Subgroups — C₃₁×#19679 #21476438

Order 7936.

806 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1722	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1719	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	—	2	2	no	—
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
17	C₃₁×#501	—	2	2	no	—
18	C₃₁×#500	C₂²	1	2	yes	complemented
19	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#500	C₂²	1	2	yes	complemented
21	C₃₁×#498	C₂²	1	2	yes	complemented
22	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#492	—	2	2	no	—
24	C₃₁×#492	C₂²	1	2	yes	—
25	C₃₁×#492	C₂²	1	2	yes	—
26	C₃₁×#492	C₂²	1	2	yes	—
27	C₃₁×#490	—	2	2	no	—
28	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#487	C₂²	1	2	yes	complemented
31	C₃₁×#486	C₂²	1	2	yes	—
32	C₃₁×#486	C₂²	1	2	yes	—
33	C₃₁×#486	C₂²	1	2	yes	—
34	C₃₁×#484	—	2	2	no	—
35	C₃₁×#477	—	2	2	no	—
36	C₃₁×#467	—	2	2	no	—
37	C₃₁×#460	—	2	2	no	—
38	C₃₁×#450	—	2	2	no	—
39	C₄×Q₁₆×C₃₁	—	2	2	no	—
40	C₃₁×#431	C₂²	1	1	yes	characteristic
41	C₃₁×#428	—	2	2	no	—
42	C₃₁×#418	—	2	2	no	—
43	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	complemented
46	C₃₁×#322	C₂²	1	2	yes	complemented
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂×Q₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×SD₁₆×C₃₁	—	2	4	no	—
55	C₂×SD₁₆×C₃₁	—	4	8	no	—
56	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
57	C₂²×C₈×C₃₁	D₄	1	2	yes	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	—	2	1	no	—
68	C₃₁×#127	—	2	1	no	—
69	C₃₁×#127	C₂³	1	2	yes	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	1	no	—
73	C₃₁×C₂²⋊Q₈	—	2	1	no	—
74	C₃₁×C₂²⋊Q₈	—	2	1	no	—
75	C₃₁×C₂²⋊Q₈	—	2	1	no	—
76	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₄×Q₈×C₃₁	—	2	1	no	—
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₃₁×#117	—	2	1	no	—
81	C₃₁×#117	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#107	—	2	4	no	—
86	C₃₁×#107	—	2	2	no	—
87	C₃₁×#107	—	2	4	no	—
88	C₃₁×#106	—	2	4	no	—
89	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#103	—	2	2	no	—
92	C₃₁×#103	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₃₁×#97	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	D₄	1	2	yes	—
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	C₂⁴×C₃₁	—	4	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	4	4	no	—
110	C₂×Q₈×C₃₁	—	4	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	4	4	no	—
115	C₂×D₄×C₃₁	—	4	1	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂×D₄×C₃₁	—	4	1	no	—
118	C₂×D₄×C₃₁	—	4	4	no	—
119	C₂×D₄×C₃₁	—	8	2	no	—
120	C₂²×C₄×C₃₁	—	4	1	no	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	Q₁₆×C₃₁	—	4	4	no	—
125	SD₁₆×C₃₁	—	4	8	no	—
126	SD₁₆×C₃₁	—	4	8	no	—
127	SD₁₆×C₃₁	—	8	4	no	—
128	M₁₆×C₃₁	—	2	4	no	—
129	C₂×C₈×C₃₁	—	2	4	no	—
130	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
132	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	4	1	no	—
145	C₄⋊C₄×C₃₁	—	4	1	no	—
146	C₄⋊C₄×C₃₁	—	4	1	no	—
147	C₄⋊C₄×C₃₁	—	4	1	no	—
148	C₄⋊C₄×C₃₁	—	4	1	no	—
149	C₄⋊C₄×C₃₁	—	4	1	no	—
150	C₂²⋊C₄×C₃₁	—	4	1	no	—
151	C₂²⋊C₄×C₃₁	—	4	1	no	—
152	C₄²×C₃₁	—	4	1	no	—
153	C₄²×C₃₁	—	4	1	no	—
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
158	#19679	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	8	2	no	—
161	C₂³×C₃₁	—	8	2	no	—
162	C₂³×C₃₁	—	8	2	no	—
163	C₂³×C₃₁	#127	1	1	yes	characteristic
164	Q₈×C₃₁	—	4	4	no	—
165	Q₈×C₃₁	—	4	2	no	—
166	Q₈×C₃₁	—	4	2	no	—
167	Q₈×C₃₁	—	4	2	no	—
168	Q₈×C₃₁	—	4	2	no	—
169	Q₈×C₃₁	—	8	2	no	—
170	D₄×C₃₁	—	4	4	no	—
171	D₄×C₃₁	—	4	4	no	—
172	D₄×C₃₁	—	4	4	no	—
173	D₄×C₃₁	—	8	2	no	—
174	D₄×C₃₁	—	8	2	no	—
175	D₄×C₃₁	—	8	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	8	2	no	—
184	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
185	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
186	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
187	C₂×C₄×C₃₁	#137	1	4	yes	—
188	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	2	4	no	—
193	C₈×C₃₁	—	2	4	no	—
194	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
195	#3230	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#3208	C₂×C₃₁	1	1	yes	characteristic
197	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
200	#3138	C₂×C₃₁	1	1	yes	characteristic
201	#1722	C₂×C₃₁	1	2	yes	complemented
202	#1719	C₂×C₃₁	1	2	yes	complemented
203	#1714	C₂×C₃₁	1	2	yes	complemented
204	#1714	C₂×C₃₁	1	2	yes	complemented
205	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	2	1	no	—
208	C₂²×C₃₁	—	8	2	no	—
209	C₂²×C₃₁	—	8	2	no	—
210	C₂²×C₃₁	—	8	2	no	—
211	C₂²×C₃₁	—	16	4	no	—
212	C₂²×C₃₁	#497	1	2	yes	—
213	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
214	C₄×C₃₁	—	2	1	no	—
215	C₄×C₃₁	—	2	1	no	—
216	C₄×C₃₁	—	8	2	no	—
217	C₄×C₃₁	—	8	1	no	—
218	C₄×C₃₁	—	8	1	no	—
219	C₄×C₃₁	—	8	1	no	—
220	C₄×C₃₁	—	8	1	no	—
221	C₄×C₃₁	#497	1	2	yes	—
222	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
223	C₄×C₃₁	C₂×#137	1	2	yes	—
224	C₄×C₃₁	#577	1	2	yes	—
225	C₂²×SD₁₆	—	2	2	no	—
226	#533	C₂²×C₃₁	1	1	yes	characteristic
227	#533	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
230	#501	—	2	2	no	—
231	#500	C₂²×C₃₁	1	2	yes	complemented
232	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
233	#500	C₂²×C₃₁	1	2	yes	complemented
234	#498	C₂²×C₃₁	1	2	yes	complemented
235	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
236	#492	—	2	2	no	—
237	#492	C₂²×C₃₁	1	2	yes	—
238	#492	C₂²×C₃₁	1	2	yes	—
239	#492	C₂²×C₃₁	1	2	yes	—
240	#490	—	2	2	no	—
241	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
242	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
243	#487	C₂²×C₃₁	1	2	yes	complemented
244	#486	C₂²×C₃₁	1	2	yes	—
245	#486	C₂²×C₃₁	1	2	yes	—
246	#486	C₂²×C₃₁	1	2	yes	—
247	#484	—	2	2	no	—
248	#477	—	2	2	no	—
249	#467	—	2	2	no	—
250	#460	—	2	2	no	—
251	#450	—	2	2	no	—
252	C₄×Q₁₆	—	2	2	no	—
253	#431	C₂²×C₃₁	1	1	yes	characteristic
254	#428	—	2	2	no	—
255	#418	—	2	2	no	—
256	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
258	#322	C₂²×C₃₁	1	2	yes	complemented
259	#322	C₂²×C₃₁	1	2	yes	complemented
260	C₂×C₃₁	—	2	2	no	—
261	C₂×C₃₁	—	16	2	no	—
262	C₂×C₃₁	C₂×#497	1	1	yes	characteristic
263	C₂×C₃₁	#3151	1	1	yes	characteristic
264	C₂×C₃₁	#3154	1	1	yes	characteristic
265	C₂²×Q₈	—	2	1	no	—
266	C₂²×D₄	—	2	1	no	—
267	C₂²×D₄	—	2	1	no	—
268	C₂²×D₄	—	2	1	no	—
269	C₂×Q₁₆	—	2	2	no	—
270	C₂×SD₁₆	—	2	4	no	—
271	C₂×SD₁₆	—	2	4	no	—
272	C₂×SD₁₆	—	2	4	no	—
273	C₂×SD₁₆	—	4	8	no	—
274	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
275	C₂²×C₈	D₄×C₃₁	1	2	yes	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	C₄⋊Q₈	—	2	1	no	—
279	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
280	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
283	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
284	#127	—	2	1	no	—
285	#127	—	2	1	no	—
286	#127	—	2	1	no	—
287	#127	C₂³×C₃₁	1	2	yes	—
288	#125	—	2	1	no	—
289	#125	—	2	1	no	—
290	C₂²⋊Q₈	—	2	1	no	—
291	C₂²⋊Q₈	—	2	1	no	—
292	C₂²⋊Q₈	—	2	1	no	—
293	C₂²⋊Q₈	—	2	1	no	—
294	C₄×Q₈	—	2	1	no	—
295	C₄×Q₈	—	2	1	no	—
296	C₄×Q₈	—	2	1	no	—
297	C₄×Q₈	—	2	1	no	—
298	#117	—	2	1	no	—
299	#117	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
303	#107	—	2	4	no	—
304	#107	—	2	2	no	—
305	#107	—	2	4	no	—
306	#106	—	2	4	no	—
307	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
308	#103	—	2	2	no	—
309	#103	—	2	2	no	—
310	#103	—	2	4	no	—
311	#102	—	2	4	no	—
312	#102	—	2	4	no	—
313	#102	—	2	4	no	—
314	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
315	#97	D₄×C₃₁	1	2	yes	—
316	C₄×C₈	D₄×C₃₁	1	2	yes	—
317	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄×C₈	C₂³×C₃₁	1	2	yes	—
320	C₄×C₈	C₂³×C₃₁	1	2	yes	—
321	C₃₁	#19679	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	—	4	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	4	4	no	—
329	C₂×Q₈	—	4	1	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	—	4	1	no	—
335	C₂×D₄	—	4	1	no	—
336	C₂×D₄	—	4	1	no	—
337	C₂×D₄	—	4	4	no	—
338	C₂×D₄	—	8	2	no	—
339	C₂²×C₄	—	4	1	no	—
340	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	Q₁₆	—	4	4	no	—
344	SD₁₆	—	4	8	no	—
345	SD₁₆	—	4	8	no	—
346	SD₁₆	—	8	4	no	—
347	M₁₆	—	2	4	no	—
348	C₂×C₈	—	2	4	no	—
349	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
350	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
351	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
352	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	4	1	no	—
364	C₄⋊C₄	—	4	1	no	—
365	C₄⋊C₄	—	4	1	no	—
366	C₄⋊C₄	—	4	1	no	—
367	C₄⋊C₄	—	4	1	no	—
368	C₄⋊C₄	—	4	1	no	—
369	C₂²⋊C₄	—	4	1	no	—
370	C₂²⋊C₄	—	4	1	no	—
371	C₄²	—	4	1	no	—
372	C₄²	—	4	1	no	—
373	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
377	C₂³	—	4	2	no	—
378	C₂³	—	8	2	no	—
379	C₂³	—	8	2	no	—
380	C₂³	—	8	2	no	—
381	C₂³	C₃₁×#127	1	1	yes	characteristic
382	Q₈	—	4	4	no	—
383	Q₈	—	4	2	no	—
384	Q₈	—	4	2	no	—
385	Q₈	—	4	2	no	—
386	Q₈	—	4	2	no	—
387	Q₈	—	8	2	no	—
388	D₄	—	4	4	no	—
389	D₄	—	4	4	no	—
390	D₄	—	4	4	no	—
391	D₄	—	8	2	no	—
392	D₄	—	8	2	no	—
393	D₄	—	8	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	8	2	no	—
402	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
403	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
404	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
405	C₂×C₄	C₃₁×#137	1	4	yes	—
406	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	2	4	no	—
411	C₈	—	2	4	no	—
412	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	8	2	no	—
418	C₂²	—	16	4	no	—
419	C₂²	C₃₁×#497	1	2	yes	—
420	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
421	C₄	—	2	1	no	—
422	C₄	—	2	1	no	—
423	C₄	—	8	2	no	—
424	C₄	—	8	1	no	—
425	C₄	—	8	1	no	—
426	C₄	—	8	1	no	—
427	C₄	—	8	1	no	—
428	C₄	C₃₁×#497	1	2	yes	—
429	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
430	C₄	C₂×C₃₁×#137	1	2	yes	—
431	C₄	C₃₁×#577	1	2	yes	—
432	C₂	—	2	2	no	—
433	C₂	—	16	2	no	—
434	C₂	C₂×C₃₁×#497	1	1	yes	characteristic
435	C₂	C₃₁×#3151	1	1	yes	characteristic
436	C₂	C₃₁×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3230	C₂	semidirect
3. C₃₁×#3170	C₂	semidirect
4. C₃₁×#3165	C₂	semidirect
7. C₃₁×#1722	C₂	semidirect
8. C₃₁×#1719	C₂	semidirect
9. C₃₁×#1714	C₂	semidirect
10. C₃₁×#1714	C₂	semidirect
11. C₃₁×#1451	C₂	semidirect
18. C₃₁×#500	C₂²	semidirect
19. C₃₁×#500	C₂²	semidirect
20. C₃₁×#500	C₂²	semidirect
21. C₃₁×#498	C₂²	semidirect
22. C₃₁×#494	C₂²	semidirect
28. C₃₁×#487	C₂²	semidirect
29. C₃₁×#487	C₂²	semidirect
30. C₃₁×#487	C₂²	semidirect
45. C₃₁×#322	C₂²	semidirect
46. C₃₁×#322	C₂²	semidirect
158. #19679	C₃₁	direct (N × H)
195. #3230	C₂×C₃₁	semidirect
197. #3170	C₂×C₃₁	semidirect
198. #3165	C₂×C₃₁	semidirect
201. #1722	C₂×C₃₁	semidirect
202. #1719	C₂×C₃₁	semidirect
203. #1714	C₂×C₃₁	semidirect
204. #1714	C₂×C₃₁	semidirect
205. #1451	C₂×C₃₁	semidirect
231. #500	C₂²×C₃₁	semidirect
232. #500	C₂²×C₃₁	semidirect
233. #500	C₂²×C₃₁	semidirect
234. #498	C₂²×C₃₁	semidirect
235. #494	C₂²×C₃₁	semidirect
241. #487	C₂²×C₃₁	semidirect
242. #487	C₂²×C₃₁	semidirect
243. #487	C₂²×C₃₁	semidirect
258. #322	C₂²×C₃₁	semidirect
259. #322	C₂²×C₃₁	semidirect
321. C₃₁	#19679	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3230	2
C₃₁×#3208	2
C₃₁×#3170	2
C₃₁×#3165	2
C₂×C₃₁×#492	2
C₃₁×#3138	2
C₃₁×#1722	2
C₃₁×#1719	2
C₃₁×#1714	2
C₃₁×#1451	2
#19679	31

Subgroups — C31×#19679 #21476438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19679 #21476438