Subgroups — C₃₁×#10062 #21466821

Order 7936.

806 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2541	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₁	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#127	—	2	1	no	—
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#121	—	2	2	no	—
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#486	—	2	2	no	—
15	C₃₁×#486	—	2	2	no	—
16	C₃₁×#461	—	2	2	no	—
17	C₃₁×#461	—	2	2	no	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	—	2	1	no	—
21	C₂×C₃₁×#102	—	2	2	no	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#392	—	2	2	no	—
24	C₃₁×#390	—	2	1	no	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#340	—	2	2	no	—
27	C₃₁×#331	—	2	2	no	—
28	C₃₁×D₄⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₃₁	—	2	1	no	—
30	C₂²×D₄×C₃₁	—	2	4	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	—	2	1	no	—
33	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×D₄×C₃₁	D₄	1	2	yes	—
35	C₂³×C₄×C₃₁	—	2	1	no	—
36	C₂×SD₁₆×C₃₁	—	2	4	no	—
37	C₂×SD₁₆×C₃₁	—	4	2	no	—
38	C₂×SD₁₆×C₃₁	—	4	2	no	—
39	C₂²×C₈×C₃₁	—	2	1	no	—
40	C₂²×C₈×C₃₁	—	2	1	no	—
41	C₂²×C₈×C₃₁	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×C₄⋊Q₈	—	2	1	no	—
45	C₃₁×C₄⋊Q₈	D₄	1	2	yes	complemented
46	C₃₁×#127	—	2	1	no	—
47	C₃₁×#127	—	2	1	no	—
48	C₃₁×#127	—	2	1	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#127	—	4	2	no	—
51	C₃₁×#127	—	4	2	no	—
52	C₃₁×#121	—	4	4	no	—
53	C₃₁×#121	—	4	4	no	—
54	C₄×D₄×C₃₁	—	2	1	no	—
55	C₄×D₄×C₃₁	—	2	1	no	—
56	C₄×D₄×C₃₁	—	2	1	no	—
57	C₄×D₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊C₈	—	2	1	no	—
67	C₃₁×C₄⋊C₈	—	2	1	no	—
68	C₃₁×C₄⋊C₈	—	2	1	no	—
69	C₃₁×C₄⋊C₈	D₄	1	2	yes	complemented
70	C₃₁×#103	—	2	4	no	—
71	C₃₁×#102	—	2	8	no	—
72	C₄×C₈×C₃₁	—	2	1	no	—
73	C₄×C₈×C₃₁	D₄	1	2	yes	complemented
74	C₃₁×#95	—	2	1	no	—
75	C₂⁴×C₃₁	—	2	1	no	—
76	C₂⁴×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	2	2	no	—
78	C₂×Q₈×C₃₁	—	4	1	no	—
79	C₂×Q₈×C₃₁	—	4	1	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×Q₈×C₃₁	—	4	1	no	—
82	C₂×Q₈×C₃₁	—	4	1	no	—
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	2	1	no	—
89	C₂×D₄×C₃₁	—	2	1	no	—
90	C₂×D₄×C₃₁	—	4	4	no	—
91	C₂×D₄×C₃₁	—	4	8	no	—
92	C₂×D₄×C₃₁	—	4	8	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	1	no	—
96	C₂×D₄×C₃₁	—	4	1	no	—
97	C₂×D₄×C₃₁	—	4	1	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	1	no	—
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	—	4	1	no	—
102	C₂×D₄×C₃₁	—	4	1	no	—
103	C₂×D₄×C₃₁	—	8	4	no	—
104	C₂×D₄×C₃₁	D₈	1	2	yes	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	4	1	no	—
108	C₂²×C₄×C₃₁	—	4	1	no	—
109	C₂²×C₄×C₃₁	—	4	1	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	SD₁₆×C₃₁	—	4	8	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	4	2	no	—
119	C₂×C₈×C₃₁	—	4	1	no	—
120	C₂×C₈×C₃₁	—	4	1	no	—
121	C₂×C₈×C₃₁	—	4	1	no	—
122	C₂×C₈×C₃₁	—	4	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	4	1	no	—
129	C₄⋊C₄×C₃₁	—	4	1	no	—
130	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	complemented
131	C₂²⋊C₄×C₃₁	—	4	8	no	—
132	C₂²⋊C₄×C₃₁	—	4	1	no	—
133	C₂²⋊C₄×C₃₁	—	4	1	no	—
134	C₄²×C₃₁	—	4	1	no	—
135	C₄²×C₃₁	—	4	1	no	—
136	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₃₁	#41	1	1	yes	characteristic
140	#10062	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	4	1	no	—
143	C₂³×C₃₁	—	4	1	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	—	4	4	no	—
150	C₂³×C₃₁	—	4	4	no	—
151	C₂³×C₃₁	#120	1	1	yes	characteristic
152	Q₈×C₃₁	—	4	2	no	—
153	Q₈×C₃₁	—	4	2	no	—
154	Q₈×C₃₁	—	8	1	no	—
155	Q₈×C₃₁	—	8	1	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	2	2	no	—
158	D₄×C₃₁	—	4	4	no	—
159	D₄×C₃₁	—	4	1	no	—
160	D₄×C₃₁	—	4	1	no	—
161	D₄×C₃₁	—	8	2	no	—
162	D₄×C₃₁	—	8	16	no	—
163	D₄×C₃₁	—	8	1	no	—
164	D₄×C₃₁	—	8	1	no	—
165	D₄×C₃₁	—	8	2	no	—
166	D₄×C₃₁	—	8	1	no	—
167	D₄×C₃₁	—	8	1	no	—
168	D₄×C₃₁	—	8	1	no	—
169	D₄×C₃₁	—	8	1	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	8	1	no	—
178	C₂×C₄×C₃₁	—	8	1	no	—
179	C₂×C₄×C₃₁	—	8	2	no	—
180	C₂×C₄×C₃₁	—	8	1	no	—
181	C₂×C₄×C₃₁	—	8	1	no	—
182	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
183	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
184	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
185	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
186	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
187	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
190	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
191	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
194	C₈×C₃₁	—	4	2	no	—
195	C₈×C₃₁	—	4	2	no	—
196	C₈×C₃₁	—	8	1	no	—
197	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#2541	C₂×C₃₁	1	1	yes	characteristic
200	#1988	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂×#331	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂×D₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1279	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂²×C₃₁	—	4	4	no	—
205	C₂²×C₃₁	—	4	2	no	—
206	C₂²×C₃₁	—	8	2	no	—
207	C₂²×C₃₁	—	8	1	no	—
208	C₂²×C₃₁	—	8	1	no	—
209	C₂²×C₃₁	—	8	4	no	—
210	C₂²×C₃₁	—	8	4	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	#392	1	1	yes	characteristic
215	C₂²×C₃₁	#447	1	1	yes	characteristic
216	C₂²×C₃₁	#449	1	1	yes	characteristic
217	C₂²×C₃₁	#450	1	2	yes	—
218	C₂²×C₃₁	#455	1	2	yes	—
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	4	4	no	—
222	C₄×C₃₁	—	8	2	no	—
223	C₄×C₃₁	—	8	1	no	—
224	C₄×C₃₁	#447	1	1	yes	characteristic
225	C₄×C₃₁	#449	1	1	yes	characteristic
226	C₄×C₃₁	#460	1	2	yes	—
227	C₄×C₃₁	#466	1	1	yes	characteristic
228	C₄×C₃₁	#469	1	1	yes	characteristic
229	C₄×C₃₁	#492	1	1	yes	characteristic
230	C₄×C₃₁	#496	1	1	yes	characteristic
231	C₂²×SD₁₆	—	2	1	no	—
232	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#127	—	2	1	no	—
234	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#121	—	2	2	no	—
236	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
237	#486	—	2	2	no	—
238	#486	—	2	2	no	—
239	#461	—	2	2	no	—
240	#461	—	2	2	no	—
241	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#103	—	2	1	no	—
244	C₂×#102	—	2	2	no	—
245	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
246	#392	—	2	2	no	—
247	#390	—	2	1	no	—
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
249	#340	—	2	2	no	—
250	#331	—	2	2	no	—
251	D₄⋊C₈	—	2	2	no	—
252	C₂×C₃₁	—	8	2	no	—
253	C₂×C₃₁	—	16	2	no	—
254	C₂×C₃₁	#1639	1	2	yes	—
255	C₂×C₃₁	#1642	1	2	yes	—
256	C₂×C₃₁	#1999	1	1	yes	characteristic
257	C₂×C₃₁	#2004	1	1	yes	characteristic
258	C₂×C₃₁	#2014	1	1	yes	characteristic
259	C₂²×Q₈	—	2	1	no	—
260	C₂²×D₄	—	2	4	no	—
261	C₂²×D₄	—	2	1	no	—
262	C₂²×D₄	—	2	1	no	—
263	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
264	C₂²×D₄	D₄×C₃₁	1	2	yes	—
265	C₂³×C₄	—	2	1	no	—
266	C₂×SD₁₆	—	2	4	no	—
267	C₂×SD₁₆	—	4	2	no	—
268	C₂×SD₁₆	—	4	2	no	—
269	C₂²×C₈	—	2	1	no	—
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	—	2	1	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	—	2	1	no	—
275	C₄⋊Q₈	D₄×C₃₁	1	2	yes	complemented
276	#127	—	2	1	no	—
277	#127	—	2	1	no	—
278	#127	—	2	1	no	—
279	#127	—	2	1	no	—
280	#127	—	4	2	no	—
281	#127	—	4	2	no	—
282	#121	—	4	4	no	—
283	#121	—	4	4	no	—
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	—	2	1	no	—
287	C₄×D₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
292	C₂×C₂²⋊C₄	—	2	4	no	—
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	—	2	1	no	—
298	C₄⋊C₈	—	2	1	no	—
299	C₄⋊C₈	D₄×C₃₁	1	2	yes	complemented
300	#103	—	2	4	no	—
301	#102	—	2	8	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	D₄×C₃₁	1	2	yes	complemented
304	#95	—	2	1	no	—
305	C₃₁	#10062	1	1	yes	characteristic, direct product factor, Sylow
306	C₂⁴	—	2	1	no	—
307	C₂⁴	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	4	1	no	—
310	C₂×Q₈	—	4	1	no	—
311	C₂×Q₈	—	4	1	no	—
312	C₂×Q₈	—	4	1	no	—
313	C₂×Q₈	—	4	1	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	4	4	no	—
322	C₂×D₄	—	4	8	no	—
323	C₂×D₄	—	4	8	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂×D₄	—	4	1	no	—
328	C₂×D₄	—	4	1	no	—
329	C₂×D₄	—	4	2	no	—
330	C₂×D₄	—	4	1	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂×D₄	—	8	4	no	—
335	C₂×D₄	D₈×C₃₁	1	2	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	4	1	no	—
339	C₂²×C₄	—	4	1	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	—	4	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	SD₁₆	—	4	8	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	4	2	no	—
350	C₂×C₈	—	4	1	no	—
351	C₂×C₈	—	4	1	no	—
352	C₂×C₈	—	4	1	no	—
353	C₂×C₈	—	4	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	complemented
362	C₂²⋊C₄	—	4	8	no	—
363	C₂²⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	—	4	1	no	—
367	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₄²	#41×C₃₁	1	1	yes	characteristic
371	C₂³	—	2	2	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	4	4	no	—
379	C₂³	—	4	4	no	—
380	C₂³	—	4	2	no	—
381	C₂³	C₃₁×#120	1	1	yes	characteristic
382	Q₈	—	4	2	no	—
383	Q₈	—	4	2	no	—
384	Q₈	—	8	1	no	—
385	Q₈	—	8	1	no	—
386	D₄	—	2	2	no	—
387	D₄	—	2	2	no	—
388	D₄	—	4	4	no	—
389	D₄	—	4	1	no	—
390	D₄	—	4	1	no	—
391	D₄	—	8	2	no	—
392	D₄	—	8	16	no	—
393	D₄	—	8	1	no	—
394	D₄	—	8	1	no	—
395	D₄	—	8	2	no	—
396	D₄	—	8	1	no	—
397	D₄	—	8	1	no	—
398	D₄	—	8	1	no	—
399	D₄	—	8	1	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	8	1	no	—
408	C₂×C₄	—	8	1	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	—	8	1	no	—
411	C₂×C₄	—	8	1	no	—
412	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
413	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
414	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
415	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
416	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
417	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
418	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
420	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
421	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
422	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
423	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
424	C₈	—	4	2	no	—
425	C₈	—	4	2	no	—
426	C₈	—	8	1	no	—
427	C₂²	—	4	4	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	1	no	—
432	C₂²	—	8	4	no	—
433	C₂²	—	8	4	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	C₃₁×#392	1	1	yes	characteristic
438	C₂²	C₃₁×#447	1	1	yes	characteristic
439	C₂²	C₃₁×#449	1	1	yes	characteristic
440	C₂²	C₃₁×#450	1	2	yes	—
441	C₂²	C₃₁×#455	1	2	yes	—
442	C₄	—	2	1	no	—
443	C₄	—	2	1	no	—
444	C₄	—	4	4	no	—
445	C₄	—	8	2	no	—
446	C₄	—	8	1	no	—
447	C₄	C₃₁×#447	1	1	yes	characteristic
448	C₄	C₃₁×#449	1	1	yes	characteristic
449	C₄	C₃₁×#460	1	2	yes	—
450	C₄	C₃₁×#466	1	1	yes	characteristic
451	C₄	C₃₁×#469	1	1	yes	characteristic
452	C₄	C₃₁×#492	1	1	yes	characteristic
453	C₄	C₃₁×#496	1	1	yes	characteristic
454	C₂	—	8	2	no	—
455	C₂	—	16	2	no	—
456	C₂	C₃₁×#1639	1	2	yes	—
457	C₂	C₃₁×#1642	1	2	yes	—
458	C₂	C₃₁×#1999	1	1	yes	characteristic
459	C₂	C₃₁×#2004	1	1	yes	characteristic
460	C₂	C₃₁×#2014	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#2541	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#486	C₂	semidirect
2. C₂×C₃₁×#461	C₂	semidirect
4. C₃₁×#1988	C₂	semidirect
5. C₂×C₃₁×#331	C₂	semidirect
6. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
7. C₃₁×#1279	C₂	semidirect
45. C₃₁×C₄⋊Q₈	D₄	semidirect
69. C₃₁×C₄⋊C₈	D₄	semidirect
73. C₄×C₈×C₃₁	D₄	semidirect
130. C₄⋊C₄×C₃₁	SD₁₆	semidirect
140. #10062	C₃₁	direct (N × H)
197. C₂×#486	C₂×C₃₁	semidirect
198. C₂×#461	C₂×C₃₁	semidirect
200. #1988	C₂×C₃₁	semidirect
201. C₂×#331	C₂×C₃₁	semidirect
202. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
203. #1279	C₂×C₃₁	semidirect
275. C₄⋊Q₈	D₄×C₃₁	semidirect
299. C₄⋊C₈	D₄×C₃₁	semidirect
303. C₄×C₈	D₄×C₃₁	semidirect
305. C₃₁	#10062	direct (N × H)
361. C₄⋊C₄	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#486	2
C₂×C₃₁×#461	2
C₃₁×#2541	2
C₃₁×#1988	2
C₂×C₃₁×#331	2
C₂×C₃₁×D₄⋊C₈	2
C₃₁×#1279	2
#10062	31

Subgroups — C31×#10062 #21466821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10062 #21466821