Subgroups — C₁₃₁×#22564 #45708448

Order 33536.

994 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#416×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2346×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2339×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1935×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1928×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1862×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1825×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#1808×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#1788×C₁₃₁	C₂	1	1	yes	maximal, characteristic
10	#1762×C₁₃₁	C₂	1	2	yes	maximal, complemented
11	#1752×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
17	#416×C₁₃₁	C₄	1	8	yes	complemented
18	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
19	C₂×#103×C₁₃₁	C₂²	1	1	yes	characteristic
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×#102×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
25	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
26	#391×C₁₃₁	C₂²	1	2	yes	—
27	#389×C₁₃₁	C₂²	1	2	yes	—
28	#388×C₁₃₁	—	2	2	no	—
29	#386×C₁₃₁	—	2	2	no	—
30	#382×C₁₃₁	—	2	1	no	—
31	#381×C₁₃₁	—	2	1	no	—
32	#380×C₁₃₁	C₂²	1	1	yes	characteristic
33	#380×C₁₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
36	#376×C₁₃₁	—	2	1	no	—
37	C₂×#95×C₁₃₁	—	2	1	no	—
38	#340×C₁₃₁	C₂²	1	2	yes	—
39	#340×C₁₃₁	C₂²	1	2	yes	—
40	#340×C₁₃₁	C₂²	1	2	yes	—
41	#336×C₁₃₁	C₂²	1	2	yes	—
42	C₂×#41×C₁₃₁	—	2	2	no	—
43	C₂×#41×C₁₃₁	—	2	4	no	—
44	C₂×#41×C₁₃₁	—	4	2	no	—
45	C₂×#41×C₁₃₁	C₂×C₄	1	4	yes	—
46	C₂²×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₁₃₁	—	2	1	no	—
48	C₂²×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₁₃₁	—	2	1	no	—
50	C₂³×C₄×C₁₃₁	—	2	2	no	—
51	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₁	—	4	2	no	—
53	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
54	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
57	C₄×D₄×C₁₃₁	—	2	4	no	—
58	C₄×D₄×C₁₃₁	—	4	2	no	—
59	#117×C₁₃₁	—	2	2	no	—
60	#117×C₁₃₁	C₂×C₄	1	4	yes	complemented
61	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	2	yes	—
71	C₂×C₄²×C₁₃₁	—	2	1	no	—
72	C₂×C₄²×C₁₃₁	—	2	2	no	—
73	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
74	#107×C₁₃₁	—	4	2	no	—
75	#103×C₁₃₁	—	4	2	no	—
76	#103×C₁₃₁	C₂×C₄	1	4	yes	complemented
77	#103×C₁₃₁	C₂×C₄	1	4	yes	complemented
78	#102×C₁₃₁	—	2	4	no	—
79	#102×C₁₃₁	C₂×C₄	1	4	yes	—
80	#102×C₁₃₁	C₂×C₄	1	4	yes	—
81	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
82	#97×C₁₃₁	—	4	2	no	—
83	#95×C₁₃₁	—	2	2	no	—
84	#95×C₁₃₁	—	2	2	no	—
85	#95×C₁₃₁	—	2	2	no	—
86	#95×C₁₃₁	—	2	2	no	—
87	#95×C₁₃₁	—	2	1	no	—
88	#95×C₁₃₁	—	2	1	no	—
89	#95×C₁₃₁	D₄	1	1	yes	characteristic
90	#95×C₁₃₁	D₄	1	1	yes	characteristic
91	C₂⁴×C₁₃₁	—	2	1	no	—
92	C₂⁴×C₁₃₁	C₂×D₄	1	1	yes	characteristic
93	#41×C₁₃₁	—	2	8	no	—
94	#41×C₁₃₁	—	4	4	no	—
95	#41×C₁₃₁	—	4	8	no	—
96	C₂×Q₈×C₁₃₁	—	2	1	no	—
97	C₂×Q₈×C₁₃₁	—	2	2	no	—
98	C₂×Q₈×C₁₃₁	—	2	2	no	—
99	C₂×Q₈×C₁₃₁	C₂²×C₄	1	2	yes	—
100	C₂×D₄×C₁₃₁	—	2	4	no	—
101	C₂×D₄×C₁₃₁	—	2	1	no	—
102	C₂×D₄×C₁₃₁	—	2	2	no	—
103	C₂×D₄×C₁₃₁	—	4	4	no	—
104	C₂×D₄×C₁₃₁	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₁₃₁	#41	1	4	yes	—
106	C₂²×C₄×C₁₃₁	—	2	2	no	—
107	C₂²×C₄×C₁₃₁	—	2	2	no	—
108	C₂²×C₄×C₁₃₁	—	2	1	no	—
109	C₂²×C₄×C₁₃₁	—	2	1	no	—
110	C₂²×C₄×C₁₃₁	—	2	1	no	—
111	C₂²×C₄×C₁₃₁	—	2	4	no	—
112	C₂²×C₄×C₁₃₁	—	2	2	no	—
113	C₂²×C₄×C₁₃₁	—	2	2	no	—
114	C₂²×C₄×C₁₃₁	—	4	4	no	—
115	C₂²×C₄×C₁₃₁	—	4	4	no	—
116	C₂²×C₄×C₁₃₁	—	4	4	no	—
117	C₂²×C₄×C₁₃₁	—	8	2	no	—
118	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
123	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
124	C₂×C₈×C₁₃₁	—	2	2	no	—
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	—	4	4	no	—
130	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
132	C₄⋊C₄×C₁₃₁	—	2	1	no	—
133	C₄⋊C₄×C₁₃₁	—	2	1	no	—
134	C₄⋊C₄×C₁₃₁	—	2	2	no	—
135	C₄⋊C₄×C₁₃₁	—	2	2	no	—
136	C₄⋊C₄×C₁₃₁	—	2	2	no	—
137	C₄⋊C₄×C₁₃₁	—	2	2	no	—
138	C₄⋊C₄×C₁₃₁	—	8	2	no	—
139	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
140	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
141	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
142	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
143	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
144	C₂²⋊C₄×C₁₃₁	—	8	1	no	—
145	C₂²⋊C₄×C₁₃₁	—	8	1	no	—
146	C₄²×C₁₃₁	—	2	4	no	—
147	C₄²×C₁₃₁	—	2	2	no	—
148	C₄²×C₁₃₁	—	2	2	no	—
149	C₄²×C₁₃₁	—	4	4	no	—
150	C₂³×C₁₃₁	—	2	2	no	—
151	C₂³×C₁₃₁	—	2	4	no	—
152	C₂³×C₁₃₁	—	2	2	no	—
153	C₂³×C₁₃₁	—	2	2	no	—
154	C₂³×C₁₃₁	—	2	1	no	—
155	C₂³×C₁₃₁	—	2	1	no	—
156	C₂³×C₁₃₁	—	4	2	no	—
157	C₂³×C₁₃₁	—	4	1	no	—
158	C₂³×C₁₃₁	—	4	1	no	—
159	C₂³×C₁₃₁	C₄×D₄	1	2	yes	—
160	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₁₃₁	—	2	2	no	—
162	Q₈×C₁₃₁	—	2	2	no	—
163	Q₈×C₁₃₁	—	4	2	no	—
164	D₄×C₁₃₁	—	2	2	no	—
165	D₄×C₁₃₁	—	2	4	no	—
166	D₄×C₁₃₁	—	4	8	no	—
167	D₄×C₁₃₁	#117	1	4	yes	—
168	C₂×C₄×C₁₃₁	—	2	2	no	—
169	C₂×C₄×C₁₃₁	—	2	2	no	—
170	C₂×C₄×C₁₃₁	—	2	2	no	—
171	C₂×C₄×C₁₃₁	—	2	4	no	—
172	C₂×C₄×C₁₃₁	—	2	2	no	—
173	C₂×C₄×C₁₃₁	—	2	2	no	—
174	C₂×C₄×C₁₃₁	—	2	2	no	—
175	C₂×C₄×C₁₃₁	—	2	2	no	—
176	C₂×C₄×C₁₃₁	—	2	2	no	—
177	C₂×C₄×C₁₃₁	—	2	2	no	—
178	C₂×C₄×C₁₃₁	—	2	2	no	—
179	C₂×C₄×C₁₃₁	—	2	2	no	—
180	C₂×C₄×C₁₃₁	—	2	2	no	—
181	C₂×C₄×C₁₃₁	—	4	8	no	—
182	C₂×C₄×C₁₃₁	—	4	2	no	—
183	C₂×C₄×C₁₃₁	—	4	2	no	—
184	C₂×C₄×C₁₃₁	—	4	1	no	—
185	C₂×C₄×C₁₃₁	—	4	1	no	—
186	C₂×C₄×C₁₃₁	—	4	2	no	—
187	C₂×C₄×C₁₃₁	—	4	2	no	—
188	C₂×C₄×C₁₃₁	—	4	2	no	—
189	C₂×C₄×C₁₃₁	—	4	2	no	—
190	C₂×C₄×C₁₃₁	—	4	2	no	—
191	C₂×C₄×C₁₃₁	—	4	2	no	—
192	C₂×C₄×C₁₃₁	—	8	8	no	—
193	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
196	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
198	C₈×C₁₃₁	—	2	2	no	—
199	C₈×C₁₃₁	—	2	2	no	—
200	C₈×C₁₃₁	—	4	2	no	—
201	C₂²×C₁₃₁	—	2	2	no	—
202	C₂²×C₁₃₁	—	2	4	no	—
203	C₂²×C₁₃₁	—	2	2	no	—
204	C₂²×C₁₃₁	—	2	2	no	—
205	C₂²×C₁₃₁	—	4	2	no	—
206	C₂²×C₁₃₁	—	4	2	no	—
207	C₂²×C₁₃₁	—	4	2	no	—
208	C₂²×C₁₃₁	—	4	4	no	—
209	C₂²×C₁₃₁	—	4	4	no	—
210	C₂²×C₁₃₁	—	4	2	no	—
211	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂²×C₁₃₁	#546	1	2	yes	—
213	C₂²×C₁₃₁	#575	1	2	yes	—
214	C₂²×C₁₃₁	#576	1	1	yes	characteristic
215	C₂²×C₁₃₁	#578	1	1	yes	characteristic
216	C₄×C₁₃₁	—	2	2	no	—
217	C₄×C₁₃₁	—	4	2	no	—
218	C₄×C₁₃₁	—	4	2	no	—
219	C₄×C₁₃₁	—	4	2	no	—
220	C₄×C₁₃₁	—	4	2	no	—
221	C₄×C₁₃₁	—	4	2	no	—
222	C₄×C₁₃₁	—	8	2	no	—
223	C₄×C₁₃₁	—	8	2	no	—
224	C₄×C₁₃₁	C₂×#117	1	1	yes	characteristic
225	C₄×C₁₃₁	#518	1	1	yes	characteristic
226	C₄×C₁₃₁	#551	1	2	yes	—
227	C₂×C₁₃₁	—	4	2	no	—
228	C₂×C₁₃₁	—	4	4	no	—
229	C₂×C₁₃₁	#2342	1	1	yes	characteristic
230	C₂×C₁₃₁	#2943	1	1	yes	characteristic
231	C₂×C₁₃₁	#2951	1	1	yes	characteristic
232	C₂×C₁₃₁	#3315	1	2	yes	—
233	C₂×C₁₃₁	#3319	1	2	yes	—
234	#22564	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₁₃₁	#22564	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×#416	C₂×C₁₃₁	1	1	yes	characteristic
237	#2346	C₂×C₁₃₁	1	1	yes	characteristic, complemented
238	#2339	C₂×C₁₃₁	1	1	yes	characteristic
239	#1935	C₂×C₁₃₁	1	2	yes	complemented
240	#1928	C₂×C₁₃₁	1	2	yes	complemented
241	#1862	C₂×C₁₃₁	1	2	yes	complemented
242	#1825	C₂×C₁₃₁	1	1	yes	characteristic, complemented
243	#1808	C₂×C₁₃₁	1	1	yes	characteristic, complemented
244	#1788	C₂×C₁₃₁	1	1	yes	characteristic
245	#1762	C₂×C₁₃₁	1	2	yes	complemented
246	#1752	C₂×C₁₃₁	1	1	yes	characteristic, complemented
247	C₂²×#41	C₂²×C₁₃₁	1	1	yes	characteristic
248	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
249	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
250	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
251	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
252	#416	C₄×C₁₃₁	1	8	yes	complemented
253	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
254	C₂×#103	C₂²×C₁₃₁	1	1	yes	characteristic
255	C₂×#103	C₂²×C₁₃₁	1	1	yes	characteristic
256	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
257	C₂×#102	C₂²×C₁₃₁	1	1	yes	characteristic
258	C₂×#102	C₂²×C₁₃₁	1	1	yes	characteristic
259	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
260	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
261	#391	C₂²×C₁₃₁	1	2	yes	—
262	#389	C₂²×C₁₃₁	1	2	yes	—
263	#388	—	2	2	no	—
264	#386	—	2	2	no	—
265	#382	—	2	1	no	—
266	#381	—	2	1	no	—
267	#380	C₂²×C₁₃₁	1	1	yes	characteristic
268	#380	C₂²×C₁₃₁	1	1	yes	characteristic
269	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
270	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
271	#376	—	2	1	no	—
272	C₂×#95	—	2	1	no	—
273	#340	C₂²×C₁₃₁	1	2	yes	—
274	#340	C₂²×C₁₃₁	1	2	yes	—
275	#340	C₂²×C₁₃₁	1	2	yes	—
276	#336	C₂²×C₁₃₁	1	2	yes	—
277	C₂×#41	—	2	2	no	—
278	C₂×#41	—	2	4	no	—
279	C₂×#41	—	4	2	no	—
280	C₂×#41	C₂×C₄×C₁₃₁	1	4	yes	—
281	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
282	C₂²×D₄	—	2	1	no	—
283	C₂²×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
284	C₂³×C₄	—	2	1	no	—
285	C₂³×C₄	—	2	2	no	—
286	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
287	C₂²×C₈	—	4	2	no	—
288	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
289	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
290	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
292	C₄×D₄	—	2	4	no	—
293	C₄×D₄	—	4	2	no	—
294	#117	—	2	2	no	—
295	#117	C₂×C₄×C₁₃₁	1	4	yes	complemented
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₂²⋊C₄	—	2	1	no	—
305	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	2	yes	—
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	—	2	2	no	—
308	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
309	#107	—	4	2	no	—
310	#103	—	4	2	no	—
311	#103	C₂×C₄×C₁₃₁	1	4	yes	complemented
312	#103	C₂×C₄×C₁₃₁	1	4	yes	complemented
313	#102	—	2	4	no	—
314	#102	C₂×C₄×C₁₃₁	1	4	yes	—
315	#102	C₂×C₄×C₁₃₁	1	4	yes	—
316	C₂²⋊C₈	—	2	2	no	—
317	#97	—	4	2	no	—
318	#95	—	2	1	no	—
319	#95	—	2	2	no	—
320	#95	—	2	1	no	—
321	#95	—	2	2	no	—
322	#95	—	2	2	no	—
323	#95	—	2	2	no	—
324	#95	D₄×C₁₃₁	1	1	yes	characteristic
325	#95	D₄×C₁₃₁	1	1	yes	characteristic
326	C₂⁴	—	2	1	no	—
327	C₂⁴	C₂×D₄×C₁₃₁	1	1	yes	characteristic
328	#41	—	2	8	no	—
329	#41	—	4	4	no	—
330	#41	—	4	8	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	C₂²×C₄×C₁₃₁	1	2	yes	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	4	4	no	—
339	C₂×D₄	C₂²×C₄×C₁₃₁	1	2	yes	—
340	C₂×D₄	#41×C₁₃₁	1	4	yes	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	4	4	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	—	8	2	no	—
353	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
357	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
358	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	4	4	no	—
365	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
366	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	8	2	no	—
374	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
375	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	4	2	no	—
378	C₂²⋊C₄	—	4	4	no	—
379	C₂²⋊C₄	—	8	1	no	—
380	C₂²⋊C₄	—	8	1	no	—
381	C₄²	—	2	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	4	4	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	1	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	C₄×D₄×C₁₃₁	1	2	yes	—
395	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
396	Q₈	—	2	2	no	—
397	Q₈	—	2	2	no	—
398	Q₈	—	4	2	no	—
399	D₄	—	2	2	no	—
400	D₄	—	2	4	no	—
401	D₄	—	4	8	no	—
402	D₄	#117×C₁₃₁	1	4	yes	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	4	8	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	8	8	no	—
428	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
429	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
431	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
432	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
433	C₈	—	2	2	no	—
434	C₈	—	2	2	no	—
435	C₈	—	4	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	2	no	—
446	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
447	C₂²	#546×C₁₃₁	1	2	yes	—
448	C₂²	#575×C₁₃₁	1	2	yes	—
449	C₂²	#576×C₁₃₁	1	1	yes	characteristic
450	C₂²	#578×C₁₃₁	1	1	yes	characteristic
451	C₄	—	2	2	no	—
452	C₄	—	4	2	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	2	no	—
459	C₄	C₂×#117×C₁₃₁	1	1	yes	characteristic
460	C₄	#518×C₁₃₁	1	1	yes	characteristic
461	C₄	#551×C₁₃₁	1	2	yes	—
462	C₂	—	4	2	no	—
463	C₂	—	4	4	no	—
464	C₂	#2342×C₁₃₁	1	1	yes	characteristic
465	C₂	#2943×C₁₃₁	1	1	yes	characteristic
466	C₂	#2951×C₁₃₁	1	1	yes	characteristic
467	C₂	#3315×C₁₃₁	1	2	yes	—
468	C₂	#3319×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×#41×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄×C₂²⋊C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2346×C₁₃₁	C₂	semidirect
4. #1935×C₁₃₁	C₂	semidirect
5. #1928×C₁₃₁	C₂	semidirect
6. #1862×C₁₃₁	C₂	semidirect
7. #1825×C₁₃₁	C₂	semidirect
8. #1808×C₁₃₁	C₂	semidirect
10. #1762×C₁₃₁	C₂	semidirect
11. #1752×C₁₃₁	C₂	semidirect
17. #416×C₁₃₁	C₄	semidirect
53. C₂²×C₈×C₁₃₁	C₂×C₄	semidirect
60. #117×C₁₃₁	C₂×C₄	semidirect
76. #103×C₁₃₁	C₂×C₄	semidirect
77. #103×C₁₃₁	C₂×C₄	semidirect
234. #22564	C₁₃₁	direct (N × H)
235. C₁₃₁	#22564	direct (N × H)
237. #2346	C₂×C₁₃₁	semidirect
239. #1935	C₂×C₁₃₁	semidirect
240. #1928	C₂×C₁₃₁	semidirect
241. #1862	C₂×C₁₃₁	semidirect
242. #1825	C₂×C₁₃₁	semidirect
243. #1808	C₂×C₁₃₁	semidirect
245. #1762	C₂×C₁₃₁	semidirect
246. #1752	C₂×C₁₃₁	semidirect
252. #416	C₄×C₁₃₁	semidirect
288. C₂²×C₈	C₂×C₄×C₁₃₁	semidirect
295. #117	C₂×C₄×C₁₃₁	semidirect
311. #103	C₂×C₄×C₁₃₁	semidirect
312. #103	C₂×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×#416×C₁₃₁	2
#2346×C₁₃₁	2
#2339×C₁₃₁	2
#1935×C₁₃₁	2
#1928×C₁₃₁	2
#1862×C₁₃₁	2
#1825×C₁₃₁	2
#1808×C₁₃₁	2
#1788×C₁₃₁	2
#1762×C₁₃₁	2
#1752×C₁₃₁	2
#22564	131

Subgroups — C131×#22564 #45708448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#22564 #45708448