Subgroups — C₁₃₁×#22811 #45708695

Order 33536.

846 conjugacy classes of subgroups.

The subgroup lattice has 480 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	#2355×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2354×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1973×C₁₃₁	C₂	1	2	yes	maximal
5	#1928×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1925×C₁₃₁	C₂	1	2	yes	maximal
7	#1845×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#1803×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#1802×C₁₃₁	C₂	1	1	yes	maximal, characteristic
10	#1776×C₁₃₁	C₂	1	2	yes	maximal, complemented
11	#1770×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₁₃₁	—	2	1	no	—
16	C₄×SD₁₆×C₁₃₁	C₄	1	8	yes	complemented
17	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
18	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
19	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
22	C₂×#103×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
24	C₂×#102×C₁₃₁	C₂²	1	1	yes	characteristic
25	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
27	#391×C₁₃₁	C₂²	1	2	yes	—
28	#390×C₁₃₁	C₂²	1	2	yes	—
29	#388×C₁₃₁	—	2	2	no	—
30	#387×C₁₃₁	—	2	2	no	—
31	#384×C₁₃₁	C₂²	1	1	yes	characteristic
32	#384×C₁₃₁	C₂²	1	1	yes	characteristic
33	#382×C₁₃₁	—	2	1	no	—
34	#380×C₁₃₁	—	2	1	no	—
35	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
36	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
39	#340×C₁₃₁	C₂²	1	2	yes	—
40	#340×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	2	no	—
46	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
47	C₂×SD₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₁₃₁	—	2	2	no	—
52	C₄×Q₈×C₁₃₁	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₁₃₁	—	2	2	no	—
54	C₄×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
55	#117×C₁₃₁	—	4	4	no	—
56	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
70	C₂×C₄²×C₁₃₁	—	2	1	no	—
71	C₂×C₄²×C₁₃₁	—	2	2	no	—
72	C₂×C₄²×C₁₃₁	—	2	2	no	—
73	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
74	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
75	#107×C₁₃₁	—	2	4	no	—
76	#107×C₁₃₁	—	4	2	no	—
77	#106×C₁₃₁	C₂×C₄	1	4	yes	—
78	C₄⋊C₈×C₁₃₁	—	2	2	no	—
79	#103×C₁₃₁	—	4	2	no	—
80	#103×C₁₃₁	C₂×C₄	1	4	yes	—
81	#102×C₁₃₁	—	4	2	no	—
82	#102×C₁₃₁	C₂×C₄	1	4	yes	—
83	#97×C₁₃₁	—	2	4	no	—
84	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
85	#95×C₁₃₁	—	2	2	no	—
86	#95×C₁₃₁	—	2	2	no	—
87	#95×C₁₃₁	—	2	2	no	—
88	#95×C₁₃₁	—	2	2	no	—
89	#95×C₁₃₁	D₄	1	2	yes	—
90	C₂⁴×C₁₃₁	—	2	1	no	—
91	C₂×Q₈×C₁₃₁	—	2	1	no	—
92	C₂×Q₈×C₁₃₁	—	2	1	no	—
93	C₂×Q₈×C₁₃₁	—	2	1	no	—
94	C₂×Q₈×C₁₃₁	—	2	1	no	—
95	C₂×Q₈×C₁₃₁	—	2	1	no	—
96	C₂×Q₈×C₁₃₁	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₁₃₁	—	2	1	no	—
98	C₂×D₄×C₁₃₁	—	2	1	no	—
99	C₂×D₄×C₁₃₁	—	2	1	no	—
100	C₂×D₄×C₁₃₁	—	2	1	no	—
101	C₂×D₄×C₁₃₁	—	2	1	no	—
102	C₂×D₄×C₁₃₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₁₃₁	—	2	1	no	—
104	C₂²×C₄×C₁₃₁	—	2	1	no	—
105	C₂²×C₄×C₁₃₁	—	2	1	no	—
106	C₂²×C₄×C₁₃₁	—	2	4	no	—
107	C₂²×C₄×C₁₃₁	—	2	2	no	—
108	C₂²×C₄×C₁₃₁	—	2	2	no	—
109	C₂²×C₄×C₁₃₁	—	4	2	no	—
110	C₂²×C₄×C₁₃₁	—	4	2	no	—
111	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
116	SD₁₆×C₁₃₁	—	2	8	no	—
117	C₂×C₈×C₁₃₁	—	2	1	no	—
118	C₂×C₈×C₁₃₁	—	2	1	no	—
119	C₂×C₈×C₁₃₁	—	2	2	no	—
120	C₂×C₈×C₁₃₁	—	2	2	no	—
121	C₂×C₈×C₁₃₁	—	2	2	no	—
122	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₁₃₁	#41	1	2	yes	—
125	C₂×C₈×C₁₃₁	#41	1	2	yes	—
126	C₄⋊C₄×C₁₃₁	—	2	4	no	—
127	C₄⋊C₄×C₁₃₁	—	2	2	no	—
128	C₄⋊C₄×C₁₃₁	—	2	2	no	—
129	C₄⋊C₄×C₁₃₁	—	2	1	no	—
130	C₄⋊C₄×C₁₃₁	—	2	1	no	—
131	C₄⋊C₄×C₁₃₁	—	2	2	no	—
132	C₄⋊C₄×C₁₃₁	—	2	2	no	—
133	C₄⋊C₄×C₁₃₁	—	2	2	no	—
134	C₄⋊C₄×C₁₃₁	—	2	2	no	—
135	C₄⋊C₄×C₁₃₁	—	4	2	no	—
136	C₄⋊C₄×C₁₃₁	—	4	2	no	—
137	C₄⋊C₄×C₁₃₁	—	8	1	no	—
138	C₄⋊C₄×C₁₃₁	—	8	1	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	4	no	—
143	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
144	C₄²×C₁₃₁	—	2	4	no	—
145	C₄²×C₁₃₁	—	2	2	no	—
146	C₄²×C₁₃₁	—	2	2	no	—
147	C₄²×C₁₃₁	—	4	4	no	—
148	C₄²×C₁₃₁	—	8	2	no	—
149	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
150	C₄²×C₁₃₁	#41	1	2	yes	—
151	C₂³×C₁₃₁	—	2	2	no	—
152	C₂³×C₁₃₁	—	4	1	no	—
153	C₂³×C₁₃₁	—	4	1	no	—
154	C₂³×C₁₃₁	—	4	1	no	—
155	C₂³×C₁₃₁	—	4	1	no	—
156	C₂³×C₁₃₁	—	4	1	no	—
157	C₂³×C₁₃₁	—	4	1	no	—
158	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₁₃₁	—	2	2	no	—
160	Q₈×C₁₃₁	—	2	2	no	—
161	Q₈×C₁₃₁	—	4	1	no	—
162	Q₈×C₁₃₁	—	4	1	no	—
163	D₄×C₁₃₁	—	2	2	no	—
164	D₄×C₁₃₁	—	2	2	no	—
165	D₄×C₁₃₁	—	4	1	no	—
166	D₄×C₁₃₁	—	4	1	no	—
167	C₂×C₄×C₁₃₁	—	2	1	no	—
168	C₂×C₄×C₁₃₁	—	2	1	no	—
169	C₂×C₄×C₁₃₁	—	2	2	no	—
170	C₂×C₄×C₁₃₁	—	2	2	no	—
171	C₂×C₄×C₁₃₁	—	2	2	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	—	4	1	no	—
179	C₂×C₄×C₁₃₁	—	4	1	no	—
180	C₂×C₄×C₁₃₁	—	4	1	no	—
181	C₂×C₄×C₁₃₁	—	4	1	no	—
182	C₂×C₄×C₁₃₁	—	4	1	no	—
183	C₂×C₄×C₁₃₁	—	4	1	no	—
184	C₂×C₄×C₁₃₁	—	4	8	no	—
185	C₂×C₄×C₁₃₁	—	4	2	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	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
194	C₂×C₄×C₁₃₁	#137	1	2	yes	—
195	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
201	C₈×C₁₃₁	—	2	2	no	—
202	C₈×C₁₃₁	—	4	2	no	—
203	C₈×C₁₃₁	#117	1	4	yes	complemented
204	C₂²×C₁₃₁	—	4	2	no	—
205	C₂²×C₁₃₁	—	4	2	no	—
206	C₂²×C₁₃₁	—	4	2	no	—
207	C₂²×C₁₃₁	—	8	1	no	—
208	C₂²×C₁₃₁	—	8	1	no	—
209	C₂²×C₁₃₁	—	8	1	no	—
210	C₂²×C₁₃₁	—	8	1	no	—
211	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂²×C₁₃₁	#547	1	1	yes	characteristic
213	C₂²×C₁₃₁	#550	1	1	yes	characteristic
214	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₂²×C₁₃₁	C₂×#137	1	1	yes	characteristic
216	C₂²×C₁₃₁	#576	1	1	yes	characteristic
217	C₂²×C₁₃₁	#578	1	1	yes	characteristic
218	C₄×C₁₃₁	—	2	2	no	—
219	C₄×C₁₃₁	—	4	2	no	—
220	C₄×C₁₃₁	—	4	2	no	—
221	C₄×C₁₃₁	—	4	2	no	—
222	C₄×C₁₃₁	—	4	2	no	—
223	C₄×C₁₃₁	—	4	2	no	—
224	C₄×C₁₃₁	—	8	2	no	—
225	C₄×C₁₃₁	—	8	2	no	—
226	C₄×C₁₃₁	#440	1	4	yes	—
227	C₄×C₁₃₁	C₂×#117	1	1	yes	characteristic
228	C₄×C₁₃₁	#520	1	1	yes	characteristic
229	C₄×C₁₃₁	#555	1	1	yes	characteristic
230	C₄×C₁₃₁	#556	1	1	yes	characteristic
231	C₂×C₁₃₁	—	8	2	no	—
232	C₂×C₁₃₁	#2354	1	1	yes	characteristic
233	C₂×C₁₃₁	C₂×#440	1	1	yes	characteristic
234	C₂×C₁₃₁	#2951	1	1	yes	characteristic
235	C₂×C₁₃₁	#3353	1	1	yes	characteristic
236	C₂×C₁₃₁	#3357	1	1	yes	characteristic
237	C₂×C₁₃₁	#3398	1	1	yes	characteristic
238	C₂×C₁₃₁	#3399	1	1	yes	characteristic
239	#22811	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₁₃₁	#22811	1	1	yes	characteristic, direct product factor, Sylow
241	C₂×C₄×SD₁₆	C₂×C₁₃₁	1	1	yes	characteristic
242	#2355	C₂×C₁₃₁	1	1	yes	characteristic, complemented
243	#2354	C₂×C₁₃₁	1	1	yes	characteristic
244	#1973	C₂×C₁₃₁	1	2	yes	—
245	#1928	C₂×C₁₃₁	1	2	yes	complemented
246	#1925	C₂×C₁₃₁	1	2	yes	—
247	#1845	C₂×C₁₃₁	1	1	yes	characteristic, complemented
248	#1803	C₂×C₁₃₁	1	1	yes	characteristic, complemented
249	#1802	C₂×C₁₃₁	1	1	yes	characteristic
250	#1776	C₂×C₁₃₁	1	2	yes	complemented
251	#1770	C₂×C₁₃₁	1	1	yes	characteristic, complemented
252	C₂²×SD₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
253	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
254	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
255	C₂×#117	—	2	1	no	—
256	C₄×SD₁₆	C₄×C₁₃₁	1	8	yes	complemented
257	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
258	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
259	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
260	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
261	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
262	C₂×#103	C₂²×C₁₃₁	1	1	yes	characteristic
263	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
264	C₂×#102	C₂²×C₁₃₁	1	1	yes	characteristic
265	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
266	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
267	#391	C₂²×C₁₃₁	1	2	yes	—
268	#390	C₂²×C₁₃₁	1	2	yes	—
269	#388	—	2	2	no	—
270	#387	—	2	2	no	—
271	#384	C₂²×C₁₃₁	1	1	yes	characteristic
272	#384	C₂²×C₁₃₁	1	1	yes	characteristic
273	#382	—	2	1	no	—
274	#380	—	2	1	no	—
275	C₄×C₄⋊C₄	—	2	1	no	—
276	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
277	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
278	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
279	#340	C₂²×C₁₃₁	1	2	yes	—
280	#340	C₂²×C₁₃₁	1	2	yes	—
281	#336	C₂²×C₁₃₁	1	2	yes	—
282	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
283	C₂²×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
284	C₂³×C₄	—	2	1	no	—
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×SD₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
288	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
289	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
291	C₄×Q₈	—	2	2	no	—
292	C₄×Q₈	C₂×C₄×C₁₃₁	1	4	yes	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
295	#117	—	4	4	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
302	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
305	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	—	2	2	no	—
312	C₂×C₄²	—	2	2	no	—
313	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
314	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
315	#107	—	2	4	no	—
316	#107	—	4	2	no	—
317	#106	C₂×C₄×C₁₃₁	1	4	yes	—
318	C₄⋊C₈	—	2	2	no	—
319	#103	—	4	2	no	—
320	#103	C₂×C₄×C₁₃₁	1	4	yes	—
321	#102	—	4	2	no	—
322	#102	C₂×C₄×C₁₃₁	1	4	yes	—
323	#97	—	2	4	no	—
324	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	#95	—	2	2	no	—
328	#95	—	2	2	no	—
329	#95	D₄×C₁₃₁	1	2	yes	—
330	C₂⁴	—	2	1	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	C₂²×C₄×C₁₃₁	1	2	yes	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	C₂²×C₄×C₁₃₁	1	2	yes	—
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	2	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
356	SD₁₆	—	2	8	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
363	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
364	C₂×C₈	#41×C₁₃₁	1	2	yes	—
365	C₂×C₈	#41×C₁₃₁	1	2	yes	—
366	C₄⋊C₄	—	2	4	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	8	1	no	—
378	C₄⋊C₄	—	8	1	no	—
379	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
380	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	8	2	no	—
384	C₄²	—	2	4	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	4	4	no	—
388	C₄²	—	8	2	no	—
389	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
390	C₄²	#41×C₁₃₁	1	2	yes	—
391	C₂³	—	2	2	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
399	Q₈	—	2	2	no	—
400	Q₈	—	2	2	no	—
401	Q₈	—	4	1	no	—
402	Q₈	—	4	1	no	—
403	D₄	—	2	2	no	—
404	D₄	—	2	2	no	—
405	D₄	—	4	1	no	—
406	D₄	—	4	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	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	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	8	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
432	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
433	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
434	C₂×C₄	#137×C₁₃₁	1	2	yes	—
435	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
439	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
440	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
441	C₈	—	2	2	no	—
442	C₈	—	4	2	no	—
443	C₈	#117×C₁₃₁	1	4	yes	complemented
444	C₂²	—	4	2	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	8	1	no	—
448	C₂²	—	8	1	no	—
449	C₂²	—	8	1	no	—
450	C₂²	—	8	1	no	—
451	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
452	C₂²	#547×C₁₃₁	1	1	yes	characteristic
453	C₂²	#550×C₁₃₁	1	1	yes	characteristic
454	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
455	C₂²	C₂×#137×C₁₃₁	1	1	yes	characteristic
456	C₂²	#576×C₁₃₁	1	1	yes	characteristic
457	C₂²	#578×C₁₃₁	1	1	yes	characteristic
458	C₄	—	2	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	2	no	—
463	C₄	—	4	2	no	—
464	C₄	—	8	2	no	—
465	C₄	—	8	2	no	—
466	C₄	#440×C₁₃₁	1	4	yes	—
467	C₄	C₂×#117×C₁₃₁	1	1	yes	characteristic
468	C₄	#520×C₁₃₁	1	1	yes	characteristic
469	C₄	#555×C₁₃₁	1	1	yes	characteristic
470	C₄	#556×C₁₃₁	1	1	yes	characteristic
471	C₂	—	8	2	no	—
472	C₂	#2354×C₁₃₁	1	1	yes	characteristic
473	C₂	C₂×#440×C₁₃₁	1	1	yes	characteristic
474	C₂	#2951×C₁₃₁	1	1	yes	characteristic
475	C₂	#3353×C₁₃₁	1	1	yes	characteristic
476	C₂	#3357×C₁₃₁	1	1	yes	characteristic
477	C₂	#3398×C₁₃₁	1	1	yes	characteristic
478	C₂	#3399×C₁₃₁	1	1	yes	characteristic

	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 ◁ C₄×C₄⋊C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2355×C₁₃₁	C₂	semidirect
5. #1928×C₁₃₁	C₂	semidirect
7. #1845×C₁₃₁	C₂	semidirect
8. #1803×C₁₃₁	C₂	semidirect
10. #1776×C₁₃₁	C₂	semidirect
11. #1770×C₁₃₁	C₂	semidirect
16. C₄×SD₁₆×C₁₃₁	C₄	semidirect
203. C₈×C₁₃₁	#117	semidirect
239. #22811	C₁₃₁	direct (N × H)
240. C₁₃₁	#22811	direct (N × H)
242. #2355	C₂×C₁₃₁	semidirect
245. #1928	C₂×C₁₃₁	semidirect
247. #1845	C₂×C₁₃₁	semidirect
248. #1803	C₂×C₁₃₁	semidirect
250. #1776	C₂×C₁₃₁	semidirect
251. #1770	C₂×C₁₃₁	semidirect
256. C₄×SD₁₆	C₄×C₁₃₁	semidirect
443. C₈	#117×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
#2355×C₁₃₁	2
#2354×C₁₃₁	2
#1973×C₁₃₁	2
#1928×C₁₃₁	2
#1925×C₁₃₁	2
#1845×C₁₃₁	2
#1803×C₁₃₁	2
#1802×C₁₃₁	2
#1776×C₁₃₁	2
#1770×C₁₃₁	2
#22811	131

Subgroups — C131×#22811 #45708695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#22811 #45708695