Subgroups — C₃₁×#19631 #21476390

Order 7936.

922 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3226	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1722	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1721	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1709	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1706	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	—	2	2	no	—
13	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#500	C₂²	1	2	yes	complemented
20	C₃₁×#497	—	2	2	no	—
21	C₃₁×#496	—	2	2	no	—
22	C₃₁×#494	C₂²	1	2	yes	complemented
23	C₃₁×#493	C₂²	1	2	yes	—
24	C₃₁×#493	C₂²	1	2	yes	—
25	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
26	C₃₁×#492	C₂²	1	2	yes	—
27	C₃₁×#492	C₂²	1	2	yes	complemented
28	C₃₁×#486	—	2	2	no	—
29	C₃₁×#486	C₂²	1	2	yes	—
30	C₃₁×#486	C₂²	1	2	yes	—
31	C₃₁×#486	C₂²	1	2	yes	—
32	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#486	C₂²	1	2	yes	complemented
35	C₃₁×#480	—	2	2	no	—
36	C₃₁×#474	—	2	2	no	—
37	C₃₁×#466	—	2	2	no	—
38	C₃₁×#459	—	2	2	no	—
39	C₃₁×#447	—	2	2	no	—
40	C₄×D₈×C₃₁	—	2	2	no	—
41	C₃₁×#431	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#102	—	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	C₈²×C₃₁	C₂²	1	2	yes	complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂²×Q₈×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₃₁×#137	—	4	8	no	—
54	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
55	C₂×Q₁₆×C₃₁	—	2	4	no	—
56	C₂×SD₁₆×C₃₁	—	2	4	no	—
57	C₂×SD₁₆×C₃₁	—	2	4	no	—
58	C₂×D₈×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	4	no	—
60	C₂×D₈×C₃₁	—	2	4	no	—
61	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
62	C₂²×C₈×C₃₁	D₄	1	2	yes	—
63	C₃₁×C₄⋊Q₈	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	—	2	1	no	—
65	C₃₁×C₄⋊Q₈	—	2	1	no	—
66	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₃₁×#127	—	2	1	no	—
70	C₃₁×#127	—	2	1	no	—
71	C₃₁×#127	—	2	1	no	—
72	C₃₁×#127	C₂³	1	2	yes	—
73	C₃₁×#127	C₂³	1	1	yes	characteristic
74	C₃₁×#127	C₂³	1	1	yes	characteristic
75	C₃₁×#124	—	2	1	no	—
76	C₃₁×#124	—	2	1	no	—
77	C₃₁×#121	—	2	1	no	—
78	C₃₁×#121	—	2	1	no	—
79	C₃₁×#121	—	2	1	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	—	2	1	no	—
84	C₄×D₄×C₃₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	C₃₁×#107	—	2	2	no	—
89	C₃₁×#107	—	2	4	no	—
90	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×#102	—	2	4	no	—
97	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
98	C₃₁×#97	D₄	1	2	yes	—
99	C₄×C₈×C₃₁	D₄	1	2	yes	—
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	C₄×C₈×C₃₁	C₂³	1	2	yes	—
104	C₂⁴×C₃₁	—	4	1	no	—
105	#41×C₃₁	—	4	4	no	—
106	#41×C₃₁	—	4	4	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	4	1	no	—
111	C₂×Q₈×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	4	4	no	—
122	C₂×D₄×C₃₁	—	4	1	no	—
123	C₂×D₄×C₃₁	—	4	1	no	—
124	C₂×D₄×C₃₁	—	4	1	no	—
125	C₂×D₄×C₃₁	—	4	1	no	—
126	C₂×D₄×C₃₁	—	4	1	no	—
127	C₂×D₄×C₃₁	—	4	4	no	—
128	C₂×D₄×C₃₁	—	8	2	no	—
129	C₂²×C₄×C₃₁	—	4	1	no	—
130	C₂²×C₄×C₃₁	—	4	1	no	—
131	C₂²×C₄×C₃₁	—	4	1	no	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	Q₁₆×C₃₁	—	4	8	no	—
136	SD₁₆×C₃₁	—	4	8	no	—
137	SD₁₆×C₃₁	—	4	8	no	—
138	D₈×C₃₁	—	4	4	no	—
139	D₈×C₃₁	—	4	8	no	—
140	D₈×C₃₁	—	8	4	no	—
141	M₁₆×C₃₁	D₈	1	8	yes	complemented
142	C₂×C₈×C₃₁	—	4	2	no	—
143	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
146	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	4	1	no	—
154	C₄⋊C₄×C₃₁	—	4	1	no	—
155	C₂²⋊C₄×C₃₁	—	4	1	no	—
156	C₂²⋊C₄×C₃₁	—	4	1	no	—
157	C₂²⋊C₄×C₃₁	—	4	1	no	—
158	C₂²⋊C₄×C₃₁	—	4	1	no	—
159	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
163	#19631	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₃₁	—	4	1	no	—
165	C₂³×C₃₁	—	4	1	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	—	8	2	no	—
168	C₂³×C₃₁	—	8	2	no	—
169	C₂³×C₃₁	—	8	2	no	—
170	C₂³×C₃₁	#127	1	1	yes	characteristic
171	Q₈×C₃₁	—	4	2	no	—
172	Q₈×C₃₁	—	4	4	no	—
173	Q₈×C₃₁	—	4	2	no	—
174	Q₈×C₃₁	—	8	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	4	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	4	4	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	4	4	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	8	2	no	—
185	D₄×C₃₁	—	8	2	no	—
186	D₄×C₃₁	—	8	2	no	—
187	D₄×C₃₁	—	8	2	no	—
188	D₄×C₃₁	—	8	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	8	2	no	—
195	C₂×C₄×C₃₁	—	8	2	no	—
196	C₂×C₄×C₃₁	—	8	2	no	—
197	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
198	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
199	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
201	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	2	4	no	—
207	C₈×C₃₁	C₂×D₈	1	8	yes	—
208	#3226	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#1722	C₂×C₃₁	1	2	yes	complemented
215	#1721	C₂×C₃₁	1	2	yes	complemented
216	#1709	C₂×C₃₁	1	2	yes	complemented
217	#1706	C₂×C₃₁	1	2	yes	complemented
218	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂²×C₃₁	—	2	1	no	—
220	C₂²×C₃₁	—	2	1	no	—
221	C₂²×C₃₁	—	8	1	no	—
222	C₂²×C₃₁	—	8	1	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	—	8	2	no	—
225	C₂²×C₃₁	—	8	1	no	—
226	C₂²×C₃₁	—	8	1	no	—
227	C₂²×C₃₁	—	8	2	no	—
228	C₂²×C₃₁	—	8	1	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	—	16	4	no	—
231	C₂²×C₃₁	#493	1	2	yes	—
232	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
233	C₄×C₃₁	—	2	1	no	—
234	C₄×C₃₁	—	2	1	no	—
235	C₄×C₃₁	—	8	1	no	—
236	C₄×C₃₁	—	8	1	no	—
237	C₄×C₃₁	—	8	2	no	—
238	C₄×C₃₁	#493	1	2	yes	—
239	C₄×C₃₁	C₂²×D₈	1	2	yes	—
240	C₄×C₃₁	#575	1	2	yes	—
241	C₄×C₃₁	#577	1	2	yes	—
242	C₂×#137	—	2	2	no	—
243	C₂×AGL(1,ℤ/8)	—	2	2	no	—
244	#532	C₂²×C₃₁	1	1	yes	characteristic
245	#532	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
248	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	#500	C₂²×C₃₁	1	2	yes	complemented
250	#497	—	2	2	no	—
251	#496	—	2	2	no	—
252	#494	C₂²×C₃₁	1	2	yes	complemented
253	#493	C₂²×C₃₁	1	2	yes	—
254	#493	C₂²×C₃₁	1	2	yes	—
255	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
256	#492	C₂²×C₃₁	1	2	yes	—
257	#492	C₂²×C₃₁	1	2	yes	complemented
258	#486	—	2	2	no	—
259	#486	C₂²×C₃₁	1	2	yes	—
260	#486	C₂²×C₃₁	1	2	yes	—
261	#486	C₂²×C₃₁	1	2	yes	—
262	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
263	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
264	#486	C₂²×C₃₁	1	2	yes	complemented
265	#480	—	2	2	no	—
266	#474	—	2	2	no	—
267	#466	—	2	2	no	—
268	#459	—	2	2	no	—
269	#447	—	2	2	no	—
270	C₄×D₈	—	2	2	no	—
271	#431	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#102	—	2	2	no	—
273	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
275	#322	C₂²×C₃₁	1	2	yes	complemented
276	C₈²	C₂²×C₃₁	1	2	yes	complemented
277	C₂×C₃₁	—	2	2	no	—
278	C₂×C₃₁	—	16	1	no	—
279	C₂×C₃₁	—	16	1	no	—
280	C₂×C₃₁	—	16	2	no	—
281	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
282	C₂×C₃₁	#3153	1	1	yes	characteristic
283	C₂×C₃₁	#3154	1	1	yes	characteristic
284	C₂×#41	—	2	1	no	—
285	C₂×#41	—	2	1	no	—
286	C₂²×Q₈	—	2	1	no	—
287	C₂²×D₄	—	2	1	no	—
288	C₂²×D₄	—	2	1	no	—
289	C₂²×D₄	—	2	1	no	—
290	#137	—	4	8	no	—
291	AGL(1,ℤ/8)	—	4	8	no	—
292	C₂×Q₁₆	—	2	4	no	—
293	C₂×SD₁₆	—	2	4	no	—
294	C₂×SD₁₆	—	2	4	no	—
295	C₂×D₈	—	2	4	no	—
296	C₂×D₈	—	2	2	no	—
297	C₂×D₈	—	2	4	no	—
298	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
299	C₂²×C₈	D₄×C₃₁	1	2	yes	—
300	C₄⋊Q₈	—	2	1	no	—
301	C₄⋊Q₈	—	2	1	no	—
302	C₄⋊Q₈	—	2	1	no	—
303	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
304	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
306	#127	—	2	1	no	—
307	#127	—	2	1	no	—
308	#127	—	2	1	no	—
309	#127	C₂³×C₃₁	1	2	yes	—
310	#127	C₂³×C₃₁	1	1	yes	characteristic
311	#127	C₂³×C₃₁	1	1	yes	characteristic
312	#124	—	2	1	no	—
313	#124	—	2	1	no	—
314	#121	—	2	1	no	—
315	#121	—	2	1	no	—
316	#121	—	2	1	no	—
317	#121	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	—	2	1	no	—
321	C₄×D₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
325	#107	—	2	2	no	—
326	#107	—	2	4	no	—
327	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
328	#102	—	2	4	no	—
329	#102	—	2	4	no	—
330	#102	—	2	2	no	—
331	#102	—	2	2	no	—
332	#102	—	2	4	no	—
333	#102	—	2	4	no	—
334	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
335	#97	D₄×C₃₁	1	2	yes	—
336	C₄×C₈	D₄×C₃₁	1	2	yes	—
337	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
338	C₄×C₈	C₂³×C₃₁	1	2	yes	—
339	C₄×C₈	C₂³×C₃₁	1	2	yes	—
340	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₃₁	#19631	1	1	yes	characteristic, direct product factor, Sylow
342	C₂⁴	—	4	1	no	—
343	#41	—	4	4	no	—
344	#41	—	4	4	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	4	1	no	—
349	C₂×Q₈	—	4	4	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	4	4	no	—
360	C₂×D₄	—	4	1	no	—
361	C₂×D₄	—	4	1	no	—
362	C₂×D₄	—	4	1	no	—
363	C₂×D₄	—	4	1	no	—
364	C₂×D₄	—	4	1	no	—
365	C₂×D₄	—	4	4	no	—
366	C₂×D₄	—	8	2	no	—
367	C₂²×C₄	—	4	1	no	—
368	C₂²×C₄	—	4	1	no	—
369	C₂²×C₄	—	4	1	no	—
370	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	Q₁₆	—	4	8	no	—
374	SD₁₆	—	4	8	no	—
375	SD₁₆	—	4	8	no	—
376	D₈	—	4	4	no	—
377	D₈	—	4	8	no	—
378	D₈	—	8	4	no	—
379	M₁₆	D₈×C₃₁	1	8	yes	complemented
380	C₂×C₈	—	4	2	no	—
381	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
382	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
383	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
384	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	4	1	no	—
392	C₄⋊C₄	—	4	1	no	—
393	C₂²⋊C₄	—	4	1	no	—
394	C₂²⋊C₄	—	4	1	no	—
395	C₂²⋊C₄	—	4	1	no	—
396	C₂²⋊C₄	—	4	1	no	—
397	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
401	C₂³	—	4	1	no	—
402	C₂³	—	4	1	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	8	2	no	—
405	C₂³	—	8	2	no	—
406	C₂³	—	8	2	no	—
407	C₂³	C₃₁×#127	1	1	yes	characteristic
408	Q₈	—	4	2	no	—
409	Q₈	—	4	4	no	—
410	Q₈	—	4	2	no	—
411	Q₈	—	8	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	4	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	4	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	4	no	—
420	D₄	—	4	2	no	—
421	D₄	—	8	2	no	—
422	D₄	—	8	2	no	—
423	D₄	—	8	2	no	—
424	D₄	—	8	2	no	—
425	D₄	—	8	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	8	2	no	—
432	C₂×C₄	—	8	2	no	—
433	C₂×C₄	—	8	2	no	—
434	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
435	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
436	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
437	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
438	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
441	C₈	—	2	2	no	—
442	C₈	—	2	2	no	—
443	C₈	—	2	4	no	—
444	C₈	C₂×D₈×C₃₁	1	8	yes	—
445	C₂²	—	2	1	no	—
446	C₂²	—	2	1	no	—
447	C₂²	—	8	1	no	—
448	C₂²	—	8	1	no	—
449	C₂²	—	8	2	no	—
450	C₂²	—	8	2	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	2	no	—
454	C₂²	—	8	1	no	—
455	C₂²	—	8	1	no	—
456	C₂²	—	16	4	no	—
457	C₂²	C₃₁×#493	1	2	yes	—
458	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
459	C₄	—	2	1	no	—
460	C₄	—	2	1	no	—
461	C₄	—	8	1	no	—
462	C₄	—	8	1	no	—
463	C₄	—	8	2	no	—
464	C₄	C₃₁×#493	1	2	yes	—
465	C₄	C₂²×D₈×C₃₁	1	2	yes	—
466	C₄	C₃₁×#575	1	2	yes	—
467	C₄	C₃₁×#577	1	2	yes	—
468	C₂	—	2	2	no	—
469	C₂	—	16	2	no	—
470	C₂	—	16	1	no	—
471	C₂	—	16	1	no	—
472	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
473	C₂	C₃₁×#3153	1	1	yes	characteristic
474	C₂	C₃₁×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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×#3226	C₂	semidirect
2. C₃₁×#3205	C₂	semidirect
3. C₃₁×#3169	C₂	semidirect
4. C₃₁×#3152	C₂	semidirect
5. C₃₁×#3151	C₂	semidirect
6. C₂×C₃₁×#486	C₂	semidirect
7. C₃₁×#1722	C₂	semidirect
8. C₃₁×#1721	C₂	semidirect
9. C₃₁×#1709	C₂	semidirect
10. C₃₁×#1706	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
18. C₃₁×#500	C₂²	semidirect
19. C₃₁×#500	C₂²	semidirect
22. C₃₁×#494	C₂²	semidirect
25. C₃₁×#493	C₂²	semidirect
27. C₃₁×#492	C₂²	semidirect
32. C₃₁×#486	C₂²	semidirect
33. C₃₁×#486	C₂²	semidirect
34. C₃₁×#486	C₂²	semidirect
45. C₃₁×#322	C₂²	semidirect
46. C₈²×C₃₁	C₂²	semidirect
141. M₁₆×C₃₁	D₈	semidirect
163. #19631	C₃₁	direct (N × H)
208. #3226	C₂×C₃₁	semidirect
209. #3205	C₂×C₃₁	semidirect
210. #3169	C₂×C₃₁	semidirect
211. #3152	C₂×C₃₁	semidirect
212. #3151	C₂×C₃₁	semidirect
213. C₂×#486	C₂×C₃₁	semidirect
214. #1722	C₂×C₃₁	semidirect
215. #1721	C₂×C₃₁	semidirect
216. #1709	C₂×C₃₁	semidirect
217. #1706	C₂×C₃₁	semidirect
218. C₈×M₁₆	C₂×C₃₁	semidirect
248. #500	C₂²×C₃₁	semidirect
249. #500	C₂²×C₃₁	semidirect
252. #494	C₂²×C₃₁	semidirect
255. #493	C₂²×C₃₁	semidirect
257. #492	C₂²×C₃₁	semidirect
262. #486	C₂²×C₃₁	semidirect
263. #486	C₂²×C₃₁	semidirect
264. #486	C₂²×C₃₁	semidirect
275. #322	C₂²×C₃₁	semidirect
276. C₈²	C₂²×C₃₁	semidirect
341. C₃₁	#19631	direct (N × H)
379. M₁₆	D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3226	2
C₃₁×#3205	2
C₃₁×#3169	2
C₃₁×#3152	2
C₃₁×#3151	2
C₂×C₃₁×#486	2
C₃₁×#1722	2
C₃₁×#1721	2
C₃₁×#1709	2
C₃₁×#1706	2
C₈×M₁₆×C₃₁	2
#19631	31

Subgroups — C31×#19631 #21476390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19631 #21476390