Subgroups — C₁₃₁×#13255 #45699139

Order 33536.

498 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#2150×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#360×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	C₄×#106×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135×C₁₃₁	—	2	1	no	—
6	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
7	#532×C₁₃₁	—	2	1	no	—
8	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
9	#495×C₁₃₁	—	2	1	no	—
10	#495×C₁₃₁	C₄	1	2	yes	complemented
11	#494×C₁₃₁	C₄	1	1	yes	characteristic, complemented
12	#493×C₁₃₁	C₄	1	1	yes	characteristic, complemented
13	#492×C₁₃₁	—	2	1	no	—
14	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
17	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
18	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
19	#360×C₁₃₁	—	2	8	no	—
20	#340×C₁₃₁	—	2	1	no	—
21	C₂×#41×C₁₃₁	—	2	2	no	—
22	#135×C₁₃₁	—	2	4	no	—
23	#135×C₁₃₁	—	2	4	no	—
24	#135×C₁₃₁	—	4	4	no	—
25	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
26	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
28	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
29	C₂×D₈×C₁₃₁	—	2	1	no	—
30	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
34	#127×C₁₃₁	—	2	1	no	—
35	#124×C₁₃₁	—	2	2	no	—
36	#121×C₁₃₁	—	4	1	no	—
37	#121×C₁₃₁	—	4	1	no	—
38	C₄×D₄×C₁₃₁	—	4	1	no	—
39	C₄×D₄×C₁₃₁	—	4	1	no	—
40	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
41	C₂×C₄²×C₁₃₁	—	4	1	no	—
42	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
43	M₃₂×C₁₃₁	—	2	4	no	—
44	C₂×C₁₆×C₁₃₁	—	2	2	no	—
45	#106×C₁₃₁	—	2	2	no	—
46	#106×C₁₃₁	D₄	1	4	yes	complemented
47	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
51	#95×C₁₃₁	—	4	1	no	—
52	#41×C₁₃₁	—	4	4	no	—
53	#41×C₁₃₁	—	4	4	no	—
54	C₂×Q₈×C₁₃₁	—	2	2	no	—
55	C₂×D₄×C₁₃₁	—	2	2	no	—
56	C₂×D₄×C₁₃₁	—	4	2	no	—
57	C₂×D₄×C₁₃₁	—	4	1	no	—
58	C₂×D₄×C₁₃₁	—	4	1	no	—
59	C₂²×C₄×C₁₃₁	—	4	2	no	—
60	C₂²×C₄×C₁₃₁	—	4	2	no	—
61	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
64	Q₁₆×C₁₃₁	—	2	4	no	—
65	Q₁₆×C₁₃₁	—	4	2	no	—
66	SD₁₆×C₁₃₁	—	4	4	no	—
67	SD₁₆×C₁₃₁	—	4	2	no	—
68	SD₁₆×C₁₃₁	—	4	2	no	—
69	D₈×C₁₃₁	—	2	4	no	—
70	D₈×C₁₃₁	—	4	2	no	—
71	C₂×C₈×C₁₃₁	—	2	2	no	—
72	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃₁	#41	1	2	yes	—
78	C₄⋊C₄×C₁₃₁	—	2	4	no	—
79	C₄⋊C₄×C₁₃₁	—	4	1	no	—
80	C₄⋊C₄×C₁₃₁	—	4	1	no	—
81	C₄⋊C₄×C₁₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
83	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
84	C₄²×C₁₃₁	—	8	2	no	—
85	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₃₁	SD₁₆	1	1	yes	characteristic
88	C₄²×C₁₃₁	SD₁₆	1	1	yes	characteristic
89	C₁₆×C₁₃₁	—	4	2	no	—
90	C₂³×C₁₃₁	—	4	2	no	—
91	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₁₃₁	—	4	2	no	—
93	Q₈×C₁₃₁	—	4	2	no	—
94	D₄×C₁₃₁	—	4	2	no	—
95	D₄×C₁₃₁	—	4	2	no	—
96	D₄×C₁₃₁	—	8	2	no	—
97	D₄×C₁₃₁	—	8	4	no	—
98	D₄×C₁₃₁	—	8	2	no	—
99	C₂×C₄×C₁₃₁	—	2	2	no	—
100	C₂×C₄×C₁₃₁	—	4	1	no	—
101	C₂×C₄×C₁₃₁	—	4	2	no	—
102	C₂×C₄×C₁₃₁	—	4	4	no	—
103	C₂×C₄×C₁₃₁	—	8	2	no	—
104	C₂×C₄×C₁₃₁	—	8	2	no	—
105	C₂×C₄×C₁₃₁	—	8	4	no	—
106	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	#121	1	2	yes	—
112	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
114	C₈×C₁₃₁	—	2	1	no	—
115	C₈×C₁₃₁	—	2	1	no	—
116	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
117	C₈×C₁₃₁	#124	1	1	yes	characteristic
118	C₈×C₁₃₁	#127	1	1	yes	characteristic
119	C₂²×C₁₃₁	—	2	1	no	—
120	C₂²×C₁₃₁	—	2	1	no	—
121	C₂²×C₁₃₁	—	8	2	no	—
122	C₂²×C₁₃₁	—	8	2	no	—
123	C₂²×C₁₃₁	—	8	2	no	—
124	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
125	C₂²×C₁₃₁	#459	1	2	yes	—
126	C₄×C₁₃₁	—	2	1	no	—
127	C₄×C₁₃₁	—	2	1	no	—
128	C₄×C₁₃₁	—	2	1	no	—
129	C₄×C₁₃₁	—	8	4	no	—
130	C₄×C₁₃₁	—	8	2	no	—
131	C₄×C₁₃₁	#360	1	2	yes	complemented
132	C₄×C₁₃₁	#390	1	1	yes	characteristic
133	C₄×C₁₃₁	#418	1	1	yes	characteristic
134	C₄×C₁₃₁	#460	1	2	yes	—
135	C₂×C₁₃₁	—	2	2	no	—
136	C₂×C₁₃₁	—	16	2	no	—
137	C₂×C₁₃₁	#1982	1	1	yes	characteristic
138	C₂×C₁₃₁	C₂×#360	1	1	yes	characteristic
139	C₂×C₁₃₁	#2145	1	1	yes	characteristic
140	#13255	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₃₁	#13255	1	1	yes	characteristic, direct product factor, Sylow
142	#3147	C₂×C₁₃₁	1	1	yes	characteristic
143	#2150	C₂×C₁₃₁	1	1	yes	characteristic, complemented
144	C₂×#360	C₂×C₁₃₁	1	4	yes	complemented
145	C₄×#106	C₂×C₁₃₁	1	1	yes	characteristic, complemented
146	C₂×#135	—	2	1	no	—
147	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
148	#532	—	2	1	no	—
149	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
150	#495	—	2	1	no	—
151	#495	C₄×C₁₃₁	1	2	yes	complemented
152	#494	C₄×C₁₃₁	1	1	yes	characteristic, complemented
153	#493	C₄×C₁₃₁	1	1	yes	characteristic, complemented
154	#492	—	2	1	no	—
155	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
157	C₄×C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₁₆	—	2	1	no	—
159	C₄⋊C₁₆	—	2	1	no	—
160	#360	—	2	8	no	—
161	#340	—	2	1	no	—
162	C₂×#41	—	2	2	no	—
163	#135	—	2	4	no	—
164	#135	—	2	4	no	—
165	#135	—	4	4	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×SD₁₆	—	2	2	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
172	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	1	no	—
175	#127	—	2	1	no	—
176	#124	—	2	2	no	—
177	#121	—	4	1	no	—
178	#121	—	4	1	no	—
179	C₄×D₄	—	4	1	no	—
180	C₄×D₄	—	4	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄²	—	4	1	no	—
183	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
184	M₃₂	—	2	4	no	—
185	C₂×C₁₆	—	2	2	no	—
186	#106	—	2	2	no	—
187	#106	D₄×C₁₃₁	1	4	yes	complemented
188	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
189	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
190	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
191	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
192	#95	—	4	1	no	—
193	#41	—	4	4	no	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
203	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	2	no	—
207	SD₁₆	—	4	4	no	—
208	SD₁₆	—	4	2	no	—
209	SD₁₆	—	4	2	no	—
210	D₈	—	2	4	no	—
211	D₈	—	4	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
215	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
218	C₂×C₈	#41×C₁₃₁	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	8	2	no	—
226	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
227	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
228	C₄²	SD₁₆×C₁₃₁	1	1	yes	characteristic
229	C₄²	SD₁₆×C₁₃₁	1	1	yes	characteristic
230	C₁₆	—	4	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	4	no	—
239	D₄	—	8	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
248	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
249	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
250	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
252	C₂×C₄	#121×C₁₃₁	1	2	yes	—
253	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
254	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
258	C₈	#124×C₁₃₁	1	1	yes	characteristic
259	C₈	#127×C₁₃₁	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
266	C₂²	#459×C₁₃₁	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	2	no	—
272	C₄	#360×C₁₃₁	1	2	yes	complemented
273	C₄	#390×C₁₃₁	1	1	yes	characteristic
274	C₄	#418×C₁₃₁	1	1	yes	characteristic
275	C₄	#460×C₁₃₁	1	2	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	16	2	no	—
278	C₂	#1982×C₁₃₁	1	1	yes	characteristic
279	C₂	C₂×#360×C₁₃₁	1	1	yes	characteristic
280	C₂	#2145×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	#3147×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2150×C₁₃₁	C₂	semidirect
3. C₂×#360×C₁₃₁	C₂	semidirect
4. C₄×#106×C₁₃₁	C₂	semidirect
10. #495×C₁₃₁	C₄	semidirect
11. #494×C₁₃₁	C₄	semidirect
12. #493×C₁₃₁	C₄	semidirect
46. #106×C₁₃₁	D₄	semidirect
131. C₄×C₁₃₁	#360	semidirect
140. #13255	C₁₃₁	direct (N × H)
141. C₁₃₁	#13255	direct (N × H)
143. #2150	C₂×C₁₃₁	semidirect
144. C₂×#360	C₂×C₁₃₁	semidirect
145. C₄×#106	C₂×C₁₃₁	semidirect
151. #495	C₄×C₁₃₁	semidirect
152. #494	C₄×C₁₃₁	semidirect
153. #493	C₄×C₁₃₁	semidirect
187. #106	D₄×C₁₃₁	semidirect
272. C₄	#360×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
#2150×C₁₃₁	2
C₂×#360×C₁₃₁	2
C₄×#106×C₁₃₁	2
#13255	131

Subgroups — C131×#13255 #45699139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13255 #45699139