Subgroups — C₃₁×#33045 #21489804

Order 7936.

930 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2628	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2619	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1895	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1893	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#121	C₂²	1	2	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₃₁×#486	—	2	4	no	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₃₁×#405	—	2	4	no	—
24	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#400	—	2	1	no	—
29	C₃₁×#399	—	2	1	no	—
30	C₃₁×#395	—	2	1	no	—
31	C₃₁×#394	—	2	1	no	—
32	C₃₁×#394	—	2	1	no	—
33	C₃₁×#393	—	2	1	no	—
34	C₃₁×#392	—	2	1	no	—
35	C₃₁×#392	—	2	1	no	—
36	C₃₁×#390	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#382	C₂²	1	1	yes	characteristic
39	C₃₁×#382	C₂²	1	1	yes	characteristic
40	C₃₁×#380	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	2	yes	—
44	C₃₁×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	—	2	2	no	—
63	C₃₁×#127	—	2	2	no	—
64	C₃₁×#127	—	2	2	no	—
65	C₃₁×#121	—	2	4	no	—
66	C₃₁×#121	—	2	4	no	—
67	C₃₁×#117	—	2	2	no	—
68	C₃₁×#117	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#102	—	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	2	no	—
89	C₃₁×#97	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	2	no	—
91	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
92	C₃₁×#95	—	2	2	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×Q₈×C₃₁	—	4	2	no	—
101	C₂×Q₈×C₃₁	—	8	2	no	—
102	C₂×D₄×C₃₁	—	2	2	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	4	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	—	8	2	no	—
112	C₂×D₄×C₃₁	—	8	2	no	—
113	C₂×D₄×C₃₁	—	8	2	no	—
114	C₂×D₄×C₃₁	—	8	2	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
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	#41	1	2	yes	—
128	C₂²×C₄×C₃₁	#41	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
130	M₁₆×C₃₁	—	2	8	no	—
131	C₂×C₈×C₃₁	—	2	8	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₃₁	#41	1	2	yes	—
143	C₂×C₈×C₃₁	#41	1	2	yes	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	—	8	2	no	—
154	C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂²⋊C₄×C₃₁	—	2	2	no	—
156	C₂²⋊C₄×C₃₁	—	4	4	no	—
157	C₂²⋊C₄×C₃₁	—	8	2	no	—
158	C₂²⋊C₄×C₃₁	—	8	2	no	—
159	C₂²⋊C₄×C₃₁	—	8	2	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	C₂×D₄	1	2	yes	—
163	C₄²×C₃₁	#41	1	2	yes	—
164	#33045	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₃₁	—	2	1	no	—
166	C₂³×C₃₁	—	2	1	no	—
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	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	#124	1	2	yes	—
177	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₃₁	—	8	4	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	8	4	no	—
184	D₄×C₃₁	—	8	4	no	—
185	D₄×C₃₁	—	8	8	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	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	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	#121	1	2	yes	—
208	C₂×C₄×C₃₁	#121	1	2	yes	—
209	C₂×C₄×C₃₁	#124	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₈×C₃₁	—	2	2	no	—
217	C₈×C₃₁	—	2	2	no	—
218	C₈×C₃₁	—	2	2	no	—
219	C₈×C₃₁	—	2	2	no	—
220	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#2628	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#2619	C₂×C₃₁	1	1	yes	characteristic
224	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1961	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1936	C₂×C₃₁	1	2	yes	complemented
227	#1934	C₂×C₃₁	1	2	yes	complemented
228	#1895	C₂×C₃₁	1	2	yes	complemented
229	#1893	C₂×C₃₁	1	2	yes	complemented
230	#1835	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	4	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	1	no	—
235	C₂²×C₃₁	—	4	1	no	—
236	C₂²×C₃₁	—	8	2	no	—
237	C₂²×C₃₁	—	8	2	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	—	8	2	no	—
240	C₂²×C₃₁	—	8	2	no	—
241	C₂²×C₃₁	—	8	2	no	—
242	C₂²×C₃₁	—	8	2	no	—
243	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
244	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
245	C₂²×C₃₁	#532	1	1	yes	characteristic
246	C₂²×C₃₁	#576	1	2	yes	—
247	C₂²×C₃₁	#577	1	2	yes	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	4	1	no	—
252	C₄×C₃₁	—	4	1	no	—
253	C₄×C₃₁	—	8	2	no	—
254	C₄×C₃₁	—	8	2	no	—
255	C₄×C₃₁	—	8	2	no	—
256	C₄×C₃₁	#525	1	1	yes	characteristic
257	C₄×C₃₁	#525	1	1	yes	characteristic
258	C₄×C₃₁	#529	1	1	yes	characteristic
259	C₄×C₃₁	#529	1	1	yes	characteristic
260	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
261	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×#121	C₂²×C₃₁	1	2	yes	—
265	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
266	#486	—	2	4	no	—
267	C₂×#102	C₂²×C₃₁	1	2	yes	—
268	C₂×#102	C₂²×C₃₁	1	2	yes	—
269	C₂×#102	C₂²×C₃₁	1	2	yes	—
270	C₂×#102	C₂²×C₃₁	1	2	yes	—
271	#405	—	2	4	no	—
272	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
276	#400	—	2	1	no	—
277	#399	—	2	1	no	—
278	#395	—	2	1	no	—
279	#394	—	2	1	no	—
280	#394	—	2	1	no	—
281	#393	—	2	1	no	—
282	#392	—	2	1	no	—
283	#392	—	2	1	no	—
284	#390	C₂²×C₃₁	1	2	yes	—
285	#389	C₂²×C₃₁	1	2	yes	—
286	#382	C₂²×C₃₁	1	1	yes	characteristic
287	#382	C₂²×C₃₁	1	1	yes	characteristic
288	#380	C₂²×C₃₁	1	2	yes	—
289	#340	C₂²×C₃₁	1	2	yes	—
290	#340	C₂²×C₃₁	1	2	yes	—
291	#340	C₂²×C₃₁	1	2	yes	—
292	#340	C₂²×C₃₁	1	2	yes	—
293	C₂×C₃₁	—	4	2	no	—
294	C₂×C₃₁	—	16	2	no	—
295	C₂×C₃₁	#2850	1	1	yes	characteristic
296	C₂×C₃₁	#3057	1	1	yes	characteristic
297	C₂×C₃₁	#3058	1	1	yes	characteristic
298	C₂×C₃₁	#3136	1	2	yes	—
299	C₂×C₃₁	#3179	1	2	yes	—
300	C₂²×Q₈	—	2	1	no	—
301	C₂²×D₄	—	2	2	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
305	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×M₁₆	—	2	2	no	—
307	C₂×M₁₆	—	2	2	no	—
308	C₂×M₁₆	—	2	2	no	—
309	C₂²×C₈	—	2	2	no	—
310	C₂²×C₈	—	2	2	no	—
311	C₂²×C₈	—	2	2	no	—
312	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₄⋊Q₈	—	2	2	no	—
317	C₄⋊Q₈	—	2	2	no	—
318	#127	—	2	2	no	—
319	#127	—	2	2	no	—
320	#121	—	2	4	no	—
321	#121	—	2	4	no	—
322	#117	—	2	2	no	—
323	#117	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
336	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
339	#102	—	2	4	no	—
340	#102	—	2	4	no	—
341	#102	—	2	4	no	—
342	#102	—	2	4	no	—
343	#97	—	2	2	no	—
344	#97	—	2	2	no	—
345	C₄×C₈	—	2	2	no	—
346	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
347	#95	—	2	2	no	—
348	#95	—	2	1	no	—
349	#95	—	2	1	no	—
350	#95	—	2	1	no	—
351	#95	—	2	1	no	—
352	#95	—	2	2	no	—
353	C₃₁	#33045	1	1	yes	characteristic, direct product factor, Sylow
354	C₂⁴	—	2	2	no	—
355	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂×Q₈	—	4	2	no	—
357	C₂×Q₈	—	8	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	4	4	no	—
365	C₂×D₄	—	4	2	no	—
366	C₂×D₄	—	4	2	no	—
367	C₂×D₄	—	8	2	no	—
368	C₂×D₄	—	8	2	no	—
369	C₂×D₄	—	8	2	no	—
370	C₂×D₄	—	8	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	#41×C₃₁	1	2	yes	—
384	C₂²×C₄	#41×C₃₁	1	2	yes	—
385	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
386	M₁₆	—	2	8	no	—
387	C₂×C₈	—	2	8	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
397	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
398	C₂×C₈	#41×C₃₁	1	2	yes	—
399	C₂×C₈	#41×C₃₁	1	2	yes	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	4	2	no	—
409	C₄⋊C₄	—	8	2	no	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	4	4	no	—
413	C₂²⋊C₄	—	8	2	no	—
414	C₂²⋊C₄	—	8	2	no	—
415	C₂²⋊C₄	—	8	2	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	C₂×D₄×C₃₁	1	2	yes	—
419	C₄²	#41×C₃₁	1	2	yes	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	C₃₁×#124	1	2	yes	—
432	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	Q₈	—	8	4	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	8	4	no	—
439	D₄	—	8	4	no	—
440	D₄	—	8	8	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	C₃₁×#121	1	2	yes	—
463	C₂×C₄	C₃₁×#121	1	2	yes	—
464	C₂×C₄	C₃₁×#124	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
471	C₈	—	2	2	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	2	no	—
474	C₈	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	1	no	—
479	C₂²	—	4	1	no	—
480	C₂²	—	8	2	no	—
481	C₂²	—	8	2	no	—
482	C₂²	—	8	2	no	—
483	C₂²	—	8	2	no	—
484	C₂²	—	8	2	no	—
485	C₂²	—	8	2	no	—
486	C₂²	—	8	2	no	—
487	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
488	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
489	C₂²	C₃₁×#532	1	1	yes	characteristic
490	C₂²	C₃₁×#576	1	2	yes	—
491	C₂²	C₃₁×#577	1	2	yes	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	8	2	no	—
498	C₄	—	8	2	no	—
499	C₄	—	8	2	no	—
500	C₄	C₃₁×#525	1	1	yes	characteristic
501	C₄	C₃₁×#525	1	1	yes	characteristic
502	C₄	C₃₁×#529	1	1	yes	characteristic
503	C₄	C₃₁×#529	1	1	yes	characteristic
504	C₂	—	4	2	no	—
505	C₂	—	16	2	no	—
506	C₂	C₃₁×#2850	1	1	yes	characteristic
507	C₂	C₃₁×#3057	1	1	yes	characteristic
508	C₂	C₃₁×#3058	1	1	yes	characteristic
509	C₂	C₃₁×#3136	1	2	yes	—
510	C₂	C₃₁×#3179	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#486	C₂	semidirect
2. C₂×C₃₁×#405	C₂	semidirect
3. C₃₁×#2628	C₂	semidirect
5. C₃₁×#1993	C₂	semidirect
6. C₃₁×#1961	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1934	C₂	semidirect
9. C₃₁×#1895	C₂	semidirect
10. C₃₁×#1893	C₂	semidirect
11. C₃₁×#1835	C₂	semidirect
91. C₄×C₈×C₃₁	D₄	semidirect
164. #33045	C₃₁	direct (N × H)
220. C₂×#486	C₂×C₃₁	semidirect
221. C₂×#405	C₂×C₃₁	semidirect
222. #2628	C₂×C₃₁	semidirect
224. #1993	C₂×C₃₁	semidirect
225. #1961	C₂×C₃₁	semidirect
226. #1936	C₂×C₃₁	semidirect
227. #1934	C₂×C₃₁	semidirect
228. #1895	C₂×C₃₁	semidirect
229. #1893	C₂×C₃₁	semidirect
230. #1835	C₂×C₃₁	semidirect
346. C₄×C₈	D₄×C₃₁	semidirect
353. C₃₁	#33045	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×#486	2
C₂×C₃₁×#405	2
C₃₁×#2628	2
C₃₁×#2619	2
C₃₁×#1993	2
C₃₁×#1961	2
C₃₁×#1936	2
C₃₁×#1934	2
C₃₁×#1895	2
C₃₁×#1893	2
C₃₁×#1835	2
#33045	31

Subgroups — C31×#33045 #21489804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33045 #21489804