Subgroups — C₁₃₁×#23705 #45709589

Order 33536.

820 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#433×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2323×C₁₃₁	C₂	1	2	yes	maximal
4	#2086×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#2075×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#2068×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2056×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#2036×C₁₃₁	C₂	1	2	yes	maximal
9	#2001×C₁₃₁	C₂	1	1	yes	maximal, characteristic
10	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#1763×C₁₃₁	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆×C₁₃₁	—	2	4	no	—
17	#433×C₁₃₁	—	2	4	no	—
18	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
19	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
23	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
24	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
29	#391×C₁₃₁	C₂²	1	2	yes	—
30	#389×C₁₃₁	—	2	2	no	—
31	#389×C₁₃₁	C₂²	1	2	yes	—
32	#388×C₁₃₁	—	2	2	no	—
33	#388×C₁₃₁	—	2	2	no	—
34	#387×C₁₃₁	—	2	2	no	—
35	#380×C₁₃₁	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
37	#340×C₁₃₁	C₂²	1	1	yes	characteristic
38	#340×C₁₃₁	C₂²	1	2	yes	—
39	#340×C₁₃₁	C₂²	1	2	yes	—
40	#340×C₁₃₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
44	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
45	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂²⋊Q₈×C₁₃₁	—	2	4	no	—
50	C₂²⋊Q₈×C₁₃₁	—	2	4	no	—
51	C₄×Q₈×C₁₃₁	—	2	4	no	—
52	C₄×Q₈×C₁₃₁	—	2	4	no	—
53	#117×C₁₃₁	—	2	2	no	—
54	#117×C₁₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₁₃₁	—	2	2	no	—
68	C₂×C₄²×C₁₃₁	—	2	2	no	—
69	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
71	#107×C₁₃₁	—	2	2	no	—
72	#107×C₁₃₁	—	2	4	no	—
73	C₄⋊C₈×C₁₃₁	—	2	2	no	—
74	C₄⋊C₈×C₁₃₁	—	2	2	no	—
75	#103×C₁₃₁	—	2	4	no	—
76	#103×C₁₃₁	—	2	4	no	—
77	#103×C₁₃₁	—	2	4	no	—
78	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
79	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
80	#97×C₁₃₁	—	2	2	no	—
81	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
82	#95×C₁₃₁	—	2	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	C₂⁴×C₁₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×Q₈×C₁₃₁	—	2	2	no	—
89	C₂×Q₈×C₁₃₁	—	2	2	no	—
90	C₂×Q₈×C₁₃₁	—	2	2	no	—
91	C₂×Q₈×C₁₃₁	—	2	2	no	—
92	C₂×Q₈×C₁₃₁	—	2	2	no	—
93	C₂×Q₈×C₁₃₁	#41	1	4	yes	—
94	C₂²×C₄×C₁₃₁	—	2	2	no	—
95	C₂²×C₄×C₁₃₁	—	2	2	no	—
96	C₂²×C₄×C₁₃₁	—	2	1	no	—
97	C₂²×C₄×C₁₃₁	—	2	1	no	—
98	C₂²×C₄×C₁₃₁	—	2	2	no	—
99	C₂²×C₄×C₁₃₁	—	2	2	no	—
100	C₂²×C₄×C₁₃₁	—	2	2	no	—
101	C₂²×C₄×C₁₃₁	—	2	2	no	—
102	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
108	Q₁₆×C₁₃₁	—	2	8	no	—
109	C₂×C₈×C₁₃₁	—	2	2	no	—
110	C₂×C₈×C₁₃₁	—	2	2	no	—
111	C₂×C₈×C₁₃₁	—	2	1	no	—
112	C₂×C₈×C₁₃₁	—	2	1	no	—
113	C₂×C₈×C₁₃₁	—	2	2	no	—
114	C₂×C₈×C₁₃₁	—	2	2	no	—
115	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
117	C₄⋊C₄×C₁₃₁	—	2	8	no	—
118	C₄⋊C₄×C₁₃₁	—	2	2	no	—
119	C₄⋊C₄×C₁₃₁	—	2	2	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	4	no	—
126	C₄⋊C₄×C₁₃₁	—	8	2	no	—
127	C₄⋊C₄×C₁₃₁	#41	1	4	yes	—
128	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
129	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
130	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
131	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
132	C₄²×C₁₃₁	—	2	1	no	—
133	C₄²×C₁₃₁	—	2	1	no	—
134	C₄²×C₁₃₁	—	4	4	no	—
135	C₄²×C₁₃₁	—	8	2	no	—
136	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
137	C₄²×C₁₃₁	#41	1	2	yes	—
138	C₂³×C₁₃₁	—	2	1	no	—
139	C₂³×C₁₃₁	—	2	2	no	—
140	C₂³×C₁₃₁	—	2	1	no	—
141	C₂³×C₁₃₁	—	2	2	no	—
142	C₂³×C₁₃₁	—	2	1	no	—
143	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
144	Q₈×C₁₃₁	—	2	4	no	—
145	Q₈×C₁₃₁	—	2	4	no	—
146	Q₈×C₁₃₁	—	4	2	no	—
147	Q₈×C₁₃₁	—	4	2	no	—
148	C₂×C₄×C₁₃₁	—	2	1	no	—
149	C₂×C₄×C₁₃₁	—	2	2	no	—
150	C₂×C₄×C₁₃₁	—	2	1	no	—
151	C₂×C₄×C₁₃₁	—	2	1	no	—
152	C₂×C₄×C₁₃₁	—	2	1	no	—
153	C₂×C₄×C₁₃₁	—	2	1	no	—
154	C₂×C₄×C₁₃₁	—	2	4	no	—
155	C₂×C₄×C₁₃₁	—	2	4	no	—
156	C₂×C₄×C₁₃₁	—	2	2	no	—
157	C₂×C₄×C₁₃₁	—	2	1	no	—
158	C₂×C₄×C₁₃₁	—	2	2	no	—
159	C₂×C₄×C₁₃₁	—	2	1	no	—
160	C₂×C₄×C₁₃₁	—	4	2	no	—
161	C₂×C₄×C₁₃₁	—	4	2	no	—
162	C₂×C₄×C₁₃₁	—	4	2	no	—
163	C₂×C₄×C₁₃₁	—	4	2	no	—
164	C₂×C₄×C₁₃₁	—	4	1	no	—
165	C₂×C₄×C₁₃₁	—	4	2	no	—
166	C₂×C₄×C₁₃₁	—	4	1	no	—
167	C₂×C₄×C₁₃₁	—	4	2	no	—
168	C₂×C₄×C₁₃₁	—	4	2	no	—
169	C₂×C₄×C₁₃₁	—	4	2	no	—
170	C₂×C₄×C₁₃₁	—	4	2	no	—
171	C₂×C₄×C₁₃₁	—	4	2	no	—
172	C₂×C₄×C₁₃₁	—	4	2	no	—
173	C₂×C₄×C₁₃₁	#121	1	2	yes	—
174	C₂×C₄×C₁₃₁	#121	1	2	yes	—
175	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
177	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
179	C₈×C₁₃₁	—	2	2	no	—
180	C₈×C₁₃₁	—	2	2	no	—
181	C₈×C₁₃₁	—	4	1	no	—
182	C₈×C₁₃₁	—	4	1	no	—
183	C₂²×C₁₃₁	—	4	2	no	—
184	C₂²×C₁₃₁	—	4	2	no	—
185	C₂²×C₁₃₁	—	4	1	no	—
186	C₂²×C₁₃₁	—	4	1	no	—
187	C₂²×C₁₃₁	—	4	1	no	—
188	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
189	C₂²×C₁₃₁	#525	1	2	yes	—
190	C₂²×C₁₃₁	#576	1	1	yes	characteristic
191	C₂²×C₁₃₁	#578	1	2	yes	—
192	C₂²×C₁₃₁	#578	1	1	yes	characteristic
193	C₄×C₁₃₁	—	2	2	no	—
194	C₄×C₁₃₁	—	2	2	no	—
195	C₄×C₁₃₁	—	4	4	no	—
196	C₄×C₁₃₁	—	4	1	no	—
197	C₄×C₁₃₁	—	4	4	no	—
198	C₄×C₁₃₁	—	4	1	no	—
199	C₄×C₁₃₁	—	4	1	no	—
200	C₄×C₁₃₁	—	8	2	no	—
201	C₄×C₁₃₁	—	8	2	no	—
202	C₄×C₁₃₁	#525	1	1	yes	characteristic
203	C₄×C₁₃₁	#529	1	2	yes	—
204	C₄×C₁₃₁	#532	1	1	yes	characteristic
205	C₂×C₁₃₁	—	8	1	no	—
206	C₂×C₁₃₁	#2392	1	1	yes	characteristic
207	C₂×C₁₃₁	#3059	1	1	yes	characteristic
208	C₂×C₁₃₁	#3061	1	1	yes	characteristic
209	C₂×C₁₃₁	#3124	1	2	yes	—
210	C₂×C₁₃₁	#3127	1	2	yes	—
211	#23705	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₁₃₁	#23705	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×C₄×Q₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
214	C₂×#433	C₂×C₁₃₁	1	1	yes	characteristic
215	#2323	C₂×C₁₃₁	1	2	yes	—
216	#2086	C₂×C₁₃₁	1	1	yes	characteristic, complemented
217	#2075	C₂×C₁₃₁	1	1	yes	characteristic
218	#2068	C₂×C₁₃₁	1	2	yes	complemented
219	#2056	C₂×C₁₃₁	1	1	yes	characteristic, complemented
220	#2036	C₂×C₁₃₁	1	2	yes	—
221	#2001	C₂×C₁₃₁	1	1	yes	characteristic
222	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic, complemented
223	#1763	C₂×C₁₃₁	1	2	yes	complemented
224	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
225	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
226	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
227	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
228	C₄×Q₁₆	—	2	4	no	—
229	#433	—	2	4	no	—
230	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
231	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
232	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
234	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
235	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
236	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
237	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
238	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
239	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
240	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
241	#391	C₂²×C₁₃₁	1	2	yes	—
242	#389	—	2	2	no	—
243	#389	C₂²×C₁₃₁	1	2	yes	—
244	#388	—	2	2	no	—
245	#388	—	2	2	no	—
246	#387	—	2	2	no	—
247	#380	C₂²×C₁₃₁	1	2	yes	—
248	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
249	#340	C₂²×C₁₃₁	1	1	yes	characteristic
250	#340	C₂²×C₁₃₁	1	2	yes	—
251	#340	C₂²×C₁₃₁	1	1	yes	characteristic
252	#340	C₂²×C₁₃₁	1	2	yes	—
253	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
254	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
255	C₂×Q₁₆	—	2	4	no	—
256	C₂×Q₁₆	—	2	2	no	—
257	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
261	C₂²⋊Q₈	—	2	4	no	—
262	C₂²⋊Q₈	—	2	4	no	—
263	C₄×Q₈	—	2	4	no	—
264	C₄×Q₈	—	2	4	no	—
265	#117	—	2	2	no	—
266	#117	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
278	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
279	C₂×C₄²	—	2	2	no	—
280	C₂×C₄²	—	2	2	no	—
281	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
283	#107	—	2	2	no	—
284	#107	—	2	4	no	—
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	—	2	2	no	—
287	#103	—	2	4	no	—
288	#103	—	2	4	no	—
289	#103	—	2	4	no	—
290	C₂²⋊C₈	—	2	2	no	—
291	C₂²⋊C₈	—	2	2	no	—
292	#97	—	2	2	no	—
293	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	—	2	2	no	—
297	#95	—	2	2	no	—
298	#95	—	2	2	no	—
299	C₂⁴	C₂×D₄×C₁₃₁	1	1	yes	characteristic
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	#41×C₁₃₁	1	4	yes	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
320	Q₁₆	—	2	8	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
328	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	4	no	—
338	C₄⋊C₄	—	8	2	no	—
339	C₄⋊C₄	#41×C₁₃₁	1	4	yes	—
340	C₂²⋊C₄	—	2	8	no	—
341	C₂²⋊C₄	—	2	2	no	—
342	C₂²⋊C₄	—	2	2	no	—
343	C₂²⋊C₄	—	8	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	4	4	no	—
347	C₄²	—	8	2	no	—
348	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
349	C₄²	#41×C₁₃₁	1	2	yes	—
350	C₂³	—	2	1	no	—
351	C₂³	—	2	2	no	—
352	C₂³	—	2	1	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	1	no	—
355	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
356	Q₈	—	2	4	no	—
357	Q₈	—	2	4	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	#121×C₁₃₁	1	2	yes	—
386	C₂×C₄	#121×C₁₃₁	1	2	yes	—
387	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
388	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
389	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
391	C₈	—	2	2	no	—
392	C₈	—	2	2	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	4	2	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	1	no	—
400	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
401	C₂²	#525×C₁₃₁	1	2	yes	—
402	C₂²	#576×C₁₃₁	1	1	yes	characteristic
403	C₂²	#578×C₁₃₁	1	2	yes	—
404	C₂²	#578×C₁₃₁	1	1	yes	characteristic
405	C₄	—	2	2	no	—
406	C₄	—	2	2	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	—	4	4	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	4	no	—
412	C₄	—	8	2	no	—
413	C₄	—	8	2	no	—
414	C₄	#525×C₁₃₁	1	1	yes	characteristic
415	C₄	#529×C₁₃₁	1	2	yes	—
416	C₄	#532×C₁₃₁	1	1	yes	characteristic
417	C₂	—	8	1	no	—
418	C₂	#2392×C₁₃₁	1	1	yes	characteristic
419	C₂	#3059×C₁₃₁	1	1	yes	characteristic
420	C₂	#3061×C₁₃₁	1	1	yes	characteristic
421	C₂	#3124×C₁₃₁	1	2	yes	—
422	C₂	#3127×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂⁴×C₁₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×#117×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₁₆×C₁₃₁	C₂	semidirect
4. #2086×C₁₃₁	C₂	semidirect
6. #2068×C₁₃₁	C₂	semidirect
7. #2056×C₁₃₁	C₂	semidirect
10. C₄×#107×C₁₃₁	C₂	semidirect
11. #1763×C₁₃₁	C₂	semidirect
211. #23705	C₁₃₁	direct (N × H)
212. C₁₃₁	#23705	direct (N × H)
213. C₂×C₄×Q₁₆	C₂×C₁₃₁	semidirect
216. #2086	C₂×C₁₃₁	semidirect
218. #2068	C₂×C₁₃₁	semidirect
219. #2056	C₂×C₁₃₁	semidirect
222. C₄×#107	C₂×C₁₃₁	semidirect
223. #1763	C₂×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×Q₁₆×C₁₃₁	2
C₂×#433×C₁₃₁	2
#2323×C₁₃₁	2
#2086×C₁₃₁	2
#2075×C₁₃₁	2
#2068×C₁₃₁	2
#2056×C₁₃₁	2
#2036×C₁₃₁	2
#2001×C₁₃₁	2
C₄×#107×C₁₃₁	2
#1763×C₁₃₁	2
#23705	131

Subgroups — C131×#23705 #45709589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23705 #45709589