Subgroups — C₁₃₁×#19410 #45705294

Order 33536.

758 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	#3205×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3038×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#3035×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2988×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1682×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1681×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1668×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1667×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#1571×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#1547×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#331×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₁₃₁	—	2	1	no	—
13	C₂×#135×C₁₃₁	—	2	1	no	—
14	#532×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄×#41×C₁₃₁	C₂²	1	1	yes	characteristic
17	#493×C₁₃₁	C₂²	1	2	yes	—
18	#486×C₁₃₁	C₂²	1	2	yes	—
19	#464×C₁₃₁	—	2	1	no	—
20	#463×C₁₃₁	—	2	1	no	—
21	#462×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
22	#461×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
23	#460×C₁₃₁	—	2	1	no	—
24	#460×C₁₃₁	C₂²	1	2	yes	—
25	#460×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
26	#459×C₁₃₁	—	2	1	no	—
27	#459×C₁₃₁	—	2	1	no	—
28	#459×C₁₃₁	—	2	1	no	—
29	#459×C₁₃₁	C₂²	1	2	yes	—
30	#459×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
31	#447×C₁₃₁	—	2	2	no	—
32	C₈×Q₈×C₁₃₁	C₂²	1	2	yes	complemented
33	C₈×D₄×C₁₃₁	—	2	2	no	—
34	C₈×D₄×C₁₃₁	C₂²	1	2	yes	complemented
35	#431×C₁₃₁	—	2	1	no	—
36	#431×C₁₃₁	C₂²	1	1	yes	characteristic
37	#423×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
39	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
41	#418×C₁₃₁	—	2	1	no	—
42	#416×C₁₃₁	—	2	1	no	—
43	#335×C₁₃₁	C₂²	1	2	yes	complemented
44	#335×C₁₃₁	C₂²	1	2	yes	complemented
45	#332×C₁₃₁	C₂²	1	2	yes	complemented
46	#331×C₁₃₁	C₂²	1	2	yes	—
47	#331×C₁₃₁	C₂²	1	2	yes	—
48	#331×C₁₃₁	C₂²	1	2	yes	complemented
49	C₂×#41×C₁₃₁	—	2	1	no	—
50	C₂×#41×C₁₃₁	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₁₃₁	—	2	1	no	—
52	C₂²×D₄×C₁₃₁	—	2	1	no	—
53	C₂²×D₄×C₁₃₁	—	2	1	no	—
54	AGL(1,ℤ/8)×C₁₃₁	—	4	4	no	—
55	#135×C₁₃₁	—	4	4	no	—
56	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
57	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₁₃₁	—	2	1	no	—
59	C₂×SD₁₆×C₁₃₁	—	2	1	no	—
60	C₂×D₈×C₁₃₁	—	2	2	no	—
61	C₂×D₈×C₁₃₁	—	2	1	no	—
62	C₂×D₈×C₁₃₁	—	2	2	no	—
63	C₂×D₈×C₁₃₁	—	2	2	no	—
64	C₂×M₁₆×C₁₃₁	—	2	1	no	—
65	C₂²×C₈×C₁₃₁	—	2	1	no	—
66	C₂²×C₈×C₁₃₁	—	2	1	no	—
67	C₂²×C₈×C₁₃₁	—	2	2	no	—
68	C₄⋊Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
69	#127×C₁₃₁	—	2	1	no	—
70	#127×C₁₃₁	—	2	1	no	—
71	#127×C₁₃₁	—	2	1	no	—
72	#127×C₁₃₁	C₂³	1	2	yes	—
73	#127×C₁₃₁	C₂³	1	1	yes	characteristic
74	#124×C₁₃₁	—	2	1	no	—
75	#121×C₁₃₁	—	2	1	no	—
76	#121×C₁₃₁	—	2	1	no	—
77	C₄×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
78	C₄×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
79	C₄×D₄×C₁₃₁	—	2	1	no	—
80	C₄×D₄×C₁₃₁	—	2	1	no	—
81	C₄×D₄×C₁₃₁	—	2	1	no	—
82	C₄×D₄×C₁₃₁	—	2	1	no	—
83	C₄×D₄×C₁₃₁	D₄	1	1	yes	characteristic
84	C₄×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
85	#117×C₁₃₁	—	2	1	no	—
86	#117×C₁₃₁	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₁₃₁	—	2	1	no	—
88	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₁₃₁	—	2	2	no	—
90	C₄⋊C₈×C₁₃₁	—	2	1	no	—
91	C₄⋊C₈×C₁₃₁	—	2	1	no	—
92	C₄⋊C₈×C₁₃₁	—	2	1	no	—
93	C₄⋊C₈×C₁₃₁	D₄	1	1	yes	characteristic
94	C₄⋊C₈×C₁₃₁	D₄	1	1	yes	characteristic
95	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
96	C₄⋊C₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
99	#103×C₁₃₁	—	2	1	no	—
100	#103×C₁₃₁	—	2	1	no	—
101	#102×C₁₃₁	—	2	2	no	—
102	#102×C₁₃₁	—	2	2	no	—
103	#102×C₁₃₁	—	2	1	no	—
104	#102×C₁₃₁	—	2	1	no	—
105	#102×C₁₃₁	—	2	2	no	—
106	#102×C₁₃₁	—	2	2	no	—
107	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
108	C₂²⋊C₈×C₁₃₁	D₄	1	2	yes	—
109	C₄×C₈×C₁₃₁	—	2	2	no	—
110	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
111	C₂⁴×C₁₃₁	—	4	1	no	—
112	#41×C₁₃₁	—	2	2	no	—
113	#41×C₁₃₁	—	4	2	no	—
114	#41×C₁₃₁	—	4	2	no	—
115	#41×C₁₃₁	D₈	1	4	yes	complemented
116	C₂×Q₈×C₁₃₁	—	2	1	no	—
117	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×D₄×C₁₃₁	—	2	1	no	—
119	C₂×D₄×C₁₃₁	—	2	1	no	—
120	C₂×D₄×C₁₃₁	—	2	2	no	—
121	C₂×D₄×C₁₃₁	—	2	2	no	—
122	C₂×D₄×C₁₃₁	—	2	1	no	—
123	C₂×D₄×C₁₃₁	—	2	1	no	—
124	C₂×D₄×C₁₃₁	—	2	2	no	—
125	C₂×D₄×C₁₃₁	—	4	1	no	—
126	C₂×D₄×C₁₃₁	—	4	1	no	—
127	C₂×D₄×C₁₃₁	—	4	1	no	—
128	C₂×D₄×C₁₃₁	—	4	1	no	—
129	C₂×D₄×C₁₃₁	—	4	2	no	—
130	C₂×D₄×C₁₃₁	—	4	2	no	—
131	C₂×D₄×C₁₃₁	—	8	1	no	—
132	C₂×D₄×C₁₃₁	—	8	1	no	—
133	C₂×D₄×C₁₃₁	—	8	2	no	—
134	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₃₁	—	2	1	no	—
136	C₂²×C₄×C₁₃₁	—	2	1	no	—
137	C₂²×C₄×C₁₃₁	—	2	1	no	—
138	C₂²×C₄×C₁₃₁	—	4	1	no	—
139	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
142	Q₁₆×C₁₃₁	—	4	2	no	—
143	SD₁₆×C₁₃₁	—	4	2	no	—
144	SD₁₆×C₁₃₁	—	4	4	no	—
145	SD₁₆×C₁₃₁	—	4	2	no	—
146	D₈×C₁₃₁	—	4	2	no	—
147	D₈×C₁₃₁	—	4	4	no	—
148	D₈×C₁₃₁	—	8	2	no	—
149	D₈×C₁₃₁	—	8	2	no	—
150	M₁₆×C₁₃₁	—	4	2	no	—
151	C₂×C₈×C₁₃₁	—	2	2	no	—
152	C₂×C₈×C₁₃₁	—	2	2	no	—
153	C₂×C₈×C₁₃₁	—	2	1	no	—
154	C₂×C₈×C₁₃₁	—	2	1	no	—
155	C₂×C₈×C₁₃₁	—	4	2	no	—
156	C₂×C₈×C₁₃₁	—	8	2	no	—
157	C₂×C₈×C₁₃₁	—	8	1	no	—
158	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₁₃₁	—	2	1	no	—
161	C₄⋊C₄×C₁₃₁	—	2	1	no	—
162	C₄⋊C₄×C₁₃₁	—	2	1	no	—
163	C₄⋊C₄×C₁₃₁	—	2	1	no	—
164	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
165	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
167	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
168	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
170	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
171	C₄²×C₁₃₁	—	2	1	no	—
172	C₄²×C₁₃₁	—	2	1	no	—
173	C₄²×C₁₃₁	—	2	1	no	—
174	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₁₃₁	#41	1	1	yes	characteristic
177	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
178	C₂³×C₁₃₁	—	2	1	no	—
179	C₂³×C₁₃₁	—	4	1	no	—
180	C₂³×C₁₃₁	—	4	2	no	—
181	C₂³×C₁₃₁	—	8	2	no	—
182	C₂³×C₁₃₁	—	8	2	no	—
183	C₂³×C₁₃₁	—	8	2	no	—
184	C₂³×C₁₃₁	#121	1	1	yes	characteristic
185	Q₈×C₁₃₁	—	2	1	no	—
186	Q₈×C₁₃₁	—	4	1	no	—
187	Q₈×C₁₃₁	—	4	1	no	—
188	Q₈×C₁₃₁	C₂×D₈	1	2	yes	—
189	D₄×C₁₃₁	—	2	4	no	—
190	D₄×C₁₃₁	—	2	1	no	—
191	D₄×C₁₃₁	—	4	2	no	—
192	D₄×C₁₃₁	—	4	1	no	—
193	D₄×C₁₃₁	—	4	2	no	—
194	D₄×C₁₃₁	—	4	1	no	—
195	D₄×C₁₃₁	—	4	1	no	—
196	D₄×C₁₃₁	—	4	2	no	—
197	D₄×C₁₃₁	—	4	1	no	—
198	D₄×C₁₃₁	—	4	2	no	—
199	D₄×C₁₃₁	—	8	1	no	—
200	D₄×C₁₃₁	—	8	2	no	—
201	D₄×C₁₃₁	—	8	2	no	—
202	D₄×C₁₃₁	—	8	2	no	—
203	D₄×C₁₃₁	—	8	2	no	—
204	D₄×C₁₃₁	—	8	2	no	—
205	D₄×C₁₃₁	C₂×D₈	1	2	yes	—
206	C₂×C₄×C₁₃₁	—	2	1	no	—
207	C₂×C₄×C₁₃₁	—	2	1	no	—
208	C₂×C₄×C₁₃₁	—	2	1	no	—
209	C₂×C₄×C₁₃₁	—	2	4	no	—
210	C₂×C₄×C₁₃₁	—	2	1	no	—
211	C₂×C₄×C₁₃₁	—	2	1	no	—
212	C₂×C₄×C₁₃₁	—	4	1	no	—
213	C₂×C₄×C₁₃₁	—	4	1	no	—
214	C₂×C₄×C₁₃₁	—	4	1	no	—
215	C₂×C₄×C₁₃₁	—	4	1	no	—
216	C₂×C₄×C₁₃₁	—	8	1	no	—
217	C₂×C₄×C₁₃₁	—	8	2	no	—
218	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
219	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
220	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
221	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
222	C₂×C₄×C₁₃₁	#135	1	2	yes	—
223	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
226	C₈×C₁₃₁	—	2	2	no	—
227	C₈×C₁₃₁	—	2	2	no	—
228	C₈×C₁₃₁	—	4	1	no	—
229	C₈×C₁₃₁	—	4	1	no	—
230	C₈×C₁₃₁	—	4	2	no	—
231	C₈×C₁₃₁	—	4	2	no	—
232	C₂²×C₁₃₁	—	2	1	no	—
233	C₂²×C₁₃₁	—	2	1	no	—
234	C₂²×C₁₃₁	—	2	2	no	—
235	C₂²×C₁₃₁	—	4	1	no	—
236	C₂²×C₁₃₁	—	4	1	no	—
237	C₂²×C₁₃₁	—	8	2	no	—
238	C₂²×C₁₃₁	—	8	1	no	—
239	C₂²×C₁₃₁	—	8	1	no	—
240	C₂²×C₁₃₁	—	8	2	no	—
241	C₂²×C₁₃₁	—	8	2	no	—
242	C₂²×C₁₃₁	—	8	1	no	—
243	C₂²×C₁₃₁	—	16	4	no	—
244	C₂²×C₁₃₁	#466	1	2	yes	—
245	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
246	C₄×C₁₃₁	—	2	1	no	—
247	C₄×C₁₃₁	—	2	2	no	—
248	C₄×C₁₃₁	—	2	1	no	—
249	C₄×C₁₃₁	—	2	1	no	—
250	C₄×C₁₃₁	—	4	1	no	—
251	C₄×C₁₃₁	—	4	1	no	—
252	C₄×C₁₃₁	—	8	1	no	—
253	C₄×C₁₃₁	#466	1	2	yes	—
254	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
255	C₄×C₁₃₁	C₂×#135	1	1	yes	characteristic
256	C₄×C₁₃₁	#576	1	1	yes	characteristic
257	C₄×C₁₃₁	#577	1	1	yes	characteristic
258	C₂×C₁₃₁	—	2	2	no	—
259	C₂×C₁₃₁	—	4	2	no	—
260	C₂×C₁₃₁	—	16	1	no	—
261	C₂×C₁₃₁	—	16	2	no	—
262	C₂×C₁₃₁	#3045	1	1	yes	characteristic
263	C₂×C₁₃₁	C₂×#466	1	1	yes	characteristic
264	C₂×C₁₃₁	#3068	1	1	yes	characteristic
265	#19410	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₁₃₁	#19410	1	1	yes	characteristic, direct product factor, Sylow
267	#3205	C₂×C₁₃₁	1	1	yes	characteristic, complemented
268	#3038	C₂×C₁₃₁	1	1	yes	characteristic, complemented
269	#3035	C₂×C₁₃₁	1	1	yes	characteristic, complemented
270	#2988	C₂×C₁₃₁	1	1	yes	characteristic, complemented
271	#1682	C₂×C₁₃₁	1	2	yes	complemented
272	#1681	C₂×C₁₃₁	1	2	yes	complemented
273	#1668	C₂×C₁₃₁	1	2	yes	complemented
274	#1667	C₂×C₁₃₁	1	2	yes	complemented
275	#1571	C₂×C₁₃₁	1	1	yes	characteristic, complemented
276	#1547	C₂×C₁₃₁	1	1	yes	characteristic, complemented
277	C₂×#331	C₂×C₁₃₁	1	1	yes	characteristic, complemented
278	C₂×AGL(1,ℤ/8)	—	2	1	no	—
279	C₂×#135	—	2	1	no	—
280	#532	C₂²×C₁₃₁	1	1	yes	characteristic
281	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
282	C₄×#41	C₂²×C₁₃₁	1	1	yes	characteristic
283	#493	C₂²×C₁₃₁	1	2	yes	—
284	#486	C₂²×C₁₃₁	1	2	yes	—
285	#464	—	2	1	no	—
286	#463	—	2	1	no	—
287	#462	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
288	#461	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
289	#460	—	2	1	no	—
290	#460	C₂²×C₁₃₁	1	2	yes	—
291	#460	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
292	#459	—	2	1	no	—
293	#459	—	2	1	no	—
294	#459	—	2	1	no	—
295	#459	C₂²×C₁₃₁	1	2	yes	—
296	#459	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
297	#447	—	2	2	no	—
298	C₈×Q₈	C₂²×C₁₃₁	1	2	yes	complemented
299	C₈×D₄	—	2	2	no	—
300	C₈×D₄	C₂²×C₁₃₁	1	2	yes	complemented
301	#431	—	2	1	no	—
302	#431	C₂²×C₁₃₁	1	1	yes	characteristic
303	#423	C₂²×C₁₃₁	1	1	yes	characteristic
304	C₂×C₄⋊C₈	—	2	1	no	—
305	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
306	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
307	#418	—	2	1	no	—
308	#416	—	2	1	no	—
309	#335	C₂²×C₁₃₁	1	2	yes	complemented
310	#335	C₂²×C₁₃₁	1	2	yes	complemented
311	#332	C₂²×C₁₃₁	1	2	yes	complemented
312	#331	C₂²×C₁₃₁	1	2	yes	—
313	#331	C₂²×C₁₃₁	1	2	yes	—
314	#331	C₂²×C₁₃₁	1	2	yes	complemented
315	C₂×#41	—	2	1	no	—
316	C₂×#41	D₄×C₁₃₁	1	1	yes	characteristic
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	—	2	1	no	—
320	AGL(1,ℤ/8)	—	4	4	no	—
321	#135	—	4	4	no	—
322	C₂×Q₁₆	—	2	1	no	—
323	C₂×SD₁₆	—	2	1	no	—
324	C₂×SD₁₆	—	2	2	no	—
325	C₂×SD₁₆	—	2	1	no	—
326	C₂×D₈	—	2	1	no	—
327	C₂×D₈	—	2	2	no	—
328	C₂×D₈	—	2	2	no	—
329	C₂×D₈	—	2	2	no	—
330	C₂×M₁₆	—	2	1	no	—
331	C₂²×C₈	—	2	1	no	—
332	C₂²×C₈	—	2	1	no	—
333	C₂²×C₈	—	2	2	no	—
334	C₄⋊Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
335	#127	—	2	1	no	—
336	#127	—	2	1	no	—
337	#127	—	2	1	no	—
338	#127	C₂³×C₁₃₁	1	2	yes	—
339	#127	C₂³×C₁₃₁	1	1	yes	characteristic
340	#124	—	2	1	no	—
341	#121	—	2	1	no	—
342	#121	—	2	1	no	—
343	C₄×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
344	C₄×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	1	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	—	2	1	no	—
349	C₄×D₄	D₄×C₁₃₁	1	1	yes	characteristic
350	C₄×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
351	#117	—	2	1	no	—
352	#117	D₄×C₁₃₁	1	1	yes	characteristic
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
355	C₄⋊C₈	—	2	2	no	—
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	D₄×C₁₃₁	1	1	yes	characteristic
360	C₄⋊C₈	D₄×C₁₃₁	1	1	yes	characteristic
361	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
362	C₄⋊C₈	C₂³×C₁₃₁	1	1	yes	characteristic
363	C₄⋊C₈	C₂³×C₁₃₁	1	1	yes	characteristic
364	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
365	#103	—	2	1	no	—
366	#103	—	2	1	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	#102	—	2	1	no	—
370	#102	—	2	1	no	—
371	#102	—	2	2	no	—
372	#102	—	2	2	no	—
373	C₂²⋊C₈	—	2	2	no	—
374	C₂²⋊C₈	D₄×C₁₃₁	1	2	yes	—
375	C₄×C₈	—	2	2	no	—
376	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
377	C₂⁴	—	4	1	no	—
378	#41	—	2	2	no	—
379	#41	—	4	2	no	—
380	#41	—	4	2	no	—
381	#41	D₈×C₁₃₁	1	4	yes	complemented
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	4	1	no	—
392	C₂×D₄	—	4	1	no	—
393	C₂×D₄	—	4	1	no	—
394	C₂×D₄	—	4	1	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	—	4	2	no	—
397	C₂×D₄	—	8	1	no	—
398	C₂×D₄	—	8	1	no	—
399	C₂×D₄	—	8	2	no	—
400	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	4	1	no	—
405	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
407	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
408	Q₁₆	—	4	2	no	—
409	SD₁₆	—	4	2	no	—
410	SD₁₆	—	4	4	no	—
411	SD₁₆	—	4	2	no	—
412	D₈	—	4	2	no	—
413	D₈	—	4	4	no	—
414	D₈	—	8	2	no	—
415	D₈	—	8	2	no	—
416	M₁₆	—	4	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	4	2	no	—
422	C₂×C₈	—	8	2	no	—
423	C₂×C₈	—	8	1	no	—
424	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
425	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
431	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
432	C₂²⋊C₄	—	2	1	no	—
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	4	1	no	—
436	C₂²⋊C₄	—	4	1	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
441	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
442	C₄²	#41×C₁₃₁	1	1	yes	characteristic
443	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
444	C₂³	—	2	1	no	—
445	C₂³	—	4	1	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	8	2	no	—
448	C₂³	—	8	2	no	—
449	C₂³	—	8	2	no	—
450	C₂³	#121×C₁₃₁	1	1	yes	characteristic
451	Q₈	—	2	1	no	—
452	Q₈	—	4	1	no	—
453	Q₈	—	4	1	no	—
454	Q₈	C₂×D₈×C₁₃₁	1	2	yes	—
455	D₄	—	2	4	no	—
456	D₄	—	2	1	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	1	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	2	no	—
465	D₄	—	8	2	no	—
466	D₄	—	8	1	no	—
467	D₄	—	8	2	no	—
468	D₄	—	8	2	no	—
469	D₄	—	8	2	no	—
470	D₄	—	8	2	no	—
471	D₄	C₂×D₈×C₁₃₁	1	2	yes	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	8	1	no	—
483	C₂×C₄	—	8	2	no	—
484	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
485	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
486	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
487	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
488	C₂×C₄	#135×C₁₃₁	1	2	yes	—
489	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
492	C₈	—	2	2	no	—
493	C₈	—	2	2	no	—
494	C₈	—	4	1	no	—
495	C₈	—	4	1	no	—
496	C₈	—	4	2	no	—
497	C₈	—	4	2	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	8	2	no	—
504	C₂²	—	8	1	no	—
505	C₂²	—	8	1	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	1	no	—
509	C₂²	—	16	4	no	—
510	C₂²	#466×C₁₃₁	1	2	yes	—
511	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
512	C₄	—	2	1	no	—
513	C₄	—	2	2	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	8	1	no	—
519	C₄	#466×C₁₃₁	1	2	yes	—
520	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
521	C₄	C₂×#135×C₁₃₁	1	1	yes	characteristic
522	C₄	#576×C₁₃₁	1	1	yes	characteristic
523	C₄	#577×C₁₃₁	1	1	yes	characteristic
524	C₂	—	2	2	no	—
525	C₂	—	4	2	no	—
526	C₂	—	16	2	no	—
527	C₂	—	16	1	no	—
528	C₂	#3045×C₁₃₁	1	1	yes	characteristic
529	C₂	C₂×#466×C₁₃₁	1	1	yes	characteristic
530	C₂	#3068×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
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
1. #3205×C₁₃₁	C₂	semidirect
2. #3038×C₁₃₁	C₂	semidirect
3. #3035×C₁₃₁	C₂	semidirect
4. #2988×C₁₃₁	C₂	semidirect
5. #1682×C₁₃₁	C₂	semidirect
6. #1681×C₁₃₁	C₂	semidirect
7. #1668×C₁₃₁	C₂	semidirect
8. #1667×C₁₃₁	C₂	semidirect
9. #1571×C₁₃₁	C₂	semidirect
10. #1547×C₁₃₁	C₂	semidirect
11. C₂×#331×C₁₃₁	C₂	semidirect
21. #462×C₁₃₁	C₂²	semidirect
22. #461×C₁₃₁	C₂²	semidirect
25. #460×C₁₃₁	C₂²	semidirect
30. #459×C₁₃₁	C₂²	semidirect
32. C₈×Q₈×C₁₃₁	C₂²	semidirect
34. C₈×D₄×C₁₃₁	C₂²	semidirect
43. #335×C₁₃₁	C₂²	semidirect
44. #335×C₁₃₁	C₂²	semidirect
45. #332×C₁₃₁	C₂²	semidirect
48. #331×C₁₃₁	C₂²	semidirect
115. #41×C₁₃₁	D₈	semidirect
265. #19410	C₁₃₁	direct (N × H)
266. C₁₃₁	#19410	direct (N × H)
267. #3205	C₂×C₁₃₁	semidirect
268. #3038	C₂×C₁₃₁	semidirect
269. #3035	C₂×C₁₃₁	semidirect
270. #2988	C₂×C₁₃₁	semidirect
271. #1682	C₂×C₁₃₁	semidirect
272. #1681	C₂×C₁₃₁	semidirect
273. #1668	C₂×C₁₃₁	semidirect
274. #1667	C₂×C₁₃₁	semidirect
275. #1571	C₂×C₁₃₁	semidirect
276. #1547	C₂×C₁₃₁	semidirect
277. C₂×#331	C₂×C₁₃₁	semidirect
287. #462	C₂²×C₁₃₁	semidirect
288. #461	C₂²×C₁₃₁	semidirect
291. #460	C₂²×C₁₃₁	semidirect
296. #459	C₂²×C₁₃₁	semidirect
298. C₈×Q₈	C₂²×C₁₃₁	semidirect
300. C₈×D₄	C₂²×C₁₃₁	semidirect
309. #335	C₂²×C₁₃₁	semidirect
310. #335	C₂²×C₁₃₁	semidirect
311. #332	C₂²×C₁₃₁	semidirect
314. #331	C₂²×C₁₃₁	semidirect
381. #41	D₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3205×C₁₃₁	2
#3038×C₁₃₁	2
#3035×C₁₃₁	2
#2988×C₁₃₁	2
#1682×C₁₃₁	2
#1681×C₁₃₁	2
#1668×C₁₃₁	2
#1667×C₁₃₁	2
#1571×C₁₃₁	2
#1547×C₁₃₁	2
C₂×#331×C₁₃₁	2
#19410	131

Subgroups — C131×#19410 #45705294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#19410 #45705294