Subgroups — C₃₁×#23684 #21480443

Order 7936.

1004 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#440	C₂	1	2	yes	maximal, complemented
2	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2083	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2055	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2041	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#2014	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1762	C₂	1	2	yes	maximal, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₃₁	—	2	1	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#440	—	2	8	no	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#97	C₂²	1	2	yes	—
26	C₃₁×#391	C₂²	1	1	yes	characteristic
27	C₃₁×#391	C₂²	1	1	yes	characteristic
28	C₃₁×#390	—	2	1	no	—
29	C₃₁×#390	—	2	1	no	—
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	1	yes	characteristic
32	C₃₁×#389	C₂²	1	1	yes	characteristic
33	C₃₁×#387	—	2	1	no	—
34	C₃₁×#387	—	2	1	no	—
35	C₃₁×#384	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#376	—	2	1	no	—
39	C₄³×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₃₁×#336	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	4	no	—
50	C₂×SD₁₆×C₃₁	—	2	4	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₃₁×#127	—	2	4	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	D₄	1	4	yes	complemented
57	C₄×D₄×C₃₁	—	2	4	no	—
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	4	4	no	—
60	C₄×D₄×C₃₁	—	4	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#107	—	2	4	no	—
83	C₃₁×C₄⋊C₈	—	2	4	no	—
84	C₃₁×#103	—	2	4	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#97	—	2	4	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₂⁴×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	1	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×D₄×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	2	4	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	4	2	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	#41	1	4	yes	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	4	4	no	—
119	C₂²×C₄×C₃₁	—	4	4	no	—
120	C₂²×C₄×C₃₁	—	4	4	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	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
128	SD₁₆×C₃₁	—	2	16	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	#41	1	4	yes	—
135	C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	—	4	2	no	—
144	C₄⋊C₄×C₃₁	—	4	2	no	—
145	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
146	C₂²⋊C₄×C₃₁	—	4	4	no	—
147	C₂²⋊C₄×C₃₁	—	4	4	no	—
148	C₂²⋊C₄×C₃₁	—	4	4	no	—
149	C₄²×C₃₁	—	2	4	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	2	2	no	—
152	C₄²×C₃₁	—	2	2	no	—
153	C₄²×C₃₁	—	2	2	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	4	1	no	—
156	C₄²×C₃₁	—	4	2	no	—
157	C₄²×C₃₁	C₂×D₄	1	2	yes	—
158	C₄²×C₃₁	#41	1	2	yes	—
159	#23684	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	4	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
168	Q₈×C₃₁	—	2	4	no	—
169	Q₈×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	2	4	no	—
171	D₄×C₃₁	—	2	4	no	—
172	D₄×C₃₁	—	4	8	no	—
173	D₄×C₃₁	—	4	8	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	8	8	no	—
205	C₂×C₄×C₃₁	#121	1	2	yes	—
206	C₂×C₄×C₃₁	#121	1	2	yes	—
207	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
208	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
209	C₂×C₄×C₃₁	#137	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
212	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
213	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
214	C₈×C₃₁	—	2	4	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₂×#440	C₂×C₃₁	1	2	yes	complemented
217	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
218	#2295	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#2083	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#2072	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#2067	C₂×C₃₁	1	2	yes	complemented
222	#2055	C₂×C₃₁	1	2	yes	complemented
223	#2041	C₂×C₃₁	1	1	yes	characteristic
224	#2014	C₂×C₃₁	1	1	yes	characteristic
225	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1762	C₂×C₃₁	1	2	yes	complemented
227	C₂²×C₃₁	—	4	4	no	—
228	C₂²×C₃₁	—	4	4	no	—
229	C₂²×C₃₁	—	4	4	no	—
230	C₂²×C₃₁	—	8	2	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	—	8	2	no	—
233	C₂²×C₃₁	—	8	2	no	—
234	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
235	C₂²×C₃₁	#525	1	1	yes	characteristic
236	C₂²×C₃₁	#532	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
240	C₂²×C₃₁	#575	1	1	yes	characteristic
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	2	2	no	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	4	1	no	—
248	C₄×C₃₁	—	4	2	no	—
249	C₄×C₃₁	—	8	1	no	—
250	C₄×C₃₁	—	8	1	no	—
251	C₄×C₃₁	—	8	2	no	—
252	C₄×C₃₁	#525	1	2	yes	—
253	C₄×C₃₁	#529	1	2	yes	—
254	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
255	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄×D₄	—	2	1	no	—
258	C₂×C₄×D₄	—	2	1	no	—
259	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
260	#440	—	2	8	no	—
261	C₂×#107	C₂²×C₃₁	1	2	yes	—
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
263	C₂×#103	C₂²×C₃₁	1	2	yes	—
264	C₂×#102	C₂²×C₃₁	1	2	yes	—
265	C₂×#102	C₂²×C₃₁	1	2	yes	—
266	C₂×#102	C₂²×C₃₁	1	2	yes	—
267	C₂×#97	C₂²×C₃₁	1	2	yes	—
268	#391	C₂²×C₃₁	1	1	yes	characteristic
269	#391	C₂²×C₃₁	1	1	yes	characteristic
270	#390	—	2	1	no	—
271	#390	—	2	1	no	—
272	#390	C₂²×C₃₁	1	2	yes	—
273	#389	C₂²×C₃₁	1	1	yes	characteristic
274	#389	C₂²×C₃₁	1	1	yes	characteristic
275	#387	—	2	1	no	—
276	#387	—	2	1	no	—
277	#384	C₂²×C₃₁	1	1	yes	characteristic
278	C₄×C₄⋊C₄	—	2	1	no	—
279	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
280	#376	—	2	1	no	—
281	C₄³	C₂²×C₃₁	1	1	yes	characteristic
282	#340	C₂²×C₃₁	1	2	yes	—
283	#340	C₂²×C₃₁	1	2	yes	—
284	#336	C₂²×C₃₁	1	1	yes	characteristic
285	#336	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×C₃₁	—	8	4	no	—
287	C₂×C₃₁	#2392	1	1	yes	characteristic
288	C₂×C₃₁	#3037	1	1	yes	characteristic
289	C₂×C₃₁	#3038	1	1	yes	characteristic
290	C₂×C₃₁	#3110	1	1	yes	characteristic
291	C₂×C₃₁	#3114	1	1	yes	characteristic
292	C₂×C₃₁	#3180	1	1	yes	characteristic
293	C₂×C₃₁	#3182	1	1	yes	characteristic
294	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×D₄	—	2	1	no	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
298	C₂³×C₄	—	2	2	no	—
299	C₂×SD₁₆	—	2	4	no	—
300	C₂×SD₁₆	—	2	4	no	—
301	C₂×SD₁₆	—	2	4	no	—
302	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
303	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
304	#127	—	2	4	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	D₄×C₃₁	1	4	yes	complemented
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	—	4	4	no	—
310	C₄×D₄	—	4	4	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₄²	—	2	1	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
332	#107	—	2	4	no	—
333	C₄⋊C₈	—	2	4	no	—
334	#103	—	2	4	no	—
335	#102	—	2	4	no	—
336	#102	—	2	4	no	—
337	#102	—	2	4	no	—
338	#97	—	2	4	no	—
339	#95	—	2	1	no	—
340	#95	—	2	1	no	—
341	#95	—	2	1	no	—
342	#95	—	2	1	no	—
343	#95	—	2	1	no	—
344	#95	—	2	1	no	—
345	#95	—	2	1	no	—
346	C₃₁	#23684	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	—	2	2	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
352	C₂×D₄	—	2	4	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	4	2	no	—
360	C₂×D₄	—	4	2	no	—
361	C₂×D₄	—	4	2	no	—
362	C₂×D₄	—	4	2	no	—
363	C₂×D₄	#41×C₃₁	1	4	yes	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	4	4	no	—
370	C₂²×C₄	—	4	4	no	—
371	C₂²×C₄	—	4	4	no	—
372	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
379	SD₁₆	—	2	16	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	#41×C₃₁	1	4	yes	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₄⋊C₄	—	4	2	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
397	C₂²⋊C₄	—	4	4	no	—
398	C₂²⋊C₄	—	4	4	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₄²	—	2	4	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	4	1	no	—
407	C₄²	—	4	2	no	—
408	C₄²	C₂×D₄×C₃₁	1	2	yes	—
409	C₄²	#41×C₃₁	1	2	yes	—
410	C₂³	—	2	4	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	—	4	2	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
418	Q₈	—	2	4	no	—
419	Q₈	—	4	2	no	—
420	D₄	—	2	4	no	—
421	D₄	—	2	4	no	—
422	D₄	—	4	8	no	—
423	D₄	—	4	8	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	2	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	8	8	no	—
455	C₂×C₄	C₃₁×#121	1	2	yes	—
456	C₂×C₄	C₃₁×#121	1	2	yes	—
457	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
458	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
459	C₂×C₄	C₃₁×#137	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
462	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
463	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
464	C₈	—	2	4	no	—
465	C₈	—	4	2	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	4	4	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	8	2	no	—
470	C₂²	—	8	2	no	—
471	C₂²	—	8	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
474	C₂²	C₃₁×#525	1	1	yes	characteristic
475	C₂²	C₃₁×#532	1	1	yes	characteristic
476	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
477	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
478	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
479	C₂²	C₃₁×#575	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	1	no	—
487	C₄	—	4	2	no	—
488	C₄	—	8	1	no	—
489	C₄	—	8	1	no	—
490	C₄	—	8	2	no	—
491	C₄	C₃₁×#525	1	2	yes	—
492	C₄	C₃₁×#529	1	2	yes	—
493	C₂	—	8	4	no	—
494	C₂	C₃₁×#2392	1	1	yes	characteristic
495	C₂	C₃₁×#3037	1	1	yes	characteristic
496	C₂	C₃₁×#3038	1	1	yes	characteristic
497	C₂	C₃₁×#3110	1	1	yes	characteristic
498	C₂	C₃₁×#3114	1	1	yes	characteristic
499	C₂	C₃₁×#3180	1	1	yes	characteristic
500	C₂	C₃₁×#3182	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#440	C₂	semidirect
3. C₃₁×#2295	C₂	semidirect
4. C₃₁×#2083	C₂	semidirect
5. C₃₁×#2072	C₂	semidirect
6. C₃₁×#2067	C₂	semidirect
7. C₃₁×#2055	C₂	semidirect
10. C₃₁×#1767	C₂	semidirect
11. C₃₁×#1762	C₂	semidirect
56. C₄×Q₈×C₃₁	D₄	semidirect
159. #23684	C₃₁	direct (N × H)
216. C₂×#440	C₂×C₃₁	semidirect
218. #2295	C₂×C₃₁	semidirect
219. #2083	C₂×C₃₁	semidirect
220. #2072	C₂×C₃₁	semidirect
221. #2067	C₂×C₃₁	semidirect
222. #2055	C₂×C₃₁	semidirect
225. #1767	C₂×C₃₁	semidirect
226. #1762	C₂×C₃₁	semidirect
306. C₄×Q₈	D₄×C₃₁	semidirect
346. C₃₁	#23684	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×#440	2
C₄×C₃₁×#127	2
C₃₁×#2295	2
C₃₁×#2083	2
C₃₁×#2072	2
C₃₁×#2067	2
C₃₁×#2055	2
C₃₁×#2041	2
C₃₁×#2014	2
C₃₁×#1767	2
C₃₁×#1762	2
#23684	31

Subgroups — C31×#23684 #21480443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23684 #21480443