Subgroups — C₃₁×#33046 #21489805

Order 7936.

830 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2628	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2621	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1960	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1935	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1933	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1894	C₂	1	2	yes	maximal
10	C₃₁×#1892	C₂	1	2	yes	maximal
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×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₃₁×#487	—	2	4	no	—
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	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×#398	—	2	1	no	—
31	C₃₁×#398	—	2	1	no	—
32	C₃₁×#397	—	2	1	no	—
33	C₃₁×#397	—	2	1	no	—
34	C₃₁×#395	—	2	1	no	—
35	C₃₁×#393	—	2	1	no	—
36	C₃₁×#391	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×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₂²×C₈×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	4	no	—
63	C₃₁×C₂²⋊Q₈	—	2	4	no	—
64	C₃₁×#117	—	2	2	no	—
65	C₃₁×#117	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	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	D₄	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#103	—	2	4	no	—
83	C₃₁×#103	—	2	4	no	—
84	C₃₁×#103	—	2	4	no	—
85	C₃₁×#103	—	2	4	no	—
86	C₃₁×#97	—	2	2	no	—
87	C₃₁×#97	—	2	2	no	—
88	C₄×C₈×C₃₁	—	2	2	no	—
89	C₄×C₈×C₃₁	D₄	1	4	yes	—
90	C₃₁×#95	—	2	2	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	2	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	4	2	no	—
106	C₂×Q₈×C₃₁	—	8	2	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	#41	1	2	yes	—
121	C₂²×C₄×C₃₁	#41	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₃₁	—	2	8	no	—
124	C₂×C₈×C₃₁	—	2	8	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₃₁	#41	1	2	yes	—
136	C₂×C₈×C₃₁	#41	1	2	yes	—
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	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	—	4	2	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	—	4	4	no	—
149	C₄⋊C₄×C₃₁	—	8	2	no	—
150	C₂²⋊C₄×C₃₁	—	2	2	no	—
151	C₂²⋊C₄×C₃₁	—	2	2	no	—
152	C₂²⋊C₄×C₃₁	—	4	4	no	—
153	C₂²⋊C₄×C₃₁	—	8	2	no	—
154	C₄²×C₃₁	—	2	1	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	C₂×D₄	1	2	yes	—
157	C₄²×C₃₁	#41	1	2	yes	—
158	#33046	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	2	1	no	—
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	#124	1	2	yes	—
164	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	Q₈×C₃₁	—	4	2	no	—
166	Q₈×C₃₁	—	4	2	no	—
167	Q₈×C₃₁	—	4	2	no	—
168	Q₈×C₃₁	—	4	2	no	—
169	Q₈×C₃₁	—	8	4	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	#121	1	2	yes	—
195	C₂×C₄×C₃₁	#121	1	2	yes	—
196	C₂×C₄×C₃₁	#124	1	2	yes	—
197	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
203	C₈×C₃₁	—	2	2	no	—
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	2	2	no	—
207	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
208	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
209	#2628	C₂×C₃₁	1	1	yes	characteristic
210	#2621	C₂×C₃₁	1	1	yes	characteristic
211	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1960	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1935	C₂×C₃₁	1	2	yes	complemented
214	#1933	C₂×C₃₁	1	2	yes	complemented
215	#1894	C₂×C₃₁	1	2	yes	—
216	#1892	C₂×C₃₁	1	2	yes	—
217	#1835	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₂²×C₃₁	—	2	2	no	—
219	C₂²×C₃₁	—	2	4	no	—
220	C₂²×C₃₁	—	4	2	no	—
221	C₂²×C₃₁	—	4	1	no	—
222	C₂²×C₃₁	—	4	1	no	—
223	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
225	C₂²×C₃₁	#532	1	1	yes	characteristic
226	C₂²×C₃₁	#577	1	2	yes	—
227	C₂²×C₃₁	#578	1	2	yes	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	4	1	no	—
232	C₄×C₃₁	—	4	1	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	—	8	2	no	—
237	C₄×C₃₁	#525	1	1	yes	characteristic
238	C₄×C₃₁	#525	1	1	yes	characteristic
239	C₄×C₃₁	#529	1	1	yes	characteristic
240	C₄×C₃₁	#529	1	1	yes	characteristic
241	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
242	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
246	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
247	#487	—	2	4	no	—
248	C₂×#103	C₂²×C₃₁	1	2	yes	—
249	C₂×#103	C₂²×C₃₁	1	2	yes	—
250	C₂×#103	C₂²×C₃₁	1	2	yes	—
251	C₂×#103	C₂²×C₃₁	1	2	yes	—
252	#405	—	2	4	no	—
253	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
257	#400	—	2	1	no	—
258	#399	—	2	1	no	—
259	#398	—	2	1	no	—
260	#398	—	2	1	no	—
261	#397	—	2	1	no	—
262	#397	—	2	1	no	—
263	#395	—	2	1	no	—
264	#393	—	2	1	no	—
265	#391	C₂²×C₃₁	1	2	yes	—
266	#389	C₂²×C₃₁	1	2	yes	—
267	#382	C₂²×C₃₁	1	1	yes	characteristic
268	#382	C₂²×C₃₁	1	1	yes	characteristic
269	#380	C₂²×C₃₁	1	2	yes	—
270	#340	C₂²×C₃₁	1	2	yes	—
271	#340	C₂²×C₃₁	1	2	yes	—
272	#340	C₂²×C₃₁	1	2	yes	—
273	#340	C₂²×C₃₁	1	2	yes	—
274	C₂×C₃₁	—	4	2	no	—
275	C₂×C₃₁	#2850	1	1	yes	characteristic
276	C₂×C₃₁	#3057	1	1	yes	characteristic
277	C₂×C₃₁	#3059	1	1	yes	characteristic
278	C₂×C₃₁	#3137	1	2	yes	—
279	C₂×C₃₁	#3178	1	2	yes	—
280	C₂²×Q₈	—	2	1	no	—
281	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
282	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×M₁₆	—	2	2	no	—
284	C₂×M₁₆	—	2	2	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂²×C₈	—	2	2	no	—
287	C₂²×C₈	—	2	2	no	—
288	C₂²×C₈	—	2	2	no	—
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₄⋊Q₈	—	2	2	no	—
294	C₄⋊Q₈	—	2	2	no	—
295	C₄⋊Q₈	—	2	2	no	—
296	C₄⋊Q₈	—	2	2	no	—
297	C₂²⋊Q₈	—	2	4	no	—
298	C₂²⋊Q₈	—	2	4	no	—
299	#117	—	2	2	no	—
300	#117	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
314	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
316	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
317	#103	—	2	4	no	—
318	#103	—	2	4	no	—
319	#103	—	2	4	no	—
320	#103	—	2	4	no	—
321	#97	—	2	2	no	—
322	#97	—	2	2	no	—
323	C₄×C₈	—	2	2	no	—
324	C₄×C₈	D₄×C₃₁	1	4	yes	—
325	#95	—	2	2	no	—
326	#95	—	2	1	no	—
327	#95	—	2	1	no	—
328	#95	—	2	1	no	—
329	#95	—	2	1	no	—
330	#95	—	2	2	no	—
331	#95	—	2	2	no	—
332	#95	—	2	2	no	—
333	C₃₁	#33046	1	1	yes	characteristic, direct product factor, Sylow
334	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	4	2	no	—
342	C₂×Q₈	—	8	2	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	#41×C₃₁	1	2	yes	—
357	C₂²×C₄	#41×C₃₁	1	2	yes	—
358	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
359	M₁₆	—	2	8	no	—
360	C₂×C₈	—	2	8	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
370	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
371	C₂×C₈	#41×C₃₁	1	2	yes	—
372	C₂×C₈	#41×C₃₁	1	2	yes	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₄⋊C₄	—	4	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₄⋊C₄	—	8	2	no	—
386	C₂²⋊C₄	—	2	2	no	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	4	4	no	—
389	C₂²⋊C₄	—	8	2	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	2	1	no	—
392	C₄²	C₂×D₄×C₃₁	1	2	yes	—
393	C₄²	#41×C₃₁	1	2	yes	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	C₃₁×#124	1	2	yes	—
399	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
400	Q₈	—	4	2	no	—
401	Q₈	—	4	2	no	—
402	Q₈	—	4	2	no	—
403	Q₈	—	4	2	no	—
404	Q₈	—	8	4	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	C₃₁×#121	1	2	yes	—
430	C₂×C₄	C₃₁×#121	1	2	yes	—
431	C₂×C₄	C₃₁×#124	1	2	yes	—
432	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
438	C₈	—	2	2	no	—
439	C₈	—	2	2	no	—
440	C₈	—	2	2	no	—
441	C₈	—	2	2	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
448	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
449	C₂²	C₃₁×#532	1	1	yes	characteristic
450	C₂²	C₃₁×#577	1	2	yes	—
451	C₂²	C₃₁×#578	1	2	yes	—
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	2	2	no	—
455	C₄	—	4	1	no	—
456	C₄	—	4	1	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	2	no	—
459	C₄	—	8	2	no	—
460	C₄	—	8	2	no	—
461	C₄	C₃₁×#525	1	1	yes	characteristic
462	C₄	C₃₁×#525	1	1	yes	characteristic
463	C₄	C₃₁×#529	1	1	yes	characteristic
464	C₄	C₃₁×#529	1	1	yes	characteristic
465	C₂	—	4	2	no	—
466	C₂	C₃₁×#2850	1	1	yes	characteristic
467	C₂	C₃₁×#3057	1	1	yes	characteristic
468	C₂	C₃₁×#3059	1	1	yes	characteristic
469	C₂	C₃₁×#3137	1	2	yes	—
470	C₂	C₃₁×#3178	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
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×#487	C₂	semidirect
5. C₃₁×#1993	C₂	semidirect
6. C₃₁×#1960	C₂	semidirect
7. C₃₁×#1935	C₂	semidirect
8. C₃₁×#1933	C₂	semidirect
11. C₃₁×#1835	C₂	semidirect
158. #33046	C₃₁	direct (N × H)
207. C₂×#487	C₂×C₃₁	semidirect
211. #1993	C₂×C₃₁	semidirect
212. #1960	C₂×C₃₁	semidirect
213. #1935	C₂×C₃₁	semidirect
214. #1933	C₂×C₃₁	semidirect
217. #1835	C₂×C₃₁	semidirect
333. C₃₁	#33046	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×#487	2
C₂×C₃₁×#405	2
C₃₁×#2628	2
C₃₁×#2621	2
C₃₁×#1993	2
C₃₁×#1960	2
C₃₁×#1935	2
C₃₁×#1933	2
C₃₁×#1894	2
C₃₁×#1892	2
C₃₁×#1835	2
#33046	31

Subgroups — C31×#33046 #21489805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33046 #21489805