
91	C₂²×C₄×C₁₃₁	—	2	2	no	—
92	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
93	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
94	SD₁₆×C₁₃₁	—	4	4	no	—
95	D₈×C₁₃₁	—	4	4	no	—
96	M₁₆×C₁₃₁	—	2	2	no	—
97	M₁₆×C₁₃₁	—	2	4	no	—
98	M₁₆×C₁₃₁	—	2	4	no	—
99	M₁₆×C₁₃₁	—	2	2	no	—
100	C₂×C₈×C₁₃₁	—	2	2	no	—
101	C₂×C₈×C₁₃₁	—	2	2	no	—
102	C₂×C₈×C₁₃₁	—	2	2	no	—
103	C₂×C₈×C₁₃₁	—	2	2	no	—
104	C₂×C₈×C₁₃₁	—	2	2	no	—
105	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
106	C₄⋊C₄×C₁₃₁	—	2	2	no	—
107	C₄⋊C₄×C₁₃₁	—	2	2	no	—
108	C₄⋊C₄×C₁₃₁	—	2	2	no	—
109	C₄⋊C₄×C₁₃₁	—	4	2	no	—
110	C₄⋊C₄×C₁₃₁	—	4	2	no	—
111	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
113	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
114	C₄²×C₁₃₁	—	2	2	no	—
115	C₄²×C₁₃₁	—	2	1	no	—
116	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
119	C₂³×C₁₃₁	—	2	1	no	—
120	C₂³×C₁₃₁	—	4	2	no	—
121	C₂³×C₁₃₁	—	4	4	no	—
122	C₂³×C₁₃₁	—	8	4	no	—
123	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
124	Q₈×C₁₃₁	—	2	1	no	—
125	Q₈×C₁₃₁	—	2	1	no	—
126	D₄×C₁₃₁	—	2	1	no	—
127	D₄×C₁₃₁	—	2	1	no	—
128	D₄×C₁₃₁	—	4	2	no	—
129	D₄×C₁₃₁	—	4	2	no	—
130	D₄×C₁₃₁	—	4	2	no	—
131	D₄×C₁₃₁	—	4	4	no	—
132	D₄×C₁₃₁	—	8	4	no	—
133	D₄×C₁₃₁	—	8	4	no	—
134	C₂×C₄×C₁₃₁	—	2	2	no	—
135	C₂×C₄×C₁₃₁	—	2	1	no	—
136	C₂×C₄×C₁₃₁	—	4	2	no	—
137	C₂×C₄×C₁₃₁	—	4	2	no	—
138	C₂×C₄×C₁₃₁	—	4	2	no	—
139	C₂×C₄×C₁₃₁	—	8	2	no	—
140	C₂×C₄×C₁₃₁	—	8	2	no	—
141	C₂×C₄×C₁₃₁	#121	1	2	yes	—
142	C₂×C₄×C₁₃₁	#121	1	2	yes	—
143	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
146	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
147	C₈×C₁₃₁	—	2	4	no	—
148	C₈×C₁₃₁	—	4	2	no	—
149	C₈×C₁₃₁	—	4	2	no	—
150	C₈×C₁₃₁	—	4	2	no	—
151	C₂²×C₁₃₁	—	2	2	no	—
152	C₂²×C₁₃₁	—	4	1	no	—
153	C₂²×C₁₃₁	—	4	2	no	—
154	C₂²×C₁₃₁	—	8	2	no	—
155	C₂²×C₁₃₁	—	8	4	no	—
156	C₂²×C₁₃₁	—	16	4	no	—
157	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
158	C₂²×C₁₃₁	#542	1	2	yes	—
159	C₄×C₁₃₁	—	2	2	no	—
160	C₄×C₁₃₁	—	2	2	no	—
161	C₄×C₁₃₁	—	4	1	no	—
162	C₄×C₁₃₁	—	4	2	no	—
163	C₄×C₁₃₁	—	8	2	no	—
164	C₄×C₁₃₁	#535	1	1	yes	characteristic
165	C₄×C₁₃₁	#536	1	1	yes	characteristic
166	C₄×C₁₃₁	#537	1	2	yes	—
167	C₂×C₁₃₁	—	2	2	no	—
168	C₂×C₁₃₁	—	8	1	no	—
169	C₂×C₁₃₁	—	16	2	no	—
170	C₂×C₁₃₁	#2620	1	1	yes	characteristic
171	C₂×C₁₃₁	#3068	1	2	yes	—
172	#29861	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₁₃₁	#29861	1	1	yes	characteristic, direct product factor, Sylow
174	C₂×#490	C₂×C₁₃₁	1	1	yes	characteristic, complemented
175	#3080	C₂×C₁₃₁	1	1	yes	characteristic, complemented
176	#3038	C₂×C₁₃₁	1	1	yes	characteristic
177	#2923	C₂×C₁₃₁	1	1	yes	characteristic, complemented
178	#2063	C₂×C₁₃₁	1	4	yes	complemented
179	#2051	C₂×C₁₃₁	1	4	yes	complemented
180	#1998	C₂×C₁₃₁	1	1	yes	characteristic, complemented
181	#1965	C₂×C₁₃₁	1	1	yes	characteristic, complemented
182	#1849	C₂×C₁₃₁	1	1	yes	characteristic, complemented
183	C₂×AGL(1,ℤ/8)	C₂²×C₁₃₁	1	2	yes	—
184	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
185	C₂×#125	C₂²×C₁₃₁	1	1	yes	characteristic
186	C₄×#41	C₂²×C₁₃₁	1	1	yes	characteristic
187	#490	—	2	2	no	—
188	#490	—	2	2	no	—
189	#479	—	2	2	no	—
190	#478	—	2	2	no	—
191	#460	—	2	2	no	—
192	#459	—	2	2	no	—
193	#433	—	2	2	no	—
194	#432	—	2	2	no	—
195	C₂×#108	C₂²×C₁₃₁	1	4	yes	—
196	#428	C₂²×C₁₃₁	1	2	yes	—
197	#424	C₂²×C₁₃₁	1	2	yes	—
198	#423	C₂²×C₁₃₁	1	1	yes	characteristic
199	#423	C₂²×C₁₃₁	1	1	yes	characteristic
200	#418	C₂²×C₁₃₁	1	2	yes	—
201	#417	C₂²×C₁₃₁	1	2	yes	—
202	C₂×#102	C₂²×C₁₃₁	1	4	yes	—
203	C₂×#100	C₂²×C₁₃₁	1	4	yes	—
204	#408	C₂²×C₁₃₁	1	2	yes	—
205	C₄×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
206	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
207	#343	C₂²×C₁₃₁	1	4	yes	—
208	C₂×#41	C₂³×C₁₃₁	1	1	yes	characteristic
209	C₂²×D₄	—	2	2	no	—
210	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
211	AGL(1,ℤ/8)	—	4	4	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
215	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
216	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
217	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
218	#127	—	2	2	no	—
219	#127	—	2	1	no	—
220	#127	—	2	1	no	—
221	#125	—	2	2	no	—
222	#125	—	2	1	no	—
223	#125	—	2	1	no	—
224	C₄×Q₈	D₄×C₁₃₁	1	2	yes	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	D₄×C₁₃₁	1	2	yes	—
227	#117	—	2	1	no	—
228	#117	C₂³×C₁₃₁	1	1	yes	characteristic
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
233	#108	—	4	4	no	—
234	#107	—	2	2	no	—
235	#106	—	2	2	no	—
236	C₄⋊C₈	—	2	4	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	#103	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	2	8	no	—
242	#102	—	2	4	no	—
243	#100	—	4	4	no	—
244	C₂²⋊C₈	—	2	4	no	—
245	#97	—	2	1	no	—
246	#97	D₄×C₁₃₁	1	2	yes	—
247	#97	D₄×C₁₃₁	1	2	yes	—
248	C₄×C₈	—	2	1	no	—
249	C₂⁴	—	2	2	no	—
250	#41	—	2	2	no	—
251	#41	—	2	2	no	—
252	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	8	4	no	—
261	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
267	SD₁₆	—	4	4	no	—
268	D₈	—	4	4	no	—
269	M₁₆	—	2	2	no	—
270	M₁₆	—	2	4	no	—
271	M₁₆	—	2	4	no	—
272	M₁₆	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	1	no	—
289	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
292	C₂³	—	2	1	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	8	4	no	—
296	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	4	no	—
305	D₄	—	8	4	no	—
306	D₄	—	8	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	#121×C₁₃₁	1	2	yes	—
315	C₂×C₄	#121×C₁₃₁	1	2	yes	—
316	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
318	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
319	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₈	—	4	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	16	4	no	—
330	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
331	C₂²	#542×C₁₃₁	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	8	2	no	—
337	C₄	#535×C₁₃₁	1	1	yes	characteristic
338	C₄	#536×C₁₃₁	1	1	yes	characteristic
339	C₄	#537×C₁₃₁	1	2	yes	—
340	C₂	—	2	2	no	—
341	C₂	—	8	1	no	—
342	C₂	—	16	2	no	—
343	C₂	#2620×C₁₃₁	1	1	yes	characteristic
344	C₂	#3068×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	#3038×C₁₃₁	C₂
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×#490×C₁₃₁	C₂	semidirect
2. #3080×C₁₃₁	C₂	semidirect
4. #2923×C₁₃₁	C₂	semidirect
5. #2063×C₁₃₁	C₂	semidirect
6. #2051×C₁₃₁	C₂	semidirect
7. #1998×C₁₃₁	C₂	semidirect
8. #1965×C₁₃₁	C₂	semidirect
9. #1849×C₁₃₁	C₂	semidirect
172. #29861	C₁₃₁	direct (N × H)
173. C₁₃₁	#29861	direct (N × H)
174. C₂×#490	C₂×C₁₃₁	semidirect
175. #3080	C₂×C₁₃₁	semidirect
177. #2923	C₂×C₁₃₁	semidirect
178. #2063	C₂×C₁₃₁	semidirect
179. #2051	C₂×C₁₃₁	semidirect
180. #1998	C₂×C₁₃₁	semidirect
181. #1965	C₂×C₁₃₁	semidirect
182. #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×#490×C₁₃₁	2
#3080×C₁₃₁	2
#3038×C₁₃₁	2
#2923×C₁₃₁	2
#2063×C₁₃₁	2
#2051×C₁₃₁	2
#1998×C₁₃₁	2
#1965×C₁₃₁	2
#1849×C₁₃₁	2
#29861	131

Subgroups — C131×#29861 #45715745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29861 #45715745