Subgroups — C₃₁×#10261 #21467020

Order 7936.

678 conjugacy classes of subgroups.

The subgroup lattice has 444 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	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₁	—	2	1	no	—
9	C₂²×D₈×C₃₁	—	2	1	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#486	—	2	2	no	—
14	C₃₁×#486	—	2	2	no	—
15	C₃₁×#461	—	2	2	no	—
16	C₃₁×#461	—	2	2	no	—
17	C₃₁×#459	—	2	2	no	—
18	C₃₁×#459	—	2	2	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	—	2	1	no	—
23	C₂×C₃₁×#102	—	2	2	no	—
24	C₂×C₃₁×#102	—	2	1	no	—
25	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#332	—	2	2	no	—
28	C₃₁×#331	—	2	2	no	—
29	C₂²×Q₈×C₃₁	—	2	1	no	—
30	C₂²×D₄×C₃₁	—	2	1	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	C₂³×C₄×C₃₁	—	2	1	no	—
35	C₂×SD₁₆×C₃₁	—	2	4	no	—
36	C₂×SD₁₆×C₃₁	—	4	2	no	—
37	C₂×SD₁₆×C₃₁	—	4	2	no	—
38	C₂×D₈×C₃₁	—	2	4	no	—
39	C₂×D₈×C₃₁	—	4	2	no	—
40	C₂×D₈×C₃₁	—	4	2	no	—
41	C₂²×C₈×C₃₁	—	2	1	no	—
42	C₂²×C₈×C₃₁	—	2	1	no	—
43	C₂²×C₈×C₃₁	—	2	1	no	—
44	C₂²×C₈×C₃₁	D₄	1	2	yes	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×C₄⋊Q₈	—	2	1	no	—
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#127	—	2	1	no	—
51	C₃₁×#127	—	2	1	no	—
52	C₃₁×#127	D₄	1	2	yes	—
53	C₄×D₄×C₃₁	—	2	1	no	—
54	C₄×D₄×C₃₁	—	2	1	no	—
55	C₄×D₄×C₃₁	—	2	1	no	—
56	C₄×D₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊C₈	—	2	1	no	—
63	C₃₁×C₄⋊C₈	—	2	1	no	—
64	C₃₁×C₄⋊C₈	—	2	1	no	—
65	C₃₁×C₄⋊C₈	—	2	1	no	—
66	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
67	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
68	C₃₁×#103	—	2	4	no	—
69	C₃₁×#102	—	2	4	no	—
70	C₃₁×#102	—	2	8	no	—
71	C₃₁×C₂²⋊C₈	—	4	4	no	—
72	C₄×C₈×C₃₁	—	2	1	no	—
73	C₄×C₈×C₃₁	—	2	1	no	—
74	C₂⁴×C₃₁	—	2	1	no	—
75	C₂⁴×C₃₁	—	4	1	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	4	1	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×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	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	1	no	—
91	C₂×D₄×C₃₁	—	4	1	no	—
92	C₂×D₄×C₃₁	—	4	1	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	2	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	—	8	2	no	—
100	C₂×D₄×C₃₁	D₈	1	2	yes	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	4	1	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	—	4	1	no	—
105	C₂²×C₄×C₃₁	—	4	2	no	—
106	C₂²×C₄×C₃₁	—	4	2	no	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
110	SD₁₆×C₃₁	—	4	8	no	—
111	D₈×C₃₁	—	4	8	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	4	1	no	—
118	C₂×C₈×C₃₁	—	4	1	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	1	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	4	1	no	—
128	C₄⋊C₄×C₃₁	—	4	1	no	—
129	C₂²⋊C₄×C₃₁	—	4	1	no	—
130	C₂²⋊C₄×C₃₁	—	4	1	no	—
131	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
135	#10261	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂³×C₃₁	—	2	2	no	—
137	C₂³×C₃₁	—	4	1	no	—
138	C₂³×C₃₁	—	4	2	no	—
139	C₂³×C₃₁	—	4	1	no	—
140	C₂³×C₃₁	—	4	2	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	—	8	2	no	—
143	C₂³×C₃₁	—	8	1	no	—
144	C₂³×C₃₁	—	8	1	no	—
145	C₂³×C₃₁	—	8	2	no	—
146	C₂³×C₃₁	#121	1	1	yes	characteristic
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	8	1	no	—
150	Q₈×C₃₁	—	8	1	no	—
151	D₄×C₃₁	—	2	2	no	—
152	D₄×C₃₁	—	2	2	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	1	no	—
155	D₄×C₃₁	—	4	1	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	4	4	no	—
158	D₄×C₃₁	—	8	2	no	—
159	D₄×C₃₁	—	8	2	no	—
160	D₄×C₃₁	—	8	1	no	—
161	D₄×C₃₁	—	8	1	no	—
162	D₄×C₃₁	—	8	1	no	—
163	D₄×C₃₁	—	8	1	no	—
164	C₂×C₄×C₃₁	—	2	2	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	—	8	1	no	—
169	C₂×C₄×C₃₁	—	8	1	no	—
170	C₂×C₄×C₃₁	—	8	2	no	—
171	C₂×C₄×C₃₁	—	8	1	no	—
172	C₂×C₄×C₃₁	—	8	1	no	—
173	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
174	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
175	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
176	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
177	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
178	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#135	1	2	yes	—
182	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
185	C₈×C₃₁	—	4	2	no	—
186	C₈×C₃₁	—	4	2	no	—
187	C₈×C₃₁	—	4	2	no	—
188	C₈×C₃₁	—	8	1	no	—
189	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
191	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
192	#1954	C₂×C₃₁	1	1	yes	characteristic, complemented
193	C₂×#332	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂×#331	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1277	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₂²×C₃₁	—	4	4	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	1	no	—
200	C₂²×C₃₁	—	8	1	no	—
201	C₂²×C₃₁	—	8	2	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	8	2	no	—
204	C₂²×C₃₁	—	16	2	no	—
205	C₂²×C₃₁	—	16	2	no	—
206	C₂²×C₃₁	#392	1	1	yes	characteristic
207	C₂²×C₃₁	#463	1	2	yes	—
208	C₂²×C₃₁	#466	1	1	yes	characteristic
209	C₂²×C₃₁	#469	1	1	yes	characteristic
210	C₂²×C₃₁	#471	1	2	yes	—
211	C₄×C₃₁	—	2	1	no	—
212	C₄×C₃₁	—	2	1	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	—	8	1	no	—
215	C₄×C₃₁	#447	1	1	yes	characteristic
216	C₄×C₃₁	#447	1	1	yes	characteristic
217	C₄×C₃₁	#449	1	1	yes	characteristic
218	C₄×C₃₁	#449	1	1	yes	characteristic
219	C₄×C₃₁	#449	1	2	yes	—
220	C₄×C₃₁	#495	1	1	yes	characteristic
221	C₄×C₃₁	#496	1	1	yes	characteristic
222	C₂²×SD₁₆	—	2	1	no	—
223	C₂²×D₈	—	2	1	no	—
224	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
227	#486	—	2	2	no	—
228	#486	—	2	2	no	—
229	#461	—	2	2	no	—
230	#461	—	2	2	no	—
231	#459	—	2	2	no	—
232	#459	—	2	2	no	—
233	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#103	—	2	1	no	—
237	C₂×#102	—	2	2	no	—
238	C₂×#102	—	2	1	no	—
239	C₂×C₂²⋊C₈	—	2	2	no	—
240	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
241	#332	—	2	2	no	—
242	#331	—	2	2	no	—
243	C₂×C₃₁	—	8	2	no	—
244	C₂×C₃₁	—	16	2	no	—
245	C₂×C₃₁	#1681	1	2	yes	—
246	C₂×C₃₁	#1701	1	2	yes	—
247	C₂×C₃₁	#1999	1	1	yes	characteristic
248	C₂×C₃₁	#2012	1	1	yes	characteristic
249	C₂×C₃₁	#2019	1	1	yes	characteristic
250	C₂²×Q₈	—	2	1	no	—
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	—	2	1	no	—
254	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
255	C₂³×C₄	—	2	1	no	—
256	C₂×SD₁₆	—	2	4	no	—
257	C₂×SD₁₆	—	4	2	no	—
258	C₂×SD₁₆	—	4	2	no	—
259	C₂×D₈	—	2	4	no	—
260	C₂×D₈	—	4	2	no	—
261	C₂×D₈	—	4	2	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₂²×C₈	—	2	1	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	D₄×C₃₁	1	2	yes	—
266	C₄⋊Q₈	—	2	1	no	—
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
270	#127	—	2	1	no	—
271	#127	—	2	1	no	—
272	#127	—	2	1	no	—
273	#127	D₄×C₃₁	1	2	yes	—
274	C₄×D₄	—	2	1	no	—
275	C₄×D₄	—	2	1	no	—
276	C₄×D₄	—	2	1	no	—
277	C₄×D₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₂²⋊C₄	—	2	1	no	—
282	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
288	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
289	#103	—	2	4	no	—
290	#102	—	2	4	no	—
291	#102	—	2	8	no	—
292	C₂²⋊C₈	—	4	4	no	—
293	C₄×C₈	—	2	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₃₁	#10261	1	1	yes	characteristic, direct product factor, Sylow
296	C₂⁴	—	2	1	no	—
297	C₂⁴	—	4	1	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	4	1	no	—
300	C₂×Q₈	—	4	1	no	—
301	C₂×Q₈	—	4	1	no	—
302	C₂×Q₈	—	4	1	no	—
303	C₂×Q₈	—	4	1	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	1	no	—
310	C₂×D₄	—	2	1	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	4	1	no	—
313	C₂×D₄	—	4	1	no	—
314	C₂×D₄	—	4	1	no	—
315	C₂×D₄	—	4	1	no	—
316	C₂×D₄	—	4	1	no	—
317	C₂×D₄	—	4	2	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	2	no	—
321	C₂×D₄	—	8	2	no	—
322	C₂×D₄	D₈×C₃₁	1	2	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	4	1	no	—
325	C₂²×C₄	—	4	1	no	—
326	C₂²×C₄	—	4	1	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	—	4	2	no	—
329	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
332	SD₁₆	—	4	8	no	—
333	D₈	—	4	8	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	4	1	no	—
340	C₂×C₈	—	4	1	no	—
341	C₂×C₈	—	4	1	no	—
342	C₂×C₈	—	4	1	no	—
343	C₂×C₈	—	4	1	no	—
344	C₂×C₈	—	4	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₂²⋊C₄	—	4	1	no	—
352	C₂²⋊C₄	—	4	1	no	—
353	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₂³	—	2	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	4	1	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	1	no	—
362	C₂³	—	4	2	no	—
363	C₂³	—	8	2	no	—
364	C₂³	—	8	1	no	—
365	C₂³	—	8	1	no	—
366	C₂³	—	8	2	no	—
367	C₂³	C₃₁×#121	1	1	yes	characteristic
368	Q₈	—	4	2	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	8	1	no	—
371	Q₈	—	8	1	no	—
372	D₄	—	2	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	4	1	no	—
376	D₄	—	4	1	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	4	no	—
379	D₄	—	8	2	no	—
380	D₄	—	8	2	no	—
381	D₄	—	8	1	no	—
382	D₄	—	8	1	no	—
383	D₄	—	8	1	no	—
384	D₄	—	8	1	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	8	1	no	—
390	C₂×C₄	—	8	1	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	8	1	no	—
393	C₂×C₄	—	8	1	no	—
394	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
395	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
396	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
397	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
398	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
399	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
400	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
402	C₂×C₄	C₃₁×#135	1	2	yes	—
403	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
404	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
405	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
406	C₈	—	4	2	no	—
407	C₈	—	4	2	no	—
408	C₈	—	4	2	no	—
409	C₈	—	8	1	no	—
410	C₂²	—	4	4	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	8	2	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	8	2	no	—
418	C₂²	—	16	2	no	—
419	C₂²	—	16	2	no	—
420	C₂²	C₃₁×#392	1	1	yes	characteristic
421	C₂²	C₃₁×#463	1	2	yes	—
422	C₂²	C₃₁×#466	1	1	yes	characteristic
423	C₂²	C₃₁×#469	1	1	yes	characteristic
424	C₂²	C₃₁×#471	1	2	yes	—
425	C₄	—	2	1	no	—
426	C₄	—	2	1	no	—
427	C₄	—	8	2	no	—
428	C₄	—	8	1	no	—
429	C₄	C₃₁×#447	1	1	yes	characteristic
430	C₄	C₃₁×#447	1	1	yes	characteristic
431	C₄	C₃₁×#449	1	1	yes	characteristic
432	C₄	C₃₁×#449	1	1	yes	characteristic
433	C₄	C₃₁×#449	1	2	yes	—
434	C₄	C₃₁×#495	1	1	yes	characteristic
435	C₄	C₃₁×#496	1	1	yes	characteristic
436	C₂	—	8	2	no	—
437	C₂	—	16	2	no	—
438	C₂	C₃₁×#1681	1	2	yes	—
439	C₂	C₃₁×#1701	1	2	yes	—
440	C₂	C₃₁×#1999	1	1	yes	characteristic
441	C₂	C₃₁×#2012	1	1	yes	characteristic
442	C₂	C₃₁×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#459	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×#1954	C₂	semidirect
5. C₂×C₃₁×#332	C₂	semidirect
6. C₂×C₃₁×#331	C₂	semidirect
7. C₃₁×#1277	C₂	semidirect
135. #10261	C₃₁	direct (N × H)
189. C₂×#486	C₂×C₃₁	semidirect
190. C₂×#461	C₂×C₃₁	semidirect
192. #1954	C₂×C₃₁	semidirect
193. C₂×#332	C₂×C₃₁	semidirect
194. C₂×#331	C₂×C₃₁	semidirect
195. #1277	C₂×C₃₁	semidirect
295. C₃₁	#10261	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×#486	2
C₂×C₃₁×#461	2
C₂×C₃₁×#459	2
C₃₁×#1954	2
C₂×C₃₁×#332	2
C₂×C₃₁×#331	2
C₃₁×#1277	2
#10261	31

Subgroups — C31×#10261 #21467020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10261 #21467020