Subgroups — C₃₁×#25047 #21481806

Order 7936.

792 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×C₃₁×#470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2423	C₂	1	2	yes	maximal
4	C₃₁×#2072	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2035	C₂	1	2	yes	maximal
6	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1809	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#470	—	2	4	no	—
17	C₃₁×#441	—	2	4	no	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#399	—	2	2	no	—
29	C₃₁×#398	—	2	2	no	—
30	C₃₁×#391	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#388	—	2	2	no	—
33	C₃₁×#387	—	2	2	no	—
34	C₃₁×#384	C₂²	1	2	yes	—
35	C₃₁×#380	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
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	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₃₁	—	2	2	no	—
44	C₂×Q₁₆×C₃₁	—	2	2	no	—
45	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
46	C₂×M₁₆×C₃₁	—	2	2	no	—
47	C₂×M₁₆×C₃₁	—	2	2	no	—
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₂²⋊Q₈	—	2	4	no	—
54	C₃₁×C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈×C₃₁	—	2	4	no	—
56	C₄×Q₈×C₃₁	—	4	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	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	D₄	1	2	yes	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#107	—	2	2	no	—
72	C₃₁×#106	—	2	2	no	—
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	4	2	no	—
75	C₃₁×#103	—	2	4	no	—
76	C₃₁×#103	—	2	4	no	—
77	C₃₁×#103	—	2	4	no	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₃₁×#97	—	2	2	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	D₄	1	2	yes	—
86	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	Q₁₆×C₃₁	—	2	8	no	—
108	M₁₆×C₃₁	—	4	4	no	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₃₁	#41	1	2	yes	—
119	C₂×C₈×C₃₁	#41	1	2	yes	—
120	C₂×C₈×C₃₁	#41	1	2	yes	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	4	4	no	—
128	C₄⋊C₄×C₃₁	—	4	4	no	—
129	C₄⋊C₄×C₃₁	—	4	4	no	—
130	C₄⋊C₄×C₃₁	—	8	2	no	—
131	C₄⋊C₄×C₃₁	—	8	2	no	—
132	C₂²⋊C₄×C₃₁	—	2	4	no	—
133	C₂²⋊C₄×C₃₁	—	4	4	no	—
134	C₂²⋊C₄×C₃₁	—	8	2	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	C₄²×C₃₁	—	2	1	no	—
137	C₄²×C₃₁	—	2	1	no	—
138	C₄²×C₃₁	—	2	1	no	—
139	C₄²×C₃₁	—	4	4	no	—
140	#25047	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₃₁	—	2	4	no	—
148	Q₈×C₃₁	—	2	4	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	1	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	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	#137	1	4	yes	—
176	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
182	C₈×C₃₁	—	2	2	no	—
183	C₈×C₃₁	—	4	1	no	—
184	C₈×C₃₁	—	4	1	no	—
185	C₈×C₃₁	#123	1	4	yes	—
186	C₂×#470	C₂×C₃₁	1	1	yes	characteristic
187	C₂×#441	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#2423	C₂×C₃₁	1	2	yes	—
189	#2072	C₂×C₃₁	1	2	yes	complemented
190	#2035	C₂×C₃₁	1	2	yes	—
191	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1943	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1865	C₂×C₃₁	1	2	yes	complemented
194	#1844	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1809	C₂×C₃₁	1	1	yes	characteristic
196	#1753	C₂×C₃₁	1	1	yes	characteristic
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	4	1	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	4	1	no	—
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	#532	1	1	yes	characteristic
203	C₂²×C₃₁	D₄²	1	1	yes	characteristic
204	C₂²×C₃₁	#547	1	1	yes	characteristic
205	C₂²×C₃₁	C₂×#137	1	2	yes	—
206	C₂²×C₃₁	#577	1	2	yes	—
207	C₄×C₃₁	—	2	4	no	—
208	C₄×C₃₁	—	4	1	no	—
209	C₄×C₃₁	—	4	4	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	8	2	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
215	C₄×C₃₁	#542	1	1	yes	characteristic
216	C₄×C₃₁	#547	1	1	yes	characteristic
217	C₄×C₃₁	#553	1	1	yes	characteristic
218	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
219	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
221	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
222	#470	—	2	4	no	—
223	#441	—	2	4	no	—
224	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#106	C₂²×C₃₁	1	2	yes	—
228	C₂×#103	C₂²×C₃₁	1	2	yes	—
229	C₂×#103	C₂²×C₃₁	1	2	yes	—
230	C₂×#103	C₂²×C₃₁	1	2	yes	—
231	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
234	#399	—	2	2	no	—
235	#398	—	2	2	no	—
236	#391	C₂²×C₃₁	1	2	yes	—
237	#389	C₂²×C₃₁	1	2	yes	—
238	#388	—	2	2	no	—
239	#387	—	2	2	no	—
240	#384	C₂²×C₃₁	1	2	yes	—
241	#380	C₂²×C₃₁	1	2	yes	—
242	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
243	#340	C₂²×C₃₁	1	2	yes	—
244	#340	C₂²×C₃₁	1	2	yes	—
245	#336	C₂²×C₃₁	1	1	yes	characteristic
246	#336	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₃₁	—	8	1	no	—
248	C₂×C₃₁	#2422	1	1	yes	characteristic
249	C₂×C₃₁	#3173	1	1	yes	characteristic
250	C₂×C₃₁	#3185	1	1	yes	characteristic
251	C₂×C₃₁	#3278	1	2	yes	—
252	C₂×C₃₁	#3344	1	2	yes	—
253	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
254	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
255	C₂×Q₁₆	—	2	2	no	—
256	C₂×Q₁₆	—	2	2	no	—
257	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₂²⋊Q₈	—	2	4	no	—
266	C₂²⋊Q₈	—	4	2	no	—
267	C₄×Q₈	—	2	4	no	—
268	C₄×Q₈	—	4	2	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₂²⋊C₄	—	2	2	no	—
279	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
280	C₂×C₄²	—	2	2	no	—
281	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
283	#107	—	2	2	no	—
284	#106	—	2	2	no	—
285	#106	—	2	2	no	—
286	#106	—	4	2	no	—
287	#103	—	2	4	no	—
288	#103	—	2	4	no	—
289	#103	—	2	4	no	—
290	C₂²⋊C₈	—	2	2	no	—
291	#97	—	2	2	no	—
292	C₄×C₈	—	2	2	no	—
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	—	2	2	no	—
297	#95	D₄×C₃₁	1	2	yes	—
298	C₃₁	#25047	1	1	yes	characteristic, direct product factor, Sylow
299	C₂⁴	C₂×D₄×C₃₁	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	C₂×D₄×C₃₁	1	4	yes	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
320	Q₁₆	—	2	8	no	—
321	M₁₆	—	4	4	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
331	C₂×C₈	#41×C₃₁	1	2	yes	—
332	C₂×C₈	#41×C₃₁	1	2	yes	—
333	C₂×C₈	#41×C₃₁	1	2	yes	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	4	no	—
341	C₄⋊C₄	—	4	4	no	—
342	C₄⋊C₄	—	4	4	no	—
343	C₄⋊C₄	—	8	2	no	—
344	C₄⋊C₄	—	8	2	no	—
345	C₂²⋊C₄	—	2	4	no	—
346	C₂²⋊C₄	—	4	4	no	—
347	C₂²⋊C₄	—	8	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	2	1	no	—
352	C₄²	—	4	4	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	2	1	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	2	no	—
357	C₂³	—	2	2	no	—
358	C₂³	C₂²×D₄×C₃₁	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	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	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	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	C₃₁×#137	1	4	yes	—
388	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
393	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
394	C₈	—	2	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₈	C₃₁×#123	1	4	yes	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	2	no	—
403	C₂²	C₃₁×#532	1	1	yes	characteristic
404	C₂²	D₄²×C₃₁	1	1	yes	characteristic
405	C₂²	C₃₁×#547	1	1	yes	characteristic
406	C₂²	C₂×C₃₁×#137	1	2	yes	—
407	C₂²	C₃₁×#577	1	2	yes	—
408	C₄	—	2	4	no	—
409	C₄	—	4	1	no	—
410	C₄	—	4	4	no	—
411	C₄	—	4	2	no	—
412	C₄	—	8	2	no	—
413	C₄	—	8	2	no	—
414	C₄	—	8	2	no	—
415	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
416	C₄	C₃₁×#542	1	1	yes	characteristic
417	C₄	C₃₁×#547	1	1	yes	characteristic
418	C₄	C₃₁×#553	1	1	yes	characteristic
419	C₂	—	8	1	no	—
420	C₂	C₃₁×#2422	1	1	yes	characteristic
421	C₂	C₃₁×#3173	1	1	yes	characteristic
422	C₂	C₃₁×#3185	1	1	yes	characteristic
423	C₂	C₃₁×#3278	1	2	yes	—
424	C₂	C₃₁×#3344	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#441	C₂	semidirect
4. C₃₁×#2072	C₂	semidirect
6. C₃₁×#1969	C₂	semidirect
7. C₃₁×#1943	C₂	semidirect
8. C₃₁×#1865	C₂	semidirect
9. C₃₁×#1844	C₂	semidirect
140. #25047	C₃₁	direct (N × H)
187. C₂×#441	C₂×C₃₁	semidirect
189. #2072	C₂×C₃₁	semidirect
191. #1969	C₂×C₃₁	semidirect
192. #1943	C₂×C₃₁	semidirect
193. #1865	C₂×C₃₁	semidirect
194. #1844	C₂×C₃₁	semidirect
298. C₃₁	#25047	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×#470	2
C₂×C₃₁×#441	2
C₃₁×#2423	2
C₃₁×#2072	2
C₃₁×#2035	2
C₃₁×#1969	2
C₃₁×#1943	2
C₃₁×#1865	2
C₃₁×#1844	2
C₃₁×#1809	2
C₃₁×#1753	2
#25047	31

Subgroups — C31×#25047 #21481806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25047 #21481806