Subgroups — C₁₃₁×#23183 #45709067

Order 33536.

1038 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×SD₁₆×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2367×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1983×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1968×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1879×C₁₃₁	C₂	1	2	yes	maximal
7	#1878×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₁₃₁	C₂²	1	2	yes	—
12	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×#124×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
16	#495×C₁₃₁	C₄	1	8	yes	complemented
17	C₄×SD₁₆×C₁₃₁	—	2	8	no	—
18	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
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	2	yes	—
23	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
24	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×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	#388×C₁₃₁	—	2	2	no	—
31	#385×C₁₃₁	—	2	2	no	—
32	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
35	#340×C₁₃₁	C₂²	1	2	yes	—
36	#336×C₁₃₁	C₂²	1	2	yes	—
37	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
38	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₁₃₁	—	4	1	no	—
41	C₂×Q₁₆×C₁₃₁	—	4	1	no	—
42	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	4	yes	complemented
43	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
44	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
45	C₂×SD₁₆×C₁₃₁	C₂×C₄	1	8	yes	complemented
46	C₂×D₈×C₁₃₁	—	4	1	no	—
47	C₂×D₈×C₁₃₁	—	4	1	no	—
48	C₂×D₈×C₁₃₁	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
52	#124×C₁₃₁	—	2	4	no	—
53	#124×C₁₃₁	C₂×C₄	1	8	yes	complemented
54	C₄×Q₈×C₁₃₁	—	2	4	no	—
55	C₄×Q₈×C₁₃₁	—	4	2	no	—
56	C₄×D₄×C₁₃₁	—	2	4	no	—
57	C₄×D₄×C₁₃₁	—	4	2	no	—
58	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
62	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
67	C₂×C₄²×C₁₃₁	—	2	2	no	—
68	C₂×C₄²×C₁₃₁	—	2	2	no	—
69	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
71	#106×C₁₃₁	—	2	2	no	—
72	#106×C₁₃₁	D₄	1	4	yes	—
73	C₄⋊C₈×C₁₃₁	—	2	2	no	—
74	#103×C₁₃₁	—	2	4	no	—
75	#103×C₁₃₁	—	4	2	no	—
76	#102×C₁₃₁	—	2	4	no	—
77	#102×C₁₃₁	—	4	2	no	—
78	C₄×C₈×C₁₃₁	—	2	4	no	—
79	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
80	#95×C₁₃₁	—	2	2	no	—
81	#95×C₁₃₁	—	2	2	no	—
82	#95×C₁₃₁	—	2	2	no	—
83	#95×C₁₃₁	D₄	1	1	yes	characteristic
84	#95×C₁₃₁	D₄	1	1	yes	characteristic
85	C₂⁴×C₁₃₁	—	2	2	no	—
86	C₂×Q₈×C₁₃₁	—	2	2	no	—
87	C₂×Q₈×C₁₃₁	—	2	2	no	—
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	C₂²×C₄	1	4	yes	—
92	C₂×D₄×C₁₃₁	—	2	2	no	—
93	C₂×D₄×C₁₃₁	—	2	2	no	—
94	C₂×D₄×C₁₃₁	—	2	2	no	—
95	C₂×D₄×C₁₃₁	—	2	2	no	—
96	C₂×D₄×C₁₃₁	—	2	2	no	—
97	C₂×D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
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	—	4	4	no	—
102	C₂²×C₄×C₁₃₁	—	4	4	no	—
103	C₂²×C₄×C₁₃₁	—	8	2	no	—
104	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
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	2	yes	—
108	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
109	Q₁₆×C₁₃₁	—	4	4	no	—
110	SD₁₆×C₁₃₁	—	2	16	no	—
111	D₈×C₁₃₁	—	4	4	no	—
112	C₂×C₈×C₁₃₁	—	2	1	no	—
113	C₂×C₈×C₁₃₁	—	2	1	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	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₁₃₁	#41	1	2	yes	—
121	C₄⋊C₄×C₁₃₁	—	2	4	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	C₂×D₄	1	4	yes	—
127	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
128	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
129	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
130	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
131	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
132	C₄²×C₁₃₁	—	2	2	no	—
133	C₄²×C₁₃₁	—	2	2	no	—
134	C₄²×C₁₃₁	—	4	4	no	—
135	C₄²×C₁₃₁	—	8	2	no	—
136	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
137	C₄²×C₁₃₁	#41	1	2	yes	—
138	C₂³×C₁₃₁	—	2	4	no	—
139	C₂³×C₁₃₁	—	4	2	no	—
140	C₂³×C₁₃₁	—	4	2	no	—
141	C₂³×C₁₃₁	—	4	2	no	—
142	C₂³×C₁₃₁	—	4	2	no	—
143	C₂³×C₁₃₁	—	4	2	no	—
144	C₂³×C₁₃₁	—	4	2	no	—
145	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₁₃₁	—	2	4	no	—
147	Q₈×C₁₃₁	—	2	4	no	—
148	Q₈×C₁₃₁	—	4	2	no	—
149	Q₈×C₁₃₁	—	4	2	no	—
150	D₄×C₁₃₁	—	2	4	no	—
151	D₄×C₁₃₁	—	2	4	no	—
152	D₄×C₁₃₁	—	4	2	no	—
153	D₄×C₁₃₁	—	4	2	no	—
154	C₂×C₄×C₁₃₁	—	2	2	no	—
155	C₂×C₄×C₁₃₁	—	2	2	no	—
156	C₂×C₄×C₁₃₁	—	2	2	no	—
157	C₂×C₄×C₁₃₁	—	2	2	no	—
158	C₂×C₄×C₁₃₁	—	2	4	no	—
159	C₂×C₄×C₁₃₁	—	2	2	no	—
160	C₂×C₄×C₁₃₁	—	2	4	no	—
161	C₂×C₄×C₁₃₁	—	2	2	no	—
162	C₂×C₄×C₁₃₁	—	2	2	no	—
163	C₂×C₄×C₁₃₁	—	4	2	no	—
164	C₂×C₄×C₁₃₁	—	4	2	no	—
165	C₂×C₄×C₁₃₁	—	4	2	no	—
166	C₂×C₄×C₁₃₁	—	4	2	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	—	8	8	no	—
171	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₁₃₁	#135	1	4	yes	—
174	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
180	C₈×C₁₃₁	—	2	2	no	—
181	C₈×C₁₃₁	—	4	2	no	—
182	C₈×C₁₃₁	C₄×D₄	1	4	yes	—
183	C₂²×C₁₃₁	—	4	4	no	—
184	C₂²×C₁₃₁	—	4	4	no	—
185	C₂²×C₁₃₁	—	4	4	no	—
186	C₂²×C₁₃₁	—	8	2	no	—
187	C₂²×C₁₃₁	—	8	2	no	—
188	C₂²×C₁₃₁	—	8	2	no	—
189	C₂²×C₁₃₁	—	8	2	no	—
190	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
191	C₂²×C₁₃₁	D₄²	1	1	yes	characteristic
192	C₂²×C₁₃₁	#550	1	1	yes	characteristic
193	C₂²×C₁₃₁	C₂×#135	1	2	yes	—
194	C₂²×C₁₃₁	#577	1	2	yes	—
195	C₄×C₁₃₁	—	2	2	no	—
196	C₄×C₁₃₁	—	2	2	no	—
197	C₄×C₁₃₁	—	2	4	no	—
198	C₄×C₁₃₁	—	4	4	no	—
199	C₄×C₁₃₁	—	4	2	no	—
200	C₄×C₁₃₁	—	4	4	no	—
201	C₄×C₁₃₁	—	8	2	no	—
202	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
203	C₄×C₁₃₁	#518	1	1	yes	characteristic
204	C₄×C₁₃₁	#550	1	1	yes	characteristic
205	C₄×C₁₃₁	#559	1	1	yes	characteristic
206	C₂×C₁₃₁	—	8	4	no	—
207	C₂×C₁₃₁	#2366	1	1	yes	characteristic
208	C₂×C₁₃₁	C₄×#135	1	1	yes	characteristic
209	C₂×C₁₃₁	#2952	1	1	yes	characteristic
210	C₂×C₁₃₁	#3282	1	2	yes	—
211	C₂×C₁₃₁	#3394	1	2	yes	—
212	#23183	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₁₃₁	#23183	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×#495	C₂×C₁₃₁	1	1	yes	characteristic
215	C₂×C₄×SD₁₆	C₂×C₁₃₁	1	2	yes	complemented
216	#2367	C₂×C₁₃₁	1	2	yes	complemented
217	#1983	C₂×C₁₃₁	1	2	yes	complemented
218	#1968	C₂×C₁₃₁	1	2	yes	complemented
219	#1879	C₂×C₁₃₁	1	2	yes	—
220	#1878	C₂×C₁₃₁	1	2	yes	complemented
221	C₄×#106	C₂×C₁₃₁	1	1	yes	characteristic, complemented
222	C₈×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
223	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
224	C₂²×SD₁₆	C₂²×C₁₃₁	1	2	yes	—
225	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
226	C₂×#124	C₂²×C₁₃₁	1	2	yes	—
227	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
228	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
229	#495	C₄×C₁₃₁	1	8	yes	complemented
230	C₄×SD₁₆	—	2	8	no	—
231	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
232	C₂×#106	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×#102	C₂²×C₁₃₁	1	2	yes	—
237	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
238	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
239	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
240	#391	C₂²×C₁₃₁	1	2	yes	—
241	#390	C₂²×C₁₃₁	1	2	yes	—
242	#388	—	2	2	no	—
243	#388	—	2	2	no	—
244	#385	—	2	2	no	—
245	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
246	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
247	C₄×C₂²⋊C₄	—	2	2	no	—
248	#340	C₂²×C₁₃₁	1	2	yes	—
249	#336	C₂²×C₁₃₁	1	2	yes	—
250	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
251	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
252	C₂³×C₄	—	2	2	no	—
253	C₂×Q₁₆	—	4	1	no	—
254	C₂×Q₁₆	—	4	1	no	—
255	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	4	yes	complemented
256	C₂×SD₁₆	—	2	4	no	—
257	C₂×SD₁₆	—	2	4	no	—
258	C₂×SD₁₆	C₂×C₄×C₁₃₁	1	8	yes	complemented
259	C₂×D₈	—	4	1	no	—
260	C₂×D₈	—	4	1	no	—
261	C₂×D₈	C₂×C₄×C₁₃₁	1	4	yes	—
262	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
263	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
265	#124	—	2	4	no	—
266	#124	C₂×C₄×C₁₃₁	1	8	yes	complemented
267	C₄×Q₈	—	2	4	no	—
268	C₄×Q₈	—	4	2	no	—
269	C₄×D₄	—	2	4	no	—
270	C₄×D₄	—	4	2	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	D₄×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	—	2	2	no	—
278	C₂×C₂²⋊C₄	—	2	2	no	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₄²	—	2	2	no	—
281	C₂×C₄²	—	2	2	no	—
282	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
283	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
284	#106	—	2	2	no	—
285	#106	D₄×C₁₃₁	1	4	yes	—
286	C₄⋊C₈	—	2	2	no	—
287	#103	—	2	4	no	—
288	#103	—	4	2	no	—
289	#102	—	2	4	no	—
290	#102	—	4	2	no	—
291	C₄×C₈	—	2	4	no	—
292	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	D₄×C₁₃₁	1	1	yes	characteristic
297	#95	D₄×C₁₃₁	1	1	yes	characteristic
298	C₂⁴	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
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	C₂²×C₄×C₁₃₁	1	4	yes	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	4	4	no	—
315	C₂²×C₄	—	4	4	no	—
316	C₂²×C₄	—	8	2	no	—
317	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
318	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
321	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
322	Q₁₆	—	4	4	no	—
323	SD₁₆	—	2	16	no	—
324	D₈	—	4	4	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
331	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
332	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
333	C₂×C₈	#41×C₁₃₁	1	2	yes	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	4	4	no	—
339	C₄⋊C₄	C₂×D₄×C₁₃₁	1	4	yes	—
340	C₂²⋊C₄	—	2	8	no	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₂²⋊C₄	—	4	4	no	—
343	C₂²⋊C₄	—	4	4	no	—
344	C₂²⋊C₄	—	8	2	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	4	4	no	—
348	C₄²	—	8	2	no	—
349	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
350	C₄²	#41×C₁₃₁	1	2	yes	—
351	C₂³	—	2	4	no	—
352	C₂³	—	4	2	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
359	Q₈	—	2	4	no	—
360	Q₈	—	2	4	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	4	2	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	4	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	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	—	8	8	no	—
384	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
385	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
386	C₂×C₄	#135×C₁₃₁	1	4	yes	—
387	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
388	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
391	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
392	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
393	C₈	—	2	2	no	—
394	C₈	—	4	2	no	—
395	C₈	C₄×D₄×C₁₃₁	1	4	yes	—
396	C₂²	—	4	4	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	—	8	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
404	C₂²	D₄²×C₁₃₁	1	1	yes	characteristic
405	C₂²	#550×C₁₃₁	1	1	yes	characteristic
406	C₂²	C₂×#135×C₁₃₁	1	2	yes	—
407	C₂²	#577×C₁₃₁	1	2	yes	—
408	C₄	—	2	4	no	—
409	C₄	—	2	2	no	—
410	C₄	—	2	2	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	4	no	—
413	C₄	—	4	4	no	—
414	C₄	—	8	2	no	—
415	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
416	C₄	#518×C₁₃₁	1	1	yes	characteristic
417	C₄	#550×C₁₃₁	1	1	yes	characteristic
418	C₄	#559×C₁₃₁	1	1	yes	characteristic
419	C₂	—	8	4	no	—
420	C₂	#2366×C₁₃₁	1	1	yes	characteristic
421	C₂	C₄×#135×C₁₃₁	1	1	yes	characteristic
422	C₂	#2952×C₁₃₁	1	1	yes	characteristic
423	C₂	#3282×C₁₃₁	1	2	yes	—
424	C₂	#3394×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×D₈×C₁₃₁	C₂²
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. C₂×C₄×SD₁₆×C₁₃₁	C₂	semidirect
3. #2367×C₁₃₁	C₂	semidirect
4. #1983×C₁₃₁	C₂	semidirect
5. #1968×C₁₃₁	C₂	semidirect
7. #1878×C₁₃₁	C₂	semidirect
8. C₄×#106×C₁₃₁	C₂	semidirect
9. C₈×C₄⋊C₄×C₁₃₁	C₂	semidirect
16. #495×C₁₃₁	C₄	semidirect
42. C₂×Q₁₆×C₁₃₁	C₂×C₄	semidirect
45. C₂×SD₁₆×C₁₃₁	C₂×C₄	semidirect
53. #124×C₁₃₁	C₂×C₄	semidirect
79. C₄×C₈×C₁₃₁	C₂×C₄	semidirect
212. #23183	C₁₃₁	direct (N × H)
213. C₁₃₁	#23183	direct (N × H)
215. C₂×C₄×SD₁₆	C₂×C₁₃₁	semidirect
216. #2367	C₂×C₁₃₁	semidirect
217. #1983	C₂×C₁₃₁	semidirect
218. #1968	C₂×C₁₃₁	semidirect
220. #1878	C₂×C₁₃₁	semidirect
221. C₄×#106	C₂×C₁₃₁	semidirect
222. C₈×C₄⋊C₄	C₂×C₁₃₁	semidirect
229. #495	C₄×C₁₃₁	semidirect
255. C₂×Q₁₆	C₂×C₄×C₁₃₁	semidirect
258. C₂×SD₁₆	C₂×C₄×C₁₃₁	semidirect
266. #124	C₂×C₄×C₁₃₁	semidirect
292. C₄×C₈	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×#495×C₁₃₁	2
C₂×C₄×SD₁₆×C₁₃₁	2
#2367×C₁₃₁	2
#1983×C₁₃₁	2
#1968×C₁₃₁	2
#1879×C₁₃₁	2
#1878×C₁₃₁	2
C₄×#106×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
#23183	131

Subgroups — C131×#23183 #45709067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23183 #45709067