Subgroups — C₃₁×#24807 #21481566

Order 7936.

844 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3056	C₂	1	2	yes	maximal
3	C₂×C₃₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2930	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2064	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2052	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1901	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1815	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#525	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₃₁×#473	—	2	4	no	—
20	C₃₁×#470	—	2	2	no	—
21	C₃₁×#470	—	2	2	no	—
22	C₃₁×#470	—	2	2	no	—
23	C₃₁×#468	—	2	2	no	—
24	C₃₁×#467	—	2	2	no	—
25	C₃₁×#465	—	2	2	no	—
26	C₃₁×#436	—	2	2	no	—
27	C₃₁×#435	—	2	2	no	—
28	C₃₁×#430	—	2	4	no	—
29	C₂×C₃₁×#108	C₂²	1	2	yes	—
30	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
32	C₃₁×#428	C₂²	1	2	yes	—
33	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
34	C₃₁×#424	C₂²	1	1	yes	characteristic
35	C₃₁×#419	C₂²	1	2	yes	—
36	C₃₁×#417	C₂²	1	2	yes	—
37	C₂×C₃₁×#103	C₂²	1	2	yes	—
38	C₂×C₃₁×#101	C₂²	1	2	yes	—
39	C₂×C₃₁×#101	C₂²	1	2	yes	—
40	C₃₁×#408	C₂²	1	1	yes	characteristic
41	C₃₁×#407	C₂²	1	1	yes	characteristic
42	C₃₁×#405	C₂²	1	1	yes	characteristic
43	C₃₁×#343	C₂²	1	2	yes	—
44	C₃₁×#343	C₂²	1	2	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#137	—	2	4	no	—
49	C₃₁×#137	—	4	2	no	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	—	2	1	no	—
52	C₂×Q₁₆×C₃₁	—	2	1	no	—
53	C₂×Q₁₆×C₃₁	—	2	2	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₃₁×#131	—	2	2	no	—
57	C₃₁×#131	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	—	2	1	no	—
61	C₂×M₁₆×C₃₁	—	2	2	no	—
62	C₂×M₁₆×C₃₁	—	2	4	no	—
63	C₂×M₁₆×C₃₁	—	4	2	no	—
64	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
65	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
66	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂²×C₈×C₃₁	—	2	2	no	—
71	C₂²×C₈×C₃₁	—	2	1	no	—
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#123	—	2	1	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
80	C₃₁×#121	D₄	1	2	yes	—
81	C₃₁×#120	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	2	no	—
83	C₃₁×#117	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₃₁×#108	—	2	4	no	—
88	C₃₁×#108	—	2	2	no	—
89	C₃₁×#108	D₄	1	4	yes	complemented
90	C₃₁×#107	—	2	2	no	—
91	C₃₁×#106	—	2	1	no	—
92	C₃₁×#106	—	2	1	no	—
93	C₃₁×#106	—	2	2	no	—
94	C₃₁×C₄⋊C₈	—	2	2	no	—
95	C₃₁×#103	—	2	2	no	—
96	C₃₁×#103	—	2	4	no	—
97	C₃₁×#103	—	2	2	no	—
98	C₃₁×#103	—	2	2	no	—
99	C₃₁×#102	—	2	2	no	—
100	C₃₁×#101	—	2	4	no	—
101	C₃₁×#101	—	4	2	no	—
102	C₃₁×C₂²⋊C₈	—	2	2	no	—
103	C₃₁×C₂²⋊C₈	—	2	2	no	—
104	C₃₁×#97	—	2	1	no	—
105	C₄×C₈×C₃₁	—	2	1	no	—
106	C₂⁴×C₃₁	#41	1	1	yes	characteristic
107	#41×C₃₁	—	2	1	no	—
108	#41×C₃₁	—	2	1	no	—
109	#41×C₃₁	—	2	1	no	—
110	#41×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₃₁	—	4	4	no	—
133	Q₁₆×C₃₁	—	4	2	no	—
134	Q₁₆×C₃₁	—	4	2	no	—
135	SD₁₆×C₃₁	—	4	4	no	—
136	M₁₆×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	—	2	4	no	—
138	M₁₆×C₃₁	—	2	4	no	—
139	M₁₆×C₃₁	—	2	1	no	—
140	M₁₆×C₃₁	—	2	2	no	—
141	M₁₆×C₃₁	—	2	1	no	—
142	M₁₆×C₃₁	—	2	1	no	—
143	M₁₆×C₃₁	—	2	2	no	—
144	M₁₆×C₃₁	—	2	1	no	—
145	M₁₆×C₃₁	—	4	4	no	—
146	M₁₆×C₃₁	—	4	2	no	—
147	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	4	4	no	—
155	C₂×C₈×C₃₁	—	4	2	no	—
156	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
157	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₃₁	#41	1	2	yes	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	4	2	no	—
165	C₄⋊C₄×C₃₁	—	4	2	no	—
166	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²⋊C₄×C₃₁	—	2	2	no	—
168	C₂²⋊C₄×C₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₃₁	—	2	2	no	—
170	C₂²⋊C₄×C₃₁	—	4	2	no	—
171	C₂²⋊C₄×C₃₁	—	4	2	no	—
172	C₄²×C₃₁	—	2	1	no	—
173	#24807	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	#121	1	2	yes	—
179	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
180	Q₈×C₃₁	—	2	1	no	—
181	Q₈×C₃₁	—	2	1	no	—
182	Q₈×C₃₁	—	4	2	no	—
183	Q₈×C₃₁	—	4	2	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	2	1	no	—
187	D₄×C₃₁	—	2	1	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	—	4	1	no	—
190	D₄×C₃₁	—	8	2	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	8	2	no	—
206	C₂×C₄×C₃₁	#121	1	2	yes	—
207	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₃₁	—	2	2	no	—
214	C₈×C₃₁	—	2	4	no	—
215	C₈×C₃₁	—	4	1	no	—
216	C₈×C₃₁	—	4	1	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	#121	1	2	yes	—
219	C₈×C₃₁	#121	1	2	yes	—
220	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#3056	C₂×C₃₁	1	2	yes	—
222	C₂×#470	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#2930	C₂×C₃₁	1	1	yes	characteristic
224	C₂×#430	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#2064	C₂×C₃₁	1	2	yes	complemented
226	#2052	C₂×C₃₁	1	2	yes	complemented
227	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1951	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1901	C₂×C₃₁	1	2	yes	complemented
230	#1815	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	4	1	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	8	1	no	—
238	C₂²×C₃₁	—	8	1	no	—
239	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
240	C₂²×C₃₁	#548	1	1	yes	characteristic
241	C₂²×C₃₁	#553	1	1	yes	characteristic
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	4	1	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	8	2	no	—
246	C₄×C₃₁	—	8	2	no	—
247	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
248	C₄×C₃₁	#537	1	1	yes	characteristic
249	C₄×C₃₁	D₄²	1	1	yes	characteristic
250	C₄×C₃₁	#547	1	1	yes	characteristic
251	C₂×#137	C₂²×C₃₁	1	2	yes	—
252	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
254	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
255	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
256	#525	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
258	#473	—	2	4	no	—
259	#470	—	2	2	no	—
260	#470	—	2	2	no	—
261	#470	—	2	2	no	—
262	#468	—	2	2	no	—
263	#467	—	2	2	no	—
264	#465	—	2	2	no	—
265	#436	—	2	2	no	—
266	#435	—	2	2	no	—
267	#430	—	2	4	no	—
268	C₂×#108	C₂²×C₃₁	1	2	yes	—
269	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
271	#428	C₂²×C₃₁	1	2	yes	—
272	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
273	#424	C₂²×C₃₁	1	1	yes	characteristic
274	#419	C₂²×C₃₁	1	2	yes	—
275	#417	C₂²×C₃₁	1	2	yes	—
276	C₂×#103	C₂²×C₃₁	1	2	yes	—
277	C₂×#101	C₂²×C₃₁	1	2	yes	—
278	C₂×#101	C₂²×C₃₁	1	2	yes	—
279	#408	C₂²×C₃₁	1	1	yes	characteristic
280	#407	C₂²×C₃₁	1	1	yes	characteristic
281	#405	C₂²×C₃₁	1	1	yes	characteristic
282	#343	C₂²×C₃₁	1	2	yes	—
283	#343	C₂²×C₃₁	1	2	yes	—
284	C₂×C₃₁	—	2	1	no	—
285	C₂×C₃₁	—	2	1	no	—
286	C₂×C₃₁	—	4	2	no	—
287	C₂×C₃₁	—	8	1	no	—
288	C₂×C₃₁	#2416	1	1	yes	characteristic
289	C₂×C₃₁	#3069	1	2	yes	—
290	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
291	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
292	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
293	#137	—	2	4	no	—
294	#137	—	4	2	no	—
295	C₂×Q₁₆	—	2	2	no	—
296	C₂×Q₁₆	—	2	1	no	—
297	C₂×Q₁₆	—	2	1	no	—
298	C₂×Q₁₆	—	2	2	no	—
299	C₂×Q₁₆	—	2	2	no	—
300	C₂×SD₁₆	—	2	2	no	—
301	#131	—	2	2	no	—
302	#131	—	2	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	—	2	2	no	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂×M₁₆	—	2	2	no	—
307	C₂×M₁₆	—	2	4	no	—
308	C₂×M₁₆	—	4	2	no	—
309	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
310	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
311	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
312	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
315	C₂²×C₈	—	2	2	no	—
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
319	#123	—	2	1	no	—
320	C₂²⋊Q₈	—	2	2	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	—	2	2	no	—
324	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
325	#121	D₄×C₃₁	1	2	yes	—
326	#120	—	2	1	no	—
327	C₄×D₄	—	2	2	no	—
328	#117	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	#108	—	2	4	no	—
333	#108	—	2	2	no	—
334	#108	D₄×C₃₁	1	4	yes	complemented
335	#107	—	2	2	no	—
336	#106	—	2	1	no	—
337	#106	—	2	1	no	—
338	#106	—	2	2	no	—
339	C₄⋊C₈	—	2	2	no	—
340	#103	—	2	2	no	—
341	#103	—	2	4	no	—
342	#103	—	2	2	no	—
343	#103	—	2	2	no	—
344	#102	—	2	2	no	—
345	#101	—	2	4	no	—
346	#101	—	4	2	no	—
347	C₂²⋊C₈	—	2	2	no	—
348	C₂²⋊C₈	—	2	2	no	—
349	#97	—	2	1	no	—
350	C₄×C₈	—	2	1	no	—
351	C₃₁	#24807	1	1	yes	characteristic, direct product factor, Sylow
352	C₂⁴	#41×C₃₁	1	1	yes	characteristic
353	#41	—	2	1	no	—
354	#41	—	2	1	no	—
355	#41	—	2	1	no	—
356	#41	—	2	1	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
375	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
376	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
378	Q₁₆	—	4	4	no	—
379	Q₁₆	—	4	2	no	—
380	Q₁₆	—	4	2	no	—
381	SD₁₆	—	4	4	no	—
382	M₁₆	—	2	2	no	—
383	M₁₆	—	2	4	no	—
384	M₁₆	—	2	4	no	—
385	M₁₆	—	2	1	no	—
386	M₁₆	—	2	2	no	—
387	M₁₆	—	2	1	no	—
388	M₁₆	—	2	1	no	—
389	M₁₆	—	2	2	no	—
390	M₁₆	—	2	1	no	—
391	M₁₆	—	4	4	no	—
392	M₁₆	—	4	2	no	—
393	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	4	4	no	—
401	C₂×C₈	—	4	2	no	—
402	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
403	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
406	C₂×C₈	#41×C₃₁	1	2	yes	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	4	2	no	—
411	C₄⋊C₄	—	4	2	no	—
412	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	2	1	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	4	2	no	—
417	C₂²⋊C₄	—	4	2	no	—
418	C₄²	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	4	2	no	—
423	C₂³	C₃₁×#121	1	2	yes	—
424	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
425	Q₈	—	2	1	no	—
426	Q₈	—	2	1	no	—
427	Q₈	—	4	2	no	—
428	Q₈	—	4	2	no	—
429	Q₈	—	4	2	no	—
430	Q₈	—	4	2	no	—
431	D₄	—	2	1	no	—
432	D₄	—	2	1	no	—
433	D₄	—	4	1	no	—
434	D₄	—	4	1	no	—
435	D₄	—	8	2	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	8	2	no	—
451	C₂×C₄	C₃₁×#121	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
456	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
457	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
458	C₈	—	2	2	no	—
459	C₈	—	2	4	no	—
460	C₈	—	4	1	no	—
461	C₈	—	4	1	no	—
462	C₈	—	4	2	no	—
463	C₈	C₃₁×#121	1	2	yes	—
464	C₈	C₃₁×#121	1	2	yes	—
465	C₂²	—	2	2	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	8	1	no	—
472	C₂²	—	8	1	no	—
473	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
474	C₂²	C₃₁×#548	1	1	yes	characteristic
475	C₂²	C₃₁×#553	1	1	yes	characteristic
476	C₄	—	2	2	no	—
477	C₄	—	4	1	no	—
478	C₄	—	4	2	no	—
479	C₄	—	8	2	no	—
480	C₄	—	8	2	no	—
481	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
482	C₄	C₃₁×#537	1	1	yes	characteristic
483	C₄	D₄²×C₃₁	1	1	yes	characteristic
484	C₄	C₃₁×#547	1	1	yes	characteristic
485	C₂	—	2	1	no	—
486	C₂	—	2	1	no	—
487	C₂	—	4	2	no	—
488	C₂	—	8	1	no	—
489	C₂	C₃₁×#2416	1	1	yes	characteristic
490	C₂	C₃₁×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#473	C₂	semidirect
3. C₂×C₃₁×#470	C₂	semidirect
5. C₂×C₃₁×#430	C₂	semidirect
6. C₃₁×#2064	C₂	semidirect
7. C₃₁×#2052	C₂	semidirect
8. C₃₁×#1979	C₂	semidirect
9. C₃₁×#1951	C₂	semidirect
10. C₃₁×#1901	C₂	semidirect
11. C₃₁×#1815	C₂	semidirect
89. C₃₁×#108	D₄	semidirect
173. #24807	C₃₁	direct (N × H)
220. C₂×#473	C₂×C₃₁	semidirect
222. C₂×#470	C₂×C₃₁	semidirect
224. C₂×#430	C₂×C₃₁	semidirect
225. #2064	C₂×C₃₁	semidirect
226. #2052	C₂×C₃₁	semidirect
227. #1979	C₂×C₃₁	semidirect
228. #1951	C₂×C₃₁	semidirect
229. #1901	C₂×C₃₁	semidirect
230. #1815	C₂×C₃₁	semidirect
334. #108	D₄×C₃₁	semidirect
351. C₃₁	#24807	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×#473	2
C₃₁×#3056	2
C₂×C₃₁×#470	2
C₃₁×#2930	2
C₂×C₃₁×#430	2
C₃₁×#2064	2
C₃₁×#2052	2
C₃₁×#1979	2
C₃₁×#1951	2
C₃₁×#1901	2
C₃₁×#1815	2
#24807	31

Subgroups — C31×#24807 #21481566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24807 #21481566