Subgroups — C₁₃₁×#23781 #45709665

Order 33536.

808 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#2293×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2099×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2093×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#2087×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2065×C₁₃₁	C₂	1	2	yes	maximal
7	#2058×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#2023×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	#1776×C₁₃₁	C₂	1	2	yes	maximal, complemented
10	#1767×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#1762×C₁₃₁	C₂	1	2	yes	maximal
12	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃₁	—	2	1	no	—
14	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₁₃₁	—	2	1	no	—
16	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
19	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
20	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
21	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
22	#391×C₁₃₁	—	2	1	no	—
23	#390×C₁₃₁	C₂²	1	1	yes	characteristic
24	#390×C₁₃₁	C₂²	1	1	yes	characteristic
25	#389×C₁₃₁	—	2	1	no	—
26	#389×C₁₃₁	C₂²	1	1	yes	characteristic
27	#389×C₁₃₁	C₂²	1	1	yes	characteristic
28	#389×C₁₃₁	C₂²	1	2	yes	—
29	#388×C₁₃₁	—	2	1	no	—
30	#388×C₁₃₁	—	2	1	no	—
31	#384×C₁₃₁	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
33	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
36	C₄³×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	#340×C₁₃₁	C₂²	1	2	yes	—
41	#336×C₁₃₁	C₂²	1	1	yes	characteristic
42	#336×C₁₃₁	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₁₃₁	—	2	1	no	—
44	C₂²×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₁₃₁	—	2	1	no	—
46	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
49	C₄×Q₈×C₁₃₁	—	4	4	no	—
50	C₄×D₄×C₁₃₁	—	2	2	no	—
51	C₄×D₄×C₁₃₁	Q₈	1	4	yes	complemented
52	#117×C₁₃₁	—	4	4	no	—
53	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
69	C₂×C₄²×C₁₃₁	—	2	1	no	—
70	C₂×C₄²×C₁₃₁	—	2	2	no	—
71	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
78	#107×C₁₃₁	—	2	4	no	—
79	#106×C₁₃₁	—	2	4	no	—
80	C₄⋊C₈×C₁₃₁	—	2	4	no	—
81	#102×C₁₃₁	—	2	4	no	—
82	#102×C₁₃₁	—	2	4	no	—
83	#97×C₁₃₁	—	2	4	no	—
84	#95×C₁₃₁	—	2	1	no	—
85	#95×C₁₃₁	—	2	1	no	—
86	#95×C₁₃₁	—	2	1	no	—
87	#95×C₁₃₁	—	2	2	no	—
88	#95×C₁₃₁	—	2	2	no	—
89	C₂⁴×C₁₃₁	—	2	1	no	—
90	C₂×Q₈×C₁₃₁	—	2	4	no	—
91	C₂×Q₈×C₁₃₁	—	4	2	no	—
92	C₂×Q₈×C₁₃₁	—	4	2	no	—
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	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₁₃₁	—	2	2	no	—
98	C₂²×C₄×C₁₃₁	—	2	1	no	—
99	C₂²×C₄×C₁₃₁	—	2	4	no	—
100	C₂²×C₄×C₁₃₁	—	2	1	no	—
101	C₂²×C₄×C₁₃₁	—	2	1	no	—
102	C₂²×C₄×C₁₃₁	—	2	2	no	—
103	C₂²×C₄×C₁₃₁	—	4	2	no	—
104	C₂²×C₄×C₁₃₁	—	4	2	no	—
105	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
116	C₂×C₈×C₁₃₁	—	2	2	no	—
117	C₂×C₈×C₁₃₁	#41	1	4	yes	—
118	C₄⋊C₄×C₁₃₁	—	2	4	no	—
119	C₄⋊C₄×C₁₃₁	—	2	1	no	—
120	C₄⋊C₄×C₁₃₁	—	2	2	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	—	2	2	no	—
125	C₄⋊C₄×C₁₃₁	—	4	2	no	—
126	C₄⋊C₄×C₁₃₁	—	4	4	no	—
127	C₄⋊C₄×C₁₃₁	—	4	2	no	—
128	C₄⋊C₄×C₁₃₁	—	4	2	no	—
129	C₄⋊C₄×C₁₃₁	—	4	2	no	—
130	C₄⋊C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
131	C₄⋊C₄×C₁₃₁	#41	1	4	yes	—
132	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
133	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
134	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
136	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
137	C₄²×C₁₃₁	—	2	2	no	—
138	C₄²×C₁₃₁	—	2	2	no	—
139	C₄²×C₁₃₁	—	2	2	no	—
140	C₄²×C₁₃₁	—	2	2	no	—
141	C₄²×C₁₃₁	—	2	2	no	—
142	C₄²×C₁₃₁	—	4	1	no	—
143	C₄²×C₁₃₁	—	4	2	no	—
144	C₄²×C₁₃₁	—	4	4	no	—
145	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
146	C₄²×C₁₃₁	#41	1	2	yes	—
147	C₂³×C₁₃₁	—	2	2	no	—
148	C₂³×C₁₃₁	—	4	1	no	—
149	C₂³×C₁₃₁	—	4	1	no	—
150	C₂³×C₁₃₁	—	4	1	no	—
151	C₂³×C₁₃₁	—	4	1	no	—
152	C₂³×C₁₃₁	—	4	1	no	—
153	C₂³×C₁₃₁	—	4	1	no	—
154	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
155	Q₈×C₁₃₁	—	4	8	no	—
156	D₄×C₁₃₁	—	2	4	no	—
157	D₄×C₁₃₁	—	4	2	no	—
158	C₂×C₄×C₁₃₁	—	2	1	no	—
159	C₂×C₄×C₁₃₁	—	2	1	no	—
160	C₂×C₄×C₁₃₁	—	2	1	no	—
161	C₂×C₄×C₁₃₁	—	2	1	no	—
162	C₂×C₄×C₁₃₁	—	2	1	no	—
163	C₂×C₄×C₁₃₁	—	2	1	no	—
164	C₂×C₄×C₁₃₁	—	2	1	no	—
165	C₂×C₄×C₁₃₁	—	2	1	no	—
166	C₂×C₄×C₁₃₁	—	2	1	no	—
167	C₂×C₄×C₁₃₁	—	2	1	no	—
168	C₂×C₄×C₁₃₁	—	2	2	no	—
169	C₂×C₄×C₁₃₁	—	2	4	no	—
170	C₂×C₄×C₁₃₁	—	2	1	no	—
171	C₂×C₄×C₁₃₁	—	2	1	no	—
172	C₂×C₄×C₁₃₁	—	2	1	no	—
173	C₂×C₄×C₁₃₁	—	4	8	no	—
174	C₂×C₄×C₁₃₁	—	4	2	no	—
175	C₂×C₄×C₁₃₁	—	4	1	no	—
176	C₂×C₄×C₁₃₁	—	4	1	no	—
177	C₂×C₄×C₁₃₁	—	4	2	no	—
178	C₂×C₄×C₁₃₁	—	4	1	no	—
179	C₂×C₄×C₁₃₁	—	4	2	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	2	no	—
184	C₂×C₄×C₁₃₁	—	4	1	no	—
185	C₂×C₄×C₁₃₁	—	4	2	no	—
186	C₂×C₄×C₁₃₁	—	4	1	no	—
187	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
188	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
189	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
190	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
191	C₂×C₄×C₁₃₁	#137	1	2	yes	—
192	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
193	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
195	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
196	C₈×C₁₃₁	—	2	4	no	—
197	C₈×C₁₃₁	—	4	2	no	—
198	C₂²×C₁₃₁	—	4	2	no	—
199	C₂²×C₁₃₁	—	4	2	no	—
200	C₂²×C₁₃₁	—	4	2	no	—
201	C₂²×C₁₃₁	—	8	1	no	—
202	C₂²×C₁₃₁	—	8	1	no	—
203	C₂²×C₁₃₁	—	8	1	no	—
204	C₂²×C₁₃₁	—	8	1	no	—
205	C₂²×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
206	C₂²×C₁₃₁	#525	1	1	yes	characteristic
207	C₂²×C₁₃₁	#532	1	1	yes	characteristic
208	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₂²×C₁₃₁	C₂×#137	1	1	yes	characteristic
211	C₂²×C₁₃₁	#575	1	1	yes	characteristic
212	C₄×C₁₃₁	—	2	2	no	—
213	C₄×C₁₃₁	—	2	2	no	—
214	C₄×C₁₃₁	—	2	2	no	—
215	C₄×C₁₃₁	—	2	2	no	—
216	C₄×C₁₃₁	—	2	2	no	—
217	C₄×C₁₃₁	—	4	2	no	—
218	C₄×C₁₃₁	—	4	4	no	—
219	C₄×C₁₃₁	—	4	1	no	—
220	C₄×C₁₃₁	—	8	1	no	—
221	C₄×C₁₃₁	—	8	1	no	—
222	C₄×C₁₃₁	—	8	2	no	—
223	C₄×C₁₃₁	#529	1	2	yes	—
224	C₄×C₁₃₁	#533	1	2	yes	—
225	C₂×C₁₃₁	—	8	2	no	—
226	C₂×C₁₃₁	#2396	1	1	yes	characteristic
227	C₂×C₁₃₁	#3077	1	1	yes	characteristic
228	C₂×C₁₃₁	#3079	1	1	yes	characteristic
229	C₂×C₁₃₁	#3112	1	1	yes	characteristic
230	C₂×C₁₃₁	#3114	1	1	yes	characteristic
231	C₂×C₁₃₁	#3181	1	1	yes	characteristic
232	C₂×C₁₃₁	#3184	1	1	yes	characteristic
233	#23781	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₁₃₁	#23781	1	1	yes	characteristic, direct product factor, Sylow
235	C₄×C₄⋊Q₈	C₂×C₁₃₁	1	1	yes	characteristic, complemented
236	#2293	C₂×C₁₃₁	1	1	yes	characteristic
237	#2099	C₂×C₁₃₁	1	1	yes	characteristic, complemented
238	#2093	C₂×C₁₃₁	1	1	yes	characteristic
239	#2087	C₂×C₁₃₁	1	2	yes	complemented
240	#2065	C₂×C₁₃₁	1	2	yes	—
241	#2058	C₂×C₁₃₁	1	1	yes	characteristic, complemented
242	#2023	C₂×C₁₃₁	1	1	yes	characteristic
243	#1776	C₂×C₁₃₁	1	2	yes	complemented
244	#1767	C₂×C₁₃₁	1	1	yes	characteristic, complemented
245	#1762	C₂×C₁₃₁	1	2	yes	—
246	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
247	C₂×C₄×Q₈	—	2	1	no	—
248	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
249	C₂×#117	—	2	1	no	—
250	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
251	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
252	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
253	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
254	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
255	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
256	#391	—	2	1	no	—
257	#390	C₂²×C₁₃₁	1	1	yes	characteristic
258	#390	C₂²×C₁₃₁	1	1	yes	characteristic
259	#389	—	2	1	no	—
260	#389	C₂²×C₁₃₁	1	2	yes	—
261	#389	C₂²×C₁₃₁	1	1	yes	characteristic
262	#389	C₂²×C₁₃₁	1	1	yes	characteristic
263	#388	—	2	1	no	—
264	#388	—	2	1	no	—
265	#384	C₂²×C₁₃₁	1	1	yes	characteristic
266	C₄×C₄⋊C₄	—	2	1	no	—
267	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
268	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
269	C₄×C₂²⋊C₄	—	2	1	no	—
270	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
271	#340	C₂²×C₁₃₁	1	2	yes	—
272	#340	C₂²×C₁₃₁	1	2	yes	—
273	#340	C₂²×C₁₃₁	1	2	yes	—
274	#340	C₂²×C₁₃₁	1	2	yes	—
275	#336	C₂²×C₁₃₁	1	1	yes	characteristic
276	#336	C₂²×C₁₃₁	1	1	yes	characteristic
277	C₂²×Q₈	—	2	1	no	—
278	C₂²×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
279	C₂³×C₄	—	2	1	no	—
280	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
281	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
282	C₄⋊Q₈	—	2	4	no	—
283	C₄×Q₈	—	4	4	no	—
284	C₄×D₄	—	2	2	no	—
285	C₄×D₄	Q₈×C₁₃₁	1	4	yes	complemented
286	#117	—	4	4	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
298	C₂×C₂²⋊C₄	—	2	1	no	—
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₄²	—	2	1	no	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
306	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
307	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
308	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
310	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
312	#107	—	2	4	no	—
313	#106	—	2	4	no	—
314	C₄⋊C₈	—	2	4	no	—
315	#102	—	2	4	no	—
316	#102	—	2	4	no	—
317	#97	—	2	4	no	—
318	#95	—	2	1	no	—
319	#95	—	2	1	no	—
320	#95	—	2	2	no	—
321	#95	—	2	1	no	—
322	#95	—	2	2	no	—
323	C₂⁴	—	2	1	no	—
324	C₂×Q₈	—	2	4	no	—
325	C₂×Q₈	—	4	2	no	—
326	C₂×Q₈	—	4	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	C₂×Q₈×C₁₃₁	1	2	yes	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	2	no	—
338	C₂²×C₄	—	4	2	no	—
339	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×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	C₂⁴×C₁₃₁	1	1	yes	characteristic
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	#41×C₁₃₁	1	4	yes	—
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	—	4	4	no	—
361	C₄⋊C₄	—	4	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₄⋊C₄	—	4	2	no	—
364	C₄⋊C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
365	C₄⋊C₄	#41×C₁₃₁	1	4	yes	—
366	C₂²⋊C₄	—	2	4	no	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₂²⋊C₄	—	4	2	no	—
369	C₂²⋊C₄	—	4	2	no	—
370	C₂²⋊C₄	—	4	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	4	1	no	—
377	C₄²	—	4	2	no	—
378	C₄²	—	4	4	no	—
379	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
380	C₄²	#41×C₁₃₁	1	2	yes	—
381	C₂³	—	2	2	no	—
382	C₂³	—	4	1	no	—
383	C₂³	—	4	1	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	1	no	—
388	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
389	Q₈	—	4	8	no	—
390	D₄	—	2	4	no	—
391	D₄	—	4	2	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	4	8	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
422	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
423	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
424	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
425	C₂×C₄	#137×C₁₃₁	1	2	yes	—
426	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
429	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
430	C₈	—	2	4	no	—
431	C₈	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	8	1	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	1	no	—
439	C₂²	C₂×C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
440	C₂²	#525×C₁₃₁	1	1	yes	characteristic
441	C₂²	#532×C₁₃₁	1	1	yes	characteristic
442	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
443	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
444	C₂²	C₂×#137×C₁₃₁	1	1	yes	characteristic
445	C₂²	#575×C₁₃₁	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	2	no	—
448	C₄	—	2	2	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	4	no	—
453	C₄	—	4	1	no	—
454	C₄	—	8	1	no	—
455	C₄	—	8	1	no	—
456	C₄	—	8	2	no	—
457	C₄	#529×C₁₃₁	1	2	yes	—
458	C₄	#533×C₁₃₁	1	2	yes	—
459	C₂	—	8	2	no	—
460	C₂	#2396×C₁₃₁	1	1	yes	characteristic
461	C₂	#3077×C₁₃₁	1	1	yes	characteristic
462	C₂	#3079×C₁₃₁	1	1	yes	characteristic
463	C₂	#3112×C₁₃₁	1	1	yes	characteristic
464	C₂	#3114×C₁₃₁	1	1	yes	characteristic
465	C₂	#3181×C₁₃₁	1	1	yes	characteristic
466	C₂	#3184×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊Q₈×C₁₃₁	C₂	semidirect
3. #2099×C₁₃₁	C₂	semidirect
5. #2087×C₁₃₁	C₂	semidirect
7. #2058×C₁₃₁	C₂	semidirect
9. #1776×C₁₃₁	C₂	semidirect
10. #1767×C₁₃₁	C₂	semidirect
51. C₄×D₄×C₁₃₁	Q₈	semidirect
233. #23781	C₁₃₁	direct (N × H)
234. C₁₃₁	#23781	direct (N × H)
235. C₄×C₄⋊Q₈	C₂×C₁₃₁	semidirect
237. #2099	C₂×C₁₃₁	semidirect
239. #2087	C₂×C₁₃₁	semidirect
241. #2058	C₂×C₁₃₁	semidirect
243. #1776	C₂×C₁₃₁	semidirect
244. #1767	C₂×C₁₃₁	semidirect
285. C₄×D₄	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₃₁	2
#2293×C₁₃₁	2
#2099×C₁₃₁	2
#2093×C₁₃₁	2
#2087×C₁₃₁	2
#2065×C₁₃₁	2
#2058×C₁₃₁	2
#2023×C₁₃₁	2
#1776×C₁₃₁	2
#1767×C₁₃₁	2
#1762×C₁₃₁	2
#23781	131

Subgroups — C131×#23781 #45709665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23781 #45709665