Subgroups — C₃₁×#33122 #21489881

Order 7936.

782 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	C₃₁×#3095	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3094	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#465	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1977	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1951	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1931	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1930	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1901	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1900	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₃₁×#485	—	2	2	no	—
19	C₃₁×#484	—	2	1	no	—
20	C₃₁×#483	—	2	1	no	—
21	C₃₁×#482	—	2	2	no	—
22	C₃₁×#481	—	2	1	no	—
23	C₃₁×#480	—	2	1	no	—
24	C₃₁×#465	—	2	1	no	—
25	C₃₁×#465	—	2	1	no	—
26	C₃₁×#465	—	2	1	no	—
27	C₃₁×#465	—	2	1	no	—
28	C₂×C₃₁×#108	C₂²	1	2	yes	—
29	C₃₁×#428	C₂²	1	2	yes	—
30	C₃₁×#428	C₂²	1	2	yes	—
31	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
32	C₃₁×#419	C₂²	1	2	yes	—
33	C₃₁×#418	C₂²	1	2	yes	—
34	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#101	C₂²	1	2	yes	—
37	C₂×C₃₁×#100	C₂²	1	2	yes	—
38	C₃₁×#407	C₂²	1	1	yes	characteristic
39	C₃₁×#407	C₂²	1	1	yes	characteristic
40	C₃₁×#405	—	2	1	no	—
41	C₃₁×#405	—	2	1	no	—
42	C₃₁×#405	C₂²	1	2	yes	—
43	C₃₁×#405	C₂²	1	2	yes	—
44	C₂×C₃₁×#97	—	2	1	no	—
45	C₂×C₄×C₈×C₃₁	—	2	1	no	—
46	C₃₁×#343	C₂²	1	2	yes	—
47	C₃₁×#343	C₂²	1	2	yes	—
48	C₃₁×#341	C₂²	1	2	yes	—
49	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₁	—	2	1	no	—
54	C₂×SD₁₆×C₃₁	—	2	1	no	—
55	C₂×SD₁₆×C₃₁	—	2	4	no	—
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂×M₁₆×C₃₁	—	2	4	no	—
58	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	—	2	1	no	—
62	C₂²×C₈×C₃₁	—	2	4	no	—
63	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	—	2	1	no	—
77	C₃₁×#117	—	2	1	no	—
78	C₃₁×#117	—	2	1	no	—
79	C₃₁×#117	D₄	1	2	yes	—
80	C₃₁×#117	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₄²×C₃₁	—	2	1	no	—
90	C₃₁×#108	—	2	4	no	—
91	C₃₁×#107	—	2	2	no	—
92	C₃₁×#107	—	2	2	no	—
93	C₃₁×#106	—	2	1	no	—
94	C₃₁×#106	—	2	1	no	—
95	C₃₁×#106	—	2	2	no	—
96	C₃₁×#106	—	2	2	no	—
97	C₃₁×#103	—	2	1	no	—
98	C₃₁×#103	—	2	1	no	—
99	C₃₁×#103	—	2	2	no	—
100	C₃₁×#103	—	2	2	no	—
101	C₃₁×#102	—	2	1	no	—
102	C₃₁×#102	—	2	1	no	—
103	C₃₁×#102	—	2	2	no	—
104	C₃₁×#102	—	2	2	no	—
105	C₃₁×#101	—	2	4	no	—
106	C₃₁×#100	—	2	4	no	—
107	C₃₁×C₂²⋊C₈	—	2	1	no	—
108	C₃₁×C₂²⋊C₈	—	2	1	no	—
109	C₃₁×C₂²⋊C₈	—	2	1	no	—
110	C₃₁×C₂²⋊C₈	—	2	1	no	—
111	C₃₁×#97	—	2	2	no	—
112	C₃₁×#97	—	2	2	no	—
113	C₄×C₈×C₃₁	—	2	2	no	—
114	C₄×C₈×C₃₁	—	2	2	no	—
115	C₂⁴×C₃₁	—	2	1	no	—
116	C₂⁴×C₃₁	#41	1	1	yes	characteristic
117	C₂×Q₈×C₃₁	—	2	1	no	—
118	C₂×Q₈×C₃₁	—	2	1	no	—
119	C₂×Q₈×C₃₁	—	2	2	no	—
120	C₂×Q₈×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	4	1	no	—
126	C₂×D₄×C₃₁	—	4	1	no	—
127	C₂×D₄×C₃₁	—	8	2	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
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	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
141	SD₁₆×C₃₁	—	4	2	no	—
142	SD₁₆×C₃₁	—	4	2	no	—
143	M₁₆×C₃₁	—	2	4	no	—
144	M₁₆×C₃₁	—	2	2	no	—
145	M₁₆×C₃₁	—	2	2	no	—
146	M₁₆×C₃₁	—	2	2	no	—
147	M₁₆×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	#41	1	4	yes	—
150	C₂×C₈×C₃₁	—	2	4	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₃₁	#41	1	4	yes	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	4	1	no	—
171	C₄⋊C₄×C₃₁	—	4	1	no	—
172	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²⋊C₄×C₃₁	—	2	2	no	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	4	1	no	—
177	C₂²⋊C₄×C₃₁	—	4	1	no	—
178	C₂²⋊C₄×C₃₁	—	4	1	no	—
179	C₂²⋊C₄×C₃₁	—	4	1	no	—
180	C₄²×C₃₁	—	2	2	no	—
181	C₄²×C₃₁	—	2	2	no	—
182	#33122	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	2	no	—
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	4	1	no	—
186	C₂³×C₃₁	—	4	1	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	8	1	no	—
189	C₂³×C₃₁	—	8	1	no	—
190	C₂³×C₃₁	#121	1	1	yes	characteristic
191	C₂³×C₃₁	#121	1	1	yes	characteristic
192	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
193	Q₈×C₃₁	—	4	1	no	—
194	Q₈×C₃₁	—	4	1	no	—
195	Q₈×C₃₁	—	4	2	no	—
196	D₄×C₃₁	—	4	1	no	—
197	D₄×C₃₁	—	4	1	no	—
198	D₄×C₃₁	—	4	2	no	—
199	D₄×C₃₁	—	8	2	no	—
200	D₄×C₃₁	—	8	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	8	1	no	—
217	C₂×C₄×C₃₁	—	8	1	no	—
218	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
219	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
223	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
224	C₈×C₃₁	—	2	1	no	—
225	C₈×C₃₁	—	2	1	no	—
226	C₈×C₃₁	—	2	4	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₈×C₃₁	—	4	2	no	—
229	C₈×C₃₁	#123	1	2	yes	—
230	C₈×C₃₁	#123	1	2	yes	—
231	#3095	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#3094	C₂×C₃₁	1	1	yes	characteristic
233	C₂×#465	C₂×C₃₁	1	1	yes	characteristic
234	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1977	C₂×C₃₁	1	2	yes	complemented
236	#1951	C₂×C₃₁	1	2	yes	complemented
237	#1931	C₂×C₃₁	1	2	yes	complemented
238	#1930	C₂×C₃₁	1	2	yes	complemented
239	#1901	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1900	C₂×C₃₁	1	1	yes	characteristic
241	#1810	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂²×C₃₁	—	2	1	no	—
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	2	1	no	—
249	C₂²×C₃₁	—	4	4	no	—
250	C₂²×C₃₁	—	8	1	no	—
251	C₂²×C₃₁	—	8	1	no	—
252	C₂²×C₃₁	—	8	1	no	—
253	C₂²×C₃₁	—	16	2	no	—
254	C₂²×C₃₁	#471	1	2	yes	—
255	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
256	C₂²×C₃₁	#529	1	2	yes	—
257	C₄×C₃₁	—	2	1	no	—
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	—	8	1	no	—
262	C₄×C₃₁	—	8	1	no	—
263	C₄×C₃₁	—	8	1	no	—
264	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
265	C₄×C₃₁	#525	1	1	yes	characteristic
266	C₄×C₃₁	#532	1	2	yes	—
267	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
269	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
273	#485	—	2	2	no	—
274	#484	—	2	1	no	—
275	#483	—	2	1	no	—
276	#482	—	2	2	no	—
277	#481	—	2	1	no	—
278	#480	—	2	1	no	—
279	#465	—	2	1	no	—
280	#465	—	2	1	no	—
281	#465	—	2	1	no	—
282	#465	—	2	1	no	—
283	C₂×#108	C₂²×C₃₁	1	2	yes	—
284	#428	C₂²×C₃₁	1	2	yes	—
285	#428	C₂²×C₃₁	1	2	yes	—
286	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
287	#419	C₂²×C₃₁	1	2	yes	—
288	#418	C₂²×C₃₁	1	2	yes	—
289	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×#101	C₂²×C₃₁	1	2	yes	—
292	C₂×#100	C₂²×C₃₁	1	2	yes	—
293	#407	C₂²×C₃₁	1	1	yes	characteristic
294	#407	C₂²×C₃₁	1	1	yes	characteristic
295	#405	—	2	1	no	—
296	#405	—	2	1	no	—
297	#405	C₂²×C₃₁	1	2	yes	—
298	#405	C₂²×C₃₁	1	2	yes	—
299	C₂×#97	—	2	1	no	—
300	C₂×C₄×C₈	—	2	1	no	—
301	#343	C₂²×C₃₁	1	2	yes	—
302	#343	C₂²×C₃₁	1	2	yes	—
303	#341	C₂²×C₃₁	1	2	yes	—
304	C₂×C₃₁	—	2	2	no	—
305	C₂×C₃₁	—	2	2	no	—
306	C₂×C₃₁	—	4	1	no	—
307	C₂×C₃₁	—	16	1	no	—
308	C₂×C₃₁	#2854	1	1	yes	characteristic
309	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
310	C₂×C₃₁	#3067	1	1	yes	characteristic
311	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×D₄	—	2	1	no	—
313	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×SD₁₆	—	2	1	no	—
316	C₂×SD₁₆	—	2	1	no	—
317	C₂×SD₁₆	—	2	4	no	—
318	C₂×M₁₆	—	2	1	no	—
319	C₂×M₁₆	—	2	4	no	—
320	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
321	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
322	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
323	C₂²×C₈	—	2	1	no	—
324	C₂²×C₈	—	2	4	no	—
325	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₂²⋊Q₈	—	2	1	no	—
332	C₂²⋊Q₈	—	2	1	no	—
333	C₂²⋊Q₈	—	2	1	no	—
334	C₂²⋊Q₈	—	2	1	no	—
335	#121	—	2	1	no	—
336	#121	—	2	1	no	—
337	#121	—	2	1	no	—
338	#121	—	2	1	no	—
339	#117	—	2	1	no	—
340	#117	—	2	1	no	—
341	#117	D₄×C₃₁	1	2	yes	—
342	#117	C₂³×C₃₁	1	2	yes	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
347	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₂²⋊C₄	—	2	1	no	—
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₄²	—	2	1	no	—
352	#108	—	2	4	no	—
353	#107	—	2	2	no	—
354	#107	—	2	2	no	—
355	#106	—	2	1	no	—
356	#106	—	2	1	no	—
357	#106	—	2	2	no	—
358	#106	—	2	2	no	—
359	#103	—	2	1	no	—
360	#103	—	2	1	no	—
361	#103	—	2	2	no	—
362	#103	—	2	2	no	—
363	#102	—	2	1	no	—
364	#102	—	2	1	no	—
365	#102	—	2	2	no	—
366	#102	—	2	2	no	—
367	#101	—	2	4	no	—
368	#100	—	2	4	no	—
369	C₂²⋊C₈	—	2	1	no	—
370	C₂²⋊C₈	—	2	1	no	—
371	C₂²⋊C₈	—	2	1	no	—
372	C₂²⋊C₈	—	2	1	no	—
373	#97	—	2	2	no	—
374	#97	—	2	2	no	—
375	C₄×C₈	—	2	2	no	—
376	C₄×C₈	—	2	2	no	—
377	C₃₁	#33122	1	1	yes	characteristic, direct product factor, Sylow
378	C₂⁴	—	2	1	no	—
379	C₂⁴	#41×C₃₁	1	1	yes	characteristic
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	4	1	no	—
389	C₂×D₄	—	4	1	no	—
390	C₂×D₄	—	8	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
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	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
404	SD₁₆	—	4	2	no	—
405	SD₁₆	—	4	2	no	—
406	M₁₆	—	2	4	no	—
407	M₁₆	—	2	2	no	—
408	M₁₆	—	2	2	no	—
409	M₁₆	—	2	2	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	#41×C₃₁	1	4	yes	—
413	C₂×C₈	—	2	4	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂×C₈	#41×C₃₁	1	4	yes	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	4	1	no	—
434	C₄⋊C₄	—	4	1	no	—
435	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
436	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	4	1	no	—
440	C₂²⋊C₄	—	4	1	no	—
441	C₂²⋊C₄	—	4	1	no	—
442	C₂²⋊C₄	—	4	1	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	4	1	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	8	1	no	—
451	C₂³	—	8	1	no	—
452	C₂³	C₃₁×#121	1	1	yes	characteristic
453	C₂³	C₃₁×#121	1	1	yes	characteristic
454	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
455	Q₈	—	4	1	no	—
456	Q₈	—	4	1	no	—
457	Q₈	—	4	2	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	2	no	—
461	D₄	—	8	2	no	—
462	D₄	—	8	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	8	1	no	—
479	C₂×C₄	—	8	1	no	—
480	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
481	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
485	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
486	C₈	—	2	1	no	—
487	C₈	—	2	1	no	—
488	C₈	—	2	4	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	2	no	—
491	C₈	C₃₁×#123	1	2	yes	—
492	C₈	C₃₁×#123	1	2	yes	—
493	C₂²	—	2	1	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	8	1	no	—
502	C₂²	—	8	1	no	—
503	C₂²	—	8	1	no	—
504	C₂²	—	16	2	no	—
505	C₂²	C₃₁×#471	1	2	yes	—
506	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
507	C₂²	C₃₁×#529	1	2	yes	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	8	1	no	—
513	C₄	—	8	1	no	—
514	C₄	—	8	1	no	—
515	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
516	C₄	C₃₁×#525	1	1	yes	characteristic
517	C₄	C₃₁×#532	1	2	yes	—
518	C₂	—	2	2	no	—
519	C₂	—	2	2	no	—
520	C₂	—	4	1	no	—
521	C₂	—	16	1	no	—
522	C₂	C₃₁×#2854	1	1	yes	characteristic
523	C₂	C₂×C₃₁×#471	1	1	yes	characteristic
524	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	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. C₃₁×#3095	C₂	semidirect
4. C₂×C₃₁×#405	C₂	semidirect
5. C₃₁×#1977	C₂	semidirect
6. C₃₁×#1951	C₂	semidirect
7. C₃₁×#1931	C₂	semidirect
8. C₃₁×#1930	C₂	semidirect
9. C₃₁×#1901	C₂	semidirect
11. C₃₁×#1810	C₂	semidirect
182. #33122	C₃₁	direct (N × H)
231. #3095	C₂×C₃₁	semidirect
234. C₂×#405	C₂×C₃₁	semidirect
235. #1977	C₂×C₃₁	semidirect
236. #1951	C₂×C₃₁	semidirect
237. #1931	C₂×C₃₁	semidirect
238. #1930	C₂×C₃₁	semidirect
239. #1901	C₂×C₃₁	semidirect
241. #1810	C₂×C₃₁	semidirect
377. C₃₁	#33122	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3095	2
C₃₁×#3094	2
C₂×C₃₁×#465	2
C₂×C₃₁×#405	2
C₃₁×#1977	2
C₃₁×#1951	2
C₃₁×#1931	2
C₃₁×#1930	2
C₃₁×#1901	2
C₃₁×#1900	2
C₃₁×#1810	2
#33122	31

Subgroups — C31×#33122 #21489881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33122 #21489881