Subgroups — C₁₃₁×#18669 #45704553

Order 33536.

882 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3507×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#3246×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#3170×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#3078×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1650×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1643×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1638×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1632×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#1550×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#1492×C₁₃₁	C₂	1	1	yes	maximal, characteristic
11	C₂×Q₈⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#559×C₁₃₁	—	2	1	no	—
13	#557×C₁₃₁	—	2	1	no	—
14	#556×C₁₃₁	—	2	2	no	—
15	#556×C₁₃₁	C₂²	1	2	yes	—
16	#554×C₁₃₁	C₂²	1	2	yes	—
17	#552×C₁₃₁	—	2	2	no	—
18	#539×C₁₃₁	—	2	2	no	—
19	#536×C₁₃₁	—	2	1	no	—
20	#533×C₁₃₁	C₂²	1	1	yes	characteristic
21	#533×C₁₃₁	C₂²	1	1	yes	characteristic
22	#532×C₁₃₁	C₂²	1	1	yes	characteristic
23	#529×C₁₃₁	—	2	2	no	—
24	#520×C₁₃₁	—	2	1	no	—
25	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
26	#500×C₁₃₁	C₂²	1	1	yes	characteristic
27	#494×C₁₃₁	C₂²	1	1	yes	characteristic
28	#487×C₁₃₁	C₂²	1	1	yes	characteristic
29	#487×C₁₃₁	C₂²	1	1	yes	characteristic
30	#479×C₁₃₁	—	2	2	no	—
31	#477×C₁₃₁	C₂²	1	2	yes	—
32	#476×C₁₃₁	C₂²	1	2	yes	—
33	#475×C₁₃₁	—	2	1	no	—
34	#475×C₁₃₁	C₂²	1	2	yes	—
35	#467×C₁₃₁	—	2	2	no	—
36	#465×C₁₃₁	—	2	2	no	—
37	#461×C₁₃₁	C₂²	1	2	yes	—
38	#460×C₁₃₁	—	2	1	no	—
39	C₄×Q₁₆×C₁₃₁	—	2	2	no	—
40	#428×C₁₃₁	—	2	1	no	—
41	#423×C₁₃₁	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
43	#419×C₁₃₁	—	2	1	no	—
44	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
45	#333×C₁₃₁	C₂²	1	2	yes	—
46	#332×C₁₃₁	C₂²	1	2	yes	—
47	Q₈⋊C₈×C₁₃₁	C₂²	1	2	yes	—
48	Q₈⋊C₈×C₁₃₁	C₂²	1	2	yes	—
49	D₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
50	D₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
51	C₂×#41×C₁₃₁	—	2	1	no	—
52	C₂²×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
55	C₂×M₁₆×C₁₃₁	—	2	1	no	—
56	C₂²×C₈×C₁₃₁	—	2	1	no	—
57	C₂²×C₈×C₁₃₁	—	2	1	no	—
58	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
59	C₄⋊Q₈×C₁₃₁	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
64	#127×C₁₃₁	D₄	1	1	yes	characteristic
65	#126×C₁₃₁	—	2	4	no	—
66	#126×C₁₃₁	—	2	4	no	—
67	#125×C₁₃₁	—	2	4	no	—
68	#125×C₁₃₁	—	2	2	no	—
69	#125×C₁₃₁	—	2	1	no	—
70	#125×C₁₃₁	—	2	1	no	—
71	#124×C₁₃₁	—	2	2	no	—
72	#124×C₁₃₁	—	2	2	no	—
73	#124×C₁₃₁	—	2	1	no	—
74	#123×C₁₃₁	—	2	4	no	—
75	#123×C₁₃₁	—	2	4	no	—
76	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
77	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
78	C₂²⋊Q₈×C₁₃₁	—	2	4	no	—
79	C₂²⋊Q₈×C₁₃₁	—	2	1	no	—
80	C₂²⋊Q₈×C₁₃₁	—	2	1	no	—
81	C₂²⋊Q₈×C₁₃₁	—	2	1	no	—
82	C₂²⋊Q₈×C₁₃₁	D₄	1	2	yes	—
83	#121×C₁₃₁	—	2	1	no	—
84	#121×C₁₃₁	D₄	1	2	yes	—
85	C₄×Q₈×C₁₃₁	—	2	2	no	—
86	C₄×Q₈×C₁₃₁	—	2	2	no	—
87	C₄×Q₈×C₁₃₁	—	2	1	no	—
88	C₄×Q₈×C₁₃₁	—	2	1	no	—
89	C₄×Q₈×C₁₃₁	—	2	1	no	—
90	C₄×Q₈×C₁₃₁	—	2	1	no	—
91	C₄×Q₈×C₁₃₁	—	2	1	no	—
92	C₄×Q₈×C₁₃₁	D₄	1	2	yes	—
93	C₄×Q₈×C₁₃₁	D₄	1	2	yes	—
94	C₄×Q₈×C₁₃₁	C₂³	1	2	yes	—
95	C₄×D₄×C₁₃₁	—	2	4	no	—
96	C₄×D₄×C₁₃₁	—	2	2	no	—
97	C₄×D₄×C₁₃₁	—	2	2	no	—
98	C₄×D₄×C₁₃₁	—	2	1	no	—
99	C₄×D₄×C₁₃₁	C₂³	1	2	yes	—
100	#117×C₁₃₁	—	2	4	no	—
101	#117×C₁₃₁	—	2	1	no	—
102	#117×C₁₃₁	—	2	2	no	—
103	#117×C₁₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
105	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₁₃₁	—	2	1	no	—
108	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
109	#107×C₁₃₁	—	2	2	no	—
110	#107×C₁₃₁	—	2	2	no	—
111	#106×C₁₃₁	—	2	2	no	—
112	#106×C₁₃₁	—	2	2	no	—
113	C₄⋊C₈×C₁₃₁	—	2	1	no	—
114	C₄⋊C₈×C₁₃₁	—	2	1	no	—
115	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
116	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
117	#103×C₁₃₁	—	2	2	no	—
118	#103×C₁₃₁	—	2	2	no	—
119	#103×C₁₃₁	—	2	2	no	—
120	#103×C₁₃₁	—	2	2	no	—
121	#103×C₁₃₁	—	2	2	no	—
122	#102×C₁₃₁	—	2	2	no	—
123	C₄×C₈×C₁₃₁	—	2	1	no	—
124	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
126	#41×C₁₃₁	—	4	2	no	—
127	#41×C₁₃₁	—	4	2	no	—
128	C₂×Q₈×C₁₃₁	—	2	1	no	—
129	C₂×Q₈×C₁₃₁	—	2	1	no	—
130	C₂×Q₈×C₁₃₁	—	2	1	no	—
131	C₂×Q₈×C₁₃₁	—	2	2	no	—
132	C₂×Q₈×C₁₃₁	—	2	2	no	—
133	C₂×Q₈×C₁₃₁	—	2	1	no	—
134	C₂×Q₈×C₁₃₁	—	2	1	no	—
135	C₂×Q₈×C₁₃₁	C₂×D₄	1	2	yes	—
136	C₂×Q₈×C₁₃₁	C₂×D₄	1	2	yes	—
137	C₂×D₄×C₁₃₁	—	2	2	no	—
138	C₂×D₄×C₁₃₁	—	2	1	no	—
139	C₂×D₄×C₁₃₁	—	2	1	no	—
140	C₂×D₄×C₁₃₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₁₃₁	—	2	1	no	—
142	C₂²×C₄×C₁₃₁	—	2	4	no	—
143	C₂²×C₄×C₁₃₁	—	2	1	no	—
144	C₂²×C₄×C₁₃₁	—	2	2	no	—
145	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
148	Q₁₆×C₁₃₁	—	4	4	no	—
149	SD₁₆×C₁₃₁	—	8	2	no	—
150	M₁₆×C₁₃₁	—	4	2	no	—
151	C₂×C₈×C₁₃₁	—	2	1	no	—
152	C₂×C₈×C₁₃₁	—	2	1	no	—
153	C₂×C₈×C₁₃₁	—	2	2	no	—
154	C₂×C₈×C₁₃₁	—	2	2	no	—
155	C₂×C₈×C₁₃₁	—	4	2	no	—
156	C₂×C₈×C₁₃₁	—	8	1	no	—
157	C₄⋊C₄×C₁₃₁	—	2	4	no	—
158	C₄⋊C₄×C₁₃₁	—	2	4	no	—
159	C₄⋊C₄×C₁₃₁	—	2	4	no	—
160	C₄⋊C₄×C₁₃₁	—	2	2	no	—
161	C₄⋊C₄×C₁₃₁	—	2	1	no	—
162	C₄⋊C₄×C₁₃₁	—	2	2	no	—
163	C₄⋊C₄×C₁₃₁	—	2	1	no	—
164	C₄⋊C₄×C₁₃₁	—	2	2	no	—
165	C₄⋊C₄×C₁₃₁	—	2	1	no	—
166	C₄⋊C₄×C₁₃₁	—	2	2	no	—
167	C₄⋊C₄×C₁₃₁	—	2	1	no	—
168	C₄⋊C₄×C₁₃₁	—	2	1	no	—
169	C₄⋊C₄×C₁₃₁	—	2	1	no	—
170	C₄⋊C₄×C₁₃₁	—	2	1	no	—
171	C₄⋊C₄×C₁₃₁	—	2	1	no	—
172	C₄⋊C₄×C₁₃₁	—	2	1	no	—
173	C₄⋊C₄×C₁₃₁	—	4	1	no	—
174	C₄⋊C₄×C₁₃₁	—	4	1	no	—
175	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
177	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
178	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
181	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
185	C₄²×C₁₃₁	—	2	4	no	—
186	C₄²×C₁₃₁	—	2	2	no	—
187	C₄²×C₁₃₁	—	2	2	no	—
188	C₄²×C₁₃₁	—	2	2	no	—
189	C₄²×C₁₃₁	—	4	1	no	—
190	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
194	C₂³×C₁₃₁	—	2	2	no	—
195	C₂³×C₁₃₁	#120	1	1	yes	characteristic
196	Q₈×C₁₃₁	—	2	2	no	—
197	Q₈×C₁₃₁	—	2	2	no	—
198	Q₈×C₁₃₁	—	4	2	no	—
199	Q₈×C₁₃₁	—	4	2	no	—
200	Q₈×C₁₃₁	—	4	1	no	—
201	Q₈×C₁₃₁	—	4	1	no	—
202	Q₈×C₁₃₁	—	4	2	no	—
203	Q₈×C₁₃₁	—	4	2	no	—
204	Q₈×C₁₃₁	—	4	1	no	—
205	Q₈×C₁₃₁	—	4	1	no	—
206	D₄×C₁₃₁	—	2	2	no	—
207	D₄×C₁₃₁	—	2	2	no	—
208	D₄×C₁₃₁	—	4	4	no	—
209	D₄×C₁₃₁	—	4	2	no	—
210	D₄×C₁₃₁	—	8	1	no	—
211	C₂×C₄×C₁₃₁	—	2	1	no	—
212	C₂×C₄×C₁₃₁	—	2	1	no	—
213	C₂×C₄×C₁₃₁	—	2	2	no	—
214	C₂×C₄×C₁₃₁	—	2	2	no	—
215	C₂×C₄×C₁₃₁	—	2	2	no	—
216	C₂×C₄×C₁₃₁	—	2	2	no	—
217	C₂×C₄×C₁₃₁	—	2	2	no	—
218	C₂×C₄×C₁₃₁	—	4	1	no	—
219	C₂×C₄×C₁₃₁	—	4	2	no	—
220	C₂×C₄×C₁₃₁	—	4	2	no	—
221	C₂×C₄×C₁₃₁	—	4	2	no	—
222	C₂×C₄×C₁₃₁	—	4	1	no	—
223	C₂×C₄×C₁₃₁	—	4	1	no	—
224	C₂×C₄×C₁₃₁	—	8	1	no	—
225	C₂×C₄×C₁₃₁	—	8	4	no	—
226	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
227	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
228	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
229	C₂×C₄×C₁₃₁	#135	1	2	yes	—
230	C₂×C₄×C₁₃₁	#137	1	2	yes	—
231	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
234	C₈×C₁₃₁	—	2	2	no	—
235	C₈×C₁₃₁	—	4	1	no	—
236	C₈×C₁₃₁	—	4	2	no	—
237	C₈×C₁₃₁	—	4	2	no	—
238	C₂²×C₁₃₁	—	2	1	no	—
239	C₂²×C₁₃₁	—	2	1	no	—
240	C₂²×C₁₃₁	—	4	2	no	—
241	C₂²×C₁₃₁	—	4	2	no	—
242	C₂²×C₁₃₁	—	4	2	no	—
243	C₂²×C₁₃₁	#452	1	2	yes	—
244	C₂²×C₁₃₁	C₂×#120	1	1	yes	characteristic
245	C₄×C₁₃₁	—	2	1	no	—
246	C₄×C₁₃₁	—	2	1	no	—
247	C₄×C₁₃₁	—	2	1	no	—
248	C₄×C₁₃₁	—	4	2	no	—
249	C₄×C₁₃₁	—	4	2	no	—
250	C₄×C₁₃₁	—	4	2	no	—
251	C₄×C₁₃₁	—	4	2	no	—
252	C₄×C₁₃₁	—	4	2	no	—
253	C₄×C₁₃₁	—	8	1	no	—
254	C₄×C₁₃₁	—	8	1	no	—
255	C₄×C₁₃₁	#452	1	2	yes	—
256	C₄×C₁₃₁	C₂×#135	1	1	yes	characteristic
257	C₄×C₁₃₁	C₂×#137	1	1	yes	characteristic
258	C₄×C₁₃₁	#577	1	1	yes	characteristic
259	C₄×C₁₃₁	#578	1	1	yes	characteristic
260	C₂×C₁₃₁	—	2	2	no	—
261	C₂×C₁₃₁	—	8	2	no	—
262	C₂×C₁₃₁	C₂×#452	1	1	yes	characteristic
263	C₂×C₁₃₁	#3013	1	1	yes	characteristic
264	C₂×C₁₃₁	#3018	1	1	yes	characteristic
265	#18669	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₁₃₁	#18669	1	1	yes	characteristic, direct product factor, Sylow
267	#3507	C₂×C₁₃₁	1	1	yes	characteristic
268	#3246	C₂×C₁₃₁	1	1	yes	characteristic
269	#3170	C₂×C₁₃₁	1	1	yes	characteristic, complemented
270	#3078	C₂×C₁₃₁	1	1	yes	characteristic, complemented
271	#1650	C₂×C₁₃₁	1	2	yes	complemented
272	#1643	C₂×C₁₃₁	1	2	yes	complemented
273	#1638	C₂×C₁₃₁	1	2	yes	complemented
274	#1632	C₂×C₁₃₁	1	2	yes	complemented
275	#1550	C₂×C₁₃₁	1	1	yes	characteristic, complemented
276	#1492	C₂×C₁₃₁	1	1	yes	characteristic
277	C₂×Q₈⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic, complemented
278	#559	—	2	1	no	—
279	#557	—	2	1	no	—
280	#556	—	2	2	no	—
281	#556	C₂²×C₁₃₁	1	2	yes	—
282	#554	C₂²×C₁₃₁	1	2	yes	—
283	#552	—	2	2	no	—
284	#539	—	2	2	no	—
285	#536	—	2	1	no	—
286	#533	C₂²×C₁₃₁	1	1	yes	characteristic
287	#533	C₂²×C₁₃₁	1	1	yes	characteristic
288	#532	C₂²×C₁₃₁	1	1	yes	characteristic
289	#529	—	2	2	no	—
290	#520	—	2	1	no	—
291	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
292	#500	C₂²×C₁₃₁	1	1	yes	characteristic
293	#494	C₂²×C₁₃₁	1	1	yes	characteristic
294	#487	C₂²×C₁₃₁	1	1	yes	characteristic
295	#487	C₂²×C₁₃₁	1	1	yes	characteristic
296	#479	—	2	2	no	—
297	#477	C₂²×C₁₃₁	1	2	yes	—
298	#476	C₂²×C₁₃₁	1	2	yes	—
299	#475	—	2	1	no	—
300	#475	C₂²×C₁₃₁	1	2	yes	—
301	#467	—	2	2	no	—
302	#465	—	2	2	no	—
303	#461	C₂²×C₁₃₁	1	2	yes	—
304	#460	—	2	1	no	—
305	C₄×Q₁₆	—	2	2	no	—
306	#428	—	2	1	no	—
307	#423	C₂²×C₁₃₁	1	1	yes	characteristic
308	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
309	#419	—	2	1	no	—
310	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
311	#333	C₂²×C₁₃₁	1	2	yes	—
312	#332	C₂²×C₁₃₁	1	2	yes	—
313	Q₈⋊C₈	C₂²×C₁₃₁	1	2	yes	—
314	Q₈⋊C₈	C₂²×C₁₃₁	1	2	yes	—
315	D₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
316	D₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
317	C₂×#41	—	2	1	no	—
318	C₂²×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
319	C₂×Q₁₆	—	2	2	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂²×C₈	—	2	1	no	—
323	C₂²×C₈	—	2	1	no	—
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	D₄×C₁₃₁	1	1	yes	characteristic
326	C₄⋊Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
327	C₄⋊Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
328	C₄⋊Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
329	C₄⋊Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
330	#127	D₄×C₁₃₁	1	1	yes	characteristic
331	#126	—	2	4	no	—
332	#126	—	2	4	no	—
333	#125	—	2	4	no	—
334	#125	—	2	2	no	—
335	#125	—	2	1	no	—
336	#125	—	2	1	no	—
337	#124	—	2	2	no	—
338	#124	—	2	2	no	—
339	#124	—	2	1	no	—
340	#123	—	2	4	no	—
341	#123	—	2	4	no	—
342	C₂²⋊Q₈	—	2	2	no	—
343	C₂²⋊Q₈	—	2	2	no	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	1	no	—
346	C₂²⋊Q₈	—	2	1	no	—
347	C₂²⋊Q₈	—	2	1	no	—
348	C₂²⋊Q₈	D₄×C₁₃₁	1	2	yes	—
349	#121	—	2	1	no	—
350	#121	D₄×C₁₃₁	1	2	yes	—
351	C₄×Q₈	—	2	2	no	—
352	C₄×Q₈	—	2	2	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	—	2	1	no	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	D₄×C₁₃₁	1	2	yes	—
359	C₄×Q₈	D₄×C₁₃₁	1	2	yes	—
360	C₄×Q₈	C₂³×C₁₃₁	1	2	yes	—
361	C₄×D₄	—	2	4	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	C₂³×C₁₃₁	1	2	yes	—
366	#117	—	2	4	no	—
367	#117	—	2	1	no	—
368	#117	—	2	2	no	—
369	#117	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	2	no	—
372	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
373	C₂×C₄²	—	2	1	no	—
374	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
375	#107	—	2	2	no	—
376	#107	—	2	2	no	—
377	#106	—	2	2	no	—
378	#106	—	2	2	no	—
379	C₄⋊C₈	—	2	1	no	—
380	C₄⋊C₈	—	2	1	no	—
381	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
382	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
383	#103	—	2	2	no	—
384	#103	—	2	2	no	—
385	#103	—	2	2	no	—
386	#103	—	2	2	no	—
387	#103	—	2	2	no	—
388	#102	—	2	2	no	—
389	C₄×C₈	—	2	1	no	—
390	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
391	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
392	#41	—	4	2	no	—
393	#41	—	4	2	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	C₂×D₄×C₁₃₁	1	2	yes	—
402	C₂×Q₈	C₂×D₄×C₁₃₁	1	2	yes	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	C₂×D₄×C₁₃₁	1	2	yes	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
412	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
414	Q₁₆	—	4	4	no	—
415	SD₁₆	—	8	2	no	—
416	M₁₆	—	4	2	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	4	2	no	—
422	C₂×C₈	—	8	1	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	4	1	no	—
440	C₄⋊C₄	—	4	1	no	—
441	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
442	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
443	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	4	1	no	—
451	C₄²	—	2	4	no	—
452	C₄²	—	2	2	no	—
453	C₄²	—	2	2	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	4	1	no	—
456	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
457	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
458	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
459	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
460	C₂³	—	2	2	no	—
461	C₂³	#120×C₁₃₁	1	1	yes	characteristic
462	Q₈	—	2	2	no	—
463	Q₈	—	2	2	no	—
464	Q₈	—	4	2	no	—
465	Q₈	—	4	2	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	4	1	no	—
469	Q₈	—	4	1	no	—
470	Q₈	—	4	1	no	—
471	Q₈	—	4	1	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	2	no	—
474	D₄	—	4	4	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	8	1	no	—
491	C₂×C₄	—	8	4	no	—
492	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
493	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
494	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
495	C₂×C₄	#135×C₁₃₁	1	2	yes	—
496	C₂×C₄	#137×C₁₃₁	1	2	yes	—
497	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
500	C₈	—	2	2	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	2	no	—
503	C₈	—	4	2	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	2	1	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	#452×C₁₃₁	1	2	yes	—
510	C₂²	C₂×#120×C₁₃₁	1	1	yes	characteristic
511	C₄	—	2	1	no	—
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	4	2	no	—
515	C₄	—	4	2	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	8	1	no	—
520	C₄	—	8	1	no	—
521	C₄	#452×C₁₃₁	1	2	yes	—
522	C₄	C₂×#135×C₁₃₁	1	1	yes	characteristic
523	C₄	C₂×#137×C₁₃₁	1	1	yes	characteristic
524	C₄	#577×C₁₃₁	1	1	yes	characteristic
525	C₄	#578×C₁₃₁	1	1	yes	characteristic
526	C₂	—	2	2	no	—
527	C₂	—	8	2	no	—
528	C₂	C₂×#452×C₁₃₁	1	1	yes	characteristic
529	C₂	#3013×C₁₃₁	1	1	yes	characteristic
530	C₂	#3018×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
prime radical	#532×C₁₃₁	C₂²
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
3. #3170×C₁₃₁	C₂	semidirect
4. #3078×C₁₃₁	C₂	semidirect
5. #1650×C₁₃₁	C₂	semidirect
6. #1643×C₁₃₁	C₂	semidirect
7. #1638×C₁₃₁	C₂	semidirect
8. #1632×C₁₃₁	C₂	semidirect
9. #1550×C₁₃₁	C₂	semidirect
11. C₂×Q₈⋊C₈×C₁₃₁	C₂	semidirect
265. #18669	C₁₃₁	direct (N × H)
266. C₁₃₁	#18669	direct (N × H)
269. #3170	C₂×C₁₃₁	semidirect
270. #3078	C₂×C₁₃₁	semidirect
271. #1650	C₂×C₁₃₁	semidirect
272. #1643	C₂×C₁₃₁	semidirect
273. #1638	C₂×C₁₃₁	semidirect
274. #1632	C₂×C₁₃₁	semidirect
275. #1550	C₂×C₁₃₁	semidirect
277. C₂×Q₈⋊C₈	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3507×C₁₃₁	2
#3246×C₁₃₁	2
#3170×C₁₃₁	2
#3078×C₁₃₁	2
#1650×C₁₃₁	2
#1643×C₁₃₁	2
#1638×C₁₃₁	2
#1632×C₁₃₁	2
#1550×C₁₃₁	2
#1492×C₁₃₁	2
C₂×Q₈⋊C₈×C₁₃₁	2
#18669	131

Subgroups — C131×#18669 #45704553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#18669 #45704553