Subgroups — C₃₁×#34081 #21490840

Order 7936.

2202 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2337	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2336	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2335	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2334	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2333	C₂	1	2	yes	maximal
7	C₃₁×#2332	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2298	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2295	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2293	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2290	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2290	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₃₁×#117	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#529	C₄	1	16	yes	complemented
17	C₂×C₃₁×#126	C₂²	1	2	yes	—
18	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#123	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₃₁×#117	C₂²	1	2	yes	—
27	C₂×C₃₁×#117	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	C₂²	1	2	yes	—
29	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
30	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#391	C₂²	1	4	yes	—
32	C₃₁×#390	C₂²	1	4	yes	—
33	C₃₁×#389	C₂²	1	4	yes	—
34	C₃₁×#389	C₂²	1	4	yes	—
35	C₃₁×#388	C₂²	1	4	yes	—
36	C₃₁×#388	C₂²	1	4	yes	—
37	C₃₁×#388	C₂²	1	4	yes	—
38	C₃₁×#388	C₂²	1	4	yes	—
39	C₃₁×#388	C₂²	1	4	yes	—
40	C₃₁×#388	C₂²	1	4	yes	—
41	C₃₁×#387	C₂²	1	4	yes	—
42	C₃₁×#387	C₂²	1	4	yes	—
43	C₃₁×#387	C₂²	1	4	yes	—
44	C₃₁×#387	C₂²	1	4	yes	—
45	C₃₁×#387	C₂²	1	4	yes	—
46	C₃₁×#387	C₂²	1	4	yes	—
47	C₃₁×#386	C₂²	1	4	yes	—
48	C₃₁×#386	C₂²	1	4	yes	—
49	C₃₁×#385	C₂²	1	4	yes	—
50	C₃₁×#385	C₂²	1	4	yes	—
51	C₃₁×#383	C₂²	1	2	yes	—
52	C₃₁×#382	C₂²	1	2	yes	—
53	C₃₁×#381	C₂²	1	2	yes	—
54	C₃₁×#380	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
62	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
67	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
68	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
69	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
70	C₃₁×#376	C₂²	1	2	yes	—
71	C₃₁×#376	C₂²	1	2	yes	—
72	C₃₁×#376	C₂²	1	2	yes	—
73	C₄³×C₃₁	C₂²	1	1	yes	characteristic
74	C₄³×C₃₁	C₂²	1	2	yes	—
75	C₄³×C₃₁	C₂²	1	1	yes	characteristic
76	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
78	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
80	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#126	C₂×C₄	1	16	yes	complemented
85	C₃₁×#125	C₂×C₄	1	8	yes	complemented
86	C₃₁×#124	C₂×C₄	1	8	yes	—
87	C₃₁×#123	C₂×C₄	1	16	yes	complemented
88	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
89	C₃₁×#121	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
91	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
93	C₃₁×#117	—	2	8	no	—
94	C₃₁×#117	—	2	8	no	—
95	C₃₁×#117	C₂×C₄	1	16	yes	complemented
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄²×C₃₁	—	2	8	no	—
119	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
120	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
128	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
130	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
135	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
136	C₃₁×#95	C₂³	1	4	yes	—
137	C₃₁×#95	C₂³	1	4	yes	—
138	C₃₁×#95	C₂³	1	4	yes	—
139	C₃₁×#95	C₂³	1	4	yes	—
140	C₃₁×#95	C₂³	1	4	yes	—
141	C₃₁×#95	C₂³	1	4	yes	—
142	C₃₁×#95	C₂³	1	2	yes	—
143	C₃₁×#95	C₂³	1	2	yes	—
144	C₃₁×#95	C₂³	1	4	yes	—
145	C₃₁×#95	C₂³	1	4	yes	—
146	C₃₁×#95	C₂³	1	4	yes	—
147	C₃₁×#95	C₂³	1	4	yes	—
148	C₃₁×#95	C₂³	1	4	yes	—
149	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
150	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
151	C₂×Q₈×C₃₁	—	4	2	no	—
152	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
153	C₂×D₄×C₃₁	—	4	4	no	—
154	C₂×D₄×C₃₁	—	4	2	no	—
155	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
156	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
157	C₂²×C₄×C₃₁	—	2	4	no	—
158	C₂²×C₄×C₃₁	—	2	4	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	8	no	—
161	C₂²×C₄×C₃₁	—	4	2	no	—
162	C₂²×C₄×C₃₁	—	4	2	no	—
163	C₂²×C₄×C₃₁	—	4	4	no	—
164	C₂²×C₄×C₃₁	—	4	4	no	—
165	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₄⋊C₄×C₃₁	—	2	8	no	—
183	C₄⋊C₄×C₃₁	—	2	8	no	—
184	C₄⋊C₄×C₃₁	—	4	4	no	—
185	C₄⋊C₄×C₃₁	—	4	4	no	—
186	C₄⋊C₄×C₃₁	—	4	4	no	—
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
191	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
192	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
193	C₂²⋊C₄×C₃₁	—	2	8	no	—
194	C₂²⋊C₄×C₃₁	—	2	8	no	—
195	C₂²⋊C₄×C₃₁	—	4	2	no	—
196	C₂²⋊C₄×C₃₁	—	4	4	no	—
197	C₂²⋊C₄×C₃₁	—	4	4	no	—
198	C₂²⋊C₄×C₃₁	—	4	2	no	—
199	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
200	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
201	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
202	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
203	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
204	C₄²×C₃₁	—	2	8	no	—
205	C₄²×C₃₁	—	2	4	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	4	no	—
208	C₄²×C₃₁	—	4	2	no	—
209	C₄²×C₃₁	—	4	2	no	—
210	C₄²×C₃₁	—	4	4	no	—
211	C₄²×C₃₁	—	4	4	no	—
212	C₄²×C₃₁	—	4	2	no	—
213	C₄²×C₃₁	—	4	2	no	—
214	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
215	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
216	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
217	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
218	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
219	C₄²×C₃₁	#41	1	8	yes	—
220	#34081	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂³×C₃₁	—	2	2	no	—
222	C₂³×C₃₁	—	2	2	no	—
223	C₂³×C₃₁	—	2	2	no	—
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	—	2	4	no	—
226	C₂³×C₃₁	—	2	4	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
229	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
230	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
231	Q₈×C₃₁	—	4	4	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	8	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	16	no	—
246	C₂×C₄×C₃₁	—	2	8	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	4	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	4	4	no	—
257	C₂×C₄×C₃₁	—	4	4	no	—
258	C₂×C₄×C₃₁	—	4	16	no	—
259	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
261	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
262	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
263	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
265	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
266	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
267	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
269	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
270	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
271	#2337	C₂×C₃₁	1	2	yes	complemented
272	#2336	C₂×C₃₁	1	2	yes	complemented
273	#2335	C₂×C₃₁	1	4	yes	complemented
274	#2334	C₂×C₃₁	1	4	yes	complemented
275	#2333	C₂×C₃₁	1	2	yes	—
276	#2332	C₂×C₃₁	1	2	yes	complemented
277	#2298	C₂×C₃₁	1	2	yes	complemented
278	#2295	C₂×C₃₁	1	2	yes	complemented
279	#2293	C₂×C₃₁	1	2	yes	complemented
280	#2290	C₂×C₃₁	1	2	yes	complemented
281	#2290	C₂×C₃₁	1	2	yes	complemented
282	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₄×#117	C₂×C₃₁	1	2	yes	complemented
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	2	4	no	—
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	2	4	no	—
290	C₂²×C₃₁	—	2	4	no	—
291	C₂²×C₃₁	—	4	4	no	—
292	C₂²×C₃₁	—	4	4	no	—
293	C₂²×C₃₁	—	4	4	no	—
294	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
295	C₂²×C₃₁	C₂²×#41	1	2	yes	—
296	C₂²×C₃₁	#585	1	2	yes	—
297	C₂²×C₃₁	#585	1	2	yes	—
298	C₄×C₃₁	—	2	4	no	—
299	C₄×C₃₁	—	2	2	no	—
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	2	no	—
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	2	4	no	—
305	C₄×C₃₁	—	2	4	no	—
306	C₄×C₃₁	—	2	8	no	—
307	C₄×C₃₁	—	4	4	no	—
308	C₄×C₃₁	—	4	4	no	—
309	C₄×C₃₁	—	4	4	no	—
310	C₄×C₃₁	C₄×#41	1	4	yes	—
311	#529	C₄×C₃₁	1	16	yes	complemented
312	C₂×#126	C₂²×C₃₁	1	2	yes	—
313	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
315	C₂×#123	C₂²×C₃₁	1	2	yes	—
316	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
321	C₂×#117	C₂²×C₃₁	1	2	yes	—
322	C₂×#117	C₂²×C₃₁	1	2	yes	—
323	C₂×#117	C₂²×C₃₁	1	2	yes	—
324	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
325	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
326	#391	C₂²×C₃₁	1	4	yes	—
327	#390	C₂²×C₃₁	1	4	yes	—
328	#389	C₂²×C₃₁	1	4	yes	—
329	#389	C₂²×C₃₁	1	4	yes	—
330	#388	C₂²×C₃₁	1	4	yes	—
331	#388	C₂²×C₃₁	1	4	yes	—
332	#388	C₂²×C₃₁	1	4	yes	—
333	#388	C₂²×C₃₁	1	4	yes	—
334	#388	C₂²×C₃₁	1	4	yes	—
335	#388	C₂²×C₃₁	1	4	yes	—
336	#387	C₂²×C₃₁	1	4	yes	—
337	#387	C₂²×C₃₁	1	4	yes	—
338	#387	C₂²×C₃₁	1	4	yes	—
339	#387	C₂²×C₃₁	1	4	yes	—
340	#387	C₂²×C₃₁	1	4	yes	—
341	#387	C₂²×C₃₁	1	4	yes	—
342	#386	C₂²×C₃₁	1	4	yes	—
343	#386	C₂²×C₃₁	1	4	yes	—
344	#385	C₂²×C₃₁	1	4	yes	—
345	#385	C₂²×C₃₁	1	4	yes	—
346	#383	C₂²×C₃₁	1	2	yes	—
347	#382	C₂²×C₃₁	1	2	yes	—
348	#381	C₂²×C₃₁	1	2	yes	—
349	#380	C₂²×C₃₁	1	2	yes	—
350	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
351	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
352	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
353	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
354	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
355	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
356	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
357	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
358	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
359	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
360	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
361	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
362	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
363	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
364	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
365	#376	C₂²×C₃₁	1	2	yes	—
366	#376	C₂²×C₃₁	1	2	yes	—
367	#376	C₂²×C₃₁	1	2	yes	—
368	C₄³	C₂²×C₃₁	1	1	yes	characteristic
369	C₄³	C₂²×C₃₁	1	2	yes	—
370	C₄³	C₂²×C₃₁	1	1	yes	characteristic
371	C₂×C₃₁	—	4	2	no	—
372	C₂×C₃₁	—	4	4	no	—
373	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
374	C₂×C₃₁	#3428	1	1	yes	characteristic
375	C₂×C₃₁	#3428	1	1	yes	characteristic
376	C₂×C₃₁	#3455	1	4	yes	—
377	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
379	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
380	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
381	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
383	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
384	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
385	#126	C₂×C₄×C₃₁	1	16	yes	complemented
386	#125	C₂×C₄×C₃₁	1	8	yes	complemented
387	#124	C₂×C₄×C₃₁	1	8	yes	—
388	#123	C₂×C₄×C₃₁	1	16	yes	complemented
389	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
390	#121	C₂×C₄×C₃₁	1	8	yes	—
391	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
392	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
393	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
394	#117	—	2	8	no	—
395	#117	—	2	8	no	—
396	#117	C₂×C₄×C₃₁	1	16	yes	complemented
397	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
401	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
419	C₂×C₄²	—	2	8	no	—
420	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
421	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
422	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
423	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
424	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
425	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
426	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
427	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
428	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
429	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
430	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
431	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
433	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
437	#95	C₂³×C₃₁	1	4	yes	—
438	#95	C₂³×C₃₁	1	4	yes	—
439	#95	C₂³×C₃₁	1	4	yes	—
440	#95	C₂³×C₃₁	1	4	yes	—
441	#95	C₂³×C₃₁	1	4	yes	—
442	#95	C₂³×C₃₁	1	4	yes	—
443	#95	C₂³×C₃₁	1	2	yes	—
444	#95	C₂³×C₃₁	1	2	yes	—
445	#95	C₂³×C₃₁	1	4	yes	—
446	#95	C₂³×C₃₁	1	4	yes	—
447	#95	C₂³×C₃₁	1	4	yes	—
448	#95	C₂³×C₃₁	1	4	yes	—
449	#95	C₂³×C₃₁	1	4	yes	—
450	C₃₁	#34081	1	1	yes	characteristic, direct product factor, Sylow
451	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
452	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
453	C₂×Q₈	—	4	2	no	—
454	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
455	C₂×D₄	—	4	4	no	—
456	C₂×D₄	—	4	2	no	—
457	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
458	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	8	no	—
463	C₂²×C₄	—	4	2	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	—	4	4	no	—
467	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
468	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
469	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
470	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
471	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
483	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
484	C₄⋊C₄	—	2	8	no	—
485	C₄⋊C₄	—	2	8	no	—
486	C₄⋊C₄	—	4	4	no	—
487	C₄⋊C₄	—	4	4	no	—
488	C₄⋊C₄	—	4	4	no	—
489	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
490	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
491	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
492	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
493	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
494	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
495	C₂²⋊C₄	—	2	8	no	—
496	C₂²⋊C₄	—	2	8	no	—
497	C₂²⋊C₄	—	4	2	no	—
498	C₂²⋊C₄	—	4	4	no	—
499	C₂²⋊C₄	—	4	4	no	—
500	C₂²⋊C₄	—	4	2	no	—
501	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
502	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
503	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
504	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
505	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
506	C₄²	—	2	8	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	4	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	4	2	no	—
511	C₄²	—	4	2	no	—
512	C₄²	—	4	4	no	—
513	C₄²	—	4	4	no	—
514	C₄²	—	4	2	no	—
515	C₄²	—	4	2	no	—
516	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
517	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
518	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
519	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
520	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
521	C₄²	#41×C₃₁	1	8	yes	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	2	4	no	—
528	C₂³	—	2	2	no	—
529	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
530	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
531	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
532	Q₈	—	4	4	no	—
533	D₄	—	4	4	no	—
534	D₄	—	4	8	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	16	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	4	no	—
559	C₂×C₄	—	4	16	no	—
560	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
561	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
562	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
563	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
566	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
567	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
568	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
569	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
570	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
581	C₂²	C₂²×#41×C₃₁	1	2	yes	—
582	C₂²	C₃₁×#585	1	2	yes	—
583	C₂²	C₃₁×#585	1	2	yes	—
584	C₄	—	2	4	no	—
585	C₄	—	2	2	no	—
586	C₄	—	2	2	no	—
587	C₄	—	2	2	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	4	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	8	no	—
593	C₄	—	4	4	no	—
594	C₄	—	4	4	no	—
595	C₄	—	4	4	no	—
596	C₄	C₄×#41×C₃₁	1	4	yes	—
597	C₂	—	4	2	no	—
598	C₂	—	4	4	no	—
599	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
600	C₂	C₃₁×#3428	1	1	yes	characteristic
601	C₂	C₃₁×#3428	1	1	yes	characteristic
602	C₂	C₃₁×#3455	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2337	C₂	semidirect
3. C₃₁×#2336	C₂	semidirect
4. C₃₁×#2335	C₂	semidirect
5. C₃₁×#2334	C₂	semidirect
7. C₃₁×#2332	C₂	semidirect
8. C₃₁×#2298	C₂	semidirect
9. C₃₁×#2295	C₂	semidirect
10. C₃₁×#2293	C₂	semidirect
11. C₃₁×#2290	C₂	semidirect
12. C₃₁×#2290	C₂	semidirect
13. C₃₁×#2283	C₂	semidirect
14. C₄×C₃₁×#117	C₂	semidirect
15. C₄×C₃₁×#117	C₂	semidirect
16. C₃₁×#529	C₄	semidirect
84. C₃₁×#126	C₂×C₄	semidirect
85. C₃₁×#125	C₂×C₄	semidirect
87. C₃₁×#123	C₂×C₄	semidirect
88. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
90. C₄×Q₈×C₃₁	C₂×C₄	semidirect
92. C₄×D₄×C₃₁	C₂×C₄	semidirect
95. C₃₁×#117	C₂×C₄	semidirect
119. C₂×C₄²×C₃₁	C₂×C₄	semidirect
220. #34081	C₃₁	direct (N × H)
271. #2337	C₂×C₃₁	semidirect
272. #2336	C₂×C₃₁	semidirect
273. #2335	C₂×C₃₁	semidirect
274. #2334	C₂×C₃₁	semidirect
276. #2332	C₂×C₃₁	semidirect
277. #2298	C₂×C₃₁	semidirect
278. #2295	C₂×C₃₁	semidirect
279. #2293	C₂×C₃₁	semidirect
280. #2290	C₂×C₃₁	semidirect
281. #2290	C₂×C₃₁	semidirect
282. #2283	C₂×C₃₁	semidirect
283. C₄×#117	C₂×C₃₁	semidirect
284. C₄×#117	C₂×C₃₁	semidirect
311. #529	C₄×C₃₁	semidirect
385. #126	C₂×C₄×C₃₁	semidirect
386. #125	C₂×C₄×C₃₁	semidirect
388. #123	C₂×C₄×C₃₁	semidirect
389. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
391. C₄×Q₈	C₂×C₄×C₃₁	semidirect
393. C₄×D₄	C₂×C₄×C₃₁	semidirect
396. #117	C₂×C₄×C₃₁	semidirect
420. C₂×C₄²	C₂×C₄×C₃₁	semidirect
450. C₃₁	#34081	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#529	2
C₃₁×#2337	2
C₃₁×#2336	2
C₃₁×#2335	2
C₃₁×#2334	2
C₃₁×#2333	2
C₃₁×#2332	2
C₃₁×#2298	2
C₃₁×#2295	2
C₃₁×#2293	2
C₃₁×#2290	2
C₃₁×#2283	2
C₄×C₃₁×#117	2
#34081	31

Subgroups — C31×#34081 #21490840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34081 #21490840