Subgroups — C₃₁×#33228 #21489987

Order 7936.

1004 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#489	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2660	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2657	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1961	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1959	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1958	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1957	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	2	yes	—
15	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#124	C₂²	1	2	yes	—
18	C₃₁×#490	—	2	4	no	—
19	C₃₁×#489	—	2	4	no	—
20	C₃₁×#488	—	2	4	no	—
21	C₃₁×#486	—	2	4	no	—
22	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#97	C₂²	1	2	yes	—
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#398	—	2	2	no	—
37	C₃₁×#395	—	2	2	no	—
38	C₃₁×#392	—	2	2	no	—
39	C₃₁×#392	—	2	2	no	—
40	C₃₁×#384	C₂²	1	2	yes	—
41	C₃₁×#382	C₂²	1	2	yes	—
42	C₃₁×#376	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	4	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	—	2	2	no	—
54	C₃₁×C₄⋊Q₈	—	2	2	no	—
55	C₃₁×#127	—	2	2	no	—
56	C₃₁×#127	—	2	2	no	—
57	C₃₁×#127	—	2	4	no	—
58	C₃₁×#127	—	2	4	no	—
59	C₃₁×#125	—	2	2	no	—
60	C₃₁×#125	—	2	2	no	—
61	C₃₁×#124	—	2	2	no	—
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×#124	—	2	4	no	—
64	C₃₁×#124	—	2	4	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
77	C₃₁×#103	—	2	2	no	—
78	C₃₁×#103	—	2	2	no	—
79	C₃₁×#103	—	2	4	no	—
80	C₃₁×#102	—	2	4	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#97	—	2	4	no	—
88	C₃₁×#97	D₄	1	4	yes	complemented
89	C₄×C₈×C₃₁	—	2	4	no	—
90	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
91	C₃₁×#95	—	2	4	no	—
92	C₃₁×#95	D₄	1	2	yes	—
93	C₃₁×#95	D₄	1	2	yes	—
94	C₂⁴×C₃₁	—	2	2	no	—
95	C₂⁴×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	1	no	—
97	C₂×Q₈×C₃₁	—	2	1	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	4	2	no	—
101	C₂×Q₈×C₃₁	—	8	2	no	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	4	8	no	—
113	C₂×D₄×C₃₁	—	4	2	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	—	8	4	no	—
116	C₂×D₄×C₃₁	—	8	2	no	—
117	C₂×D₄×C₃₁	—	8	4	no	—
118	C₂×D₄×C₃₁	—	8	2	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	—	8	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	8	no	—
145	C₂²⋊C₄×C₃₁	—	4	2	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₄²×C₃₁	C₂×D₄	1	2	yes	—
148	C₄²×C₃₁	C₂×D₄	1	2	yes	—
149	C₄²×C₃₁	#41	1	4	yes	—
150	C₄²×C₃₁	#41	1	4	yes	—
151	#33228	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	4	2	no	—
153	C₂³×C₃₁	—	4	2	no	—
154	C₂³×C₃₁	—	4	2	no	—
155	C₂³×C₃₁	—	4	1	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	4	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	4	no	—
161	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₃₁	—	4	1	no	—
163	Q₈×C₃₁	—	4	1	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	Q₈×C₃₁	—	8	4	no	—
166	D₄×C₃₁	—	4	1	no	—
167	D₄×C₃₁	—	4	1	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	8	16	no	—
174	D₄×C₃₁	—	8	4	no	—
175	D₄×C₃₁	—	8	4	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	4	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	#124	1	8	yes	—
189	C₂×C₄×C₃₁	#127	1	2	yes	—
190	C₂×C₄×C₃₁	#127	1	2	yes	—
191	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
194	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	2	4	no	—
197	C₈×C₃₁	—	2	2	no	—
198	C₂×#490	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂×#489	C₂×C₃₁	1	1	yes	characteristic, complemented
200	C₂×#488	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2660	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#2657	C₂×C₃₁	1	1	yes	characteristic
204	#1961	C₂×C₃₁	1	2	yes	complemented
205	#1959	C₂×C₃₁	1	2	yes	complemented
206	#1958	C₂×C₃₁	1	2	yes	complemented
207	#1957	C₂×C₃₁	1	2	yes	complemented
208	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂²×C₃₁	—	8	2	no	—
210	C₂²×C₃₁	—	8	1	no	—
211	C₂²×C₃₁	—	8	4	no	—
212	C₂²×C₃₁	—	8	4	no	—
213	C₂²×C₃₁	—	8	4	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	8	1	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	1	no	—
218	C₂²×C₃₁	C₂×#124	1	2	yes	—
219	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
220	C₂²×C₃₁	#576	1	2	yes	—
221	C₂²×C₃₁	#577	1	2	yes	—
222	C₄×C₃₁	—	2	8	no	—
223	C₄×C₃₁	—	2	2	no	—
224	C₄×C₃₁	—	2	2	no	—
225	C₄×C₃₁	—	8	1	no	—
226	C₄×C₃₁	—	8	1	no	—
227	C₄×C₃₁	—	8	1	no	—
228	C₄×C₃₁	—	8	2	no	—
229	C₄×C₃₁	#532	1	1	yes	characteristic
230	C₄×C₃₁	#532	1	2	yes	—
231	C₄×C₃₁	#532	1	1	yes	characteristic
232	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#127	C₂²×C₃₁	1	2	yes	—
235	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×#124	C₂²×C₃₁	1	2	yes	—
238	#490	—	2	4	no	—
239	#489	—	2	4	no	—
240	#488	—	2	4	no	—
241	#486	—	2	4	no	—
242	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#103	C₂²×C₃₁	1	2	yes	—
245	C₂×#102	C₂²×C₃₁	1	2	yes	—
246	C₂×#102	C₂²×C₃₁	1	2	yes	—
247	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#102	C₂²×C₃₁	1	2	yes	—
250	C₂×#102	C₂²×C₃₁	1	2	yes	—
251	C₂×#102	C₂²×C₃₁	1	2	yes	—
252	C₂×#97	C₂²×C₃₁	1	2	yes	—
253	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
256	#398	—	2	2	no	—
257	#395	—	2	2	no	—
258	#392	—	2	2	no	—
259	#392	—	2	2	no	—
260	#384	C₂²×C₃₁	1	2	yes	—
261	#382	C₂²×C₃₁	1	2	yes	—
262	#376	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₃₁	—	16	1	no	—
264	C₂×C₃₁	—	16	2	no	—
265	C₂×C₃₁	#2858	1	1	yes	characteristic
266	C₂×C₃₁	#3136	1	4	yes	—
267	C₂×C₃₁	#3154	1	1	yes	characteristic
268	C₂×C₃₁	#3155	1	1	yes	characteristic
269	C₂²×Q₈	—	2	1	no	—
270	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₂²×D₄	—	2	4	no	—
272	C₂²×D₄	—	2	1	no	—
273	C₂²×D₄	—	2	1	no	—
274	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
275	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
276	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
278	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
279	C₄⋊Q₈	—	2	2	no	—
280	C₄⋊Q₈	—	2	2	no	—
281	#127	—	2	2	no	—
282	#127	—	2	2	no	—
283	#127	—	2	4	no	—
284	#127	—	2	4	no	—
285	#125	—	2	2	no	—
286	#125	—	2	2	no	—
287	#124	—	2	2	no	—
288	#124	—	2	2	no	—
289	#124	—	2	4	no	—
290	#124	—	2	4	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	4	no	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
298	C₂×C₂²⋊C₄	—	2	4	no	—
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
303	#103	—	2	2	no	—
304	#103	—	2	2	no	—
305	#103	—	2	4	no	—
306	#102	—	2	4	no	—
307	#102	—	2	4	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	2	4	no	—
311	#102	—	2	4	no	—
312	#102	—	2	4	no	—
313	#97	—	2	4	no	—
314	#97	D₄×C₃₁	1	4	yes	complemented
315	C₄×C₈	—	2	4	no	—
316	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
317	#95	—	2	4	no	—
318	#95	D₄×C₃₁	1	2	yes	—
319	#95	D₄×C₃₁	1	2	yes	—
320	C₃₁	#33228	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	2	2	no	—
322	C₂⁴	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	4	2	no	—
328	C₂×Q₈	—	8	2	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	8	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	4	2	no	—
342	C₂×D₄	—	8	4	no	—
343	C₂×D₄	—	8	2	no	—
344	C₂×D₄	—	8	4	no	—
345	C₂×D₄	—	8	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
359	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	8	4	no	—
371	C₂²⋊C₄	—	4	8	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₄²	C₂×D₄×C₃₁	1	2	yes	—
375	C₄²	C₂×D₄×C₃₁	1	2	yes	—
376	C₄²	#41×C₃₁	1	4	yes	—
377	C₄²	#41×C₃₁	1	4	yes	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	4	1	no	—
382	C₂³	—	4	2	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	4	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	4	no	—
387	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	Q₈	—	4	1	no	—
389	Q₈	—	4	1	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	8	4	no	—
392	D₄	—	4	1	no	—
393	D₄	—	4	1	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	8	16	no	—
400	D₄	—	8	4	no	—
401	D₄	—	8	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	4	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	C₃₁×#124	1	8	yes	—
415	C₂×C₄	C₃₁×#127	1	2	yes	—
416	C₂×C₄	C₃₁×#127	1	2	yes	—
417	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
420	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
421	C₈	—	2	2	no	—
422	C₈	—	2	4	no	—
423	C₈	—	2	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	4	no	—
427	C₂²	—	8	4	no	—
428	C₂²	—	8	4	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	1	no	—
433	C₂²	C₂×C₃₁×#124	1	2	yes	—
434	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
435	C₂²	C₃₁×#576	1	2	yes	—
436	C₂²	C₃₁×#577	1	2	yes	—
437	C₄	—	2	8	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	8	1	no	—
441	C₄	—	8	1	no	—
442	C₄	—	8	2	no	—
443	C₄	—	8	1	no	—
444	C₄	C₃₁×#532	1	1	yes	characteristic
445	C₄	C₃₁×#532	1	2	yes	—
446	C₄	C₃₁×#532	1	1	yes	characteristic
447	C₂	—	16	1	no	—
448	C₂	—	16	2	no	—
449	C₂	C₃₁×#2858	1	1	yes	characteristic
450	C₂	C₃₁×#3136	1	4	yes	—
451	C₂	C₃₁×#3154	1	1	yes	characteristic
452	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2657	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#490	C₂	semidirect
2. C₂×C₃₁×#489	C₂	semidirect
3. C₂×C₃₁×#488	C₂	semidirect
4. C₂×C₃₁×#486	C₂	semidirect
5. C₃₁×#2660	C₂	semidirect
7. C₃₁×#1961	C₂	semidirect
8. C₃₁×#1959	C₂	semidirect
9. C₃₁×#1958	C₂	semidirect
10. C₃₁×#1957	C₂	semidirect
11. C₃₁×#1726	C₂	semidirect
88. C₃₁×#97	D₄	semidirect
90. C₄×C₈×C₃₁	D₄	semidirect
151. #33228	C₃₁	direct (N × H)
198. C₂×#490	C₂×C₃₁	semidirect
199. C₂×#489	C₂×C₃₁	semidirect
200. C₂×#488	C₂×C₃₁	semidirect
201. C₂×#486	C₂×C₃₁	semidirect
202. #2660	C₂×C₃₁	semidirect
204. #1961	C₂×C₃₁	semidirect
205. #1959	C₂×C₃₁	semidirect
206. #1958	C₂×C₃₁	semidirect
207. #1957	C₂×C₃₁	semidirect
208. #1726	C₂×C₃₁	semidirect
314. #97	D₄×C₃₁	semidirect
316. C₄×C₈	D₄×C₃₁	semidirect
320. C₃₁	#33228	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×#490	2
C₂×C₃₁×#489	2
C₂×C₃₁×#488	2
C₂×C₃₁×#486	2
C₃₁×#2660	2
C₃₁×#2657	2
C₃₁×#1961	2
C₃₁×#1959	2
C₃₁×#1958	2
C₃₁×#1957	2
C₃₁×#1726	2
#33228	31

Subgroups — C31×#33228 #21489987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33228 #21489987