Subgroups — C₃₁×#24644 #21481403

Order 7936.

1020 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2541	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2356	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2055	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2014	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1982	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	2	yes	—
16	C₂×C₃₁×#121	—	2	2	no	—
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#460	—	2	4	no	—
20	C₃₁×#440	—	2	4	no	—
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
31	C₃₁×#392	—	2	1	no	—
32	C₃₁×#392	—	2	1	no	—
33	C₃₁×#391	C₂²	1	2	yes	—
34	C₃₁×#390	C₂²	1	2	yes	—
35	C₃₁×#387	—	2	2	no	—
36	C₃₁×#383	—	2	1	no	—
37	C₃₁×#382	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#376	—	2	1	no	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	4	2	no	—
56	C₂×SD₁₆×C₃₁	—	4	2	no	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#127	—	2	4	no	—
61	C₃₁×#127	—	2	2	no	—
62	C₃₁×#127	—	2	2	no	—
63	C₃₁×#127	—	4	2	no	—
64	C₃₁×#121	—	4	8	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	D₄	1	4	yes	complemented
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	—	2	2	no	—
79	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#107	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	4	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#102	—	4	2	no	—
89	C₃₁×#102	—	4	2	no	—
90	C₃₁×#97	—	2	2	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	2	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₃₁×#95	D₄	1	1	yes	characteristic
97	C₃₁×#95	D₄	1	1	yes	characteristic
98	C₂⁴×C₃₁	—	2	1	no	—
99	C₂⁴×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	4	4	no	—
118	C₂×D₄×C₃₁	—	4	4	no	—
119	C₂×D₄×C₃₁	—	4	4	no	—
120	C₂×D₄×C₃₁	—	4	4	no	—
121	C₂×D₄×C₃₁	—	4	2	no	—
122	C₂×D₄×C₃₁	—	4	2	no	—
123	C₂×D₄×C₃₁	—	4	4	no	—
124	C₂×D₄×C₃₁	—	8	2	no	—
125	C₂×D₄×C₃₁	—	8	2	no	—
126	C₂×D₄×C₃₁	—	8	2	no	—
127	C₂×D₄×C₃₁	#41	1	2	yes	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	4	4	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
139	SD₁₆×C₃₁	—	2	8	no	—
140	SD₁₆×C₃₁	—	4	8	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	#41	1	2	yes	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	4	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²⋊C₄×C₃₁	—	4	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	2	no	—
160	C₂²⋊C₄×C₃₁	—	4	4	no	—
161	C₄²×C₃₁	—	2	4	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	4	4	no	—
165	C₄²×C₃₁	C₂×D₄	1	2	yes	—
166	C₄²×C₃₁	#41	1	2	yes	—
167	#24644	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₃₁	—	2	2	no	—
182	Q₈×C₃₁	—	2	2	no	—
183	Q₈×C₃₁	—	4	1	no	—
184	Q₈×C₃₁	—	4	1	no	—
185	D₄×C₃₁	—	2	2	no	—
186	D₄×C₃₁	—	2	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	4	1	no	—
192	D₄×C₃₁	—	4	2	no	—
193	D₄×C₃₁	—	8	8	no	—
194	D₄×C₃₁	—	8	4	no	—
195	D₄×C₃₁	—	8	4	no	—
196	D₄×C₃₁	—	8	8	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	8	4	no	—
215	C₂×C₄×C₃₁	#121	1	2	yes	—
216	C₂×C₄×C₃₁	#121	1	2	yes	—
217	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
223	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
224	C₈×C₃₁	—	2	2	no	—
225	C₈×C₃₁	—	4	1	no	—
226	C₈×C₃₁	—	4	2	no	—
227	C₂×#460	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂×#440	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#2541	C₂×C₃₁	1	1	yes	characteristic
230	#2356	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2055	C₂×C₃₁	1	2	yes	complemented
232	#2014	C₂×C₃₁	1	2	yes	complemented
233	#1982	C₂×C₃₁	1	2	yes	complemented
234	#1961	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1925	C₂×C₃₁	1	2	yes	complemented
237	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	8	2	no	—
241	C₂²×C₃₁	—	8	1	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	—	8	2	no	—
244	C₂²×C₃₁	—	8	2	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	—	8	2	no	—
248	C₂²×C₃₁	—	8	2	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
251	C₂²×C₃₁	#546	1	1	yes	characteristic
252	C₂²×C₃₁	#550	1	1	yes	characteristic
253	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
254	C₂²×C₃₁	#577	1	2	yes	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	4	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	—	8	2	no	—
262	C₄×C₃₁	—	8	1	no	—
263	C₄×C₃₁	#525	1	1	yes	characteristic
264	C₄×C₃₁	#535	1	1	yes	characteristic
265	C₄×C₃₁	#553	1	1	yes	characteristic
266	C₄×C₃₁	#555	1	1	yes	characteristic
267	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
269	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#127	C₂²×C₃₁	1	2	yes	—
271	C₂×#121	—	2	2	no	—
272	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
274	#460	—	2	4	no	—
275	#440	—	2	4	no	—
276	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
279	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×#102	C₂²×C₃₁	1	2	yes	—
282	C₂×#102	C₂²×C₃₁	1	2	yes	—
283	C₂×#102	C₂²×C₃₁	1	2	yes	—
284	C₂×#102	C₂²×C₃₁	1	2	yes	—
285	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
286	#392	—	2	1	no	—
287	#392	—	2	1	no	—
288	#391	C₂²×C₃₁	1	2	yes	—
289	#390	C₂²×C₃₁	1	2	yes	—
290	#387	—	2	2	no	—
291	#383	—	2	1	no	—
292	#382	C₂²×C₃₁	1	1	yes	characteristic
293	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
294	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
295	#376	—	2	1	no	—
296	#340	C₂²×C₃₁	1	2	yes	—
297	#336	C₂²×C₃₁	1	2	yes	—
298	C₂×C₃₁	—	8	2	no	—
299	C₂×C₃₁	—	16	2	no	—
300	C₂×C₃₁	#2414	1	1	yes	characteristic
301	C₂×C₃₁	#3038	1	1	yes	characteristic
302	C₂×C₃₁	#3043	1	1	yes	characteristic
303	C₂×C₃₁	#3310	1	2	yes	—
304	C₂×C₃₁	#3397	1	2	yes	—
305	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	—	2	2	no	—
308	C₂²×D₄	—	2	2	no	—
309	C₂²×D₄	—	2	1	no	—
310	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
311	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	—	2	1	no	—
313	C₂×SD₁₆	—	2	2	no	—
314	C₂×SD₁₆	—	2	2	no	—
315	C₂×SD₁₆	—	2	4	no	—
316	C₂×SD₁₆	—	2	2	no	—
317	C₂×SD₁₆	—	4	2	no	—
318	C₂×SD₁₆	—	4	2	no	—
319	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
320	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	#127	—	2	4	no	—
323	#127	—	2	2	no	—
324	#127	—	2	2	no	—
325	#127	—	4	2	no	—
326	#121	—	4	8	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	D₄×C₃₁	1	4	yes	complemented
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×C₂²⋊C₄	—	2	1	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
342	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
343	#107	—	2	2	no	—
344	C₄⋊C₈	—	2	2	no	—
345	C₄⋊C₈	—	4	2	no	—
346	#103	—	2	2	no	—
347	#102	—	2	4	no	—
348	#102	—	2	2	no	—
349	#102	—	2	4	no	—
350	#102	—	4	2	no	—
351	#102	—	4	2	no	—
352	#97	—	2	2	no	—
353	#95	—	2	1	no	—
354	#95	—	2	2	no	—
355	#95	—	2	2	no	—
356	#95	—	2	1	no	—
357	#95	—	2	2	no	—
358	#95	D₄×C₃₁	1	1	yes	characteristic
359	#95	D₄×C₃₁	1	1	yes	characteristic
360	C₃₁	#24644	1	1	yes	characteristic, direct product factor, Sylow
361	C₂⁴	—	2	1	no	—
362	C₂⁴	—	2	2	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	4	4	no	—
381	C₂×D₄	—	4	4	no	—
382	C₂×D₄	—	4	4	no	—
383	C₂×D₄	—	4	4	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	2	no	—
386	C₂×D₄	—	4	4	no	—
387	C₂×D₄	—	8	2	no	—
388	C₂×D₄	—	8	2	no	—
389	C₂×D₄	—	8	2	no	—
390	C₂×D₄	#41×C₃₁	1	2	yes	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	4	4	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
402	SD₁₆	—	2	8	no	—
403	SD₁₆	—	4	8	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	#41×C₃₁	1	2	yes	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
421	C₂²⋊C₄	—	4	4	no	—
422	C₂²⋊C₄	—	4	2	no	—
423	C₂²⋊C₄	—	4	4	no	—
424	C₄²	—	2	4	no	—
425	C₄²	—	2	2	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	4	4	no	—
428	C₄²	C₂×D₄×C₃₁	1	2	yes	—
429	C₄²	#41×C₃₁	1	2	yes	—
430	C₂³	—	2	2	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	1	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	2	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
443	Q₈	—	2	2	no	—
444	Q₈	—	2	2	no	—
445	Q₈	—	4	1	no	—
446	Q₈	—	4	1	no	—
447	D₄	—	2	2	no	—
448	D₄	—	2	2	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	1	no	—
451	D₄	—	4	2	no	—
452	D₄	—	4	2	no	—
453	D₄	—	4	1	no	—
454	D₄	—	4	2	no	—
455	D₄	—	8	8	no	—
456	D₄	—	8	4	no	—
457	D₄	—	8	4	no	—
458	D₄	—	8	8	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	8	4	no	—
477	C₂×C₄	C₃₁×#121	1	2	yes	—
478	C₂×C₄	C₃₁×#121	1	2	yes	—
479	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
485	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
486	C₈	—	2	2	no	—
487	C₈	—	4	1	no	—
488	C₈	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	8	2	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	1	no	—
494	C₂²	—	8	2	no	—
495	C₂²	—	8	2	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	8	2	no	—
499	C₂²	—	8	2	no	—
500	C₂²	—	8	2	no	—
501	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
502	C₂²	C₃₁×#546	1	1	yes	characteristic
503	C₂²	C₃₁×#550	1	1	yes	characteristic
504	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
505	C₂²	C₃₁×#577	1	2	yes	—
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	4	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	8	2	no	—
513	C₄	—	8	1	no	—
514	C₄	C₃₁×#525	1	1	yes	characteristic
515	C₄	C₃₁×#535	1	1	yes	characteristic
516	C₄	C₃₁×#553	1	1	yes	characteristic
517	C₄	C₃₁×#555	1	1	yes	characteristic
518	C₂	—	8	2	no	—
519	C₂	—	16	2	no	—
520	C₂	C₃₁×#2414	1	1	yes	characteristic
521	C₂	C₃₁×#3038	1	1	yes	characteristic
522	C₂	C₃₁×#3043	1	1	yes	characteristic
523	C₂	C₃₁×#3310	1	2	yes	—
524	C₂	C₃₁×#3397	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2541	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#460	C₂	semidirect
2. C₂×C₃₁×#440	C₂	semidirect
4. C₃₁×#2356	C₂	semidirect
5. C₃₁×#2055	C₂	semidirect
6. C₃₁×#2014	C₂	semidirect
7. C₃₁×#1982	C₂	semidirect
8. C₃₁×#1961	C₂	semidirect
9. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1925	C₂	semidirect
11. C₃₁×#1804	C₂	semidirect
66. C₄×Q₈×C₃₁	D₄	semidirect
167. #24644	C₃₁	direct (N × H)
227. C₂×#460	C₂×C₃₁	semidirect
228. C₂×#440	C₂×C₃₁	semidirect
230. #2356	C₂×C₃₁	semidirect
231. #2055	C₂×C₃₁	semidirect
232. #2014	C₂×C₃₁	semidirect
233. #1982	C₂×C₃₁	semidirect
234. #1961	C₂×C₃₁	semidirect
235. #1940	C₂×C₃₁	semidirect
236. #1925	C₂×C₃₁	semidirect
237. #1804	C₂×C₃₁	semidirect
328. C₄×Q₈	D₄×C₃₁	semidirect
360. C₃₁	#24644	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×#460	2
C₂×C₃₁×#440	2
C₃₁×#2541	2
C₃₁×#2356	2
C₃₁×#2055	2
C₃₁×#2014	2
C₃₁×#1982	2
C₃₁×#1961	2
C₃₁×#1940	2
C₃₁×#1925	2
C₃₁×#1804	2
#24644	31

Subgroups — C31×#24644 #21481403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24644 #21481403