Subgroups — C₁₃₁×#21275 #45707159

Order 33536.

974 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	C₂×C₄×D₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2293×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1970×C₁₃₁	C₂	1	2	yes	maximal
4	#1942×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1934×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#102×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#117×C₁₃₁	—	2	2	no	—
12	C₄×D₈×C₁₃₁	C₄	1	8	yes	complemented
13	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
14	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#102×C₁₃₁	C₂²	1	4	yes	—
16	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	#390×C₁₃₁	C₂²	1	4	yes	—
22	#389×C₁₃₁	C₂²	1	4	yes	—
23	#388×C₁₃₁	—	2	4	no	—
24	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
25	C₄×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
26	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
27	#340×C₁₃₁	C₂²	1	4	yes	—
28	#340×C₁₃₁	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃₁	—	2	2	no	—
31	C₂×D₈×C₁₃₁	—	2	4	no	—
32	C₂×D₈×C₁₃₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₁₃₁	—	2	4	no	—
37	C₄×D₄×C₁₃₁	C₂×C₄	1	8	yes	—
38	#117×C₁₃₁	—	4	8	no	—
39	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
45	C₂×C₄²×C₁₃₁	—	2	2	no	—
46	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
51	#107×C₁₃₁	—	2	4	no	—
52	#107×C₁₃₁	C₂×C₄	1	4	yes	—
53	#102×C₁₃₁	—	2	8	no	—
54	#102×C₁₃₁	C₂×C₄	1	8	yes	—
55	C₄×C₈×C₁₃₁	—	2	4	no	—
56	C₄×C₈×C₁₃₁	—	2	4	no	—
57	C₄×C₈×C₁₃₁	—	2	2	no	—
58	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
59	#95×C₁₃₁	—	2	2	no	—
60	#95×C₁₃₁	—	2	4	no	—
61	C₂⁴×C₁₃₁	—	2	2	no	—
62	C₂×D₄×C₁₃₁	—	2	4	no	—
63	C₂×D₄×C₁₃₁	—	2	4	no	—
64	C₂×D₄×C₁₃₁	—	2	2	no	—
65	C₂×D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃₁	—	2	2	no	—
67	C₂²×C₄×C₁₃₁	—	2	8	no	—
68	C₂²×C₄×C₁₃₁	—	2	4	no	—
69	C₂²×C₄×C₁₃₁	—	4	4	no	—
70	C₂²×C₄×C₁₃₁	—	4	4	no	—
71	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
76	D₈×C₁₃₁	—	2	8	no	—
77	C₂×C₈×C₁₃₁	—	2	2	no	—
78	C₂×C₈×C₁₃₁	—	2	2	no	—
79	C₂×C₈×C₁₃₁	—	2	2	no	—
80	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃₁	#41	1	4	yes	—
83	C₂×C₈×C₁₃₁	#41	1	4	yes	—
84	C₄⋊C₄×C₁₃₁	—	2	2	no	—
85	C₄⋊C₄×C₁₃₁	—	2	4	no	—
86	C₄⋊C₄×C₁₃₁	—	2	4	no	—
87	C₄⋊C₄×C₁₃₁	—	4	8	no	—
88	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
89	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
90	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
91	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
92	C₄²×C₁₃₁	—	2	4	no	—
93	C₄²×C₁₃₁	—	2	2	no	—
94	C₄²×C₁₃₁	—	2	2	no	—
95	C₄²×C₁₃₁	—	2	2	no	—
96	C₄²×C₁₃₁	—	4	4	no	—
97	C₄²×C₁₃₁	D₈	1	4	yes	complemented
98	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₃₁	#41	1	2	yes	—
100	C₂³×C₁₃₁	—	2	4	no	—
101	C₂³×C₁₃₁	—	4	4	no	—
102	C₂³×C₁₃₁	—	4	2	no	—
103	C₂³×C₁₃₁	—	4	4	no	—
104	C₂³×C₁₃₁	—	4	2	no	—
105	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₁₃₁	—	2	4	no	—
107	D₄×C₁₃₁	—	2	4	no	—
108	D₄×C₁₃₁	—	4	4	no	—
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	2	2	no	—
111	C₂×C₄×C₁₃₁	—	2	2	no	—
112	C₂×C₄×C₁₃₁	—	2	2	no	—
113	C₂×C₄×C₁₃₁	—	2	2	no	—
114	C₂×C₄×C₁₃₁	—	2	4	no	—
115	C₂×C₄×C₁₃₁	—	4	2	no	—
116	C₂×C₄×C₁₃₁	—	4	2	no	—
117	C₂×C₄×C₁₃₁	—	4	16	no	—
118	C₂×C₄×C₁₃₁	—	4	4	no	—
119	C₂×C₄×C₁₃₁	—	4	4	no	—
120	C₂×C₄×C₁₃₁	—	4	4	no	—
121	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
124	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₁₃₁	#135	1	4	yes	—
126	C₂×C₄×C₁₃₁	#135	1	2	yes	—
127	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
130	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
131	C₈×C₁₃₁	—	2	2	no	—
132	C₈×C₁₃₁	—	2	4	no	—
133	C₈×C₁₃₁	#117	1	4	yes	complemented
134	C₂²×C₁₃₁	—	4	4	no	—
135	C₂²×C₁₃₁	—	4	4	no	—
136	C₂²×C₁₃₁	—	4	4	no	—
137	C₂²×C₁₃₁	—	8	4	no	—
138	C₂²×C₁₃₁	—	8	4	no	—
139	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₁₃₁	#532	1	2	yes	—
141	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
142	C₂²×C₁₃₁	C₂×#135	1	1	yes	characteristic
143	C₂²×C₁₃₁	C₂×#135	1	2	yes	—
144	C₄×C₁₃₁	—	2	4	no	—
145	C₄×C₁₃₁	—	2	4	no	—
146	C₄×C₁₃₁	—	2	2	no	—
147	C₄×C₁₃₁	—	4	4	no	—
148	C₄×C₁₃₁	—	8	4	no	—
149	C₄×C₁₃₁	C₄×D₈	1	4	yes	—
150	C₄×C₁₃₁	C₂×#117	1	1	yes	characteristic
151	C₄×C₁₃₁	C₄×#41	1	1	yes	characteristic
152	C₄×C₁₃₁	#529	1	2	yes	—
153	C₂×C₁₃₁	—	8	4	no	—
154	C₂×C₁₃₁	#2293	1	1	yes	characteristic
155	C₂×C₁₃₁	C₂×C₄×D₈	1	1	yes	characteristic
156	C₂×C₁₃₁	C₄×#135	1	1	yes	characteristic
157	C₂×C₁₃₁	#3168	1	2	yes	—
158	C₂×C₁₃₁	#3169	1	2	yes	—
159	#21275	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₁₃₁	#21275	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×C₄×D₈	C₂×C₁₃₁	1	1	yes	characteristic
162	#2293	C₂×C₁₃₁	1	2	yes	complemented
163	#1970	C₂×C₁₃₁	1	2	yes	—
164	#1942	C₂×C₁₃₁	1	2	yes	complemented
165	#1934	C₂×C₁₃₁	1	4	yes	complemented
166	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic, complemented
167	C₄×#102	C₂×C₁₃₁	1	2	yes	complemented
168	C₄²×C₈	C₂×C₁₃₁	1	1	yes	characteristic, complemented
169	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
170	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
171	C₂×#117	—	2	2	no	—
172	C₄×D₈	C₄×C₁₃₁	1	8	yes	complemented
173	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
174	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
175	C₂×#102	C₂²×C₁₃₁	1	4	yes	—
176	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
181	#390	C₂²×C₁₃₁	1	4	yes	—
182	#389	C₂²×C₁₃₁	1	4	yes	—
183	#388	—	2	4	no	—
184	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
185	C₄×C₂²⋊C₄	—	2	2	no	—
186	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
187	#340	C₂²×C₁₃₁	1	2	yes	—
188	#340	C₂²×C₁₃₁	1	4	yes	—
189	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
190	C₂³×C₄	—	2	2	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂×D₈	C₂×C₄×C₁₃₁	1	4	yes	—
193	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	C₂×C₄×C₁₃₁	1	8	yes	—
198	#117	—	4	8	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
208	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
210	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
211	#107	—	2	4	no	—
212	#107	C₂×C₄×C₁₃₁	1	4	yes	—
213	#102	—	2	8	no	—
214	#102	C₂×C₄×C₁₃₁	1	8	yes	—
215	C₄×C₈	—	2	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
219	#95	—	2	4	no	—
220	#95	—	2	2	no	—
221	C₂⁴	—	2	2	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
236	D₈	—	2	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
242	C₂×C₈	#41×C₁₃₁	1	4	yes	—
243	C₂×C₈	#41×C₁₃₁	1	4	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	8	no	—
248	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	4	8	no	—
251	C₂²⋊C₄	—	4	8	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	4	no	—
257	C₄²	D₈×C₁₃₁	1	4	yes	complemented
258	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
259	C₄²	#41×C₁₃₁	1	2	yes	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	2	no	—
265	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	16	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
282	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
283	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
284	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
285	C₂×C₄	#135×C₁₃₁	1	4	yes	—
286	C₂×C₄	#135×C₁₃₁	1	2	yes	—
287	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
289	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
290	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	4	no	—
293	C₈	#117×C₁₃₁	1	4	yes	complemented
294	C₂²	—	4	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
300	C₂²	#532×C₁₃₁	1	2	yes	—
301	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
302	C₂²	C₂×#135×C₁₃₁	1	1	yes	characteristic
303	C₂²	C₂×#135×C₁₃₁	1	2	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	4	no	—
309	C₄	C₄×D₈×C₁₃₁	1	4	yes	—
310	C₄	C₂×#117×C₁₃₁	1	1	yes	characteristic
311	C₄	C₄×#41×C₁₃₁	1	1	yes	characteristic
312	C₄	#529×C₁₃₁	1	2	yes	—
313	C₂	—	8	4	no	—
314	C₂	#2293×C₁₃₁	1	1	yes	characteristic
315	C₂	C₂×C₄×D₈×C₁₃₁	1	1	yes	characteristic
316	C₂	C₄×#135×C₁₃₁	1	1	yes	characteristic
317	C₂	#3168×C₁₃₁	1	2	yes	—
318	C₂	#3169×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂³×C₁₃₁	C₂²×D₄
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2293×C₁₃₁	C₂	semidirect
4. #1942×C₁₃₁	C₂	semidirect
5. #1934×C₁₃₁	C₂	semidirect
6. C₄×#107×C₁₃₁	C₂	semidirect
7. C₄×#102×C₁₃₁	C₂	semidirect
8. C₄²×C₈×C₁₃₁	C₂	semidirect
12. C₄×D₈×C₁₃₁	C₄	semidirect
97. C₄²×C₁₃₁	D₈	semidirect
133. C₈×C₁₃₁	#117	semidirect
159. #21275	C₁₃₁	direct (N × H)
160. C₁₃₁	#21275	direct (N × H)
162. #2293	C₂×C₁₃₁	semidirect
164. #1942	C₂×C₁₃₁	semidirect
165. #1934	C₂×C₁₃₁	semidirect
166. C₄×#107	C₂×C₁₃₁	semidirect
167. C₄×#102	C₂×C₁₃₁	semidirect
168. C₄²×C₈	C₂×C₁₃₁	semidirect
172. C₄×D₈	C₄×C₁₃₁	semidirect
257. C₄²	D₈×C₁₃₁	semidirect
293. C₈	#117×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₁₃₁	2
#2293×C₁₃₁	2
#1970×C₁₃₁	2
#1942×C₁₃₁	2
#1934×C₁₃₁	2
C₄×#107×C₁₃₁	2
C₄×#102×C₁₃₁	2
C₄²×C₈×C₁₃₁	2
#21275	131

Subgroups — C131×#21275 #45707159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#21275 #45707159