Subgroups — C₃₁×#25442 #21482201

Order 7936.

890 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3156	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3063	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3061	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2038	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2007	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1908	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂×C₃₁×#137	C₂²	1	2	yes	—
14	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#131	C₂²	1	2	yes	—
18	C₃₁×#537	C₂²	1	2	yes	—
19	C₃₁×#536	C₂²	1	2	yes	—
20	C₃₁×#497	—	2	2	no	—
21	C₃₁×#495	—	2	1	no	—
22	C₃₁×#494	—	2	1	no	—
23	C₃₁×#473	—	2	4	no	—
24	C₃₁×#470	—	2	2	no	—
25	C₃₁×#468	—	2	2	no	—
26	C₃₁×#468	—	2	2	no	—
27	C₃₁×#467	—	2	1	no	—
28	C₃₁×#467	—	2	2	no	—
29	C₃₁×#466	—	2	1	no	—
30	C₃₁×#465	—	2	2	no	—
31	C₃₁×#443	—	2	4	no	—
32	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
33	C₃₁×#428	C₂²	1	2	yes	—
34	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#104	C₂²	1	2	yes	—
36	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
37	C₃₁×#417	C₂²	1	2	yes	—
38	C₃₁×#416	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#103	C₂²	1	2	yes	—
40	C₂×C₃₁×#101	C₂²	1	2	yes	—
41	C₃₁×#405	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#391	C₂²	1	2	yes	—
44	C₃₁×#343	C₂²	1	2	yes	—
45	C₃₁×#341	C₂²	1	1	yes	characteristic
46	C₃₁×#339	C₂²	1	1	yes	characteristic
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₃₁	C₂³	1	2	yes	—
49	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#137	—	2	4	no	—
51	C₃₁×#137	—	2	4	no	—
52	C₃₁×#137	—	4	2	no	—
53	C₃₁×#137	—	4	2	no	—
54	C₃₁×#135	—	2	2	no	—
55	C₃₁×#135	—	2	2	no	—
56	C₂×Q₁₆×C₃₁	—	2	2	no	—
57	C₂×Q₁₆×C₃₁	—	2	2	no	—
58	C₂×Q₁₆×C₃₁	—	2	2	no	—
59	C₂×Q₁₆×C₃₁	—	2	1	no	—
60	C₂×Q₁₆×C₃₁	—	2	1	no	—
61	C₂×Q₁₆×C₃₁	—	2	2	no	—
62	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×SD₁₆×C₃₁	—	2	2	no	—
65	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
66	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
67	C₃₁×#131	—	2	4	no	—
68	C₃₁×#131	—	2	2	no	—
69	C₃₁×#131	—	2	2	no	—
70	C₃₁×#131	—	4	2	no	—
71	C₂×M₁₆×C₃₁	—	2	1	no	—
72	C₂×M₁₆×C₃₁	—	2	2	no	—
73	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
75	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	—	2	2	no	—
77	C₂²×C₈×C₃₁	—	2	1	no	—
78	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	—	2	2	no	—
81	C₃₁×#124	—	2	2	no	—
82	C₃₁×#123	—	4	2	no	—
83	C₃₁×C₂²⋊Q₈	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	—	2	2	no	—
85	C₃₁×C₂²⋊Q₈	—	2	2	no	—
86	C₃₁×C₂²⋊Q₈	—	2	2	no	—
87	C₃₁×#121	—	2	2	no	—
88	C₃₁×#121	—	2	2	no	—
89	C₃₁×#121	—	4	2	no	—
90	C₃₁×#117	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#108	D₄	1	4	yes	complemented
94	C₃₁×#107	—	2	1	no	—
95	C₃₁×#107	—	2	1	no	—
96	C₃₁×#107	—	2	2	no	—
97	C₃₁×#106	—	2	2	no	—
98	C₃₁×#104	—	2	4	no	—
99	C₃₁×#104	—	2	2	no	—
100	C₃₁×#103	—	2	2	no	—
101	C₃₁×#103	—	2	2	no	—
102	C₃₁×#103	—	2	2	no	—
103	C₃₁×#103	D₄	1	1	yes	characteristic
104	C₃₁×#103	D₄	1	1	yes	characteristic
105	C₃₁×#102	—	2	2	no	—
106	C₃₁×#102	D₄	1	1	yes	characteristic
107	C₃₁×#102	D₄	1	1	yes	characteristic
108	C₃₁×#101	—	2	4	no	—
109	C₃₁×#97	—	2	1	no	—
110	C₄×C₈×C₃₁	—	2	2	no	—
111	C₄×C₈×C₃₁	—	2	1	no	—
112	C₃₁×#95	—	4	2	no	—
113	#41×C₃₁	—	2	2	no	—
114	#41×C₃₁	—	2	1	no	—
115	#41×C₃₁	—	2	1	no	—
116	#41×C₃₁	—	2	1	no	—
117	#41×C₃₁	—	2	2	no	—
118	#41×C₃₁	—	2	2	no	—
119	#41×C₃₁	—	2	2	no	—
120	#41×C₃₁	—	2	1	no	—
121	C₂×Q₈×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	—	2	2	no	—
124	C₂×Q₈×C₃₁	—	2	2	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	2	2	no	—
127	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
128	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₃₁	—	2	1	no	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	4	2	no	—
132	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₃₁	—	2	2	no	—
141	Q₁₆×C₃₁	—	4	4	no	—
142	Q₁₆×C₃₁	—	4	4	no	—
143	Q₁₆×C₃₁	—	4	2	no	—
144	Q₁₆×C₃₁	—	4	2	no	—
145	SD₁₆×C₃₁	—	2	4	no	—
146	SD₁₆×C₃₁	—	4	4	no	—
147	SD₁₆×C₃₁	—	4	4	no	—
148	D₈×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	M₁₆×C₃₁	—	2	1	no	—
151	M₁₆×C₃₁	—	2	1	no	—
152	M₁₆×C₃₁	—	2	1	no	—
153	M₁₆×C₃₁	—	4	4	no	—
154	M₁₆×C₃₁	—	4	2	no	—
155	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
156	M₁₆×C₃₁	#41	1	2	yes	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	4	4	no	—
163	C₂×C₈×C₃₁	—	4	2	no	—
164	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂×C₈×C₃₁	#41	1	2	yes	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	4	2	no	—
173	C₄⋊C₄×C₃₁	—	4	2	no	—
174	C₄⋊C₄×C₃₁	—	4	2	no	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²⋊C₄×C₃₁	—	2	1	no	—
178	C₂²⋊C₄×C₃₁	—	4	2	no	—
179	C₂²⋊C₄×C₃₁	—	4	2	no	—
180	C₂²⋊C₄×C₃₁	—	4	2	no	—
181	C₂²⋊C₄×C₃₁	—	4	2	no	—
182	C₄²×C₃₁	—	2	1	no	—
183	C₄²×C₃₁	—	2	1	no	—
184	C₄²×C₃₁	—	2	1	no	—
185	#25442	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	2	1	no	—
188	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
189	Q₈×C₃₁	—	2	2	no	—
190	Q₈×C₃₁	—	2	1	no	—
191	Q₈×C₃₁	—	2	1	no	—
192	Q₈×C₃₁	—	2	2	no	—
193	Q₈×C₃₁	—	4	2	no	—
194	Q₈×C₃₁	—	4	2	no	—
195	Q₈×C₃₁	—	4	2	no	—
196	Q₈×C₃₁	—	4	2	no	—
197	D₄×C₃₁	—	2	2	no	—
198	D₄×C₃₁	—	2	1	no	—
199	D₄×C₃₁	—	2	1	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	4	1	no	—
202	D₄×C₃₁	—	4	2	no	—
203	D₄×C₃₁	—	4	1	no	—
204	D₄×C₃₁	—	4	2	no	—
205	D₄×C₃₁	—	16	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	2	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	8	2	no	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
227	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
228	C₈×C₃₁	—	2	1	no	—
229	C₈×C₃₁	—	2	1	no	—
230	C₈×C₃₁	—	2	4	no	—
231	C₈×C₃₁	—	2	2	no	—
232	C₈×C₃₁	—	4	1	no	—
233	C₈×C₃₁	#121	1	2	yes	—
234	C₈×C₃₁	#121	1	2	yes	—
235	#3156	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#3063	C₂×C₃₁	1	2	yes	complemented
238	#3061	C₂×C₃₁	1	1	yes	characteristic
239	C₂×#443	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#2038	C₂×C₃₁	1	2	yes	complemented
241	#2007	C₂×C₃₁	1	2	yes	complemented
242	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#1908	C₂×C₃₁	1	2	yes	complemented
245	#1764	C₂×C₃₁	1	1	yes	characteristic, complemented
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	4	1	no	—
249	C₂²×C₃₁	—	4	2	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	1	no	—
254	C₂²×C₃₁	#540	1	1	yes	characteristic
255	C₂²×C₃₁	D₄²	1	1	yes	characteristic
256	C₂²×C₃₁	#550	1	1	yes	characteristic
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	1	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	1	no	—
261	C₄×C₃₁	—	4	1	no	—
262	C₄×C₃₁	—	8	2	no	—
263	C₄×C₃₁	—	8	2	no	—
264	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
265	C₄×C₃₁	#532	1	1	yes	characteristic
266	C₄×C₃₁	D₄²	1	1	yes	characteristic
267	C₄×C₃₁	#546	1	1	yes	characteristic
268	C₂×#137	C₂²×C₃₁	1	2	yes	—
269	C₂×#137	C₂²×C₃₁	1	2	yes	—
270	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
271	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#131	C₂²×C₃₁	1	2	yes	—
274	#537	C₂²×C₃₁	1	2	yes	—
275	#536	C₂²×C₃₁	1	2	yes	—
276	#497	—	2	2	no	—
277	#495	—	2	1	no	—
278	#494	—	2	1	no	—
279	#473	—	2	4	no	—
280	#470	—	2	2	no	—
281	#468	—	2	2	no	—
282	#468	—	2	2	no	—
283	#467	—	2	1	no	—
284	#467	—	2	2	no	—
285	#466	—	2	1	no	—
286	#465	—	2	2	no	—
287	#443	—	2	4	no	—
288	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
289	#428	C₂²×C₃₁	1	2	yes	—
290	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×#104	C₂²×C₃₁	1	2	yes	—
292	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
293	#417	C₂²×C₃₁	1	2	yes	—
294	#416	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#103	C₂²×C₃₁	1	2	yes	—
296	C₂×#101	C₂²×C₃₁	1	2	yes	—
297	#405	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
299	#391	C₂²×C₃₁	1	2	yes	—
300	#343	C₂²×C₃₁	1	2	yes	—
301	#341	C₂²×C₃₁	1	1	yes	characteristic
302	#339	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₃₁	—	2	1	no	—
304	C₂×C₃₁	—	2	1	no	—
305	C₂×C₃₁	—	8	1	no	—
306	C₂×C₃₁	—	8	2	no	—
307	C₂×C₃₁	#2428	1	1	yes	characteristic
308	C₂×C₃₁	#3290	1	1	yes	characteristic
309	C₂×C₃₁	#3293	1	1	yes	characteristic
310	C₂×#41	C₂³×C₃₁	1	2	yes	—
311	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
313	#137	—	2	4	no	—
314	#137	—	2	4	no	—
315	#137	—	4	2	no	—
316	#137	—	4	2	no	—
317	#135	—	2	2	no	—
318	#135	—	2	2	no	—
319	C₂×Q₁₆	—	2	2	no	—
320	C₂×Q₁₆	—	2	2	no	—
321	C₂×Q₁₆	—	2	2	no	—
322	C₂×Q₁₆	—	2	1	no	—
323	C₂×Q₁₆	—	2	1	no	—
324	C₂×Q₁₆	—	2	2	no	—
325	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
326	C₂×SD₁₆	—	2	2	no	—
327	C₂×SD₁₆	—	2	2	no	—
328	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
329	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
330	#131	—	2	4	no	—
331	#131	—	2	2	no	—
332	#131	—	2	2	no	—
333	#131	—	4	2	no	—
334	C₂×M₁₆	—	2	1	no	—
335	C₂×M₁₆	—	2	2	no	—
336	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
337	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
338	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
339	C₂²×C₈	—	2	2	no	—
340	C₂²×C₈	—	2	1	no	—
341	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
342	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₄⋊Q₈	—	2	2	no	—
344	#124	—	2	2	no	—
345	#123	—	4	2	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	2	no	—
349	C₂²⋊Q₈	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	4	2	no	—
353	#117	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
355	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	#108	D₄×C₃₁	1	4	yes	complemented
357	#107	—	2	1	no	—
358	#107	—	2	1	no	—
359	#107	—	2	2	no	—
360	#106	—	2	2	no	—
361	#104	—	2	4	no	—
362	#104	—	2	2	no	—
363	#103	—	2	2	no	—
364	#103	—	2	2	no	—
365	#103	—	2	2	no	—
366	#103	D₄×C₃₁	1	1	yes	characteristic
367	#103	D₄×C₃₁	1	1	yes	characteristic
368	#102	—	2	2	no	—
369	#102	D₄×C₃₁	1	1	yes	characteristic
370	#102	D₄×C₃₁	1	1	yes	characteristic
371	#101	—	2	4	no	—
372	#97	—	2	1	no	—
373	C₄×C₈	—	2	2	no	—
374	C₄×C₈	—	2	1	no	—
375	#95	—	4	2	no	—
376	C₃₁	#25442	1	1	yes	characteristic, direct product factor, Sylow
377	#41	—	2	2	no	—
378	#41	—	2	1	no	—
379	#41	—	2	1	no	—
380	#41	—	2	1	no	—
381	#41	—	2	2	no	—
382	#41	—	2	2	no	—
383	#41	—	2	2	no	—
384	#41	—	2	1	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
392	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
397	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
404	Q₁₆	—	2	2	no	—
405	Q₁₆	—	4	4	no	—
406	Q₁₆	—	4	4	no	—
407	Q₁₆	—	4	2	no	—
408	Q₁₆	—	4	2	no	—
409	SD₁₆	—	2	4	no	—
410	SD₁₆	—	4	4	no	—
411	SD₁₆	—	4	4	no	—
412	D₈	—	2	2	no	—
413	M₁₆	—	2	2	no	—
414	M₁₆	—	2	1	no	—
415	M₁₆	—	2	1	no	—
416	M₁₆	—	2	1	no	—
417	M₁₆	—	4	4	no	—
418	M₁₆	—	4	2	no	—
419	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
420	M₁₆	#41×C₃₁	1	2	yes	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	4	4	no	—
427	C₂×C₈	—	4	2	no	—
428	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
432	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₈	#41×C₃₁	1	2	yes	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	4	2	no	—
437	C₄⋊C₄	—	4	2	no	—
438	C₄⋊C₄	—	4	2	no	—
439	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂²⋊C₄	—	2	1	no	—
442	C₂²⋊C₄	—	4	2	no	—
443	C₂²⋊C₄	—	4	2	no	—
444	C₂²⋊C₄	—	4	2	no	—
445	C₂²⋊C₄	—	4	2	no	—
446	C₄²	—	2	1	no	—
447	C₄²	—	2	1	no	—
448	C₄²	—	2	1	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	1	no	—
451	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
452	Q₈	—	2	2	no	—
453	Q₈	—	2	1	no	—
454	Q₈	—	2	1	no	—
455	Q₈	—	2	2	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	2	no	—
458	Q₈	—	4	2	no	—
459	Q₈	—	4	2	no	—
460	D₄	—	2	2	no	—
461	D₄	—	2	1	no	—
462	D₄	—	2	1	no	—
463	D₄	—	2	2	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	2	no	—
466	D₄	—	4	1	no	—
467	D₄	—	4	2	no	—
468	D₄	—	16	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	8	2	no	—
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
491	C₈	—	2	1	no	—
492	C₈	—	2	1	no	—
493	C₈	—	2	4	no	—
494	C₈	—	2	2	no	—
495	C₈	—	4	1	no	—
496	C₈	C₃₁×#121	1	2	yes	—
497	C₈	C₃₁×#121	1	2	yes	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	1	no	—
506	C₂²	C₃₁×#540	1	1	yes	characteristic
507	C₂²	D₄²×C₃₁	1	1	yes	characteristic
508	C₂²	C₃₁×#550	1	1	yes	characteristic
509	C₄	—	4	2	no	—
510	C₄	—	4	1	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	8	2	no	—
515	C₄	—	8	2	no	—
516	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
517	C₄	C₃₁×#532	1	1	yes	characteristic
518	C₄	D₄²×C₃₁	1	1	yes	characteristic
519	C₄	C₃₁×#546	1	1	yes	characteristic
520	C₂	—	2	1	no	—
521	C₂	—	2	1	no	—
522	C₂	—	8	1	no	—
523	C₂	—	8	2	no	—
524	C₂	C₃₁×#2428	1	1	yes	characteristic
525	C₂	C₃₁×#3290	1	1	yes	characteristic
526	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3156	C₂	semidirect
2. C₂×C₃₁×#473	C₂	semidirect
3. C₃₁×#3063	C₂	semidirect
5. C₂×C₃₁×#443	C₂	semidirect
6. C₃₁×#2038	C₂	semidirect
7. C₃₁×#2007	C₂	semidirect
8. C₃₁×#1969	C₂	semidirect
9. C₃₁×#1952	C₂	semidirect
10. C₃₁×#1908	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
93. C₃₁×#108	D₄	semidirect
185. #25442	C₃₁	direct (N × H)
235. #3156	C₂×C₃₁	semidirect
236. C₂×#473	C₂×C₃₁	semidirect
237. #3063	C₂×C₃₁	semidirect
239. C₂×#443	C₂×C₃₁	semidirect
240. #2038	C₂×C₃₁	semidirect
241. #2007	C₂×C₃₁	semidirect
242. #1969	C₂×C₃₁	semidirect
243. #1952	C₂×C₃₁	semidirect
244. #1908	C₂×C₃₁	semidirect
245. #1764	C₂×C₃₁	semidirect
356. #108	D₄×C₃₁	semidirect
376. C₃₁	#25442	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×#3156	2
C₂×C₃₁×#473	2
C₃₁×#3063	2
C₃₁×#3061	2
C₂×C₃₁×#443	2
C₃₁×#2038	2
C₃₁×#2007	2
C₃₁×#1969	2
C₃₁×#1952	2
C₃₁×#1908	2
C₃₁×#1764	2
#25442	31

Subgroups — C31×#25442 #21482201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25442 #21482201