Subgroups — C₃₁×#26262 #21483021

Order 7936.

882 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2447	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2055	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2042	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1941	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1877	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1771	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#495	—	2	4	no	—
18	C₃₁×#442	—	2	4	no	—
19	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#397	—	2	2	no	—
30	C₃₁×#397	—	2	2	no	—
31	C₃₁×#391	C₂²	1	2	yes	—
32	C₃₁×#391	C₂²	1	2	yes	—
33	C₃₁×#390	C₂²	1	2	yes	—
34	C₃₁×#389	C₂²	1	2	yes	—
35	C₃₁×#385	—	2	2	no	—
36	C₃₁×#381	—	2	2	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
43	C₂³×C₄×C₃₁	—	2	2	no	—
44	C₂×Q₁₆×C₃₁	—	2	2	no	—
45	C₂×Q₁₆×C₃₁	—	2	2	no	—
46	C₂×Q₁₆×C₃₁	—	2	2	no	—
47	C₂×SD₁₆×C₃₁	—	2	4	no	—
48	C₂×SD₁₆×C₃₁	—	2	4	no	—
49	C₂×SD₁₆×C₃₁	—	4	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	D₄	1	4	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×#124	—	2	4	no	—
58	C₃₁×#124	—	4	2	no	—
59	C₄×D₄×C₃₁	—	2	4	no	—
60	C₄×D₄×C₃₁	—	4	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×C₄⋊C₈	—	2	2	no	—
76	C₃₁×#103	—	4	2	no	—
77	C₃₁×#103	—	4	2	no	—
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×#102	—	4	2	no	—
80	C₃₁×#97	—	2	2	no	—
81	C₃₁×#97	—	2	2	no	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	D₄	1	2	yes	—
89	C₂⁴×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	—	4	4	no	—
107	C₂²×C₄×C₃₁	—	8	2	no	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
113	Q₁₆×C₃₁	—	4	4	no	—
114	SD₁₆×C₃₁	—	4	8	no	—
115	D₈×C₃₁	—	2	8	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₃₁	#41	1	2	yes	—
126	C₂×C₈×C₃₁	#41	1	2	yes	—
127	C₂×C₈×C₃₁	#41	1	2	yes	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	8	2	no	—
134	C₂²⋊C₄×C₃₁	—	4	4	no	—
135	C₂²⋊C₄×C₃₁	—	4	4	no	—
136	C₂²⋊C₄×C₃₁	—	4	4	no	—
137	C₂²⋊C₄×C₃₁	—	4	4	no	—
138	C₂²⋊C₄×C₃₁	—	8	2	no	—
139	C₄²×C₃₁	—	2	1	no	—
140	C₄²×C₃₁	—	2	1	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	#26262	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	4	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	—	4	2	no	—
153	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	2	4	no	—
159	D₄×C₃₁	—	2	4	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	8	8	no	—
182	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	4	1	no	—
190	C₈×C₃₁	—	4	1	no	—
191	C₈×C₃₁	#123	1	4	yes	—
192	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
193	C₂×#442	C₂×C₃₁	1	1	yes	characteristic
194	#2447	C₂×C₃₁	1	2	yes	complemented
195	#2055	C₂×C₃₁	1	2	yes	complemented
196	#2042	C₂×C₃₁	1	2	yes	complemented
197	#1972	C₂×C₃₁	1	1	yes	characteristic
198	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1941	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1877	C₂×C₃₁	1	2	yes	complemented
202	#1771	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	4	4	no	—
204	C₂²×C₃₁	—	4	4	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	8	2	no	—
207	C₂²×C₃₁	—	8	2	no	—
208	C₂²×C₃₁	—	8	2	no	—
209	C₂²×C₃₁	—	8	2	no	—
210	C₂²×C₃₁	#532	1	1	yes	characteristic
211	C₂²×C₃₁	D₄²	1	1	yes	characteristic
212	C₂²×C₃₁	#550	1	1	yes	characteristic
213	C₂²×C₃₁	#575	1	2	yes	—
214	C₂²×C₃₁	#577	1	2	yes	—
215	C₄×C₃₁	—	4	1	no	—
216	C₄×C₃₁	—	4	1	no	—
217	C₄×C₃₁	—	4	2	no	—
218	C₄×C₃₁	—	4	2	no	—
219	C₄×C₃₁	—	8	2	no	—
220	C₄×C₃₁	—	8	2	no	—
221	C₄×C₃₁	—	8	2	no	—
222	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
223	C₄×C₃₁	#540	1	1	yes	characteristic
224	C₄×C₃₁	#546	1	1	yes	characteristic
225	C₄×C₃₁	#559	1	1	yes	characteristic
226	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
227	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
228	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#124	C₂²×C₃₁	1	2	yes	—
230	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
231	#495	—	2	4	no	—
232	#442	—	2	4	no	—
233	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#103	C₂²×C₃₁	1	2	yes	—
237	C₂×#103	C₂²×C₃₁	1	2	yes	—
238	C₂×#102	C₂²×C₃₁	1	2	yes	—
239	C₂×#102	C₂²×C₃₁	1	2	yes	—
240	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
243	#397	—	2	2	no	—
244	#397	—	2	2	no	—
245	#391	C₂²×C₃₁	1	2	yes	—
246	#391	C₂²×C₃₁	1	2	yes	—
247	#390	C₂²×C₃₁	1	2	yes	—
248	#389	C₂²×C₃₁	1	2	yes	—
249	#385	—	2	2	no	—
250	#381	—	2	2	no	—
251	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
252	#340	C₂²×C₃₁	1	2	yes	—
253	#336	C₂²×C₃₁	1	1	yes	characteristic
254	#336	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₃₁	—	8	4	no	—
256	C₂×C₃₁	#2445	1	1	yes	characteristic
257	C₂×C₃₁	#3173	1	1	yes	characteristic
258	C₂×C₃₁	#3183	1	1	yes	characteristic
259	C₂×C₃₁	#3267	1	2	yes	—
260	C₂×C₃₁	#3401	1	2	yes	—
261	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
262	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
263	C₂³×C₄	—	2	2	no	—
264	C₂×Q₁₆	—	2	2	no	—
265	C₂×Q₁₆	—	2	2	no	—
266	C₂×Q₁₆	—	2	2	no	—
267	C₂×SD₁₆	—	2	4	no	—
268	C₂×SD₁₆	—	2	4	no	—
269	C₂×SD₁₆	—	4	2	no	—
270	C₂×D₈	—	2	2	no	—
271	C₂×D₈	—	2	2	no	—
272	C₂×D₈	D₄×C₃₁	1	4	yes	—
273	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
277	#124	—	2	4	no	—
278	#124	—	4	2	no	—
279	C₄×D₄	—	2	4	no	—
280	C₄×D₄	—	4	2	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
293	#106	—	2	2	no	—
294	#106	—	2	2	no	—
295	C₄⋊C₈	—	2	2	no	—
296	#103	—	4	2	no	—
297	#103	—	4	2	no	—
298	#102	—	2	4	no	—
299	#102	—	4	2	no	—
300	#97	—	2	2	no	—
301	#97	—	2	2	no	—
302	C₄×C₈	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	#95	—	2	2	no	—
308	#95	D₄×C₃₁	1	2	yes	—
309	C₃₁	#26262	1	1	yes	characteristic, direct product factor, Sylow
310	C₂⁴	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	4	4	no	—
327	C₂²×C₄	—	4	4	no	—
328	C₂²×C₄	—	8	2	no	—
329	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
333	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
334	Q₁₆	—	4	4	no	—
335	SD₁₆	—	4	8	no	—
336	D₈	—	2	8	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
346	C₂×C₈	#41×C₃₁	1	2	yes	—
347	C₂×C₈	#41×C₃₁	1	2	yes	—
348	C₂×C₈	#41×C₃₁	1	2	yes	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	8	2	no	—
355	C₂²⋊C₄	—	4	4	no	—
356	C₂²⋊C₄	—	4	4	no	—
357	C₂²⋊C₄	—	4	4	no	—
358	C₂²⋊C₄	—	4	4	no	—
359	C₂²⋊C₄	—	8	2	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	2	1	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	1	no	—
366	C₂³	—	2	4	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	4	2	no	—
369	C₂³	—	4	2	no	—
370	C₂³	—	4	2	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	2	no	—
373	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	Q₈	—	4	2	no	—
375	Q₈	—	4	2	no	—
376	Q₈	—	4	2	no	—
377	Q₈	—	4	2	no	—
378	D₄	—	2	4	no	—
379	D₄	—	2	4	no	—
380	D₄	—	4	2	no	—
381	D₄	—	4	2	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	8	8	no	—
402	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
403	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
407	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
408	C₈	—	2	2	no	—
409	C₈	—	4	1	no	—
410	C₈	—	4	1	no	—
411	C₈	C₃₁×#123	1	4	yes	—
412	C₂²	—	4	4	no	—
413	C₂²	—	4	4	no	—
414	C₂²	—	4	4	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	8	2	no	—
418	C₂²	—	8	2	no	—
419	C₂²	C₃₁×#532	1	1	yes	characteristic
420	C₂²	D₄²×C₃₁	1	1	yes	characteristic
421	C₂²	C₃₁×#550	1	1	yes	characteristic
422	C₂²	C₃₁×#575	1	2	yes	—
423	C₂²	C₃₁×#577	1	2	yes	—
424	C₄	—	4	1	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	8	2	no	—
429	C₄	—	8	2	no	—
430	C₄	—	8	2	no	—
431	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
432	C₄	C₃₁×#540	1	1	yes	characteristic
433	C₄	C₃₁×#546	1	1	yes	characteristic
434	C₄	C₃₁×#559	1	1	yes	characteristic
435	C₂	—	8	4	no	—
436	C₂	C₃₁×#2445	1	1	yes	characteristic
437	C₂	C₃₁×#3173	1	1	yes	characteristic
438	C₂	C₃₁×#3183	1	1	yes	characteristic
439	C₂	C₃₁×#3267	1	2	yes	—
440	C₂	C₃₁×#3401	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2447	C₂	semidirect
4. C₃₁×#2055	C₂	semidirect
5. C₃₁×#2042	C₂	semidirect
7. C₃₁×#1971	C₂	semidirect
8. C₃₁×#1971	C₂	semidirect
9. C₃₁×#1941	C₂	semidirect
10. C₃₁×#1877	C₂	semidirect
11. C₃₁×#1771	C₂	semidirect
145. #26262	C₃₁	direct (N × H)
194. #2447	C₂×C₃₁	semidirect
195. #2055	C₂×C₃₁	semidirect
196. #2042	C₂×C₃₁	semidirect
198. #1971	C₂×C₃₁	semidirect
199. #1971	C₂×C₃₁	semidirect
200. #1941	C₂×C₃₁	semidirect
201. #1877	C₂×C₃₁	semidirect
202. #1771	C₂×C₃₁	semidirect
309. C₃₁	#26262	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×#495	2
C₂×C₃₁×#442	2
C₃₁×#2447	2
C₃₁×#2055	2
C₃₁×#2042	2
C₃₁×#1972	2
C₃₁×#1971	2
C₃₁×#1941	2
C₃₁×#1877	2
C₃₁×#1771	2
#26262	31

Subgroups — C31×#26262 #21483021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26262 #21483021