Subgroups — C₁₃₁×#13259 #45699143

Order 33536.

472 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3147×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2145×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2143×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1772×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135×C₁₃₁	—	2	1	no	—
7	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
8	#532×C₁₃₁	—	2	1	no	—
9	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
10	#495×C₁₃₁	—	2	1	no	—
11	#495×C₁₃₁	C₄	1	2	yes	complemented
12	#494×C₁₃₁	C₄	1	1	yes	characteristic, complemented
13	#493×C₁₃₁	C₄	1	1	yes	characteristic, complemented
14	#492×C₁₃₁	—	2	1	no	—
15	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
16	#427×C₁₃₁	C₂²	1	1	yes	characteristic
17	#405×C₁₃₁	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#364×C₁₃₁	—	2	1	no	—
20	#364×C₁₃₁	—	2	1	no	—
21	#360×C₁₃₁	—	2	4	no	—
22	#359×C₁₃₁	—	2	4	no	—
23	#339×C₁₃₁	—	2	1	no	—
24	C₂×#41×C₁₃₁	—	2	2	no	—
25	#135×C₁₃₁	—	2	2	no	—
26	#135×C₁₃₁	—	2	2	no	—
27	#135×C₁₃₁	—	2	2	no	—
28	#135×C₁₃₁	—	2	2	no	—
29	#135×C₁₃₁	—	4	1	no	—
30	#135×C₁₃₁	—	4	1	no	—
31	#135×C₁₃₁	—	4	1	no	—
32	#135×C₁₃₁	—	4	1	no	—
33	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
34	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
36	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
37	C₂×D₈×C₁₃₁	—	2	1	no	—
38	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₁₃₁	—	2	1	no	—
40	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
43	#127×C₁₃₁	—	2	1	no	—
44	#124×C₁₃₁	—	2	2	no	—
45	#121×C₁₃₁	—	4	2	no	—
46	C₄×D₄×C₁₃₁	—	4	2	no	—
47	#117×C₁₃₁	—	2	1	no	—
48	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
49	M₃₂×C₁₃₁	—	2	2	no	—
50	M₃₂×C₁₃₁	—	2	2	no	—
51	C₂×C₁₆×C₁₃₁	—	2	2	no	—
52	#108×C₁₃₁	—	2	1	no	—
53	#108×C₁₃₁	D₄	1	2	yes	complemented
54	#107×C₁₃₁	—	2	1	no	—
55	#106×C₁₃₁	D₄	1	2	yes	complemented
56	#97×C₁₃₁	—	4	1	no	—
57	C₄×C₈×C₁₃₁	—	4	1	no	—
58	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
62	#41×C₁₃₁	—	4	2	no	—
63	#41×C₁₃₁	—	4	2	no	—
64	#41×C₁₃₁	—	4	2	no	—
65	#41×C₁₃₁	—	4	2	no	—
66	C₂×Q₈×C₁₃₁	—	2	2	no	—
67	C₂×D₄×C₁₃₁	—	2	2	no	—
68	C₂×D₄×C₁₃₁	—	4	2	no	—
69	C₂×D₄×C₁₃₁	—	4	2	no	—
70	C₂²×C₄×C₁₃₁	—	2	1	no	—
71	C₂²×C₄×C₁₃₁	—	4	2	no	—
72	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₁₃₁	—	2	2	no	—
74	Q₁₆×C₁₃₁	—	2	2	no	—
75	Q₁₆×C₁₃₁	—	4	1	no	—
76	Q₁₆×C₁₃₁	—	4	1	no	—
77	SD₁₆×C₁₃₁	—	4	2	no	—
78	SD₁₆×C₁₃₁	—	4	2	no	—
79	SD₁₆×C₁₃₁	—	4	2	no	—
80	SD₁₆×C₁₃₁	—	4	2	no	—
81	D₈×C₁₃₁	—	2	2	no	—
82	D₈×C₁₃₁	—	2	2	no	—
83	D₈×C₁₃₁	—	4	1	no	—
84	D₈×C₁₃₁	—	4	1	no	—
85	M₁₆×C₁₃₁	—	2	2	no	—
86	M₁₆×C₁₃₁	—	4	1	no	—
87	C₂×C₈×C₁₃₁	—	4	1	no	—
88	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
91	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
92	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
93	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₁	—	2	2	no	—
101	C₄⋊C₄×C₁₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
103	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
104	C₄²×C₁₃₁	—	2	1	no	—
105	C₄²×C₁₃₁	—	4	1	no	—
106	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
107	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
108	C₁₆×C₁₃₁	—	4	2	no	—
109	C₂³×C₁₃₁	—	4	2	no	—
110	C₂³×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
111	Q₈×C₁₃₁	—	4	2	no	—
112	Q₈×C₁₃₁	—	4	2	no	—
113	D₄×C₁₃₁	—	4	2	no	—
114	D₄×C₁₃₁	—	4	2	no	—
115	D₄×C₁₃₁	—	8	4	no	—
116	D₄×C₁₃₁	—	8	2	no	—
117	D₄×C₁₃₁	—	8	2	no	—
118	C₂×C₄×C₁₃₁	—	2	1	no	—
119	C₂×C₄×C₁₃₁	—	2	1	no	—
120	C₂×C₄×C₁₃₁	—	4	2	no	—
121	C₂×C₄×C₁₃₁	—	4	2	no	—
122	C₂×C₄×C₁₃₁	—	4	2	no	—
123	C₂×C₄×C₁₃₁	—	8	2	no	—
124	C₂×C₄×C₁₃₁	—	8	2	no	—
125	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₈×C₁₃₁	—	4	2	no	—
132	C₈×C₁₃₁	#102	1	1	yes	characteristic
133	C₈×C₁₃₁	#102	1	1	yes	characteristic
134	C₈×C₁₃₁	#102	1	1	yes	characteristic
135	C₈×C₁₃₁	#102	1	1	yes	characteristic
136	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
137	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
138	C₈×C₁₃₁	#124	1	1	yes	characteristic
139	C₈×C₁₃₁	#127	1	1	yes	characteristic
140	C₂²×C₁₃₁	—	4	1	no	—
141	C₂²×C₁₃₁	—	8	4	no	—
142	C₂²×C₁₃₁	—	8	2	no	—
143	C₂²×C₁₃₁	#418	1	1	yes	characteristic
144	C₂²×C₁₃₁	#459	1	1	yes	characteristic
145	C₂²×C₁₃₁	#460	1	1	yes	characteristic
146	C₄×C₁₃₁	—	2	2	no	—
147	C₄×C₁₃₁	—	4	1	no	—
148	C₄×C₁₃₁	—	8	2	no	—
149	C₄×C₁₃₁	—	8	2	no	—
150	C₄×C₁₃₁	#390	1	1	yes	characteristic
151	C₄×C₁₃₁	C₂×#102	1	1	yes	characteristic
152	C₄×C₁₃₁	#459	1	1	yes	characteristic
153	C₄×C₁₃₁	#460	1	1	yes	characteristic
154	C₂×C₁₃₁	—	2	1	no	—
155	C₂×C₁₃₁	—	2	1	no	—
156	C₂×C₁₃₁	—	2	1	no	—
157	C₂×C₁₃₁	—	16	2	no	—
158	C₂×C₁₃₁	#1982	1	1	yes	characteristic
159	#13259	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₁₃₁	#13259	1	1	yes	characteristic, direct product factor, Sylow
161	#3147	C₂×C₁₃₁	1	1	yes	characteristic
162	C₂×#364	C₂×C₁₃₁	1	1	yes	characteristic, complemented
163	#2145	C₂×C₁₃₁	1	2	yes	complemented
164	#2143	C₂×C₁₃₁	1	2	yes	complemented
165	#1772	C₂×C₁₃₁	1	1	yes	characteristic, complemented
166	C₂×#135	—	2	1	no	—
167	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
168	#532	—	2	1	no	—
169	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
170	#495	—	2	1	no	—
171	#495	C₄×C₁₃₁	1	2	yes	complemented
172	#494	C₄×C₁₃₁	1	1	yes	characteristic, complemented
173	#493	C₄×C₁₃₁	1	1	yes	characteristic, complemented
174	#492	—	2	1	no	—
175	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
176	#427	C₂²×C₁₃₁	1	1	yes	characteristic
177	#405	—	2	1	no	—
178	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
179	#364	—	2	1	no	—
180	#364	—	2	1	no	—
181	#360	—	2	4	no	—
182	#359	—	2	4	no	—
183	#339	—	2	1	no	—
184	C₂×#41	—	2	2	no	—
185	#135	—	2	2	no	—
186	#135	—	2	2	no	—
187	#135	—	2	2	no	—
188	#135	—	2	2	no	—
189	#135	—	4	1	no	—
190	#135	—	4	1	no	—
191	#135	—	4	1	no	—
192	#135	—	4	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
195	C₂×SD₁₆	—	2	2	no	—
196	C₂×SD₁₆	—	2	2	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	#127	—	2	1	no	—
204	#124	—	2	2	no	—
205	#121	—	4	2	no	—
206	C₄×D₄	—	4	2	no	—
207	#117	—	2	1	no	—
208	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
209	M₃₂	—	2	2	no	—
210	M₃₂	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	#108	—	2	1	no	—
213	#108	D₄×C₁₃₁	1	2	yes	complemented
214	#107	—	2	1	no	—
215	#106	D₄×C₁₃₁	1	2	yes	complemented
216	#97	—	4	1	no	—
217	C₄×C₈	—	4	1	no	—
218	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
219	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
220	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
221	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	#41	—	4	2	no	—
225	#41	—	4	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	1	no	—
236	Q₁₆	—	4	1	no	—
237	SD₁₆	—	4	2	no	—
238	SD₁₆	—	4	2	no	—
239	SD₁₆	—	4	2	no	—
240	SD₁₆	—	4	2	no	—
241	D₈	—	2	2	no	—
242	D₈	—	2	2	no	—
243	D₈	—	4	1	no	—
244	D₈	—	4	1	no	—
245	M₁₆	—	2	2	no	—
246	M₁₆	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
250	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
251	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
252	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
253	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
258	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
259	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	4	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	4	1	no	—
266	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
267	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
268	C₁₆	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	8	4	no	—
276	D₄	—	8	2	no	—
277	D₄	—	8	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
286	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
287	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
290	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	#102×C₁₃₁	1	1	yes	characteristic
293	C₈	#102×C₁₃₁	1	1	yes	characteristic
294	C₈	#102×C₁₃₁	1	1	yes	characteristic
295	C₈	#102×C₁₃₁	1	1	yes	characteristic
296	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
297	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
298	C₈	#124×C₁₃₁	1	1	yes	characteristic
299	C₈	#127×C₁₃₁	1	1	yes	characteristic
300	C₂²	—	4	1	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#418×C₁₃₁	1	1	yes	characteristic
304	C₂²	#459×C₁₃₁	1	1	yes	characteristic
305	C₂²	#460×C₁₃₁	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	#390×C₁₃₁	1	1	yes	characteristic
311	C₄	C₂×#102×C₁₃₁	1	1	yes	characteristic
312	C₄	#459×C₁₃₁	1	1	yes	characteristic
313	C₄	#460×C₁₃₁	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#1982×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₁	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂×C₁₃₁	#1982
prime radical	#3147×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₃₁	C₂	semidirect
3. #2145×C₁₃₁	C₂	semidirect
4. #2143×C₁₃₁	C₂	semidirect
5. #1772×C₁₃₁	C₂	semidirect
11. #495×C₁₃₁	C₄	semidirect
12. #494×C₁₃₁	C₄	semidirect
13. #493×C₁₃₁	C₄	semidirect
53. #108×C₁₃₁	D₄	semidirect
55. #106×C₁₃₁	D₄	semidirect
159. #13259	C₁₃₁	direct (N × H)
160. C₁₃₁	#13259	direct (N × H)
162. C₂×#364	C₂×C₁₃₁	semidirect
163. #2145	C₂×C₁₃₁	semidirect
164. #2143	C₂×C₁₃₁	semidirect
165. #1772	C₂×C₁₃₁	semidirect
171. #495	C₄×C₁₃₁	semidirect
172. #494	C₄×C₁₃₁	semidirect
173. #493	C₄×C₁₃₁	semidirect
213. #108	D₄×C₁₃₁	semidirect
215. #106	D₄×C₁₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#3147×C₁₃₁	2
C₂×#364×C₁₃₁	2
#2145×C₁₃₁	2
#2143×C₁₃₁	2
#1772×C₁₃₁	2
#13259	131

Subgroups — C131×#13259 #45699143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13259 #45699143