Subgroups — C₁₃₁×#19230 #45705114

Order 33536.

790 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3245×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3205×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#3151×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2935×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1687×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1685×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1667×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1665×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#1565×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#1540×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#331×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₁₃₁	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₁₃₁	—	2	1	no	—
14	C₂×#131×C₁₃₁	—	2	1	no	—
15	#532×C₁₃₁	C₂²	1	1	yes	characteristic
16	#532×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
18	#496×C₁₃₁	—	2	2	no	—
19	#493×C₁₃₁	—	2	1	no	—
20	#493×C₁₃₁	C₂²	1	2	yes	—
21	#492×C₁₃₁	C₂²	1	2	yes	—
22	#486×C₁₃₁	—	2	1	no	—
23	#474×C₁₃₁	—	2	1	no	—
24	#474×C₁₃₁	C₂²	1	2	yes	complemented
25	#466×C₁₃₁	—	2	2	no	—
26	#459×C₁₃₁	—	2	1	no	—
27	#459×C₁₃₁	C₂²	1	2	yes	—
28	#459×C₁₃₁	C₂²	1	2	yes	—
29	#459×C₁₃₁	C₂²	1	2	yes	complemented
30	#447×C₁₃₁	—	2	2	no	—
31	#436×C₁₃₁	C₂²	1	2	yes	complemented
32	#435×C₁₃₁	—	2	2	no	—
33	C₈×D₄×C₁₃₁	C₂²	1	2	yes	complemented
34	#431×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
37	#408×C₁₃₁	—	2	1	no	—
38	C₄×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
39	#335×C₁₃₁	C₂²	1	2	yes	complemented
40	#334×C₁₃₁	C₂²	1	2	yes	complemented
41	#332×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
42	#332×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
43	#331×C₁₃₁	C₂²	1	2	yes	—
44	#331×C₁₃₁	C₂²	1	2	yes	—
45	#331×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
46	#331×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
47	C₂×#41×C₁₃₁	—	2	1	no	—
48	C₂×#41×C₁₃₁	—	2	1	no	—
49	C₂²×D₄×C₁₃₁	—	2	1	no	—
50	C₂²×D₄×C₁₃₁	—	2	1	no	—
51	C₂²×D₄×C₁₃₁	—	2	1	no	—
52	AGL(1,ℤ/8)×C₁₃₁	—	4	4	no	—
53	AGL(1,ℤ/8)×C₁₃₁	—	4	4	no	—
54	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
56	C₂×D₈×C₁₃₁	—	2	2	no	—
57	C₂×D₈×C₁₃₁	—	2	2	no	—
58	C₂×D₈×C₁₃₁	—	2	2	no	—
59	C₂×D₈×C₁₃₁	—	2	2	no	—
60	C₂×D₈×C₁₃₁	—	2	2	no	—
61	#131×C₁₃₁	—	4	4	no	—
62	C₂×M₁₆×C₁₃₁	—	2	2	no	—
63	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
64	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₁₃₁	—	2	1	no	—
66	C₂²×C₈×C₁₃₁	—	2	1	no	—
67	C₂²×C₈×C₁₃₁	—	2	2	no	—
68	C₄⋊Q₈×C₁₃₁	D₄	1	1	yes	characteristic
69	C₄⋊Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
70	#127×C₁₃₁	—	2	1	no	—
71	#127×C₁₃₁	—	2	1	no	—
72	#127×C₁₃₁	—	2	1	no	—
73	#127×C₁₃₁	D₄	1	1	yes	characteristic
74	#127×C₁₃₁	C₂³	1	2	yes	—
75	#127×C₁₃₁	C₂³	1	1	yes	characteristic
76	#124×C₁₃₁	—	2	1	no	—
77	#124×C₁₃₁	—	2	1	no	—
78	#121×C₁₃₁	—	2	1	no	—
79	#121×C₁₃₁	—	2	1	no	—
80	#121×C₁₃₁	—	2	1	no	—
81	#121×C₁₃₁	D₄	1	2	yes	—
82	C₄×D₄×C₁₃₁	—	2	1	no	—
83	C₄×D₄×C₁₃₁	—	2	1	no	—
84	C₄×D₄×C₁₃₁	—	2	1	no	—
85	C₄×D₄×C₁₃₁	C₂³	1	2	yes	—
86	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
87	#107×C₁₃₁	—	2	2	no	—
88	C₄⋊C₈×C₁₃₁	—	2	1	no	—
89	C₄⋊C₈×C₁₃₁	—	2	1	no	—
90	C₄⋊C₈×C₁₃₁	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₁₃₁	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
93	C₄⋊C₈×C₁₃₁	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₁₃₁	C₂³	1	1	yes	characteristic
95	#102×C₁₃₁	—	2	2	no	—
96	#102×C₁₃₁	—	2	2	no	—
97	#102×C₁₃₁	—	2	2	no	—
98	#102×C₁₃₁	—	2	2	no	—
99	C₂²⋊C₈×C₁₃₁	—	2	1	no	—
100	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
101	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
102	#97×C₁₃₁	D₄	1	2	yes	—
103	C₄×C₈×C₁₃₁	—	2	1	no	—
104	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
105	C₂⁴×C₁₃₁	—	4	1	no	—
106	#41×C₁₃₁	—	4	2	no	—
107	#41×C₁₃₁	—	4	2	no	—
108	#41×C₁₃₁	—	4	2	no	—
109	#41×C₁₃₁	—	4	2	no	—
110	C₂×Q₈×C₁₃₁	—	2	1	no	—
111	C₂×Q₈×C₁₃₁	—	2	1	no	—
112	C₂×D₄×C₁₃₁	—	2	2	no	—
113	C₂×D₄×C₁₃₁	—	2	1	no	—
114	C₂×D₄×C₁₃₁	—	2	1	no	—
115	C₂×D₄×C₁₃₁	—	2	1	no	—
116	C₂×D₄×C₁₃₁	—	2	2	no	—
117	C₂×D₄×C₁₃₁	—	2	1	no	—
118	C₂×D₄×C₁₃₁	—	2	2	no	—
119	C₂×D₄×C₁₃₁	—	2	1	no	—
120	C₂×D₄×C₁₃₁	—	2	2	no	—
121	C₂×D₄×C₁₃₁	—	4	1	no	—
122	C₂×D₄×C₁₃₁	—	4	1	no	—
123	C₂×D₄×C₁₃₁	—	4	1	no	—
124	C₂×D₄×C₁₃₁	—	4	1	no	—
125	C₂×D₄×C₁₃₁	—	4	2	no	—
126	C₂×D₄×C₁₃₁	—	4	2	no	—
127	C₂×D₄×C₁₃₁	—	8	1	no	—
128	C₂×D₄×C₁₃₁	—	8	1	no	—
129	C₂×D₄×C₁₃₁	—	8	2	no	—
130	C₂×D₄×C₁₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₁₃₁	—	2	2	no	—
132	C₂²×C₄×C₁₃₁	—	4	1	no	—
133	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
136	SD₁₆×C₁₃₁	—	4	4	no	—
137	SD₁₆×C₁₃₁	—	4	4	no	—
138	D₈×C₁₃₁	—	4	4	no	—
139	D₈×C₁₃₁	—	4	4	no	—
140	D₈×C₁₃₁	—	8	2	no	—
141	D₈×C₁₃₁	—	8	2	no	—
142	D₈×C₁₃₁	—	8	2	no	—
143	M₁₆×C₁₃₁	—	2	2	no	—
144	M₁₆×C₁₃₁	—	4	4	no	—
145	M₁₆×C₁₃₁	D₈	1	4	yes	complemented
146	C₂×C₈×C₁₃₁	—	2	1	no	—
147	C₂×C₈×C₁₃₁	—	2	1	no	—
148	C₂×C₈×C₁₃₁	—	2	1	no	—
149	C₂×C₈×C₁₃₁	—	2	1	no	—
150	C₂×C₈×C₁₃₁	—	2	2	no	—
151	C₂×C₈×C₁₃₁	—	4	2	no	—
152	C₂×C₈×C₁₃₁	—	4	4	no	—
153	C₂×C₈×C₁₃₁	—	8	1	no	—
154	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₁₃₁	—	2	1	no	—
157	C₄⋊C₄×C₁₃₁	—	2	1	no	—
158	C₄⋊C₄×C₁₃₁	—	2	1	no	—
159	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
160	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
161	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
162	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
163	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
164	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₁₃₁	#41	1	1	yes	characteristic
167	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
168	C₂³×C₁₃₁	—	2	2	no	—
169	C₂³×C₁₃₁	—	4	1	no	—
170	C₂³×C₁₃₁	—	4	2	no	—
171	C₂³×C₁₃₁	—	8	2	no	—
172	C₂³×C₁₃₁	—	8	2	no	—
173	C₂³×C₁₃₁	—	8	2	no	—
174	C₂³×C₁₃₁	#121	1	1	yes	characteristic
175	Q₈×C₁₃₁	—	4	2	no	—
176	Q₈×C₁₃₁	—	4	1	no	—
177	Q₈×C₁₃₁	—	4	1	no	—
178	D₄×C₁₃₁	—	2	2	no	—
179	D₄×C₁₃₁	—	2	2	no	—
180	D₄×C₁₃₁	—	4	4	no	—
181	D₄×C₁₃₁	—	4	1	no	—
182	D₄×C₁₃₁	—	4	2	no	—
183	D₄×C₁₃₁	—	4	1	no	—
184	D₄×C₁₃₁	—	4	2	no	—
185	D₄×C₁₃₁	—	4	1	no	—
186	D₄×C₁₃₁	—	4	2	no	—
187	D₄×C₁₃₁	—	4	2	no	—
188	D₄×C₁₃₁	—	4	1	no	—
189	D₄×C₁₃₁	—	4	2	no	—
190	D₄×C₁₃₁	—	8	1	no	—
191	D₄×C₁₃₁	—	8	1	no	—
192	D₄×C₁₃₁	—	8	2	no	—
193	D₄×C₁₃₁	—	8	2	no	—
194	D₄×C₁₃₁	—	8	2	no	—
195	D₄×C₁₃₁	—	8	2	no	—
196	D₄×C₁₃₁	—	8	2	no	—
197	C₂×C₄×C₁₃₁	—	2	1	no	—
198	C₂×C₄×C₁₃₁	—	2	1	no	—
199	C₂×C₄×C₁₃₁	—	2	2	no	—
200	C₂×C₄×C₁₃₁	—	4	1	no	—
201	C₂×C₄×C₁₃₁	—	4	2	no	—
202	C₂×C₄×C₁₃₁	—	4	2	no	—
203	C₂×C₄×C₁₃₁	—	4	1	no	—
204	C₂×C₄×C₁₃₁	—	8	2	no	—
205	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
206	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
207	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
208	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
209	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
210	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
213	C₈×C₁₃₁	—	2	2	no	—
214	C₈×C₁₃₁	—	4	1	no	—
215	C₈×C₁₃₁	—	4	1	no	—
216	C₈×C₁₃₁	—	4	2	no	—
217	C₈×C₁₃₁	—	4	1	no	—
218	C₈×C₁₃₁	—	4	1	no	—
219	C₈×C₁₃₁	C₂×D₈	1	4	yes	—
220	C₂²×C₁₃₁	—	2	1	no	—
221	C₂²×C₁₃₁	—	2	1	no	—
222	C₂²×C₁₃₁	—	4	2	no	—
223	C₂²×C₁₃₁	—	4	2	no	—
224	C₂²×C₁₃₁	—	4	2	no	—
225	C₂²×C₁₃₁	—	8	1	no	—
226	C₂²×C₁₃₁	—	8	2	no	—
227	C₂²×C₁₃₁	—	8	2	no	—
228	C₂²×C₁₃₁	—	8	1	no	—
229	C₂²×C₁₃₁	—	8	2	no	—
230	C₂²×C₁₃₁	—	8	1	no	—
231	C₂²×C₁₃₁	—	16	4	no	—
232	C₂²×C₁₃₁	#459	1	2	yes	—
233	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
234	C₄×C₁₃₁	—	2	1	no	—
235	C₄×C₁₃₁	—	2	1	no	—
236	C₄×C₁₃₁	—	2	1	no	—
237	C₄×C₁₃₁	—	4	2	no	—
238	C₄×C₁₃₁	—	8	1	no	—
239	C₄×C₁₃₁	#459	1	2	yes	—
240	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
241	C₄×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₄×C₁₃₁	#576	1	1	yes	characteristic
243	C₄×C₁₃₁	#577	1	1	yes	characteristic
244	C₂×C₁₃₁	—	2	2	no	—
245	C₂×C₁₃₁	—	8	2	no	—
246	C₂×C₁₃₁	—	16	1	no	—
247	C₂×C₁₃₁	—	16	2	no	—
248	C₂×C₁₃₁	C₂×#459	1	1	yes	characteristic
249	C₂×C₁₃₁	#3062	1	1	yes	characteristic
250	C₂×C₁₃₁	#3068	1	1	yes	characteristic
251	#19230	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₁₃₁	#19230	1	1	yes	characteristic, direct product factor, Sylow
253	#3245	C₂×C₁₃₁	1	1	yes	characteristic, complemented
254	#3205	C₂×C₁₃₁	1	1	yes	characteristic, complemented
255	#3151	C₂×C₁₃₁	1	1	yes	characteristic, complemented
256	#2935	C₂×C₁₃₁	1	1	yes	characteristic, complemented
257	#1687	C₂×C₁₃₁	1	2	yes	complemented
258	#1685	C₂×C₁₃₁	1	2	yes	complemented
259	#1667	C₂×C₁₃₁	1	2	yes	complemented
260	#1665	C₂×C₁₃₁	1	2	yes	complemented
261	#1565	C₂×C₁₃₁	1	1	yes	characteristic, complemented
262	#1540	C₂×C₁₃₁	1	1	yes	characteristic, complemented
263	C₂×#331	C₂×C₁₃₁	1	1	yes	characteristic, complemented
264	C₂×AGL(1,ℤ/8)	—	2	1	no	—
265	C₂×AGL(1,ℤ/8)	—	2	1	no	—
266	C₂×#131	—	2	1	no	—
267	#532	C₂²×C₁₃₁	1	1	yes	characteristic
268	#532	C₂²×C₁₃₁	1	1	yes	characteristic
269	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
270	#496	—	2	2	no	—
271	#493	—	2	1	no	—
272	#493	C₂²×C₁₃₁	1	2	yes	—
273	#492	C₂²×C₁₃₁	1	2	yes	—
274	#486	—	2	1	no	—
275	#474	—	2	1	no	—
276	#474	C₂²×C₁₃₁	1	2	yes	complemented
277	#466	—	2	2	no	—
278	#459	—	2	1	no	—
279	#459	C₂²×C₁₃₁	1	2	yes	—
280	#459	C₂²×C₁₃₁	1	2	yes	—
281	#459	C₂²×C₁₃₁	1	2	yes	complemented
282	#447	—	2	2	no	—
283	#436	C₂²×C₁₃₁	1	2	yes	complemented
284	#435	—	2	2	no	—
285	C₈×D₄	C₂²×C₁₃₁	1	2	yes	complemented
286	#431	C₂²×C₁₃₁	1	1	yes	characteristic
287	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
288	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
289	#408	—	2	1	no	—
290	C₄×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
291	#335	C₂²×C₁₃₁	1	2	yes	complemented
292	#334	C₂²×C₁₃₁	1	2	yes	complemented
293	#332	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
294	#332	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
295	#331	C₂²×C₁₃₁	1	2	yes	—
296	#331	C₂²×C₁₃₁	1	2	yes	—
297	#331	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
298	#331	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
299	C₂×#41	—	2	1	no	—
300	C₂×#41	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	1	no	—
304	AGL(1,ℤ/8)	—	4	4	no	—
305	AGL(1,ℤ/8)	—	4	4	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	2	no	—
308	C₂×D₈	—	2	2	no	—
309	C₂×D₈	—	2	2	no	—
310	C₂×D₈	—	2	2	no	—
311	C₂×D₈	—	2	2	no	—
312	C₂×D₈	—	2	2	no	—
313	#131	—	4	4	no	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
316	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
317	C₂²×C₈	—	2	1	no	—
318	C₂²×C₈	—	2	1	no	—
319	C₂²×C₈	—	2	2	no	—
320	C₄⋊Q₈	D₄×C₁₃₁	1	1	yes	characteristic
321	C₄⋊Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
322	#127	—	2	1	no	—
323	#127	—	2	1	no	—
324	#127	—	2	1	no	—
325	#127	D₄×C₁₃₁	1	1	yes	characteristic
326	#127	C₂³×C₁₃₁	1	2	yes	—
327	#127	C₂³×C₁₃₁	1	1	yes	characteristic
328	#124	—	2	1	no	—
329	#124	—	2	1	no	—
330	#121	—	2	1	no	—
331	#121	—	2	1	no	—
332	#121	—	2	1	no	—
333	#121	D₄×C₁₃₁	1	2	yes	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	C₂³×C₁₃₁	1	2	yes	—
338	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
339	#107	—	2	2	no	—
340	C₄⋊C₈	—	2	1	no	—
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	C₂³×C₁₃₁	1	1	yes	characteristic
343	C₄⋊C₈	C₂³×C₁₃₁	1	1	yes	characteristic
344	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
345	C₄⋊C₈	C₂³×C₁₃₁	1	1	yes	characteristic
346	C₄⋊C₈	C₂³×C₁₃₁	1	1	yes	characteristic
347	#102	—	2	2	no	—
348	#102	—	2	2	no	—
349	#102	—	2	2	no	—
350	#102	—	2	2	no	—
351	C₂²⋊C₈	—	2	1	no	—
352	C₂²⋊C₈	—	2	2	no	—
353	C₂²⋊C₈	—	2	2	no	—
354	#97	D₄×C₁₃₁	1	2	yes	—
355	C₄×C₈	—	2	1	no	—
356	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
357	C₂⁴	—	4	1	no	—
358	#41	—	4	2	no	—
359	#41	—	4	2	no	—
360	#41	—	4	2	no	—
361	#41	—	4	2	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	4	1	no	—
374	C₂×D₄	—	4	1	no	—
375	C₂×D₄	—	4	1	no	—
376	C₂×D₄	—	4	1	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	—	8	1	no	—
380	C₂×D₄	—	8	1	no	—
381	C₂×D₄	—	8	2	no	—
382	C₂×D₄	C₂×D₄×C₁₃₁	1	2	yes	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	1	no	—
385	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
387	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
388	SD₁₆	—	4	4	no	—
389	SD₁₆	—	4	4	no	—
390	D₈	—	4	4	no	—
391	D₈	—	4	4	no	—
392	D₈	—	8	2	no	—
393	D₈	—	8	2	no	—
394	D₈	—	8	2	no	—
395	M₁₆	—	2	2	no	—
396	M₁₆	—	4	4	no	—
397	M₁₆	D₈×C₁₃₁	1	4	yes	complemented
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	4	2	no	—
404	C₂×C₈	—	4	4	no	—
405	C₂×C₈	—	8	1	no	—
406	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
407	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	4	1	no	—
415	C₂²⋊C₄	—	4	1	no	—
416	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
417	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
418	C₄²	#41×C₁₃₁	1	1	yes	characteristic
419	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
420	C₂³	—	2	2	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	8	2	no	—
424	C₂³	—	8	2	no	—
425	C₂³	—	8	2	no	—
426	C₂³	#121×C₁₃₁	1	1	yes	characteristic
427	Q₈	—	4	2	no	—
428	Q₈	—	4	1	no	—
429	Q₈	—	4	1	no	—
430	D₄	—	2	2	no	—
431	D₄	—	2	2	no	—
432	D₄	—	4	4	no	—
433	D₄	—	4	2	no	—
434	D₄	—	4	1	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	1	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	1	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	2	no	—
441	D₄	—	4	1	no	—
442	D₄	—	8	1	no	—
443	D₄	—	8	2	no	—
444	D₄	—	8	1	no	—
445	D₄	—	8	2	no	—
446	D₄	—	8	2	no	—
447	D₄	—	8	2	no	—
448	D₄	—	8	2	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	8	2	no	—
457	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
458	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
459	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
461	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
465	C₈	—	2	2	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	1	no	—
468	C₈	—	4	2	no	—
469	C₈	—	4	1	no	—
470	C₈	—	4	1	no	—
471	C₈	C₂×D₈×C₁₃₁	1	4	yes	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	8	1	no	—
478	C₂²	—	8	2	no	—
479	C₂²	—	8	2	no	—
480	C₂²	—	8	1	no	—
481	C₂²	—	8	2	no	—
482	C₂²	—	8	1	no	—
483	C₂²	—	16	4	no	—
484	C₂²	#459×C₁₃₁	1	2	yes	—
485	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	4	2	no	—
490	C₄	—	8	1	no	—
491	C₄	#459×C₁₃₁	1	2	yes	—
492	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
493	C₄	C₂×AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
494	C₄	#576×C₁₃₁	1	1	yes	characteristic
495	C₄	#577×C₁₃₁	1	1	yes	characteristic
496	C₂	—	2	2	no	—
497	C₂	—	8	2	no	—
498	C₂	—	16	2	no	—
499	C₂	—	16	1	no	—
500	C₂	C₂×#459×C₁₃₁	1	1	yes	characteristic
501	C₂	#3062×C₁₃₁	1	1	yes	characteristic
502	C₂	#3068×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ 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. #3245×C₁₃₁	C₂	semidirect
2. #3205×C₁₃₁	C₂	semidirect
3. #3151×C₁₃₁	C₂	semidirect
4. #2935×C₁₃₁	C₂	semidirect
5. #1687×C₁₃₁	C₂	semidirect
6. #1685×C₁₃₁	C₂	semidirect
7. #1667×C₁₃₁	C₂	semidirect
8. #1665×C₁₃₁	C₂	semidirect
9. #1565×C₁₃₁	C₂	semidirect
10. #1540×C₁₃₁	C₂	semidirect
11. C₂×#331×C₁₃₁	C₂	semidirect
24. #474×C₁₃₁	C₂²	semidirect
29. #459×C₁₃₁	C₂²	semidirect
31. #436×C₁₃₁	C₂²	semidirect
33. C₈×D₄×C₁₃₁	C₂²	semidirect
39. #335×C₁₃₁	C₂²	semidirect
40. #334×C₁₃₁	C₂²	semidirect
41. #332×C₁₃₁	C₂²	semidirect
42. #332×C₁₃₁	C₂²	semidirect
45. #331×C₁₃₁	C₂²	semidirect
46. #331×C₁₃₁	C₂²	semidirect
145. M₁₆×C₁₃₁	D₈	semidirect
251. #19230	C₁₃₁	direct (N × H)
252. C₁₃₁	#19230	direct (N × H)
253. #3245	C₂×C₁₃₁	semidirect
254. #3205	C₂×C₁₃₁	semidirect
255. #3151	C₂×C₁₃₁	semidirect
256. #2935	C₂×C₁₃₁	semidirect
257. #1687	C₂×C₁₃₁	semidirect
258. #1685	C₂×C₁₃₁	semidirect
259. #1667	C₂×C₁₃₁	semidirect
260. #1665	C₂×C₁₃₁	semidirect
261. #1565	C₂×C₁₃₁	semidirect
262. #1540	C₂×C₁₃₁	semidirect
263. C₂×#331	C₂×C₁₃₁	semidirect
276. #474	C₂²×C₁₃₁	semidirect
281. #459	C₂²×C₁₃₁	semidirect
283. #436	C₂²×C₁₃₁	semidirect
285. C₈×D₄	C₂²×C₁₃₁	semidirect
291. #335	C₂²×C₁₃₁	semidirect
292. #334	C₂²×C₁₃₁	semidirect
293. #332	C₂²×C₁₃₁	semidirect
294. #332	C₂²×C₁₃₁	semidirect
297. #331	C₂²×C₁₃₁	semidirect
298. #331	C₂²×C₁₃₁	semidirect
397. M₁₆	D₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3245×C₁₃₁	2
#3205×C₁₃₁	2
#3151×C₁₃₁	2
#2935×C₁₃₁	2
#1687×C₁₃₁	2
#1685×C₁₃₁	2
#1667×C₁₃₁	2
#1665×C₁₃₁	2
#1565×C₁₃₁	2
#1540×C₁₃₁	2
C₂×#331×C₁₃₁	2
#19230	131

Subgroups — C131×#19230 #45705114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#19230 #45705114