Subgroups — C₁₃₁×#7029 #45692913

Order 33536.

448 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1797×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#336×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃₁	—	2	1	no	—
5	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₁₃₁	C₄	1	2	yes	—
7	C₂×C₂²⋊C₈×C₁₃₁	—	2	1	no	—
8	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
10	#336×C₁₃₁	—	2	1	no	—
11	#336×C₁₃₁	—	2	1	no	—
12	#336×C₁₃₁	—	2	1	no	—
13	#336×C₁₃₁	C₄	1	2	yes	—
14	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃₁	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₃₁	—	2	4	no	—
18	C₂²×C₈×C₁₃₁	—	2	2	no	—
19	C₂²×C₈×C₁₃₁	—	2	2	no	—
20	C₂²×C₈×C₁₃₁	—	2	2	no	—
21	C₂²×C₈×C₁₃₁	—	4	2	no	—
22	C₂²×C₈×C₁₃₁	—	4	1	no	—
23	C₂²×C₈×C₁₃₁	—	4	1	no	—
24	C₂×C₄²×C₁₃₁	—	4	2	no	—
25	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₁₃₁	—	4	2	no	—
30	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
31	C₂²⋊C₈×C₁₃₁	—	4	1	no	—
32	C₂²⋊C₈×C₁₃₁	—	4	1	no	—
33	C₂²⋊C₈×C₁₃₁	C₈	1	4	yes	complemented
34	C₄×C₈×C₁₃₁	—	4	2	no	—
35	C₂⁴×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₁	—	2	2	no	—
37	C₂²×C₄×C₁₃₁	—	2	2	no	—
38	C₂²×C₄×C₁₃₁	—	2	2	no	—
39	C₂²×C₄×C₁₃₁	—	2	2	no	—
40	C₂²×C₄×C₁₃₁	—	4	1	no	—
41	C₂²×C₄×C₁₃₁	—	4	1	no	—
42	C₂²×C₄×C₁₃₁	—	4	1	no	—
43	C₂²×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃₁	M₁₆	1	2	yes	—
51	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₃₁	—	2	4	no	—
56	C₂×C₈×C₁₃₁	—	2	4	no	—
57	C₂×C₈×C₁₃₁	—	4	2	no	—
58	C₂×C₈×C₁₃₁	—	4	2	no	—
59	C₂×C₈×C₁₃₁	—	4	2	no	—
60	C₂×C₈×C₁₃₁	—	4	4	no	—
61	C₂×C₈×C₁₃₁	—	4	4	no	—
62	C₂×C₈×C₁₃₁	—	4	2	no	—
63	C₂×C₈×C₁₃₁	—	4	2	no	—
64	C₂×C₈×C₁₃₁	—	4	2	no	—
65	C₂×C₈×C₁₃₁	—	4	2	no	—
66	C₄²×C₁₃₁	—	2	2	no	—
67	C₄²×C₁₃₁	—	2	2	no	—
68	C₄²×C₁₃₁	—	4	1	no	—
69	C₄²×C₁₃₁	—	4	1	no	—
70	C₂³×C₁₃₁	—	2	1	no	—
71	C₂³×C₁₃₁	—	2	1	no	—
72	C₂³×C₁₃₁	—	2	1	no	—
73	C₂³×C₁₃₁	—	2	1	no	—
74	C₂³×C₁₃₁	#95	1	1	yes	characteristic
75	C₂³×C₁₃₁	C₄⋊C₈	1	2	yes	—
76	C₂³×C₁₃₁	#106	1	1	yes	characteristic
77	C₂³×C₁₃₁	#107	1	1	yes	characteristic
78	C₂³×C₁₃₁	#108	1	2	yes	—
79	C₂×C₄×C₁₃₁	—	2	1	no	—
80	C₂×C₄×C₁₃₁	—	2	1	no	—
81	C₂×C₄×C₁₃₁	—	2	1	no	—
82	C₂×C₄×C₁₃₁	—	2	1	no	—
83	C₂×C₄×C₁₃₁	—	2	1	no	—
84	C₂×C₄×C₁₃₁	—	2	1	no	—
85	C₂×C₄×C₁₃₁	—	2	1	no	—
86	C₂×C₄×C₁₃₁	—	2	1	no	—
87	C₂×C₄×C₁₃₁	—	2	2	no	—
88	C₂×C₄×C₁₃₁	—	2	2	no	—
89	C₂×C₄×C₁₃₁	—	2	2	no	—
90	C₂×C₄×C₁₃₁	—	2	2	no	—
91	C₂×C₄×C₁₃₁	—	2	1	no	—
92	C₂×C₄×C₁₃₁	—	2	1	no	—
93	C₂×C₄×C₁₃₁	—	2	1	no	—
94	C₂×C₄×C₁₃₁	—	4	2	no	—
95	C₂×C₄×C₁₃₁	—	4	2	no	—
96	C₂×C₄×C₁₃₁	—	4	4	no	—
97	C₂×C₄×C₁₃₁	C₄×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	#97	1	1	yes	characteristic
99	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₁₃₁	#99	1	2	yes	—
101	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
103	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
104	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	#104	1	2	yes	—
106	C₂×C₄×C₁₃₁	#108	1	2	yes	—
107	C₈×C₁₃₁	—	4	4	no	—
108	C₈×C₁₃₁	—	4	4	no	—
109	C₈×C₁₃₁	—	8	2	no	—
110	C₂²×C₁₃₁	—	2	2	no	—
111	C₂²×C₁₃₁	—	2	1	no	—
112	C₂²×C₁₃₁	—	2	1	no	—
113	C₂²×C₁₃₁	—	2	2	no	—
114	C₂²×C₁₃₁	—	2	1	no	—
115	C₂²×C₁₃₁	—	2	1	no	—
116	C₂²×C₁₃₁	—	2	2	no	—
117	C₂²×C₁₃₁	—	2	2	no	—
118	C₂²×C₁₃₁	#334	1	2	yes	—
119	C₂²×C₁₃₁	#335	1	2	yes	—
120	C₂²×C₁₃₁	#336	1	1	yes	characteristic
121	C₂²×C₁₃₁	#339	1	1	yes	characteristic
122	C₂²×C₁₃₁	#340	1	1	yes	characteristic
123	C₂²×C₁₃₁	#341	1	1	yes	characteristic
124	C₂²×C₁₃₁	#341	1	1	yes	characteristic
125	C₂²×C₁₃₁	#342	1	1	yes	characteristic
126	C₂²×C₁₃₁	#344	1	1	yes	characteristic
127	C₄×C₁₃₁	—	2	2	no	—
128	C₄×C₁₃₁	—	2	1	no	—
129	C₄×C₁₃₁	—	2	1	no	—
130	C₄×C₁₃₁	—	2	1	no	—
131	C₄×C₁₃₁	—	2	1	no	—
132	C₄×C₁₃₁	—	2	2	no	—
133	C₄×C₁₃₁	—	4	1	no	—
134	C₄×C₁₃₁	#327	1	2	yes	—
135	C₄×C₁₃₁	#328	1	2	yes	—
136	C₂×C₁₃₁	—	2	2	no	—
137	C₂×C₁₃₁	—	2	2	no	—
138	C₂×C₁₃₁	#1277	1	1	yes	characteristic
139	C₂×C₁₃₁	#1278	1	1	yes	characteristic
140	C₂×C₁₃₁	#1282	1	1	yes	characteristic
141	C₂×C₁₃₁	#1294	1	1	yes	characteristic
142	C₂×C₁₃₁	#1298	1	1	yes	characteristic
143	C₂×C₁₃₁	#1301	1	1	yes	characteristic
144	C₂×C₁₃₁	#1302	1	1	yes	characteristic
145	#7029	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₁	#7029	1	1	yes	characteristic, direct product factor, Sylow
147	#1797	C₂×C₁₃₁	1	1	yes	characteristic
148	C₄×C₂²⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic
149	C₂×#336	C₂×C₁₃₁	1	1	yes	characteristic
150	C₂³×C₈	—	2	1	no	—
151	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
152	C₂×C₄⋊C₈	C₄×C₁₃₁	1	2	yes	—
153	C₂×C₂²⋊C₈	—	2	1	no	—
154	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	2	yes	—
155	C₂×C₄×C₈	—	2	1	no	—
156	#336	—	2	1	no	—
157	#336	—	2	1	no	—
158	#336	—	2	1	no	—
159	#336	C₄×C₁₃₁	1	2	yes	—
160	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
161	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
162	C₂³×C₄	Q₈×C₁₃₁	1	1	yes	characteristic
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	4	2	no	—
168	C₂²×C₈	—	4	1	no	—
169	C₂²×C₈	—	4	1	no	—
170	C₂×C₄²	—	4	2	no	—
171	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
173	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
174	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
175	C₄⋊C₈	—	4	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	4	1	no	—
178	C₂²⋊C₈	—	4	1	no	—
179	C₂²⋊C₈	C₈×C₁₃₁	1	4	yes	complemented
180	C₄×C₈	—	4	2	no	—
181	C₂⁴	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	C₄²×C₁₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
196	C₂²×C₄	M₁₆×C₁₃₁	1	2	yes	—
197	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
198	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
200	C₂²×C₄	Q₁₆×C₁₃₁	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	#95×C₁₃₁	1	1	yes	characteristic
221	C₂³	C₄⋊C₈×C₁₃₁	1	2	yes	—
222	C₂³	#106×C₁₃₁	1	1	yes	characteristic
223	C₂³	#107×C₁₃₁	1	1	yes	characteristic
224	C₂³	#108×C₁₃₁	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	C₄×C₈×C₁₃₁	1	1	yes	characteristic
244	C₂×C₄	#97×C₁₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	2	yes	—
246	C₂×C₄	#99×C₁₃₁	1	2	yes	—
247	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
248	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
249	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
250	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
251	C₂×C₄	#104×C₁₃₁	1	2	yes	—
252	C₂×C₄	#108×C₁₃₁	1	2	yes	—
253	C₈	—	4	4	no	—
254	C₈	—	4	4	no	—
255	C₈	—	8	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	#334×C₁₃₁	1	2	yes	—
265	C₂²	#335×C₁₃₁	1	2	yes	—
266	C₂²	#336×C₁₃₁	1	1	yes	characteristic
267	C₂²	#339×C₁₃₁	1	1	yes	characteristic
268	C₂²	#340×C₁₃₁	1	1	yes	characteristic
269	C₂²	#341×C₁₃₁	1	1	yes	characteristic
270	C₂²	#341×C₁₃₁	1	1	yes	characteristic
271	C₂²	#342×C₁₃₁	1	1	yes	characteristic
272	C₂²	#344×C₁₃₁	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	1	no	—
280	C₄	#327×C₁₃₁	1	2	yes	—
281	C₄	#328×C₁₃₁	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	#1277×C₁₃₁	1	1	yes	characteristic
285	C₂	#1278×C₁₃₁	1	1	yes	characteristic
286	C₂	#1282×C₁₃₁	1	1	yes	characteristic
287	C₂	#1294×C₁₃₁	1	1	yes	characteristic
288	C₂	#1298×C₁₃₁	1	1	yes	characteristic
289	C₂	#1301×C₁₃₁	1	1	yes	characteristic
290	C₂	#1302×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₁₃₁
Frattini, cosocle	C₂²×C₄²	C₂²×C₁₃₁
socle	C₂³×C₁₃₁	#95
prime radical	C₂⁴×C₁₃₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
33. C₂²⋊C₈×C₁₃₁	C₈	semidirect
145. #7029	C₁₃₁	direct (N × H)
146. C₁₃₁	#7029	direct (N × H)
179. C₂²⋊C₈	C₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#1797×C₁₃₁	2
C₄×C₂²⋊C₈×C₁₃₁	2
C₂×#336×C₁₃₁	2
#7029	131

Subgroups — C131×#7029 #45692913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7029 #45692913