Subgroups — C₃₁×#61009 #21517768

Order 7936.

1696 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3512	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3261	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3260	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3259	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3232	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3212	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3204	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3203	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3183	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3168	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3120	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3119	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#553	—	2	4	no	—
15	C₃₁×#553	—	2	4	no	—
16	C₃₁×#553	—	2	4	no	—
17	C₃₁×#552	—	2	4	no	—
18	C₃₁×#541	C₂²	1	2	yes	—
19	C₃₁×#541	C₂²	1	2	yes	—
20	C₃₁×#540	C₂²	1	2	yes	—
21	C₃₁×#540	C₂²	1	2	yes	—
22	C₃₁×#537	C₂²	1	4	yes	—
23	C₃₁×#532	C₂²	1	2	yes	—
24	C₃₁×#529	C₂²	1	4	yes	—
25	C₃₁×#529	C₂²	1	4	yes	—
26	C₂×C₃₁×#125	C₂²	1	2	yes	—
27	C₃₁×#520	C₂²	1	4	yes	—
28	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#499	C₂²	1	2	yes	—
30	C₃₁×#499	C₂²	1	2	yes	—
31	C₃₁×#495	C₂²	1	2	yes	—
32	C₃₁×#495	C₂²	1	2	yes	—
33	C₃₁×#488	C₂²	1	4	yes	—
34	C₃₁×#488	C₂²	1	4	yes	—
35	C₃₁×#484	C₂²	1	4	yes	—
36	C₃₁×#483	C₂²	1	4	yes	—
37	C₃₁×#481	C₂²	1	4	yes	—
38	C₃₁×#480	C₂²	1	4	yes	—
39	C₃₁×#479	C₂²	1	4	yes	—
40	C₃₁×#479	C₂²	1	4	yes	—
41	C₃₁×#478	C₂²	1	4	yes	—
42	C₃₁×#478	C₂²	1	4	yes	—
43	C₃₁×#470	C₂²	1	2	yes	—
44	C₃₁×#469	C₂²	1	2	yes	—
45	C₃₁×#468	C₂²	1	4	yes	—
46	C₃₁×#467	C₂²	1	2	yes	—
47	C₃₁×#466	C₂²	1	2	yes	—
48	C₃₁×#465	C₂²	1	4	yes	—
49	C₃₁×#462	C₂²	1	4	yes	—
50	C₃₁×#461	C₂²	1	4	yes	—
51	C₃₁×#460	C₂²	1	4	yes	—
52	C₃₁×#459	C₂²	1	4	yes	—
53	C₃₁×#452	C₂²	1	4	yes	—
54	C₃₁×#452	C₂²	1	4	yes	—
55	C₃₁×#449	C₂²	1	4	yes	—
56	C₃₁×#449	C₂²	1	4	yes	—
57	C₃₁×#442	C₂²	1	2	yes	—
58	C₃₁×#441	C₂²	1	2	yes	—
59	C₃₁×#440	C₂²	1	4	yes	—
60	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
61	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
62	C₄×D₈×C₃₁	C₂²	1	2	yes	—
63	C₃₁×#433	C₂²	1	4	yes	—
64	C₃₁×#431	C₂²	1	4	yes	—
65	C₃₁×#424	C₂²	1	1	yes	characteristic
66	C₃₁×#424	C₂²	1	1	yes	characteristic
67	C₃₁×#408	C₂²	1	1	yes	characteristic
68	C₃₁×#408	C₂²	1	1	yes	characteristic
69	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
70	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
71	C₂×#41×C₃₁	C₂³	1	4	yes	—
72	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
74	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
75	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
77	C₂×D₈×C₃₁	C₂³	1	2	yes	—
78	C₂×D₈×C₃₁	C₂³	1	2	yes	—
79	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₃₁×#127	C₂³	1	2	yes	—
85	C₃₁×#126	—	2	4	no	—
86	C₃₁×#126	—	2	4	no	—
87	C₃₁×#126	—	2	4	no	—
88	C₃₁×#126	—	2	4	no	—
89	C₃₁×#125	—	2	4	no	—
90	C₃₁×#125	—	2	4	no	—
91	C₃₁×#125	—	2	4	no	—
92	C₃₁×#125	C₂³	1	2	yes	—
93	C₃₁×#125	C₂³	1	2	yes	—
94	C₃₁×#125	C₂³	1	4	yes	—
95	C₃₁×#124	C₂³	1	4	yes	—
96	C₃₁×#123	—	2	4	no	—
97	C₃₁×#123	—	2	4	no	—
98	C₃₁×#123	—	2	4	no	—
99	C₃₁×#123	—	2	4	no	—
100	C₃₁×C₂²⋊Q₈	—	2	4	no	—
101	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
102	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
103	C₃₁×#121	—	2	4	no	—
104	C₃₁×#121	—	2	4	no	—
105	C₃₁×#121	—	2	4	no	—
106	C₃₁×#121	—	2	4	no	—
107	C₃₁×#121	—	2	4	no	—
108	C₃₁×#121	—	2	4	no	—
109	C₃₁×#121	C₂³	1	4	yes	—
110	C₃₁×#121	C₂³	1	4	yes	—
111	C₄×Q₈×C₃₁	D₄	1	2	yes	—
112	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
113	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
114	C₄×D₄×C₃₁	—	2	4	no	—
115	C₄×D₄×C₃₁	—	2	4	no	—
116	C₄×D₄×C₃₁	—	2	4	no	—
117	C₄×D₄×C₃₁	—	2	4	no	—
118	C₄×D₄×C₃₁	—	2	4	no	—
119	C₄×D₄×C₃₁	D₄	1	2	yes	—
120	C₄×D₄×C₃₁	D₄	1	4	yes	—
121	C₄×D₄×C₃₁	C₂³	1	4	yes	—
122	C₄×D₄×C₃₁	C₂³	1	4	yes	—
123	C₃₁×#117	—	2	4	no	—
124	C₃₁×#117	—	2	4	no	—
125	C₃₁×#117	—	2	4	no	—
126	C₃₁×#117	C₂³	1	1	yes	characteristic
127	C₃₁×#117	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
133	C₃₁×#107	C₂³	1	2	yes	—
134	C₃₁×#107	C₂³	1	2	yes	—
135	C₃₁×#107	C₂³	1	4	yes	—
136	C₃₁×#106	C₂³	1	4	yes	—
137	C₃₁×#106	C₂³	1	2	yes	—
138	C₃₁×#106	C₂³	1	2	yes	—
139	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
140	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
141	C₃₁×#103	C₂³	1	4	yes	—
142	C₃₁×#103	C₂³	1	4	yes	—
143	C₃₁×#103	C₂³	1	4	yes	—
144	C₃₁×#103	C₂³	1	4	yes	—
145	C₃₁×#102	C₂³	1	4	yes	—
146	C₃₁×#102	C₂³	1	4	yes	—
147	C₃₁×#102	C₂³	1	4	yes	—
148	C₃₁×#102	C₂³	1	4	yes	—
149	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
150	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
151	C₃₁×#97	C₂³	1	2	yes	—
152	C₄×C₈×C₃₁	C₂³	1	2	yes	—
153	C₄×C₈×C₃₁	C₂³	1	2	yes	—
154	C₄×C₈×C₃₁	C₂³	1	2	yes	—
155	#41×C₃₁	—	4	1	no	—
156	#41×C₃₁	—	4	1	no	—
157	#41×C₃₁	—	4	4	no	—
158	#41×C₃₁	—	4	4	no	—
159	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	2	4	no	—
163	C₂×D₄×C₃₁	—	2	4	no	—
164	C₂×D₄×C₃₁	—	2	4	no	—
165	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₃₁	—	2	4	no	—
169	C₂²×C₄×C₃₁	—	2	4	no	—
170	C₂²×C₄×C₃₁	—	2	4	no	—
171	C₂²×C₄×C₃₁	—	2	4	no	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₃₁	—	4	2	no	—
177	Q₁₆×C₃₁	—	4	2	no	—
178	SD₁₆×C₃₁	—	4	4	no	—
179	SD₁₆×C₃₁	—	4	4	no	—
180	D₈×C₃₁	—	4	2	no	—
181	D₈×C₃₁	—	4	2	no	—
182	M₁₆×C₃₁	—	4	1	no	—
183	M₁₆×C₃₁	—	4	1	no	—
184	C₂×C₈×C₃₁	—	4	1	no	—
185	C₂×C₈×C₃₁	—	4	1	no	—
186	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
187	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
188	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
189	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
190	C₄⋊C₄×C₃₁	—	2	4	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	—	2	4	no	—
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	4	no	—
199	C₄⋊C₄×C₃₁	—	2	4	no	—
200	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
201	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
202	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
203	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₂²⋊C₄×C₃₁	—	2	4	no	—
206	C₂²⋊C₄×C₃₁	—	2	4	no	—
207	C₂²⋊C₄×C₃₁	—	2	4	no	—
208	C₂²⋊C₄×C₃₁	—	2	4	no	—
209	C₂²⋊C₄×C₃₁	—	2	4	no	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₂²⋊C₄×C₃₁	—	2	4	no	—
212	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
213	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
214	C₄²×C₃₁	—	2	4	no	—
215	C₄²×C₃₁	—	2	4	no	—
216	C₄²×C₃₁	C₂×D₄	1	2	yes	—
217	C₄²×C₃₁	C₂×D₄	1	4	yes	—
218	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
219	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
220	C₄²×C₃₁	C₂⁴	1	2	yes	—
221	#61009	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂³×C₃₁	—	2	4	no	—
223	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
225	Q₈×C₃₁	—	2	4	no	—
226	Q₈×C₃₁	—	2	4	no	—
227	Q₈×C₃₁	—	4	1	no	—
228	D₄×C₃₁	—	2	4	no	—
229	D₄×C₃₁	—	2	4	no	—
230	D₄×C₃₁	—	4	2	no	—
231	D₄×C₃₁	—	4	1	no	—
232	D₄×C₃₁	—	4	8	no	—
233	D₄×C₃₁	—	8	4	no	—
234	D₄×C₃₁	—	8	4	no	—
235	D₄×C₃₁	—	8	4	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	4	no	—
245	C₂×C₄×C₃₁	—	4	8	no	—
246	C₂×C₄×C₃₁	—	8	4	no	—
247	C₂×C₄×C₃₁	—	8	4	no	—
248	C₂×C₄×C₃₁	—	8	4	no	—
249	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
255	C₈×C₃₁	—	2	2	no	—
256	C₈×C₃₁	—	2	2	no	—
257	C₈×C₃₁	—	2	2	no	—
258	C₈×C₃₁	—	2	2	no	—
259	#3512	C₂×C₃₁	1	2	yes	complemented
260	#3261	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#3260	C₂×C₃₁	1	2	yes	complemented
262	#3259	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#3232	C₂×C₃₁	1	4	yes	complemented
264	#3212	C₂×C₃₁	1	4	yes	complemented
265	#3204	C₂×C₃₁	1	2	yes	complemented
266	#3203	C₂×C₃₁	1	2	yes	complemented
267	#3183	C₂×C₃₁	1	2	yes	complemented
268	#3168	C₂×C₃₁	1	2	yes	complemented
269	#3120	C₂×C₃₁	1	4	yes	complemented
270	#3119	C₂×C₃₁	1	4	yes	complemented
271	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂²×C₃₁	—	2	4	no	—
273	C₂²×C₃₁	—	2	2	no	—
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	1	no	—
276	C₂²×C₃₁	—	4	4	no	—
277	C₂²×C₃₁	—	4	4	no	—
278	C₂²×C₃₁	—	4	4	no	—
279	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	2	2	no	—
282	C₄×C₃₁	—	2	4	no	—
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	2	1	no	—
285	C₄×C₃₁	—	4	4	no	—
286	C₄×C₃₁	—	4	4	no	—
287	C₄×C₃₁	—	4	4	no	—
288	C₄×C₃₁	#585	1	1	yes	characteristic
289	C₄×C₃₁	#585	1	1	yes	characteristic
290	#553	—	2	4	no	—
291	#553	—	2	4	no	—
292	#553	—	2	4	no	—
293	#552	—	2	4	no	—
294	#541	C₂²×C₃₁	1	2	yes	—
295	#541	C₂²×C₃₁	1	2	yes	—
296	#540	C₂²×C₃₁	1	2	yes	—
297	#540	C₂²×C₃₁	1	2	yes	—
298	#537	C₂²×C₃₁	1	4	yes	—
299	#532	C₂²×C₃₁	1	2	yes	—
300	#529	C₂²×C₃₁	1	4	yes	—
301	#529	C₂²×C₃₁	1	4	yes	—
302	C₂×#125	C₂²×C₃₁	1	2	yes	—
303	#520	C₂²×C₃₁	1	4	yes	—
304	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
305	#499	C₂²×C₃₁	1	2	yes	—
306	#499	C₂²×C₃₁	1	2	yes	—
307	#495	C₂²×C₃₁	1	2	yes	—
308	#495	C₂²×C₃₁	1	2	yes	—
309	#488	C₂²×C₃₁	1	4	yes	—
310	#488	C₂²×C₃₁	1	4	yes	—
311	#484	C₂²×C₃₁	1	4	yes	—
312	#483	C₂²×C₃₁	1	4	yes	—
313	#481	C₂²×C₃₁	1	4	yes	—
314	#480	C₂²×C₃₁	1	4	yes	—
315	#479	C₂²×C₃₁	1	4	yes	—
316	#479	C₂²×C₃₁	1	4	yes	—
317	#478	C₂²×C₃₁	1	4	yes	—
318	#478	C₂²×C₃₁	1	4	yes	—
319	#470	C₂²×C₃₁	1	2	yes	—
320	#469	C₂²×C₃₁	1	2	yes	—
321	#468	C₂²×C₃₁	1	4	yes	—
322	#467	C₂²×C₃₁	1	2	yes	—
323	#466	C₂²×C₃₁	1	2	yes	—
324	#465	C₂²×C₃₁	1	4	yes	—
325	#462	C₂²×C₃₁	1	4	yes	—
326	#461	C₂²×C₃₁	1	4	yes	—
327	#460	C₂²×C₃₁	1	4	yes	—
328	#459	C₂²×C₃₁	1	4	yes	—
329	#452	C₂²×C₃₁	1	4	yes	—
330	#452	C₂²×C₃₁	1	4	yes	—
331	#449	C₂²×C₃₁	1	4	yes	—
332	#449	C₂²×C₃₁	1	4	yes	—
333	#442	C₂²×C₃₁	1	2	yes	—
334	#441	C₂²×C₃₁	1	2	yes	—
335	#440	C₂²×C₃₁	1	4	yes	—
336	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
337	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
338	C₄×D₈	C₂²×C₃₁	1	2	yes	—
339	#433	C₂²×C₃₁	1	4	yes	—
340	#431	C₂²×C₃₁	1	4	yes	—
341	#424	C₂²×C₃₁	1	1	yes	characteristic
342	#424	C₂²×C₃₁	1	1	yes	characteristic
343	#408	C₂²×C₃₁	1	1	yes	characteristic
344	#408	C₂²×C₃₁	1	1	yes	characteristic
345	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
346	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
347	C₂×C₃₁	—	4	2	no	—
348	C₂×C₃₁	—	4	1	no	—
349	C₂×C₃₁	—	8	4	no	—
350	C₂×C₃₁	#3449	1	1	yes	characteristic
351	C₂×C₃₁	#3584	1	2	yes	—
352	C₂×#41	C₂³×C₃₁	1	4	yes	—
353	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
355	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
356	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
357	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
358	C₂×D₈	C₂³×C₃₁	1	2	yes	—
359	C₂×D₈	C₂³×C₃₁	1	2	yes	—
360	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
361	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
365	#127	C₂³×C₃₁	1	2	yes	—
366	#126	—	2	4	no	—
367	#126	—	2	4	no	—
368	#126	—	2	4	no	—
369	#126	—	2	4	no	—
370	#125	—	2	4	no	—
371	#125	—	2	4	no	—
372	#125	—	2	4	no	—
373	#125	C₂³×C₃₁	1	4	yes	—
374	#125	C₂³×C₃₁	1	2	yes	—
375	#125	C₂³×C₃₁	1	2	yes	—
376	#124	C₂³×C₃₁	1	4	yes	—
377	#123	—	2	4	no	—
378	#123	—	2	4	no	—
379	#123	—	2	4	no	—
380	#123	—	2	4	no	—
381	C₂²⋊Q₈	—	2	4	no	—
382	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
383	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
384	#121	—	2	4	no	—
385	#121	—	2	4	no	—
386	#121	—	2	4	no	—
387	#121	—	2	4	no	—
388	#121	—	2	4	no	—
389	#121	—	2	4	no	—
390	#121	C₂³×C₃₁	1	4	yes	—
391	#121	C₂³×C₃₁	1	4	yes	—
392	C₄×Q₈	D₄×C₃₁	1	2	yes	—
393	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
394	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	—	2	4	no	—
400	C₄×D₄	D₄×C₃₁	1	2	yes	—
401	C₄×D₄	D₄×C₃₁	1	4	yes	—
402	C₄×D₄	C₂³×C₃₁	1	4	yes	—
403	C₄×D₄	C₂³×C₃₁	1	4	yes	—
404	#117	—	2	4	no	—
405	#117	—	2	4	no	—
406	#117	—	2	4	no	—
407	#117	C₂³×C₃₁	1	2	yes	—
408	#117	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	—	2	4	no	—
410	C₂×C₄⋊C₄	—	2	4	no	—
411	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
412	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
413	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
414	#107	C₂³×C₃₁	1	2	yes	—
415	#107	C₂³×C₃₁	1	2	yes	—
416	#107	C₂³×C₃₁	1	4	yes	—
417	#106	C₂³×C₃₁	1	4	yes	—
418	#106	C₂³×C₃₁	1	2	yes	—
419	#106	C₂³×C₃₁	1	2	yes	—
420	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
421	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
422	#103	C₂³×C₃₁	1	4	yes	—
423	#103	C₂³×C₃₁	1	4	yes	—
424	#103	C₂³×C₃₁	1	4	yes	—
425	#103	C₂³×C₃₁	1	4	yes	—
426	#102	C₂³×C₃₁	1	4	yes	—
427	#102	C₂³×C₃₁	1	4	yes	—
428	#102	C₂³×C₃₁	1	4	yes	—
429	#102	C₂³×C₃₁	1	4	yes	—
430	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
431	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
432	#97	C₂³×C₃₁	1	2	yes	—
433	C₄×C₈	C₂³×C₃₁	1	2	yes	—
434	C₄×C₈	C₂³×C₃₁	1	2	yes	—
435	C₄×C₈	C₂³×C₃₁	1	2	yes	—
436	C₃₁	#61009	1	1	yes	characteristic, direct product factor, Sylow
437	#41	—	4	1	no	—
438	#41	—	4	1	no	—
439	#41	—	4	4	no	—
440	#41	—	4	4	no	—
441	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
449	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
455	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
458	Q₁₆	—	4	2	no	—
459	Q₁₆	—	4	2	no	—
460	SD₁₆	—	4	4	no	—
461	SD₁₆	—	4	4	no	—
462	D₈	—	4	2	no	—
463	D₈	—	4	2	no	—
464	M₁₆	—	4	1	no	—
465	M₁₆	—	4	1	no	—
466	C₂×C₈	—	4	1	no	—
467	C₂×C₈	—	4	1	no	—
468	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
469	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
470	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
471	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
483	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
484	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
485	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
495	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
496	C₄²	—	2	4	no	—
497	C₄²	—	2	4	no	—
498	C₄²	C₂×D₄×C₃₁	1	2	yes	—
499	C₄²	C₂×D₄×C₃₁	1	4	yes	—
500	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
501	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
502	C₄²	C₂⁴×C₃₁	1	2	yes	—
503	C₂³	—	2	4	no	—
504	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
505	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
506	Q₈	—	2	4	no	—
507	Q₈	—	2	4	no	—
508	Q₈	—	4	1	no	—
509	D₄	—	2	4	no	—
510	D₄	—	2	4	no	—
511	D₄	—	4	2	no	—
512	D₄	—	4	1	no	—
513	D₄	—	4	8	no	—
514	D₄	—	8	4	no	—
515	D₄	—	8	4	no	—
516	D₄	—	8	4	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	4	8	no	—
527	C₂×C₄	—	8	4	no	—
528	C₂×C₄	—	8	4	no	—
529	C₂×C₄	—	8	4	no	—
530	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
532	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
534	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
535	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
536	C₈	—	2	2	no	—
537	C₈	—	2	2	no	—
538	C₈	—	2	2	no	—
539	C₈	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	4	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	4	no	—
547	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	2	no	—
551	C₄	—	2	2	no	—
552	C₄	—	2	4	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	4	no	—
556	C₄	C₃₁×#585	1	1	yes	characteristic
557	C₄	C₃₁×#585	1	1	yes	characteristic
558	C₂	—	4	2	no	—
559	C₂	—	4	1	no	—
560	C₂	—	8	4	no	—
561	C₂	C₃₁×#3449	1	1	yes	characteristic
562	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3512	C₂	semidirect
2. C₃₁×#3261	C₂	semidirect
3. C₃₁×#3260	C₂	semidirect
4. C₃₁×#3259	C₂	semidirect
5. C₃₁×#3232	C₂	semidirect
6. C₃₁×#3212	C₂	semidirect
7. C₃₁×#3204	C₂	semidirect
8. C₃₁×#3203	C₂	semidirect
9. C₃₁×#3183	C₂	semidirect
10. C₃₁×#3168	C₂	semidirect
11. C₃₁×#3120	C₂	semidirect
12. C₃₁×#3119	C₂	semidirect
13. C₃₁×#2922	C₂	semidirect
221. #61009	C₃₁	direct (N × H)
259. #3512	C₂×C₃₁	semidirect
260. #3261	C₂×C₃₁	semidirect
261. #3260	C₂×C₃₁	semidirect
262. #3259	C₂×C₃₁	semidirect
263. #3232	C₂×C₃₁	semidirect
264. #3212	C₂×C₃₁	semidirect
265. #3204	C₂×C₃₁	semidirect
266. #3203	C₂×C₃₁	semidirect
267. #3183	C₂×C₃₁	semidirect
268. #3168	C₂×C₃₁	semidirect
269. #3120	C₂×C₃₁	semidirect
270. #3119	C₂×C₃₁	semidirect
271. #2922	C₂×C₃₁	semidirect
436. C₃₁	#61009	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×#3512	2
C₃₁×#3261	2
C₃₁×#3260	2
C₃₁×#3259	2
C₃₁×#3232	2
C₃₁×#3212	2
C₃₁×#3204	2
C₃₁×#3203	2
C₃₁×#3183	2
C₃₁×#3168	2
C₃₁×#3120	2
C₃₁×#3119	2
C₃₁×#2922	2
#61009	31

Subgroups — C31×#61009 #21517768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61009 #21517768