Subgroups — C₁₃₁×#21131 #45707015

Order 33536.

862 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2319×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2276×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1926×C₁₃₁	C₂	1	2	yes	maximal
5	#1920×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1880×C₁₃₁	C₂	1	2	yes	maximal
7	#1865×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#1763×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#1762×C₁₃₁	C₂	1	1	yes	maximal, characteristic
11	#1746×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄×SD₁₆×C₁₃₁	C₄	1	8	yes	complemented
17	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
18	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
20	C₂×#103×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
22	C₂×#102×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
25	#390×C₁₃₁	C₂²	1	2	yes	—
26	#389×C₁₃₁	C₂²	1	2	yes	—
27	#388×C₁₃₁	—	2	2	no	—
28	#381×C₁₃₁	—	2	1	no	—
29	#380×C₁₃₁	—	2	1	no	—
30	C₄×C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
32	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
35	#376×C₁₃₁	—	2	1	no	—
36	#376×C₁₃₁	C₂²	1	1	yes	characteristic
37	#340×C₁₃₁	C₂²	1	2	yes	—
38	#340×C₁₃₁	C₂²	1	2	yes	—
39	#340×C₁₃₁	C₂²	1	2	yes	—
40	#336×C₁₃₁	C₂²	1	2	yes	—
41	#336×C₁₃₁	C₂²	1	2	yes	—
42	C₂²×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₁₃₁	—	2	1	no	—
45	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
46	C₂×SD₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₁₃₁	—	2	4	no	—
51	C₄×Q₈×C₁₃₁	—	2	4	no	—
52	C₄×Q₈×C₁₃₁	—	2	2	no	—
53	C₄×Q₈×C₁₃₁	C₂×C₄	1	4	yes	—
54	C₄×D₄×C₁₃₁	—	2	2	no	—
55	C₄×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
56	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
66	C₂×C₄²×C₁₃₁	—	2	2	no	—
67	C₂×C₄²×C₁₃₁	—	2	1	no	—
68	C₂×C₄²×C₁₃₁	—	2	1	no	—
69	C₂×C₄²×C₁₃₁	—	2	2	no	—
70	C₂×C₄²×C₁₃₁	—	2	1	no	—
71	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
72	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
73	#106×C₁₃₁	—	2	4	no	—
74	#106×C₁₃₁	C₂×C₄	1	4	yes	—
75	#103×C₁₃₁	—	2	4	no	—
76	#103×C₁₃₁	C₂×C₄	1	4	yes	—
77	#102×C₁₃₁	—	2	4	no	—
78	#102×C₁₃₁	C₂×C₄	1	4	yes	—
79	#97×C₁₃₁	—	2	2	no	—
80	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
81	#95×C₁₃₁	—	2	2	no	—
82	#95×C₁₃₁	—	2	2	no	—
83	#95×C₁₃₁	—	2	1	no	—
84	#95×C₁₃₁	—	2	1	no	—
85	#95×C₁₃₁	D₄	1	2	yes	—
86	#95×C₁₃₁	D₄	1	1	yes	characteristic
87	#95×C₁₃₁	D₄	1	1	yes	characteristic
88	C₂⁴×C₁₃₁	—	2	1	no	—
89	C₂×Q₈×C₁₃₁	—	2	1	no	—
90	C₂×Q₈×C₁₃₁	—	2	2	no	—
91	C₂×Q₈×C₁₃₁	C₂²×C₄	1	2	yes	—
92	C₂×Q₈×C₁₃₁	#41	1	4	yes	—
93	C₂×D₄×C₁₃₁	—	2	2	no	—
94	C₂×D₄×C₁₃₁	—	2	2	no	—
95	C₂×D₄×C₁₃₁	—	2	1	no	—
96	C₂×D₄×C₁₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₁₃₁	—	2	1	no	—
98	C₂²×C₄×C₁₃₁	—	2	1	no	—
99	C₂²×C₄×C₁₃₁	—	2	2	no	—
100	C₂²×C₄×C₁₃₁	—	2	1	no	—
101	C₂²×C₄×C₁₃₁	—	2	4	no	—
102	C₂²×C₄×C₁₃₁	—	2	2	no	—
103	C₂²×C₄×C₁₃₁	—	8	2	no	—
104	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
109	SD₁₆×C₁₃₁	—	2	8	no	—
110	C₂×C₈×C₁₃₁	—	2	2	no	—
111	C₂×C₈×C₁₃₁	—	2	2	no	—
112	C₂×C₈×C₁₃₁	—	2	2	no	—
113	C₂×C₈×C₁₃₁	—	2	2	no	—
114	C₂×C₈×C₁₃₁	—	2	2	no	—
115	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
117	C₄⋊C₄×C₁₃₁	—	2	8	no	—
118	C₄⋊C₄×C₁₃₁	—	2	4	no	—
119	C₄⋊C₄×C₁₃₁	—	2	1	no	—
120	C₄⋊C₄×C₁₃₁	—	2	1	no	—
121	C₄⋊C₄×C₁₃₁	—	2	2	no	—
122	C₄⋊C₄×C₁₃₁	—	2	2	no	—
123	C₄⋊C₄×C₁₃₁	—	2	2	no	—
124	C₄⋊C₄×C₁₃₁	—	8	2	no	—
125	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
126	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₁₃₁	#41	1	4	yes	—
128	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
129	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
130	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
131	C₄²×C₁₃₁	—	2	8	no	—
132	C₄²×C₁₃₁	—	2	4	no	—
133	C₄²×C₁₃₁	—	2	2	no	—
134	C₄²×C₁₃₁	—	2	2	no	—
135	C₄²×C₁₃₁	—	8	1	no	—
136	C₄²×C₁₃₁	—	8	1	no	—
137	C₄²×C₁₃₁	—	8	2	no	—
138	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
139	C₄²×C₁₃₁	#41	1	2	yes	—
140	C₂³×C₁₃₁	—	2	2	no	—
141	C₂³×C₁₃₁	—	4	2	no	—
142	C₂³×C₁₃₁	—	4	1	no	—
143	C₂³×C₁₃₁	—	4	2	no	—
144	C₂³×C₁₃₁	—	4	1	no	—
145	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₁₃₁	—	2	2	no	—
147	Q₈×C₁₃₁	—	2	4	no	—
148	Q₈×C₁₃₁	#117	1	4	yes	—
149	D₄×C₁₃₁	—	2	2	no	—
150	D₄×C₁₃₁	—	2	2	no	—
151	D₄×C₁₃₁	—	4	2	no	—
152	C₂×C₄×C₁₃₁	—	2	2	no	—
153	C₂×C₄×C₁₃₁	—	2	4	no	—
154	C₂×C₄×C₁₃₁	—	2	1	no	—
155	C₂×C₄×C₁₃₁	—	2	1	no	—
156	C₂×C₄×C₁₃₁	—	2	2	no	—
157	C₂×C₄×C₁₃₁	—	2	2	no	—
158	C₂×C₄×C₁₃₁	—	2	2	no	—
159	C₂×C₄×C₁₃₁	—	2	2	no	—
160	C₂×C₄×C₁₃₁	—	2	2	no	—
161	C₂×C₄×C₁₃₁	—	2	4	no	—
162	C₂×C₄×C₁₃₁	—	2	2	no	—
163	C₂×C₄×C₁₃₁	—	2	2	no	—
164	C₂×C₄×C₁₃₁	—	2	2	no	—
165	C₂×C₄×C₁₃₁	—	2	2	no	—
166	C₂×C₄×C₁₃₁	—	4	2	no	—
167	C₂×C₄×C₁₃₁	—	4	1	no	—
168	C₂×C₄×C₁₃₁	—	4	2	no	—
169	C₂×C₄×C₁₃₁	—	4	2	no	—
170	C₂×C₄×C₁₃₁	—	4	1	no	—
171	C₂×C₄×C₁₃₁	—	4	2	no	—
172	C₂×C₄×C₁₃₁	—	4	2	no	—
173	C₂×C₄×C₁₃₁	—	4	2	no	—
174	C₂×C₄×C₁₃₁	—	8	4	no	—
175	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
179	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
180	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
181	C₈×C₁₃₁	—	2	2	no	—
182	C₈×C₁₃₁	—	2	2	no	—
183	C₈×C₁₃₁	—	4	2	no	—
184	C₂²×C₁₃₁	—	4	2	no	—
185	C₂²×C₁₃₁	—	4	2	no	—
186	C₂²×C₁₃₁	—	4	2	no	—
187	C₂²×C₁₃₁	—	8	2	no	—
188	C₂²×C₁₃₁	—	8	2	no	—
189	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
190	C₂²×C₁₃₁	#525	1	2	yes	—
191	C₂²×C₁₃₁	#577	1	1	yes	characteristic
192	C₂²×C₁₃₁	#577	1	2	yes	—
193	C₂²×C₁₃₁	#577	1	1	yes	characteristic
194	C₄×C₁₃₁	—	2	2	no	—
195	C₄×C₁₃₁	—	2	2	no	—
196	C₄×C₁₃₁	—	2	4	no	—
197	C₄×C₁₃₁	—	4	2	no	—
198	C₄×C₁₃₁	—	4	4	no	—
199	C₄×C₁₃₁	—	4	2	no	—
200	C₄×C₁₃₁	—	4	2	no	—
201	C₄×C₁₃₁	—	4	2	no	—
202	C₄×C₁₃₁	—	8	2	no	—
203	C₄×C₁₃₁	C₂×#117	1	1	yes	characteristic
204	C₄×C₁₃₁	C₄×#41	1	1	yes	characteristic
205	C₄×C₁₃₁	#529	1	2	yes	—
206	C₂×C₁₃₁	—	8	2	no	—
207	C₂×C₁₃₁	#2290	1	1	yes	characteristic
208	C₂×C₁₃₁	#2949	1	1	yes	characteristic
209	C₂×C₁₃₁	#2952	1	1	yes	characteristic
210	C₂×C₁₃₁	#3125	1	2	yes	—
211	C₂×C₁₃₁	#3126	1	2	yes	—
212	#21131	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₁₃₁	#21131	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×C₄×SD₁₆	C₂×C₁₃₁	1	1	yes	characteristic
215	#2319	C₂×C₁₃₁	1	1	yes	characteristic
216	#2276	C₂×C₁₃₁	1	1	yes	characteristic, complemented
217	#1926	C₂×C₁₃₁	1	2	yes	—
218	#1920	C₂×C₁₃₁	1	2	yes	complemented
219	#1880	C₂×C₁₃₁	1	2	yes	—
220	#1865	C₂×C₁₃₁	1	2	yes	complemented
221	C₄×#106	C₂×C₁₃₁	1	1	yes	characteristic, complemented
222	#1763	C₂×C₁₃₁	1	1	yes	characteristic, complemented
223	#1762	C₂×C₁₃₁	1	1	yes	characteristic
224	#1746	C₂×C₁₃₁	1	1	yes	characteristic, complemented
225	C₂²×SD₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
226	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
227	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
229	C₄×SD₁₆	C₄×C₁₃₁	1	8	yes	complemented
230	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
231	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
232	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
233	C₂×#103	C₂²×C₁₃₁	1	1	yes	characteristic
234	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
235	C₂×#102	C₂²×C₁₃₁	1	1	yes	characteristic
236	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
237	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
238	#390	C₂²×C₁₃₁	1	2	yes	—
239	#389	C₂²×C₁₃₁	1	2	yes	—
240	#388	—	2	2	no	—
241	#381	—	2	1	no	—
242	#380	—	2	1	no	—
243	C₄×C₄⋊C₄	—	2	1	no	—
244	C₄×C₄⋊C₄	—	2	2	no	—
245	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
246	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
247	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
248	#376	—	2	1	no	—
249	#376	C₂²×C₁₃₁	1	1	yes	characteristic
250	#340	C₂²×C₁₃₁	1	2	yes	—
251	#340	C₂²×C₁₃₁	1	2	yes	—
252	#340	C₂²×C₁₃₁	1	2	yes	—
253	#336	C₂²×C₁₃₁	1	2	yes	—
254	#336	C₂²×C₁₃₁	1	2	yes	—
255	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
256	C₂²×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
257	C₂³×C₄	—	2	1	no	—
258	C₂×SD₁₆	—	2	4	no	—
259	C₂×SD₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
260	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
261	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
263	C₄×Q₈	—	2	4	no	—
264	C₄×Q₈	—	2	4	no	—
265	C₄×Q₈	—	2	2	no	—
266	C₄×Q₈	C₂×C₄×C₁₃₁	1	4	yes	—
267	C₄×D₄	—	2	2	no	—
268	C₄×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
276	C₂×C₂²⋊C₄	—	2	1	no	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₂²⋊C₄	—	2	2	no	—
279	C₂×C₄²	—	2	2	no	—
280	C₂×C₄²	—	2	1	no	—
281	C₂×C₄²	—	2	2	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	—	2	1	no	—
284	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
285	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
286	#106	—	2	4	no	—
287	#106	C₂×C₄×C₁₃₁	1	4	yes	—
288	#103	—	2	4	no	—
289	#103	C₂×C₄×C₁₃₁	1	4	yes	—
290	#102	—	2	4	no	—
291	#102	C₂×C₄×C₁₃₁	1	4	yes	—
292	#97	—	2	2	no	—
293	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
294	#95	—	2	2	no	—
295	#95	—	2	1	no	—
296	#95	—	2	1	no	—
297	#95	—	2	2	no	—
298	#95	D₄×C₁₃₁	1	1	yes	characteristic
299	#95	D₄×C₁₃₁	1	1	yes	characteristic
300	#95	D₄×C₁₃₁	1	2	yes	—
301	C₂⁴	—	2	1	no	—
302	C₂×Q₈	—	2	1	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	C₂²×C₄×C₁₃₁	1	2	yes	—
305	C₂×Q₈	#41×C₁₃₁	1	4	yes	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	C₂²×C₄×C₁₃₁	1	2	yes	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	8	2	no	—
317	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
318	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
322	SD₁₆	—	2	8	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
329	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	8	2	no	—
338	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
339	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
340	C₄⋊C₄	#41×C₁₃₁	1	4	yes	—
341	C₂²⋊C₄	—	2	4	no	—
342	C₂²⋊C₄	—	8	2	no	—
343	C₂²⋊C₄	—	8	2	no	—
344	C₄²	—	2	8	no	—
345	C₄²	—	2	4	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	8	1	no	—
349	C₄²	—	8	1	no	—
350	C₄²	—	8	2	no	—
351	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
352	C₄²	#41×C₁₃₁	1	2	yes	—
353	C₂³	—	2	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	1	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	1	no	—
358	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
359	Q₈	—	2	2	no	—
360	Q₈	—	2	4	no	—
361	Q₈	#117×C₁₃₁	1	4	yes	—
362	D₄	—	2	2	no	—
363	D₄	—	2	2	no	—
364	D₄	—	4	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	8	4	no	—
388	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
389	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
392	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
393	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
394	C₈	—	2	2	no	—
395	C₈	—	2	2	no	—
396	C₈	—	4	2	no	—
397	C₂²	—	4	2	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	—	8	2	no	—
402	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
403	C₂²	#525×C₁₃₁	1	2	yes	—
404	C₂²	#577×C₁₃₁	1	1	yes	characteristic
405	C₂²	#577×C₁₃₁	1	2	yes	—
406	C₂²	#577×C₁₃₁	1	1	yes	characteristic
407	C₄	—	2	4	no	—
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	2	no	—
411	C₄	—	4	4	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	8	2	no	—
416	C₄	C₂×#117×C₁₃₁	1	1	yes	characteristic
417	C₄	C₄×#41×C₁₃₁	1	1	yes	characteristic
418	C₄	#529×C₁₃₁	1	2	yes	—
419	C₂	—	8	2	no	—
420	C₂	#2290×C₁₃₁	1	1	yes	characteristic
421	C₂	#2949×C₁₃₁	1	1	yes	characteristic
422	C₂	#2952×C₁₃₁	1	1	yes	characteristic
423	C₂	#3125×C₁₃₁	1	2	yes	—
424	C₂	#3126×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2276×C₁₃₁	C₂	semidirect
5. #1920×C₁₃₁	C₂	semidirect
7. #1865×C₁₃₁	C₂	semidirect
8. C₄×#106×C₁₃₁	C₂	semidirect
9. #1763×C₁₃₁	C₂	semidirect
11. #1746×C₁₃₁	C₂	semidirect
16. C₄×SD₁₆×C₁₃₁	C₄	semidirect
212. #21131	C₁₃₁	direct (N × H)
213. C₁₃₁	#21131	direct (N × H)
216. #2276	C₂×C₁₃₁	semidirect
218. #1920	C₂×C₁₃₁	semidirect
220. #1865	C₂×C₁₃₁	semidirect
221. C₄×#106	C₂×C₁₃₁	semidirect
222. #1763	C₂×C₁₃₁	semidirect
224. #1746	C₂×C₁₃₁	semidirect
229. C₄×SD₁₆	C₄×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×SD₁₆×C₁₃₁	2
#2319×C₁₃₁	2
#2276×C₁₃₁	2
#1926×C₁₃₁	2
#1920×C₁₃₁	2
#1880×C₁₃₁	2
#1865×C₁₃₁	2
C₄×#106×C₁₃₁	2
#1763×C₁₃₁	2
#1762×C₁₃₁	2
#1746×C₁₃₁	2
#21131	131

Subgroups — C131×#21131 #45707015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#21131 #45707015