Subgroups — C₃₁×#61401 #21518160

Order 7936.

1938 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3480	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3403	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3402	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3401	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3400	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3399	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3398	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3397	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3150	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3035	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3035	C₂	1	2	yes	maximal, complemented
13	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2943	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#135	C₂²	1	2	yes	—
19	C₂×C₃₁×#135	C₂²	1	4	yes	—
20	C₃₁×#559	C₂²	1	2	yes	—
21	C₃₁×#559	C₂²	1	4	yes	—
22	C₃₁×#557	C₂²	1	2	yes	—
23	C₃₁×#556	C₂²	1	2	yes	—
24	C₃₁×#556	C₂²	1	4	yes	—
25	C₃₁×#553	—	2	2	no	—
26	C₃₁×#553	—	2	4	no	—
27	C₃₁×#550	C₂²	1	2	yes	—
28	C₃₁×#548	—	2	2	no	—
29	C₃₁×#532	C₂²	1	2	yes	—
30	C₃₁×#532	C₂²	1	4	yes	—
31	C₃₁×#529	—	2	2	no	—
32	C₃₁×#529	—	2	4	no	—
33	C₄×#41×C₃₁	—	2	2	no	—
34	C₄×#41×C₃₁	C₂²	1	2	yes	—
35	C₄×#41×C₃₁	C₂²	1	4	yes	—
36	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#497	C₂²	1	4	yes	—
38	C₃₁×#497	C₂²	1	4	yes	—
39	C₃₁×#496	C₂²	1	4	yes	—
40	C₃₁×#496	C₂²	1	4	yes	—
41	C₃₁×#495	C₂²	1	2	yes	—
42	C₃₁×#495	C₂²	1	2	yes	—
43	C₃₁×#494	C₂²	1	1	yes	characteristic
44	C₃₁×#493	C₂²	1	1	yes	characteristic
45	C₃₁×#492	C₂²	1	2	yes	—
46	C₃₁×#464	C₂²	1	4	yes	—
47	C₃₁×#464	C₂²	1	4	yes	—
48	C₃₁×#464	C₂²	1	4	yes	—
49	C₃₁×#463	C₂²	1	4	yes	—
50	C₃₁×#463	C₂²	1	4	yes	—
51	C₃₁×#463	C₂²	1	4	yes	—
52	C₃₁×#462	C₂²	1	2	yes	—
53	C₃₁×#462	C₂²	1	4	yes	—
54	C₃₁×#461	C₂²	1	2	yes	—
55	C₃₁×#461	C₂²	1	4	yes	—
56	C₃₁×#460	C₂²	1	4	yes	—
57	C₃₁×#460	C₂²	1	2	yes	—
58	C₃₁×#459	C₂²	1	2	yes	—
59	C₃₁×#459	C₂²	1	4	yes	—
60	C₃₁×#446	C₂²	1	2	yes	—
61	C₃₁×#446	C₂²	1	2	yes	—
62	C₃₁×#446	C₂²	1	2	yes	—
63	C₃₁×#446	C₂²	1	2	yes	—
64	C₃₁×#442	C₂²	1	4	yes	—
65	C₃₁×#441	C₂²	1	4	yes	—
66	C₃₁×#440	C₂²	1	4	yes	—
67	C₃₁×#440	C₂²	1	4	yes	—
68	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
69	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
70	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
71	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
72	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
73	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
74	C₃₁×#427	C₂²	1	1	yes	characteristic
75	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
76	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
77	C₃₁×#416	C₂²	1	2	yes	—
78	C₃₁×#416	C₂²	1	4	yes	—
79	C₂×C₃₁×#97	C₂²	1	2	yes	—
80	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
81	C₂×#41×C₃₁	—	2	2	no	—
82	C₂×#41×C₃₁	C₂³	1	2	yes	—
83	C₂×#41×C₃₁	C₂³	1	4	yes	—
84	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
85	C₃₁×#135	—	2	8	no	—
86	C₃₁×#135	—	2	4	no	—
87	C₃₁×#135	—	4	4	no	—
88	C₃₁×#135	—	4	8	no	—
89	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
91	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
92	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
93	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
95	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
96	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
97	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
98	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₂×D₈×C₃₁	C₂³	1	2	yes	—
100	C₂×D₈×C₃₁	C₂³	1	2	yes	—
101	C₂×D₈×C₃₁	C₂³	1	4	yes	—
102	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
104	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
105	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
106	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
107	C₃₁×#127	C₂³	1	2	yes	—
108	C₃₁×#127	C₂³	1	4	yes	—
109	C₃₁×#126	—	2	4	no	—
110	C₃₁×#126	—	2	4	no	—
111	C₃₁×#126	—	2	4	no	—
112	C₃₁×#125	—	2	2	no	—
113	C₃₁×#125	—	2	4	no	—
114	C₃₁×#124	—	2	2	no	—
115	C₃₁×#124	—	2	4	no	—
116	C₃₁×#124	C₂³	1	4	yes	—
117	C₃₁×#124	C₂³	1	4	yes	—
118	C₃₁×#124	C₂³	1	4	yes	—
119	C₃₁×#123	—	2	4	no	—
120	C₃₁×#123	—	2	4	no	—
121	C₃₁×#123	—	2	4	no	—
122	C₃₁×C₂²⋊Q₈	—	2	2	no	—
123	C₃₁×C₂²⋊Q₈	—	2	4	no	—
124	C₃₁×#121	—	2	2	no	—
125	C₃₁×#121	—	2	4	no	—
126	C₃₁×#121	—	2	2	no	—
127	C₃₁×#121	—	2	4	no	—
128	C₃₁×#121	—	2	2	no	—
129	C₃₁×#121	—	2	4	no	—
130	C₄×Q₈×C₃₁	—	2	2	no	—
131	C₄×Q₈×C₃₁	D₄	1	2	yes	—
132	C₄×Q₈×C₃₁	D₄	1	2	yes	—
133	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
134	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
135	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
136	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
137	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
138	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
139	C₄×D₄×C₃₁	—	2	2	no	—
140	C₄×D₄×C₃₁	—	2	4	no	—
141	C₄×D₄×C₃₁	—	2	2	no	—
142	C₄×D₄×C₃₁	—	2	4	no	—
143	C₄×D₄×C₃₁	—	2	2	no	—
144	C₄×D₄×C₃₁	—	2	4	no	—
145	C₄×D₄×C₃₁	—	2	2	no	—
146	C₄×D₄×C₃₁	—	2	4	no	—
147	C₄×D₄×C₃₁	—	2	4	no	—
148	C₄×D₄×C₃₁	—	2	2	no	—
149	C₄×D₄×C₃₁	C₂³	1	2	yes	—
150	C₄×D₄×C₃₁	C₂³	1	2	yes	—
151	C₄×D₄×C₃₁	C₂³	1	4	yes	—
152	C₄×D₄×C₃₁	C₂³	1	4	yes	—
153	C₃₁×#117	—	2	2	no	—
154	C₃₁×#117	—	2	4	no	—
155	C₃₁×#117	—	2	4	no	—
156	C₃₁×#117	—	2	2	no	—
157	C₃₁×#117	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
159	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
160	C₂×C₄²×C₃₁	—	2	2	no	—
161	C₂×C₄²×C₃₁	—	2	4	no	—
162	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
163	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
164	C₃₁×#107	C₂³	1	1	yes	characteristic
165	C₃₁×#107	C₂³	1	1	yes	characteristic
166	C₃₁×#106	C₂³	1	2	yes	—
167	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
168	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
169	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
170	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
171	C₃₁×#103	C₂³	1	2	yes	—
172	C₃₁×#103	C₂³	1	2	yes	—
173	C₃₁×#103	C₂³	1	4	yes	—
174	C₃₁×#103	C₂³	1	4	yes	—
175	C₃₁×#102	C₂³	1	2	yes	—
176	C₃₁×#102	C₂³	1	2	yes	—
177	C₃₁×#102	C₂³	1	4	yes	—
178	C₃₁×#102	C₂³	1	4	yes	—
179	C₃₁×#97	C₂³	1	4	yes	—
180	C₃₁×#97	C₂³	1	4	yes	—
181	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
182	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
183	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
184	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
185	#41×C₃₁	—	2	8	no	—
186	#41×C₃₁	—	2	4	no	—
187	#41×C₃₁	—	2	8	no	—
188	#41×C₃₁	—	2	4	no	—
189	#41×C₃₁	—	4	8	no	—
190	C₂×Q₈×C₃₁	—	4	2	no	—
191	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
192	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
193	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
194	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
195	C₂×D₄×C₃₁	—	2	2	no	—
196	C₂×D₄×C₃₁	—	2	4	no	—
197	C₂×D₄×C₃₁	—	2	4	no	—
198	C₂×D₄×C₃₁	—	2	2	no	—
199	C₂×D₄×C₃₁	—	2	4	no	—
200	C₂×D₄×C₃₁	—	2	2	no	—
201	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
202	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
203	C₂²×C₄×C₃₁	—	2	4	no	—
204	C₂²×C₄×C₃₁	—	2	2	no	—
205	C₂²×C₄×C₃₁	—	2	2	no	—
206	C₂²×C₄×C₃₁	—	2	4	no	—
207	C₂²×C₄×C₃₁	—	2	2	no	—
208	C₂²×C₄×C₃₁	—	2	4	no	—
209	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
212	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
213	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
214	Q₁₆×C₃₁	—	2	4	no	—
215	Q₁₆×C₃₁	—	2	2	no	—
216	Q₁₆×C₃₁	—	4	2	no	—
217	Q₁₆×C₃₁	—	4	4	no	—
218	SD₁₆×C₃₁	—	2	4	no	—
219	SD₁₆×C₃₁	—	2	8	no	—
220	SD₁₆×C₃₁	—	4	4	no	—
221	SD₁₆×C₃₁	—	4	4	no	—
222	SD₁₆×C₃₁	—	4	4	no	—
223	D₈×C₃₁	—	2	4	no	—
224	D₈×C₃₁	—	2	2	no	—
225	D₈×C₃₁	—	4	2	no	—
226	D₈×C₃₁	—	4	4	no	—
227	C₂×C₈×C₃₁	—	2	2	no	—
228	C₂×C₈×C₃₁	—	2	4	no	—
229	C₂×C₈×C₃₁	#41	1	4	yes	complemented
230	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
231	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
232	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
233	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
234	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
235	C₄⋊C₄×C₃₁	—	2	4	no	—
236	C₄⋊C₄×C₃₁	—	2	2	no	—
237	C₄⋊C₄×C₃₁	—	2	4	no	—
238	C₄⋊C₄×C₃₁	—	2	2	no	—
239	C₄⋊C₄×C₃₁	—	2	4	no	—
240	C₄⋊C₄×C₃₁	—	2	2	no	—
241	C₄⋊C₄×C₃₁	—	2	4	no	—
242	C₄⋊C₄×C₃₁	—	2	2	no	—
243	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
245	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
246	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
247	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
248	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
249	C₂²⋊C₄×C₃₁	—	2	2	no	—
250	C₂²⋊C₄×C₃₁	—	2	4	no	—
251	C₂²⋊C₄×C₃₁	—	2	4	no	—
252	C₂²⋊C₄×C₃₁	—	2	4	no	—
253	C₂²⋊C₄×C₃₁	—	2	4	no	—
254	C₂²⋊C₄×C₃₁	—	2	2	no	—
255	C₂²⋊C₄×C₃₁	—	2	4	no	—
256	C₂²⋊C₄×C₃₁	—	2	2	no	—
257	C₂²⋊C₄×C₃₁	—	2	4	no	—
258	C₂²⋊C₄×C₃₁	—	2	4	no	—
259	C₂²⋊C₄×C₃₁	—	2	4	no	—
260	C₄²×C₃₁	—	2	2	no	—
261	C₄²×C₃₁	—	2	4	no	—
262	C₄²×C₃₁	—	2	4	no	—
263	C₄²×C₃₁	—	2	2	no	—
264	C₄²×C₃₁	—	2	2	no	—
265	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
266	C₄²×C₃₁	C₂×D₄	1	4	yes	—
267	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
268	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
269	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
270	C₄²×C₃₁	C₂⁴	1	4	yes	—
271	#61401	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂³×C₃₁	—	2	4	no	—
273	C₂³×C₃₁	—	2	2	no	—
274	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
275	Q₈×C₃₁	—	2	4	no	—
276	Q₈×C₃₁	—	2	2	no	—
277	Q₈×C₃₁	—	2	4	no	—
278	Q₈×C₃₁	—	2	2	no	—
279	Q₈×C₃₁	—	2	4	no	—
280	Q₈×C₃₁	—	4	2	no	—
281	D₄×C₃₁	—	2	4	no	—
282	D₄×C₃₁	—	2	2	no	—
283	D₄×C₃₁	—	2	4	no	—
284	D₄×C₃₁	—	2	2	no	—
285	D₄×C₃₁	—	4	8	no	—
286	D₄×C₃₁	—	4	4	no	—
287	D₄×C₃₁	—	4	8	no	—
288	D₄×C₃₁	—	4	4	no	—
289	D₄×C₃₁	—	8	4	no	—
290	D₄×C₃₁	—	8	2	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	2	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	2	4	no	—
296	C₂×C₄×C₃₁	—	4	1	no	—
297	C₂×C₄×C₃₁	—	4	2	no	—
298	C₂×C₄×C₃₁	—	4	2	no	—
299	C₂×C₄×C₃₁	—	4	8	no	—
300	C₂×C₄×C₃₁	—	4	4	no	—
301	C₂×C₄×C₃₁	—	4	8	no	—
302	C₂×C₄×C₃₁	—	4	4	no	—
303	C₂×C₄×C₃₁	—	8	2	no	—
304	C₂×C₄×C₃₁	—	8	4	no	—
305	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
307	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
308	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
313	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
314	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
315	C₈×C₃₁	—	2	1	no	—
316	C₈×C₃₁	—	2	4	no	—
317	C₈×C₃₁	—	2	1	no	—
318	C₈×C₃₁	C₂×#41	1	2	yes	—
319	C₈×C₃₁	C₂×#41	1	2	yes	—
320	#3480	C₂×C₃₁	1	2	yes	complemented
321	#3403	C₂×C₃₁	1	2	yes	complemented
322	#3402	C₂×C₃₁	1	2	yes	complemented
323	#3401	C₂×C₃₁	1	4	yes	complemented
324	#3400	C₂×C₃₁	1	2	yes	complemented
325	#3399	C₂×C₃₁	1	2	yes	complemented
326	#3398	C₂×C₃₁	1	2	yes	complemented
327	#3397	C₂×C₃₁	1	2	yes	complemented
328	#3150	C₂×C₃₁	1	2	yes	complemented
329	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
330	#3035	C₂×C₃₁	1	2	yes	complemented
331	#3035	C₂×C₃₁	1	4	yes	complemented
332	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
333	#2943	C₂×C₃₁	1	2	yes	complemented
334	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
335	C₂²×C₃₁	—	2	1	no	—
336	C₂²×C₃₁	—	2	1	no	—
337	C₂²×C₃₁	—	4	2	no	—
338	C₂²×C₃₁	—	4	4	no	—
339	C₂²×C₃₁	—	4	4	no	—
340	C₂²×C₃₁	—	4	2	no	—
341	C₂²×C₃₁	—	4	4	no	—
342	C₂²×C₃₁	—	4	2	no	—
343	C₂²×C₃₁	#550	1	2	yes	—
344	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
345	C₄×C₃₁	—	2	1	no	—
346	C₄×C₃₁	—	2	4	no	—
347	C₄×C₃₁	—	2	1	no	—
348	C₄×C₃₁	—	2	1	no	—
349	C₄×C₃₁	—	2	4	no	—
350	C₄×C₃₁	—	4	2	no	—
351	C₄×C₃₁	—	4	2	no	—
352	C₄×C₃₁	—	4	4	no	—
353	C₄×C₃₁	—	4	2	no	—
354	C₄×C₃₁	#550	1	2	yes	—
355	C₄×C₃₁	#575	1	2	yes	—
356	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
357	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
358	C₂×#135	C₂²×C₃₁	1	2	yes	—
359	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×#135	C₂²×C₃₁	1	2	yes	—
361	C₂×#135	C₂²×C₃₁	1	4	yes	—
362	#559	C₂²×C₃₁	1	2	yes	—
363	#559	C₂²×C₃₁	1	4	yes	—
364	#557	C₂²×C₃₁	1	2	yes	—
365	#556	C₂²×C₃₁	1	2	yes	—
366	#556	C₂²×C₃₁	1	4	yes	—
367	#553	—	2	2	no	—
368	#553	—	2	4	no	—
369	#550	C₂²×C₃₁	1	2	yes	—
370	#548	—	2	2	no	—
371	#532	C₂²×C₃₁	1	2	yes	—
372	#532	C₂²×C₃₁	1	4	yes	—
373	#529	—	2	2	no	—
374	#529	—	2	4	no	—
375	C₄×#41	—	2	2	no	—
376	C₄×#41	C₂²×C₃₁	1	2	yes	—
377	C₄×#41	C₂²×C₃₁	1	4	yes	—
378	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
379	#497	C₂²×C₃₁	1	4	yes	—
380	#497	C₂²×C₃₁	1	4	yes	—
381	#496	C₂²×C₃₁	1	4	yes	—
382	#496	C₂²×C₃₁	1	4	yes	—
383	#495	C₂²×C₃₁	1	2	yes	—
384	#495	C₂²×C₃₁	1	2	yes	—
385	#494	C₂²×C₃₁	1	1	yes	characteristic
386	#493	C₂²×C₃₁	1	1	yes	characteristic
387	#492	C₂²×C₃₁	1	2	yes	—
388	#464	C₂²×C₃₁	1	4	yes	—
389	#464	C₂²×C₃₁	1	4	yes	—
390	#464	C₂²×C₃₁	1	4	yes	—
391	#463	C₂²×C₃₁	1	4	yes	—
392	#463	C₂²×C₃₁	1	4	yes	—
393	#463	C₂²×C₃₁	1	4	yes	—
394	#462	C₂²×C₃₁	1	2	yes	—
395	#462	C₂²×C₃₁	1	4	yes	—
396	#461	C₂²×C₃₁	1	2	yes	—
397	#461	C₂²×C₃₁	1	4	yes	—
398	#460	C₂²×C₃₁	1	4	yes	—
399	#460	C₂²×C₃₁	1	2	yes	—
400	#459	C₂²×C₃₁	1	2	yes	—
401	#459	C₂²×C₃₁	1	4	yes	—
402	#446	C₂²×C₃₁	1	2	yes	—
403	#446	C₂²×C₃₁	1	2	yes	—
404	#446	C₂²×C₃₁	1	2	yes	—
405	#446	C₂²×C₃₁	1	2	yes	—
406	#442	C₂²×C₃₁	1	4	yes	—
407	#441	C₂²×C₃₁	1	4	yes	—
408	#440	C₂²×C₃₁	1	4	yes	—
409	#440	C₂²×C₃₁	1	4	yes	—
410	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
411	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
412	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
413	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
414	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
415	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
416	#427	C₂²×C₃₁	1	1	yes	characteristic
417	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
418	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
419	#416	C₂²×C₃₁	1	2	yes	—
420	#416	C₂²×C₃₁	1	4	yes	—
421	C₂×#97	C₂²×C₃₁	1	2	yes	—
422	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
423	C₂×C₃₁	—	2	2	no	—
424	C₂×C₃₁	—	8	4	no	—
425	C₂×C₃₁	—	8	2	no	—
426	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
427	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
428	C₂×C₃₁	#3584	1	1	yes	characteristic
429	C₂×#41	—	2	2	no	—
430	C₂×#41	C₂³×C₃₁	1	2	yes	—
431	C₂×#41	C₂³×C₃₁	1	4	yes	—
432	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
433	#135	—	2	8	no	—
434	#135	—	2	4	no	—
435	#135	—	4	4	no	—
436	#135	—	4	8	no	—
437	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
439	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
440	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
441	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
442	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
443	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
444	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
445	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
446	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
447	C₂×D₈	C₂³×C₃₁	1	2	yes	—
448	C₂×D₈	C₂³×C₃₁	1	2	yes	—
449	C₂×D₈	C₂³×C₃₁	1	4	yes	—
450	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
451	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
452	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
453	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
454	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
455	#127	C₂³×C₃₁	1	2	yes	—
456	#127	C₂³×C₃₁	1	4	yes	—
457	#126	—	2	4	no	—
458	#126	—	2	4	no	—
459	#126	—	2	4	no	—
460	#125	—	2	2	no	—
461	#125	—	2	4	no	—
462	#124	—	2	2	no	—
463	#124	—	2	4	no	—
464	#124	C₂³×C₃₁	1	4	yes	—
465	#124	C₂³×C₃₁	1	4	yes	—
466	#124	C₂³×C₃₁	1	4	yes	—
467	#123	—	2	4	no	—
468	#123	—	2	4	no	—
469	#123	—	2	4	no	—
470	C₂²⋊Q₈	—	2	2	no	—
471	C₂²⋊Q₈	—	2	4	no	—
472	#121	—	2	2	no	—
473	#121	—	2	4	no	—
474	#121	—	2	2	no	—
475	#121	—	2	4	no	—
476	#121	—	2	2	no	—
477	#121	—	2	4	no	—
478	C₄×Q₈	—	2	2	no	—
479	C₄×Q₈	D₄×C₃₁	1	2	yes	—
480	C₄×Q₈	D₄×C₃₁	1	2	yes	—
481	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
482	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
483	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
484	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
485	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
486	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
487	C₄×D₄	—	2	2	no	—
488	C₄×D₄	—	2	4	no	—
489	C₄×D₄	—	2	2	no	—
490	C₄×D₄	—	2	4	no	—
491	C₄×D₄	—	2	2	no	—
492	C₄×D₄	—	2	4	no	—
493	C₄×D₄	—	2	2	no	—
494	C₄×D₄	—	2	4	no	—
495	C₄×D₄	—	2	4	no	—
496	C₄×D₄	—	2	2	no	—
497	C₄×D₄	C₂³×C₃₁	1	2	yes	—
498	C₄×D₄	C₂³×C₃₁	1	4	yes	—
499	C₄×D₄	C₂³×C₃₁	1	2	yes	—
500	C₄×D₄	C₂³×C₃₁	1	4	yes	—
501	#117	—	2	2	no	—
502	#117	—	2	4	no	—
503	#117	—	2	4	no	—
504	#117	—	2	2	no	—
505	#117	C₂³×C₃₁	1	2	yes	—
506	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
508	C₂×C₄²	—	2	2	no	—
509	C₂×C₄²	—	2	4	no	—
510	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
511	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
512	#107	C₂³×C₃₁	1	1	yes	characteristic
513	#107	C₂³×C₃₁	1	1	yes	characteristic
514	#106	C₂³×C₃₁	1	2	yes	—
515	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
516	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
517	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
518	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
519	#103	C₂³×C₃₁	1	2	yes	—
520	#103	C₂³×C₃₁	1	2	yes	—
521	#103	C₂³×C₃₁	1	4	yes	—
522	#103	C₂³×C₃₁	1	4	yes	—
523	#102	C₂³×C₃₁	1	2	yes	—
524	#102	C₂³×C₃₁	1	2	yes	—
525	#102	C₂³×C₃₁	1	4	yes	—
526	#102	C₂³×C₃₁	1	4	yes	—
527	#97	C₂³×C₃₁	1	4	yes	—
528	#97	C₂³×C₃₁	1	4	yes	—
529	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
530	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
531	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
532	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
533	C₃₁	#61401	1	1	yes	characteristic, direct product factor, Sylow
534	#41	—	2	8	no	—
535	#41	—	2	4	no	—
536	#41	—	2	8	no	—
537	#41	—	2	4	no	—
538	#41	—	4	8	no	—
539	C₂×Q₈	—	4	2	no	—
540	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
541	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
542	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
543	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
544	C₂×D₄	—	2	2	no	—
545	C₂×D₄	—	2	4	no	—
546	C₂×D₄	—	2	4	no	—
547	C₂×D₄	—	2	2	no	—
548	C₂×D₄	—	2	4	no	—
549	C₂×D₄	—	2	2	no	—
550	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
551	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
552	C₂²×C₄	—	2	2	no	—
553	C₂²×C₄	—	2	4	no	—
554	C₂²×C₄	—	2	4	no	—
555	C₂²×C₄	—	2	2	no	—
556	C₂²×C₄	—	2	2	no	—
557	C₂²×C₄	—	2	4	no	—
558	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
559	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
560	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
561	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
562	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
563	Q₁₆	—	2	4	no	—
564	Q₁₆	—	2	2	no	—
565	Q₁₆	—	4	2	no	—
566	Q₁₆	—	4	4	no	—
567	SD₁₆	—	2	4	no	—
568	SD₁₆	—	2	8	no	—
569	SD₁₆	—	4	4	no	—
570	SD₁₆	—	4	4	no	—
571	SD₁₆	—	4	4	no	—
572	D₈	—	2	4	no	—
573	D₈	—	2	2	no	—
574	D₈	—	4	2	no	—
575	D₈	—	4	4	no	—
576	C₂×C₈	—	2	2	no	—
577	C₂×C₈	—	2	4	no	—
578	C₂×C₈	#41×C₃₁	1	4	yes	complemented
579	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
580	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
581	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
582	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
583	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	4	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	4	no	—
589	C₄⋊C₄	—	2	2	no	—
590	C₄⋊C₄	—	2	4	no	—
591	C₄⋊C₄	—	2	2	no	—
592	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
593	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
594	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
595	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
596	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
597	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
598	C₂²⋊C₄	—	2	2	no	—
599	C₂²⋊C₄	—	2	4	no	—
600	C₂²⋊C₄	—	2	4	no	—
601	C₂²⋊C₄	—	2	4	no	—
602	C₂²⋊C₄	—	2	4	no	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	2	4	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	4	no	—
607	C₂²⋊C₄	—	2	4	no	—
608	C₂²⋊C₄	—	2	4	no	—
609	C₄²	—	2	2	no	—
610	C₄²	—	2	2	no	—
611	C₄²	—	2	4	no	—
612	C₄²	—	2	4	no	—
613	C₄²	—	2	2	no	—
614	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
615	C₄²	C₂×D₄×C₃₁	1	4	yes	—
616	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
617	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
618	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
619	C₄²	C₂⁴×C₃₁	1	4	yes	—
620	C₂³	—	2	4	no	—
621	C₂³	—	2	2	no	—
622	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
623	Q₈	—	2	4	no	—
624	Q₈	—	2	2	no	—
625	Q₈	—	2	4	no	—
626	Q₈	—	2	2	no	—
627	Q₈	—	2	4	no	—
628	Q₈	—	4	2	no	—
629	D₄	—	2	4	no	—
630	D₄	—	2	2	no	—
631	D₄	—	2	4	no	—
632	D₄	—	2	2	no	—
633	D₄	—	4	8	no	—
634	D₄	—	4	4	no	—
635	D₄	—	4	8	no	—
636	D₄	—	4	4	no	—
637	D₄	—	8	4	no	—
638	D₄	—	8	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	4	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	2	no	—
646	C₂×C₄	—	4	2	no	—
647	C₂×C₄	—	4	8	no	—
648	C₂×C₄	—	4	4	no	—
649	C₂×C₄	—	4	8	no	—
650	C₂×C₄	—	4	4	no	—
651	C₂×C₄	—	8	2	no	—
652	C₂×C₄	—	8	4	no	—
653	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
654	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
655	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
656	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
657	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
658	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
659	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
660	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
661	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
662	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
663	C₈	—	2	1	no	—
664	C₈	—	2	4	no	—
665	C₈	—	2	1	no	—
666	C₈	C₂×#41×C₃₁	1	2	yes	—
667	C₈	C₂×#41×C₃₁	1	2	yes	—
668	C₂²	—	2	1	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	4	4	no	—
671	C₂²	—	4	2	no	—
672	C₂²	—	4	4	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	4	4	no	—
676	C₂²	C₃₁×#550	1	2	yes	—
677	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
678	C₄	—	2	1	no	—
679	C₄	—	2	4	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	4	no	—
682	C₄	—	2	1	no	—
683	C₄	—	4	2	no	—
684	C₄	—	4	4	no	—
685	C₄	—	4	2	no	—
686	C₄	—	4	2	no	—
687	C₄	C₃₁×#550	1	2	yes	—
688	C₄	C₃₁×#575	1	2	yes	—
689	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
690	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
691	C₂	—	2	2	no	—
692	C₂	—	8	4	no	—
693	C₂	—	8	2	no	—
694	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
695	C₂	C₂×C₃₁×#575	1	1	yes	characteristic
696	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3480	C₂	semidirect
2. C₃₁×#3403	C₂	semidirect
3. C₃₁×#3402	C₂	semidirect
4. C₃₁×#3401	C₂	semidirect
5. C₃₁×#3400	C₂	semidirect
6. C₃₁×#3399	C₂	semidirect
7. C₃₁×#3398	C₂	semidirect
8. C₃₁×#3397	C₂	semidirect
9. C₃₁×#3150	C₂	semidirect
10. C₃₁×#3147	C₂	semidirect
11. C₃₁×#3035	C₂	semidirect
12. C₃₁×#3035	C₂	semidirect
13. C₂×C₃₁×#446	C₂	semidirect
14. C₃₁×#2943	C₂	semidirect
15. C₄×C₃₁×#135	C₂	semidirect
229. C₂×C₈×C₃₁	#41	semidirect
271. #61401	C₃₁	direct (N × H)
320. #3480	C₂×C₃₁	semidirect
321. #3403	C₂×C₃₁	semidirect
322. #3402	C₂×C₃₁	semidirect
323. #3401	C₂×C₃₁	semidirect
324. #3400	C₂×C₃₁	semidirect
325. #3399	C₂×C₃₁	semidirect
326. #3398	C₂×C₃₁	semidirect
327. #3397	C₂×C₃₁	semidirect
328. #3150	C₂×C₃₁	semidirect
329. #3147	C₂×C₃₁	semidirect
330. #3035	C₂×C₃₁	semidirect
331. #3035	C₂×C₃₁	semidirect
332. C₂×#446	C₂×C₃₁	semidirect
333. #2943	C₂×C₃₁	semidirect
334. C₄×#135	C₂×C₃₁	semidirect
533. C₃₁	#61401	direct (N × H)
578. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3480	2
C₃₁×#3403	2
C₃₁×#3402	2
C₃₁×#3401	2
C₃₁×#3400	2
C₃₁×#3399	2
C₃₁×#3398	2
C₃₁×#3397	2
C₃₁×#3150	2
C₃₁×#3147	2
C₃₁×#3035	2
C₂×C₃₁×#446	2
C₃₁×#2943	2
C₄×C₃₁×#135	2
#61401	31

Subgroups — C31×#61401 #21518160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61401 #21518160