Subgroups — C₃₁×#25632 #21482391

Order 7936.

1114 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2015	C₂	1	4	yes	maximal
6	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1878	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#556	—	2	4	no	—
13	C₃₁×#556	—	2	4	no	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#126	—	2	2	no	—
17	C₂×C₃₁×#124	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#117	—	2	1	no	—
23	C₃₁×#494	—	2	4	no	—
24	C₃₁×#461	—	2	4	no	—
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#103	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	4	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#400	—	2	2	no	—
31	C₃₁×#391	—	2	1	no	—
32	C₃₁×#391	C₂²	1	4	yes	—
33	C₃₁×#382	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	4	yes	—
37	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	—	2	1	no	—
42	C₂×Q₁₆×C₃₁	—	2	8	no	—
43	C₂×Q₁₆×C₃₁	—	2	8	no	—
44	C₂×Q₁₆×C₃₁	—	4	4	no	—
45	C₂×SD₁₆×C₃₁	—	2	4	no	—
46	C₂×SD₁₆×C₃₁	—	4	4	no	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₃₁×C₄⋊Q₈	—	2	2	no	—
50	C₃₁×C₄⋊Q₈	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	—	2	2	no	—
52	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
53	C₃₁×#126	—	2	16	no	—
54	C₃₁×#124	—	2	4	no	—
55	C₃₁×#124	—	2	4	no	—
56	C₃₁×C₂²⋊Q₈	—	2	8	no	—
57	C₄×Q₈×C₃₁	—	2	4	no	—
58	C₄×Q₈×C₃₁	—	2	4	no	—
59	C₄×Q₈×C₃₁	—	2	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	D₄	1	4	yes	—
63	C₃₁×#117	—	2	8	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄²×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×#103	—	2	4	no	—
82	C₃₁×#103	—	4	4	no	—
83	C₃₁×#102	—	2	4	no	—
84	C₄×C₈×C₃₁	—	2	2	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	4	no	—
87	C₃₁×#95	—	2	4	no	—
88	C₂⁴×C₃₁	—	2	1	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	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	4	no	—
96	C₂×Q₈×C₃₁	—	2	4	no	—
97	C₂×Q₈×C₃₁	—	2	4	no	—
98	C₂×Q₈×C₃₁	—	2	4	no	—
99	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
116	Q₁₆×C₃₁	—	4	16	no	—
117	SD₁₆×C₃₁	—	4	8	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	4	no	—
122	C₂×C₈×C₃₁	#41	1	4	yes	—
123	C₄⋊C₄×C₃₁	—	2	8	no	—
124	C₄⋊C₄×C₃₁	—	2	4	no	—
125	C₄⋊C₄×C₃₁	—	2	4	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	4	no	—
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²⋊C₄×C₃₁	—	2	8	no	—
135	C₂²⋊C₄×C₃₁	—	2	8	no	—
136	C₂²⋊C₄×C₃₁	—	2	4	no	—
137	C₄²×C₃₁	—	2	8	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	4	2	no	—
141	C₄²×C₃₁	C₂×D₄	1	2	yes	—
142	C₄²×C₃₁	C₂×D₄	1	2	yes	—
143	#25632	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₃₁	—	2	8	no	—
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	4	2	no	—
153	Q₈×C₃₁	—	4	4	no	—
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	Q₈×C₃₁	—	4	4	no	—
157	D₄×C₃₁	—	2	2	no	—
158	D₄×C₃₁	—	2	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	4	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	4	no	—
168	C₂×C₄×C₃₁	—	4	4	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	8	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	4	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	#120	1	2	yes	—
178	C₂×C₄×C₃₁	#120	1	2	yes	—
179	C₂×C₄×C₃₁	#135	1	2	yes	—
180	C₂×C₄×C₃₁	#137	1	2	yes	—
181	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
184	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
185	C₈×C₃₁	—	2	4	no	—
186	C₈×C₃₁	—	4	2	no	—
187	C₂×#556	C₂×C₃₁	1	1	yes	characteristic
188	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂×#461	C₂×C₃₁	1	1	yes	characteristic
190	#2568	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#2015	C₂×C₃₁	1	4	yes	—
192	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1981	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1878	C₂×C₃₁	1	4	yes	complemented
195	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
202	C₂²×C₃₁	#550	1	2	yes	—
203	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
204	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
205	C₂²×C₃₁	#578	1	2	yes	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	4	4	no	—
208	C₄×C₃₁	—	4	2	no	—
209	C₄×C₃₁	—	4	2	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	8	2	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	#452	1	4	yes	—
214	C₄×C₃₁	#532	1	1	yes	characteristic
215	C₄×C₃₁	#535	1	1	yes	characteristic
216	C₄×C₃₁	#546	1	2	yes	—
217	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
218	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
219	#556	—	2	4	no	—
220	#556	—	2	4	no	—
221	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#126	—	2	2	no	—
224	C₂×#124	C₂²×C₃₁	1	2	yes	—
225	C₂×C₂²⋊Q₈	—	2	1	no	—
226	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
227	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#117	—	2	1	no	—
230	#494	—	2	4	no	—
231	#461	—	2	4	no	—
232	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#103	C₂²×C₃₁	1	2	yes	—
234	C₂×#103	C₂²×C₃₁	1	4	yes	—
235	C₂×#102	C₂²×C₃₁	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
237	#400	—	2	2	no	—
238	#391	—	2	1	no	—
239	#391	C₂²×C₃₁	1	4	yes	—
240	#382	—	2	1	no	—
241	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
242	#340	C₂²×C₃₁	1	2	yes	—
243	#336	C₂²×C₃₁	1	4	yes	—
244	C₂×C₃₁	—	8	2	no	—
245	C₂×C₃₁	#2431	1	1	yes	characteristic
246	C₂×C₃₁	C₂×#452	1	1	yes	characteristic
247	C₂×C₃₁	#3011	1	1	yes	characteristic
248	C₂×C₃₁	#3393	1	2	yes	—
249	C₂×C₃₁	#3398	1	2	yes	—
250	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
251	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
252	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
253	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
254	C₂³×C₄	—	2	1	no	—
255	C₂×Q₁₆	—	2	8	no	—
256	C₂×Q₁₆	—	2	8	no	—
257	C₂×Q₁₆	—	4	4	no	—
258	C₂×SD₁₆	—	2	4	no	—
259	C₂×SD₁₆	—	4	4	no	—
260	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
261	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
262	C₄⋊Q₈	—	2	2	no	—
263	C₄⋊Q₈	—	2	2	no	—
264	C₄⋊Q₈	—	2	2	no	—
265	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
266	#126	—	2	16	no	—
267	#124	—	2	4	no	—
268	#124	—	2	4	no	—
269	C₂²⋊Q₈	—	2	8	no	—
270	C₄×Q₈	—	2	4	no	—
271	C₄×Q₈	—	2	4	no	—
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	—	2	2	no	—
274	C₄×D₄	—	2	2	no	—
275	C₄×D₄	D₄×C₃₁	1	4	yes	—
276	#117	—	2	8	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	—	2	1	no	—
291	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
292	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
293	C₄⋊C₈	—	2	2	no	—
294	#103	—	2	4	no	—
295	#103	—	4	4	no	—
296	#102	—	2	4	no	—
297	C₄×C₈	—	2	2	no	—
298	#95	—	2	1	no	—
299	#95	—	2	4	no	—
300	#95	—	2	4	no	—
301	C₃₁	#25632	1	1	yes	characteristic, direct product factor, Sylow
302	C₂⁴	—	2	1	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	4	no	—
310	C₂×Q₈	—	2	4	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	—	2	4	no	—
313	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	4	2	no	—
324	C₂²×C₄	—	4	2	no	—
325	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
327	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
330	Q₁₆	—	4	16	no	—
331	SD₁₆	—	4	8	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	4	no	—
336	C₂×C₈	#41×C₃₁	1	4	yes	—
337	C₄⋊C₄	—	2	8	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	4	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	4	no	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
348	C₂²⋊C₄	—	2	8	no	—
349	C₂²⋊C₄	—	2	8	no	—
350	C₂²⋊C₄	—	2	4	no	—
351	C₄²	—	2	8	no	—
352	C₄²	—	2	2	no	—
353	C₄²	—	2	2	no	—
354	C₄²	—	4	2	no	—
355	C₄²	C₂×D₄×C₃₁	1	2	yes	—
356	C₄²	C₂×D₄×C₃₁	1	2	yes	—
357	C₂³	—	2	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	4	1	no	—
360	C₂³	—	4	1	no	—
361	C₂³	—	4	2	no	—
362	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
363	Q₈	—	2	8	no	—
364	Q₈	—	4	4	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	Q₈	—	4	4	no	—
370	D₄	—	2	2	no	—
371	D₄	—	2	2	no	—
372	D₄	—	4	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	4	4	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	8	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	C₃₁×#120	1	2	yes	—
391	C₂×C₄	C₃₁×#120	1	2	yes	—
392	C₂×C₄	C₃₁×#135	1	2	yes	—
393	C₂×C₄	C₃₁×#137	1	2	yes	—
394	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
397	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
398	C₈	—	2	4	no	—
399	C₈	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
406	C₂²	C₃₁×#550	1	2	yes	—
407	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
408	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
409	C₂²	C₃₁×#578	1	2	yes	—
410	C₄	—	2	2	no	—
411	C₄	—	4	4	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	C₃₁×#452	1	4	yes	—
418	C₄	C₃₁×#532	1	1	yes	characteristic
419	C₄	C₃₁×#535	1	1	yes	characteristic
420	C₄	C₃₁×#546	1	2	yes	—
421	C₂	—	8	2	no	—
422	C₂	C₃₁×#2431	1	1	yes	characteristic
423	C₂	C₂×C₃₁×#452	1	1	yes	characteristic
424	C₂	C₃₁×#3011	1	1	yes	characteristic
425	C₂	C₃₁×#3393	1	2	yes	—
426	C₂	C₃₁×#3398	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#494	C₂	semidirect
4. C₃₁×#2568	C₂	semidirect
6. C₃₁×#1984	C₂	semidirect
7. C₃₁×#1981	C₂	semidirect
8. C₃₁×#1878	C₂	semidirect
143. #25632	C₃₁	direct (N × H)
188. C₂×#494	C₂×C₃₁	semidirect
190. #2568	C₂×C₃₁	semidirect
192. #1984	C₂×C₃₁	semidirect
193. #1981	C₂×C₃₁	semidirect
194. #1878	C₂×C₃₁	semidirect
301. C₃₁	#25632	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×#556	2
C₂×C₃₁×#494	2
C₂×C₃₁×#461	2
C₃₁×#2568	2
C₃₁×#2015	2
C₃₁×#1984	2
C₃₁×#1981	2
C₃₁×#1878	2
C₄×C₃₁×#102	2
#25632	31

Subgroups — C31×#25632 #21482391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25632 #21482391