Subgroups — C₁₃₁×#23146 #45709030

Order 33536.

820 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3050×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2933×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1937×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1922×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1876×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1871×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1764×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#341×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂×#131×C₁₃₁	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₁₃₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	#525×C₁₃₁	C₂²	1	2	yes	—
15	#467×C₁₃₁	C₄	1	1	yes	characteristic, complemented
16	#467×C₁₃₁	C₄	1	1	yes	characteristic, complemented
17	#466×C₁₃₁	C₄	1	1	yes	characteristic, complemented
18	#466×C₁₃₁	C₄	1	1	yes	characteristic, complemented
19	#465×C₁₃₁	C₄	1	2	yes	complemented
20	#465×C₁₃₁	C₄	1	2	yes	complemented
21	#436×C₁₃₁	—	2	2	no	—
22	#435×C₁₃₁	—	2	2	no	—
23	#435×C₁₃₁	—	2	2	no	—
24	C₈×D₄×C₁₃₁	—	2	2	no	—
25	#427×C₁₃₁	C₂²	1	1	yes	characteristic
26	#424×C₁₃₁	C₂²	1	2	yes	—
27	C₂×#104×C₁₃₁	C₂²	1	2	yes	—
28	C₂×#104×C₁₃₁	C₂²	1	2	yes	—
29	#416×C₁₃₁	C₂²	1	2	yes	—
30	#408×C₁₃₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
32	#405×C₁₃₁	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
34	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
35	#341×C₁₃₁	C₂²	1	1	yes	characteristic
36	#341×C₁₃₁	C₂²	1	2	yes	—
37	#341×C₁₃₁	C₂²	1	1	yes	characteristic
38	#339×C₁₃₁	C₂²	1	2	yes	—
39	#339×C₁₃₁	C₂²	1	2	yes	—
40	#336×C₁₃₁	C₂²	1	2	yes	—
41	C₂×#41×C₁₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
43	#135×C₁₃₁	—	4	2	no	—
44	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×SD₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
46	C₂×D₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
47	#131×C₁₃₁	—	2	4	no	—
48	#131×C₁₃₁	—	4	2	no	—
49	C₂×M₁₆×C₁₃₁	—	2	2	no	—
50	C₂×M₁₆×C₁₃₁	—	2	2	no	—
51	C₂×M₁₆×C₁₃₁	—	2	4	no	—
52	C₂×M₁₆×C₁₃₁	—	4	2	no	—
53	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
54	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₁₃₁	—	2	2	no	—
56	C₂²×C₈×C₁₃₁	—	2	2	no	—
57	C₂²×C₈×C₁₃₁	—	4	2	no	—
58	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
65	#123×C₁₃₁	—	2	2	no	—
66	C₂²⋊Q₈×C₁₃₁	C₂×C₄	1	2	yes	—
67	C₂²⋊Q₈×C₁₃₁	C₂×C₄	1	2	yes	—
68	#121×C₁₃₁	C₂×C₄	1	2	yes	—
69	#121×C₁₃₁	C₂×C₄	1	2	yes	—
70	#120×C₁₃₁	—	2	2	no	—
71	C₄×D₄×C₁₃₁	—	2	2	no	—
72	C₄×D₄×C₁₃₁	—	2	2	no	—
73	#117×C₁₃₁	C₂³	1	2	yes	—
74	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
75	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
76	#107×C₁₃₁	C₂×C₄	1	1	yes	characteristic
77	#107×C₁₃₁	C₂×C₄	1	1	yes	characteristic
78	#106×C₁₃₁	C₂×C₄	1	2	yes	—
79	C₄⋊C₈×C₁₃₁	—	2	2	no	—
80	C₄⋊C₈×C₁₃₁	—	2	2	no	—
81	#104×C₁₃₁	—	2	4	no	—
82	#104×C₁₃₁	—	4	2	no	—
83	#103×C₁₃₁	C₂×C₄	1	2	yes	—
84	#103×C₁₃₁	C₂×C₄	1	2	yes	—
85	#102×C₁₃₁	C₂×C₄	1	2	yes	—
86	#102×C₁₃₁	C₂×C₄	1	2	yes	—
87	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
88	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
89	C₂²⋊C₈×C₁₃₁	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₁₃₁	D₄	1	1	yes	characteristic
91	C₂²⋊C₈×C₁₃₁	D₄	1	2	yes	—
92	#97×C₁₃₁	—	2	2	no	—
93	C₄×C₈×C₁₃₁	—	2	2	no	—
94	C₄×C₈×C₁₃₁	—	4	2	no	—
95	#95×C₁₃₁	—	2	1	no	—
96	C₂⁴×C₁₃₁	#41	1	1	yes	characteristic
97	#41×C₁₃₁	—	2	2	no	—
98	#41×C₁₃₁	—	2	2	no	—
99	#41×C₁₃₁	—	2	2	no	—
100	#41×C₁₃₁	—	2	2	no	—
101	C₂×Q₈×C₁₃₁	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₁₃₁	—	2	2	no	—
103	C₂×D₄×C₁₃₁	—	2	2	no	—
104	C₂×D₄×C₁₃₁	—	2	2	no	—
105	C₂×D₄×C₁₃₁	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₁₃₁	—	2	2	no	—
107	C₂²×C₄×C₁₃₁	—	2	2	no	—
108	C₂²×C₄×C₁₃₁	—	2	2	no	—
109	C₂²×C₄×C₁₃₁	—	2	2	no	—
110	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
115	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
116	Q₁₆×C₁₃₁	—	4	1	no	—
117	SD₁₆×C₁₃₁	—	4	2	no	—
118	D₈×C₁₃₁	—	4	1	no	—
119	M₁₆×C₁₃₁	—	2	2	no	—
120	M₁₆×C₁₃₁	—	2	2	no	—
121	M₁₆×C₁₃₁	—	2	2	no	—
122	M₁₆×C₁₃₁	—	4	4	no	—
123	M₁₆×C₁₃₁	—	4	4	no	—
124	M₁₆×C₁₃₁	#41	1	4	yes	—
125	C₂×C₈×C₁₃₁	—	2	2	no	—
126	C₂×C₈×C₁₃₁	—	2	2	no	—
127	C₂×C₈×C₁₃₁	—	2	2	no	—
128	C₂×C₈×C₁₃₁	—	2	2	no	—
129	C₂×C₈×C₁₃₁	—	2	2	no	—
130	C₂×C₈×C₁₃₁	—	2	2	no	—
131	C₂×C₈×C₁₃₁	—	4	1	no	—
132	C₂×C₈×C₁₃₁	—	4	1	no	—
133	C₂×C₈×C₁₃₁	—	4	1	no	—
134	C₂×C₈×C₁₃₁	—	4	1	no	—
135	C₂×C₈×C₁₃₁	—	4	4	no	—
136	C₂×C₈×C₁₃₁	—	4	4	no	—
137	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₁₃₁	—	2	2	no	—
144	C₄⋊C₄×C₁₃₁	—	2	2	no	—
145	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
146	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
147	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
148	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
149	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
150	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
151	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
152	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
153	C₄²×C₁₃₁	—	2	2	no	—
154	C₄²×C₁₃₁	—	2	2	no	—
155	C₄²×C₁₃₁	—	2	2	no	—
156	C₂³×C₁₃₁	—	2	2	no	—
157	C₂³×C₁₃₁	—	2	2	no	—
158	C₂³×C₁₃₁	—	2	2	no	—
159	C₂³×C₁₃₁	—	4	1	no	—
160	C₂³×C₁₃₁	C₄×D₄	1	1	yes	characteristic
161	C₂³×C₁₃₁	C₄×D₄	1	1	yes	characteristic
162	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
163	Q₈×C₁₃₁	—	2	2	no	—
164	Q₈×C₁₃₁	—	2	2	no	—
165	D₄×C₁₃₁	—	2	2	no	—
166	D₄×C₁₃₁	—	2	2	no	—
167	D₄×C₁₃₁	—	4	2	no	—
168	D₄×C₁₃₁	—	4	2	no	—
169	D₄×C₁₃₁	—	4	4	no	—
170	D₄×C₁₃₁	—	8	2	no	—
171	C₂×C₄×C₁₃₁	—	2	2	no	—
172	C₂×C₄×C₁₃₁	—	2	1	no	—
173	C₂×C₄×C₁₃₁	—	2	1	no	—
174	C₂×C₄×C₁₃₁	—	2	1	no	—
175	C₂×C₄×C₁₃₁	—	2	1	no	—
176	C₂×C₄×C₁₃₁	—	2	2	no	—
177	C₂×C₄×C₁₃₁	—	2	2	no	—
178	C₂×C₄×C₁₃₁	—	2	4	no	—
179	C₂×C₄×C₁₃₁	—	2	2	no	—
180	C₂×C₄×C₁₃₁	—	2	2	no	—
181	C₂×C₄×C₁₃₁	—	2	2	no	—
182	C₂×C₄×C₁₃₁	—	4	2	no	—
183	C₂×C₄×C₁₃₁	—	4	2	no	—
184	C₂×C₄×C₁₃₁	—	4	2	no	—
185	C₂×C₄×C₁₃₁	—	4	2	no	—
186	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
191	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
192	C₈×C₁₃₁	—	2	1	no	—
193	C₈×C₁₃₁	—	2	1	no	—
194	C₈×C₁₃₁	—	2	1	no	—
195	C₈×C₁₃₁	—	2	1	no	—
196	C₈×C₁₃₁	—	2	2	no	—
197	C₈×C₁₃₁	—	2	2	no	—
198	C₈×C₁₃₁	—	2	4	no	—
199	C₈×C₁₃₁	—	4	2	no	—
200	C₂²×C₁₃₁	—	2	1	no	—
201	C₂²×C₁₃₁	—	2	1	no	—
202	C₂²×C₁₃₁	—	2	1	no	—
203	C₂²×C₁₃₁	—	2	1	no	—
204	C₂²×C₁₃₁	—	2	1	no	—
205	C₂²×C₁₃₁	—	2	1	no	—
206	C₂²×C₁₃₁	—	2	1	no	—
207	C₂²×C₁₃₁	—	4	4	no	—
208	C₂²×C₁₃₁	—	4	2	no	—
209	C₂²×C₁₃₁	—	4	2	no	—
210	C₂²×C₁₃₁	—	4	2	no	—
211	C₂²×C₁₃₁	#443	1	2	yes	—
212	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂²×C₁₃₁	#553	1	2	yes	—
214	C₄×C₁₃₁	—	2	1	no	—
215	C₄×C₁₃₁	—	2	1	no	—
216	C₄×C₁₃₁	—	4	4	no	—
217	C₄×C₁₃₁	—	4	2	no	—
218	C₄×C₁₃₁	—	4	2	no	—
219	C₄×C₁₃₁	—	4	2	no	—
220	C₄×C₁₃₁	C₄×#41	1	1	yes	characteristic
221	C₄×C₁₃₁	#518	1	1	yes	characteristic
222	C₄×C₁₃₁	#546	1	2	yes	—
223	C₂×C₁₃₁	—	2	2	no	—
224	C₂×C₁₃₁	—	2	2	no	—
225	C₂×C₁₃₁	—	4	1	no	—
226	C₂×C₁₃₁	—	8	2	no	—
227	C₂×C₁₃₁	#2364	1	1	yes	characteristic
228	C₂×C₁₃₁	C₂×#443	1	1	yes	characteristic
229	C₂×C₁₃₁	#2957	1	1	yes	characteristic
230	#23146	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₁₃₁	#23146	1	1	yes	characteristic, direct product factor, Sylow
232	#3050	C₂×C₁₃₁	1	1	yes	characteristic
233	#2933	C₂×C₁₃₁	1	2	yes	complemented
234	#1937	C₂×C₁₃₁	1	2	yes	complemented
235	#1922	C₂×C₁₃₁	1	2	yes	complemented
236	#1876	C₂×C₁₃₁	1	2	yes	complemented
237	#1871	C₂×C₁₃₁	1	2	yes	complemented
238	#1764	C₂×C₁₃₁	1	2	yes	complemented
239	C₈×C₂²⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
240	C₂×#341	C₂×C₁₃₁	1	1	yes	characteristic, complemented
241	C₂×#135	C₂²×C₁₃₁	1	1	yes	characteristic
242	C₂×#131	C₂²×C₁₃₁	1	2	yes	—
243	C₂²×M₁₆	C₂²×C₁₃₁	1	2	yes	—
244	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
245	#525	C₂²×C₁₃₁	1	2	yes	—
246	#467	C₄×C₁₃₁	1	1	yes	characteristic, complemented
247	#467	C₄×C₁₃₁	1	1	yes	characteristic, complemented
248	#466	C₄×C₁₃₁	1	1	yes	characteristic, complemented
249	#466	C₄×C₁₃₁	1	1	yes	characteristic, complemented
250	#465	C₄×C₁₃₁	1	2	yes	complemented
251	#465	C₄×C₁₃₁	1	2	yes	complemented
252	#436	—	2	2	no	—
253	#435	—	2	2	no	—
254	#435	—	2	2	no	—
255	C₈×D₄	—	2	2	no	—
256	#427	C₂²×C₁₃₁	1	1	yes	characteristic
257	#424	C₂²×C₁₃₁	1	2	yes	—
258	C₂×#104	C₂²×C₁₃₁	1	2	yes	—
259	C₂×#104	C₂²×C₁₃₁	1	2	yes	—
260	#416	C₂²×C₁₃₁	1	2	yes	—
261	#408	C₂²×C₁₃₁	1	2	yes	—
262	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
263	#405	C₂²×C₁₃₁	1	2	yes	—
264	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
265	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
266	#341	C₂²×C₁₃₁	1	2	yes	—
267	#341	C₂²×C₁₃₁	1	1	yes	characteristic
268	#341	C₂²×C₁₃₁	1	1	yes	characteristic
269	#339	C₂²×C₁₃₁	1	2	yes	—
270	#339	C₂²×C₁₃₁	1	2	yes	—
271	#336	C₂²×C₁₃₁	1	2	yes	—
272	C₂×#41	C₂³×C₁₃₁	1	2	yes	—
273	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
274	#135	—	4	2	no	—
275	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
276	C₂×SD₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
277	C₂×D₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
278	#131	—	2	4	no	—
279	#131	—	4	2	no	—
280	C₂×M₁₆	—	2	2	no	—
281	C₂×M₁₆	—	2	2	no	—
282	C₂×M₁₆	—	2	4	no	—
283	C₂×M₁₆	—	4	2	no	—
284	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
285	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
286	C₂²×C₈	—	2	2	no	—
287	C₂²×C₈	—	2	2	no	—
288	C₂²×C₈	—	4	2	no	—
289	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
290	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
291	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
292	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
294	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
296	#123	—	2	2	no	—
297	C₂²⋊Q₈	C₂×C₄×C₁₃₁	1	2	yes	—
298	C₂²⋊Q₈	C₂×C₄×C₁₃₁	1	2	yes	—
299	#121	C₂×C₄×C₁₃₁	1	2	yes	—
300	#121	C₂×C₄×C₁₃₁	1	2	yes	—
301	#120	—	2	2	no	—
302	C₄×D₄	—	2	2	no	—
303	C₄×D₄	—	2	2	no	—
304	#117	C₂³×C₁₃₁	1	2	yes	—
305	C₂×C₂²⋊C₄	—	2	1	no	—
306	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
307	#107	C₂×C₄×C₁₃₁	1	1	yes	characteristic
308	#107	C₂×C₄×C₁₃₁	1	1	yes	characteristic
309	#106	C₂×C₄×C₁₃₁	1	2	yes	—
310	C₄⋊C₈	—	2	2	no	—
311	C₄⋊C₈	—	2	2	no	—
312	#104	—	2	4	no	—
313	#104	—	4	2	no	—
314	#103	C₂×C₄×C₁₃₁	1	2	yes	—
315	#103	C₂×C₄×C₁₃₁	1	2	yes	—
316	#102	C₂×C₄×C₁₃₁	1	2	yes	—
317	#102	C₂×C₄×C₁₃₁	1	2	yes	—
318	C₂²⋊C₈	—	2	2	no	—
319	C₂²⋊C₈	—	2	2	no	—
320	C₂²⋊C₈	D₄×C₁₃₁	1	1	yes	characteristic
321	C₂²⋊C₈	D₄×C₁₃₁	1	1	yes	characteristic
322	C₂²⋊C₈	D₄×C₁₃₁	1	2	yes	—
323	#97	—	2	2	no	—
324	C₄×C₈	—	2	2	no	—
325	C₄×C₈	—	4	2	no	—
326	#95	—	2	1	no	—
327	C₂⁴	#41×C₁₃₁	1	1	yes	characteristic
328	#41	—	2	2	no	—
329	#41	—	2	2	no	—
330	#41	—	2	2	no	—
331	#41	—	2	2	no	—
332	C₂×Q₈	C₂²×C₄×C₁₃₁	1	2	yes	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	C₂²×C₄×C₁₃₁	1	2	yes	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
345	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
347	Q₁₆	—	4	1	no	—
348	SD₁₆	—	4	2	no	—
349	D₈	—	4	1	no	—
350	M₁₆	—	2	2	no	—
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	2	no	—
353	M₁₆	—	4	4	no	—
354	M₁₆	—	4	4	no	—
355	M₁₆	#41×C₁₃₁	1	4	yes	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	4	1	no	—
363	C₂×C₈	—	4	1	no	—
364	C₂×C₈	—	4	1	no	—
365	C₂×C₈	—	4	1	no	—
366	C₂×C₈	—	4	4	no	—
367	C₂×C₈	—	4	4	no	—
368	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
369	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
370	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
371	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
372	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
373	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
377	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
378	C₂²⋊C₄	—	2	4	no	—
379	C₂²⋊C₄	—	2	2	no	—
380	C₂²⋊C₄	—	2	2	no	—
381	C₂²⋊C₄	—	2	2	no	—
382	C₂²⋊C₄	—	2	2	no	—
383	C₂²⋊C₄	—	2	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	4	1	no	—
391	C₂³	C₄×D₄×C₁₃₁	1	1	yes	characteristic
392	C₂³	C₄×D₄×C₁₃₁	1	1	yes	characteristic
393	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
394	Q₈	—	2	2	no	—
395	Q₈	—	2	2	no	—
396	D₄	—	2	2	no	—
397	D₄	—	2	2	no	—
398	D₄	—	4	4	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	8	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
418	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
421	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
422	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
423	C₈	—	2	1	no	—
424	C₈	—	2	1	no	—
425	C₈	—	2	1	no	—
426	C₈	—	2	1	no	—
427	C₈	—	2	2	no	—
428	C₈	—	2	2	no	—
429	C₈	—	2	4	no	—
430	C₈	—	4	2	no	—
431	C₂²	—	2	1	no	—
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	2	1	no	—
435	C₂²	—	2	1	no	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	#443×C₁₃₁	1	2	yes	—
443	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
444	C₂²	#553×C₁₃₁	1	2	yes	—
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	4	4	no	—
448	C₄	—	4	2	no	—
449	C₄	—	4	2	no	—
450	C₄	—	4	2	no	—
451	C₄	C₄×#41×C₁₃₁	1	1	yes	characteristic
452	C₄	#518×C₁₃₁	1	1	yes	characteristic
453	C₄	#546×C₁₃₁	1	2	yes	—
454	C₂	—	2	2	no	—
455	C₂	—	2	2	no	—
456	C₂	—	4	1	no	—
457	C₂	—	8	2	no	—
458	C₂	#2364×C₁₃₁	1	1	yes	characteristic
459	C₂	C₂×#443×C₁₃₁	1	1	yes	characteristic
460	C₂	#2957×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂×C₄×D₄
prime radical	#3050×C₁₃₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₂²⋊C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2933×C₁₃₁	C₂	semidirect
3. #1937×C₁₃₁	C₂	semidirect
4. #1922×C₁₃₁	C₂	semidirect
5. #1876×C₁₃₁	C₂	semidirect
6. #1871×C₁₃₁	C₂	semidirect
7. #1764×C₁₃₁	C₂	semidirect
8. C₈×C₂²⋊C₄×C₁₃₁	C₂	semidirect
9. C₂×#341×C₁₃₁	C₂	semidirect
15. #467×C₁₃₁	C₄	semidirect
16. #467×C₁₃₁	C₄	semidirect
17. #466×C₁₃₁	C₄	semidirect
18. #466×C₁₃₁	C₄	semidirect
19. #465×C₁₃₁	C₄	semidirect
20. #465×C₁₃₁	C₄	semidirect
230. #23146	C₁₃₁	direct (N × H)
231. C₁₃₁	#23146	direct (N × H)
233. #2933	C₂×C₁₃₁	semidirect
234. #1937	C₂×C₁₃₁	semidirect
235. #1922	C₂×C₁₃₁	semidirect
236. #1876	C₂×C₁₃₁	semidirect
237. #1871	C₂×C₁₃₁	semidirect
238. #1764	C₂×C₁₃₁	semidirect
239. C₈×C₂²⋊C₄	C₂×C₁₃₁	semidirect
240. C₂×#341	C₂×C₁₃₁	semidirect
246. #467	C₄×C₁₃₁	semidirect
247. #467	C₄×C₁₃₁	semidirect
248. #466	C₄×C₁₃₁	semidirect
249. #466	C₄×C₁₃₁	semidirect
250. #465	C₄×C₁₃₁	semidirect
251. #465	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3050×C₁₃₁	2
#2933×C₁₃₁	2
#1937×C₁₃₁	2
#1922×C₁₃₁	2
#1876×C₁₃₁	2
#1871×C₁₃₁	2
#1764×C₁₃₁	2
C₈×C₂²⋊C₄×C₁₃₁	2
C₂×#341×C₁₃₁	2
#23146	131

Subgroups — C131×#23146 #45709030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23146 #45709030