Subgroups — C₃₁×#60939 #21517698

Order 7936.

1758 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×#3495	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3263	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3261	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3258	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3256	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3255	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3252	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3084	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3082	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3056	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3054	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#3046	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3044	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#556	—	2	2	no	—
22	C₃₁×#554	—	2	2	no	—
23	C₃₁×#552	—	2	4	no	—
24	C₃₁×#552	—	2	2	no	—
25	C₃₁×#552	—	2	2	no	—
26	C₃₁×#551	—	2	4	no	—
27	C₃₁×#544	C₂²	1	2	yes	—
28	C₃₁×#542	C₂²	1	4	yes	—
29	C₃₁×#541	C₂²	1	4	yes	—
30	C₃₁×#536	C₂²	1	2	yes	—
31	C₃₁×#533	C₂²	1	4	yes	—
32	C₃₁×#529	C₂²	1	4	yes	—
33	C₃₁×#525	C₂²	1	2	yes	—
34	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₃₁×#520	C₂²	1	2	yes	—
36	C₃₁×#519	C₂²	1	2	yes	—
37	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
38	C₃₁×#479	C₂²	1	4	yes	complemented
39	C₃₁×#479	C₂²	1	4	yes	complemented
40	C₃₁×#479	C₂²	1	4	yes	complemented
41	C₃₁×#478	C₂²	1	4	yes	complemented
42	C₃₁×#477	C₂²	1	4	yes	complemented
43	C₃₁×#476	C₂²	1	4	yes	complemented
44	C₃₁×#475	C₂²	1	4	yes	complemented
45	C₃₁×#475	C₂²	1	4	yes	complemented
46	C₃₁×#470	C₂²	1	2	yes	—
47	C₃₁×#470	C₂²	1	4	yes	—
48	C₃₁×#470	C₂²	1	1	yes	characteristic
49	C₃₁×#470	C₂²	1	2	yes	—
50	C₃₁×#470	C₂²	1	2	yes	—
51	C₃₁×#470	C₂²	1	1	yes	characteristic
52	C₃₁×#470	C₂²	1	2	yes	—
53	C₃₁×#469	C₂²	1	1	yes	characteristic
54	C₃₁×#469	C₂²	1	1	yes	characteristic
55	C₃₁×#468	C₂²	1	2	yes	—
56	C₃₁×#468	C₂²	1	4	yes	—
57	C₃₁×#468	C₂²	1	2	yes	—
58	C₃₁×#467	C₂²	1	4	yes	—
59	C₃₁×#465	C₂²	1	4	yes	—
60	C₃₁×#464	C₂²	1	4	yes	complemented
61	C₃₁×#464	C₂²	1	4	yes	complemented
62	C₃₁×#464	C₂²	1	4	yes	complemented
63	C₃₁×#463	C₂²	1	4	yes	complemented
64	C₃₁×#462	C₂²	1	4	yes	complemented
65	C₃₁×#462	C₂²	1	4	yes	complemented
66	C₃₁×#462	C₂²	1	4	yes	complemented
67	C₃₁×#461	C₂²	1	4	yes	complemented
68	C₃₁×#428	C₂²	1	2	yes	—
69	C₃₁×#427	C₂²	1	1	yes	characteristic
70	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
71	C₃₁×#424	C₂²	1	2	yes	—
72	C₃₁×#424	C₂²	1	2	yes	—
73	C₃₁×#424	C₂²	1	2	yes	—
74	C₃₁×#424	C₂²	1	2	yes	—
75	C₃₁×#423	C₂²	1	2	yes	—
76	C₃₁×#423	C₂²	1	2	yes	—
77	C₃₁×#419	C₂²	1	2	yes	—
78	C₃₁×#417	C₂²	1	2	yes	—
79	C₃₁×#416	C₂²	1	2	yes	—
80	C₂×C₃₁×#103	C₂²	1	2	yes	—
81	C₂×#41×C₃₁	C₂³	1	2	yes	—
82	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#137	—	4	4	no	—
85	C₃₁×#135	—	4	2	no	—
86	C₂×Q₁₆×C₃₁	—	4	2	no	—
87	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
89	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
90	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
91	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
92	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
93	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₃₁	—	4	2	no	—
95	C₂×M₁₆×C₃₁	—	4	2	no	—
96	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
97	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
98	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
99	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
105	C₃₁×C₄⋊Q₈	—	2	2	no	—
106	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
107	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
108	C₃₁×#126	—	2	4	no	—
109	C₃₁×#126	—	2	4	no	—
110	C₃₁×#126	—	2	4	no	—
111	C₃₁×#126	—	2	4	no	—
112	C₃₁×#125	—	2	4	no	—
113	C₃₁×#125	—	2	4	no	—
114	C₃₁×#125	—	2	2	no	—
115	C₃₁×#125	—	2	4	no	—
116	C₃₁×#125	C₂³	1	4	yes	—
117	C₃₁×#124	—	2	4	no	—
118	C₃₁×#124	C₂³	1	4	yes	—
119	C₃₁×#123	—	2	2	no	—
120	C₃₁×#123	—	2	4	no	—
121	C₃₁×#123	—	2	4	no	—
122	C₃₁×#123	—	2	4	no	—
123	C₃₁×C₂²⋊Q₈	—	2	2	no	—
124	C₃₁×C₂²⋊Q₈	—	2	2	no	—
125	C₃₁×C₂²⋊Q₈	—	2	4	no	—
126	C₃₁×C₂²⋊Q₈	—	2	2	no	—
127	C₃₁×C₂²⋊Q₈	—	2	2	no	—
128	C₃₁×C₂²⋊Q₈	—	2	4	no	—
129	C₃₁×C₂²⋊Q₈	—	2	4	no	—
130	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
131	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
132	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
133	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
134	C₃₁×#121	C₂³	1	2	yes	—
135	C₃₁×#121	C₂³	1	2	yes	—
136	C₃₁×#120	—	2	2	no	—
137	C₄×Q₈×C₃₁	—	2	2	no	—
138	C₄×Q₈×C₃₁	—	2	2	no	—
139	C₄×Q₈×C₃₁	—	2	4	no	—
140	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
141	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
142	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
143	C₄×D₄×C₃₁	—	2	2	no	—
144	C₄×D₄×C₃₁	—	2	2	no	—
145	C₄×D₄×C₃₁	—	2	4	no	—
146	C₄×D₄×C₃₁	—	2	4	no	—
147	C₄×D₄×C₃₁	C₂³	1	4	yes	—
148	C₃₁×#117	—	2	4	no	—
149	C₃₁×#117	—	2	2	no	—
150	C₃₁×#117	—	2	4	no	—
151	C₃₁×#117	—	2	4	no	—
152	C₃₁×#117	D₄	1	1	yes	characteristic
153	C₃₁×#117	D₄	1	2	yes	—
154	C₃₁×#117	D₄	1	1	yes	characteristic
155	C₃₁×#117	C₂³	1	2	yes	—
156	C₃₁×#117	C₂³	1	2	yes	—
157	C₃₁×#117	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
161	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
162	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
163	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
164	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
165	C₃₁×#107	C₂³	1	2	yes	—
166	C₃₁×#107	C₂³	1	4	yes	—
167	C₃₁×#106	C₂³	1	2	yes	—
168	C₃₁×#106	C₂³	1	2	yes	—
169	C₃₁×#106	C₂³	1	1	yes	characteristic
170	C₃₁×#106	C₂³	1	1	yes	characteristic
171	C₃₁×#106	C₂³	1	4	yes	—
172	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
173	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
174	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
175	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
176	C₃₁×#103	C₂³	1	4	yes	—
177	C₃₁×#103	C₂³	1	4	yes	—
178	C₃₁×#103	C₂³	1	2	yes	—
179	C₃₁×#103	C₂³	1	2	yes	—
180	C₃₁×#103	C₂³	1	4	yes	—
181	C₃₁×#103	C₂³	1	4	yes	—
182	C₃₁×#103	C₂³	1	4	yes	—
183	C₃₁×#102	C₂³	1	2	yes	—
184	C₃₁×#102	C₂³	1	2	yes	—
185	C₃₁×#102	C₂³	1	4	yes	—
186	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
187	#41×C₃₁	—	2	4	no	—
188	#41×C₃₁	—	2	4	no	—
189	C₂×Q₈×C₃₁	—	2	2	no	—
190	C₂×Q₈×C₃₁	—	2	4	no	—
191	C₂×Q₈×C₃₁	—	2	4	no	—
192	C₂×Q₈×C₃₁	—	2	4	no	—
193	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
194	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
195	C₂×D₄×C₃₁	—	2	2	no	—
196	C₂×D₄×C₃₁	—	2	2	no	—
197	C₂×D₄×C₃₁	—	2	2	no	—
198	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₃₁	—	2	2	no	—
200	C₂²×C₄×C₃₁	—	2	2	no	—
201	C₂²×C₄×C₃₁	—	2	2	no	—
202	C₂²×C₄×C₃₁	—	2	2	no	—
203	C₂²×C₄×C₃₁	—	2	2	no	—
204	C₂²×C₄×C₃₁	—	2	4	no	—
205	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
209	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
212	Q₁₆×C₃₁	—	4	2	no	—
213	Q₁₆×C₃₁	—	4	2	no	—
214	Q₁₆×C₃₁	—	4	1	no	—
215	Q₁₆×C₃₁	—	4	4	no	—
216	SD₁₆×C₃₁	—	4	2	no	—
217	SD₁₆×C₃₁	—	4	4	no	—
218	D₈×C₃₁	—	4	1	no	—
219	M₁₆×C₃₁	—	2	4	no	—
220	M₁₆×C₃₁	—	4	1	no	—
221	M₁₆×C₃₁	—	4	1	no	—
222	M₁₆×C₃₁	—	4	1	no	—
223	M₁₆×C₃₁	—	4	1	no	—
224	M₁₆×C₃₁	—	4	2	no	—
225	C₂×C₈×C₃₁	—	2	2	no	—
226	C₂×C₈×C₃₁	—	2	2	no	—
227	C₂×C₈×C₃₁	—	4	2	no	—
228	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
229	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
230	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
231	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
232	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
233	C₄⋊C₄×C₃₁	—	2	2	no	—
234	C₄⋊C₄×C₃₁	—	2	4	no	—
235	C₄⋊C₄×C₃₁	—	2	4	no	—
236	C₄⋊C₄×C₃₁	—	2	4	no	—
237	C₄⋊C₄×C₃₁	—	2	4	no	—
238	C₄⋊C₄×C₃₁	—	2	4	no	—
239	C₄⋊C₄×C₃₁	—	2	4	no	—
240	C₄⋊C₄×C₃₁	—	2	4	no	—
241	C₄⋊C₄×C₃₁	—	2	4	no	—
242	C₄⋊C₄×C₃₁	—	2	2	no	—
243	C₄⋊C₄×C₃₁	—	2	4	no	—
244	C₄⋊C₄×C₃₁	—	2	2	no	—
245	C₄⋊C₄×C₃₁	—	2	4	no	—
246	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
247	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
248	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
249	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
250	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
251	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
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	2	no	—
256	C₂²⋊C₄×C₃₁	—	2	2	no	—
257	C₂²⋊C₄×C₃₁	—	2	2	no	—
258	C₂²⋊C₄×C₃₁	—	2	4	no	—
259	C₂²⋊C₄×C₃₁	—	2	4	no	—
260	C₂²⋊C₄×C₃₁	—	2	2	no	—
261	C₂²⋊C₄×C₃₁	—	2	4	no	—
262	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
263	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
264	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
265	C₄²×C₃₁	—	2	4	no	—
266	C₄²×C₃₁	—	2	4	no	—
267	C₄²×C₃₁	—	2	4	no	—
268	C₄²×C₃₁	—	2	2	no	—
269	C₄²×C₃₁	C₂×D₄	1	2	yes	—
270	C₄²×C₃₁	C₂×D₄	1	2	yes	—
271	C₄²×C₃₁	C₂⁴	1	4	yes	—
272	#60939	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂³×C₃₁	—	2	2	no	—
274	C₂³×C₃₁	—	2	2	no	—
275	C₂³×C₃₁	—	4	2	no	—
276	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
277	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
278	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
279	Q₈×C₃₁	—	2	4	no	—
280	Q₈×C₃₁	—	2	2	no	—
281	Q₈×C₃₁	—	2	4	no	—
282	Q₈×C₃₁	—	2	2	no	—
283	Q₈×C₃₁	—	4	4	no	—
284	Q₈×C₃₁	—	8	4	no	—
285	D₄×C₃₁	—	2	2	no	—
286	D₄×C₃₁	—	2	2	no	—
287	D₄×C₃₁	—	4	4	no	—
288	D₄×C₃₁	—	4	4	no	—
289	D₄×C₃₁	—	8	2	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	1	no	—
294	C₂×C₄×C₃₁	—	2	1	no	—
295	C₂×C₄×C₃₁	—	2	1	no	—
296	C₂×C₄×C₃₁	—	2	1	no	—
297	C₂×C₄×C₃₁	—	2	2	no	—
298	C₂×C₄×C₃₁	—	2	2	no	—
299	C₂×C₄×C₃₁	—	2	4	no	—
300	C₂×C₄×C₃₁	—	2	2	no	—
301	C₂×C₄×C₃₁	—	2	4	no	—
302	C₂×C₄×C₃₁	—	4	4	no	—
303	C₂×C₄×C₃₁	—	4	4	no	—
304	C₂×C₄×C₃₁	—	4	4	no	—
305	C₂×C₄×C₃₁	—	8	4	no	—
306	C₂×C₄×C₃₁	—	8	2	no	—
307	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
308	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
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	H(5,𝔽₂)	1	2	yes	—
313	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
314	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
315	C₈×C₃₁	—	2	1	no	—
316	C₈×C₃₁	—	2	1	no	—
317	C₈×C₃₁	—	2	1	no	—
318	C₈×C₃₁	—	2	1	no	—
319	C₈×C₃₁	—	2	4	no	—
320	#3495	C₂×C₃₁	1	2	yes	complemented
321	#3263	C₂×C₃₁	1	2	yes	complemented
322	#3261	C₂×C₃₁	1	2	yes	complemented
323	#3258	C₂×C₃₁	1	4	yes	complemented
324	#3256	C₂×C₃₁	1	2	yes	complemented
325	#3255	C₂×C₃₁	1	2	yes	complemented
326	#3252	C₂×C₃₁	1	4	yes	complemented
327	#3084	C₂×C₃₁	1	2	yes	complemented
328	#3082	C₂×C₃₁	1	2	yes	complemented
329	#3056	C₂×C₃₁	1	2	yes	complemented
330	#3054	C₂×C₃₁	1	1	yes	characteristic
331	C₂×#470	C₂×C₃₁	1	1	yes	characteristic, complemented
332	#3046	C₂×C₃₁	1	2	yes	complemented
333	#3044	C₂×C₃₁	1	2	yes	complemented
334	#2905	C₂×C₃₁	1	1	yes	characteristic, complemented
335	C₂²×C₃₁	—	2	2	no	—
336	C₂²×C₃₁	—	2	2	no	—
337	C₂²×C₃₁	—	2	2	no	—
338	C₂²×C₃₁	—	2	1	no	—
339	C₂²×C₃₁	—	2	1	no	—
340	C₂²×C₃₁	—	4	2	no	—
341	C₂²×C₃₁	—	4	2	no	—
342	C₂²×C₃₁	—	4	2	no	—
343	C₂²×C₃₁	—	8	2	no	—
344	C₂²×C₃₁	#537	1	2	yes	—
345	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
346	C₄×C₃₁	—	2	4	no	—
347	C₄×C₃₁	—	2	4	no	—
348	C₄×C₃₁	—	2	1	no	—
349	C₄×C₃₁	—	2	1	no	—
350	C₄×C₃₁	—	4	4	no	—
351	C₄×C₃₁	—	4	2	no	—
352	C₄×C₃₁	—	4	2	no	—
353	C₄×C₃₁	—	4	2	no	—
354	C₄×C₃₁	—	4	4	no	—
355	C₄×C₃₁	#537	1	2	yes	—
356	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
357	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
358	C₂×#137	C₂²×C₃₁	1	2	yes	—
359	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
360	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
361	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
362	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
363	#556	—	2	2	no	—
364	#554	—	2	2	no	—
365	#552	—	2	4	no	—
366	#552	—	2	2	no	—
367	#552	—	2	2	no	—
368	#551	—	2	4	no	—
369	#544	C₂²×C₃₁	1	2	yes	—
370	#542	C₂²×C₃₁	1	4	yes	—
371	#541	C₂²×C₃₁	1	4	yes	—
372	#536	C₂²×C₃₁	1	2	yes	—
373	#533	C₂²×C₃₁	1	4	yes	—
374	#529	C₂²×C₃₁	1	4	yes	—
375	#525	C₂²×C₃₁	1	2	yes	—
376	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
377	#520	C₂²×C₃₁	1	2	yes	—
378	#519	C₂²×C₃₁	1	2	yes	—
379	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
380	#479	C₂²×C₃₁	1	4	yes	complemented
381	#479	C₂²×C₃₁	1	4	yes	complemented
382	#479	C₂²×C₃₁	1	4	yes	complemented
383	#478	C₂²×C₃₁	1	4	yes	complemented
384	#477	C₂²×C₃₁	1	4	yes	complemented
385	#476	C₂²×C₃₁	1	4	yes	complemented
386	#475	C₂²×C₃₁	1	4	yes	complemented
387	#475	C₂²×C₃₁	1	4	yes	complemented
388	#470	C₂²×C₃₁	1	2	yes	—
389	#470	C₂²×C₃₁	1	4	yes	—
390	#470	C₂²×C₃₁	1	1	yes	characteristic
391	#470	C₂²×C₃₁	1	2	yes	—
392	#470	C₂²×C₃₁	1	2	yes	—
393	#470	C₂²×C₃₁	1	1	yes	characteristic
394	#470	C₂²×C₃₁	1	2	yes	—
395	#469	C₂²×C₃₁	1	1	yes	characteristic
396	#469	C₂²×C₃₁	1	1	yes	characteristic
397	#468	C₂²×C₃₁	1	2	yes	—
398	#468	C₂²×C₃₁	1	4	yes	—
399	#468	C₂²×C₃₁	1	2	yes	—
400	#467	C₂²×C₃₁	1	4	yes	—
401	#465	C₂²×C₃₁	1	4	yes	—
402	#464	C₂²×C₃₁	1	4	yes	complemented
403	#464	C₂²×C₃₁	1	4	yes	complemented
404	#464	C₂²×C₃₁	1	4	yes	complemented
405	#463	C₂²×C₃₁	1	4	yes	complemented
406	#462	C₂²×C₃₁	1	4	yes	complemented
407	#462	C₂²×C₃₁	1	4	yes	complemented
408	#462	C₂²×C₃₁	1	4	yes	complemented
409	#461	C₂²×C₃₁	1	4	yes	complemented
410	#428	C₂²×C₃₁	1	2	yes	—
411	#427	C₂²×C₃₁	1	1	yes	characteristic
412	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
413	#424	C₂²×C₃₁	1	2	yes	—
414	#424	C₂²×C₃₁	1	2	yes	—
415	#424	C₂²×C₃₁	1	2	yes	—
416	#424	C₂²×C₃₁	1	2	yes	—
417	#423	C₂²×C₃₁	1	2	yes	—
418	#423	C₂²×C₃₁	1	2	yes	—
419	#419	C₂²×C₃₁	1	2	yes	—
420	#417	C₂²×C₃₁	1	2	yes	—
421	#416	C₂²×C₃₁	1	2	yes	—
422	C₂×#103	C₂²×C₃₁	1	2	yes	—
423	C₂×C₃₁	—	2	2	no	—
424	C₂×C₃₁	—	4	1	no	—
425	C₂×C₃₁	—	4	1	no	—
426	C₂×C₃₁	—	8	2	no	—
427	C₂×C₃₁	C₂×#537	1	1	yes	characteristic
428	C₂×C₃₁	#3586	1	2	yes	—
429	C₂×#41	C₂³×C₃₁	1	2	yes	—
430	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
431	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
432	#137	—	4	4	no	—
433	#135	—	4	2	no	—
434	C₂×Q₁₆	—	4	2	no	—
435	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
436	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
437	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
439	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
440	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
441	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
442	C₂×M₁₆	—	4	2	no	—
443	C₂×M₁₆	—	4	2	no	—
444	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
445	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
446	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
447	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
448	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
449	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
450	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
451	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
452	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
453	C₄⋊Q₈	—	2	2	no	—
454	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
455	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
456	#126	—	2	4	no	—
457	#126	—	2	4	no	—
458	#126	—	2	4	no	—
459	#126	—	2	4	no	—
460	#125	—	2	4	no	—
461	#125	—	2	4	no	—
462	#125	—	2	2	no	—
463	#125	—	2	4	no	—
464	#125	C₂³×C₃₁	1	4	yes	—
465	#124	—	2	4	no	—
466	#124	C₂³×C₃₁	1	4	yes	—
467	#123	—	2	2	no	—
468	#123	—	2	4	no	—
469	#123	—	2	4	no	—
470	#123	—	2	4	no	—
471	C₂²⋊Q₈	—	2	2	no	—
472	C₂²⋊Q₈	—	2	2	no	—
473	C₂²⋊Q₈	—	2	4	no	—
474	C₂²⋊Q₈	—	2	2	no	—
475	C₂²⋊Q₈	—	2	2	no	—
476	C₂²⋊Q₈	—	2	4	no	—
477	C₂²⋊Q₈	—	2	4	no	—
478	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
479	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
480	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
481	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
482	#121	C₂³×C₃₁	1	2	yes	—
483	#121	C₂³×C₃₁	1	2	yes	—
484	#120	—	2	2	no	—
485	C₄×Q₈	—	2	2	no	—
486	C₄×Q₈	—	2	2	no	—
487	C₄×Q₈	—	2	4	no	—
488	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
489	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
490	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
491	C₄×D₄	—	2	2	no	—
492	C₄×D₄	—	2	2	no	—
493	C₄×D₄	—	2	4	no	—
494	C₄×D₄	—	2	4	no	—
495	C₄×D₄	C₂³×C₃₁	1	4	yes	—
496	#117	—	2	4	no	—
497	#117	—	2	2	no	—
498	#117	—	2	4	no	—
499	#117	—	2	4	no	—
500	#117	D₄×C₃₁	1	1	yes	characteristic
501	#117	D₄×C₃₁	1	2	yes	—
502	#117	D₄×C₃₁	1	1	yes	characteristic
503	#117	C₂³×C₃₁	1	2	yes	—
504	#117	C₂³×C₃₁	1	2	yes	—
505	#117	C₂³×C₃₁	1	2	yes	—
506	C₂×C₄⋊C₄	—	2	2	no	—
507	C₂×C₄⋊C₄	—	2	2	no	—
508	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
509	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
510	C₂×C₂²⋊C₄	—	2	2	no	—
511	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
512	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
513	#107	C₂³×C₃₁	1	2	yes	—
514	#107	C₂³×C₃₁	1	4	yes	—
515	#106	C₂³×C₃₁	1	2	yes	—
516	#106	C₂³×C₃₁	1	2	yes	—
517	#106	C₂³×C₃₁	1	1	yes	characteristic
518	#106	C₂³×C₃₁	1	1	yes	characteristic
519	#106	C₂³×C₃₁	1	4	yes	—
520	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
521	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
522	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
523	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
524	#103	C₂³×C₃₁	1	4	yes	—
525	#103	C₂³×C₃₁	1	4	yes	—
526	#103	C₂³×C₃₁	1	2	yes	—
527	#103	C₂³×C₃₁	1	2	yes	—
528	#103	C₂³×C₃₁	1	4	yes	—
529	#103	C₂³×C₃₁	1	4	yes	—
530	#103	C₂³×C₃₁	1	4	yes	—
531	#102	C₂³×C₃₁	1	2	yes	—
532	#102	C₂³×C₃₁	1	2	yes	—
533	#102	C₂³×C₃₁	1	4	yes	—
534	C₃₁	#60939	1	1	yes	characteristic, direct product factor, Sylow
535	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
536	#41	—	2	4	no	—
537	#41	—	2	4	no	—
538	C₂×Q₈	—	2	2	no	—
539	C₂×Q₈	—	2	4	no	—
540	C₂×Q₈	—	2	4	no	—
541	C₂×Q₈	—	2	4	no	—
542	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
543	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
544	C₂×D₄	—	2	2	no	—
545	C₂×D₄	—	2	2	no	—
546	C₂×D₄	—	2	2	no	—
547	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	2	no	—
553	C₂²×C₄	—	2	4	no	—
554	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
556	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
557	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
558	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
559	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
560	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
561	Q₁₆	—	4	2	no	—
562	Q₁₆	—	4	2	no	—
563	Q₁₆	—	4	1	no	—
564	Q₁₆	—	4	4	no	—
565	SD₁₆	—	4	2	no	—
566	SD₁₆	—	4	4	no	—
567	D₈	—	4	1	no	—
568	M₁₆	—	2	4	no	—
569	M₁₆	—	4	1	no	—
570	M₁₆	—	4	1	no	—
571	M₁₆	—	4	1	no	—
572	M₁₆	—	4	1	no	—
573	M₁₆	—	4	2	no	—
574	C₂×C₈	—	2	2	no	—
575	C₂×C₈	—	2	2	no	—
576	C₂×C₈	—	4	2	no	—
577	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
578	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
579	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
580	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
581	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	4	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	4	no	—
587	C₄⋊C₄	—	2	4	no	—
588	C₄⋊C₄	—	2	4	no	—
589	C₄⋊C₄	—	2	4	no	—
590	C₄⋊C₄	—	2	4	no	—
591	C₄⋊C₄	—	2	2	no	—
592	C₄⋊C₄	—	2	4	no	—
593	C₄⋊C₄	—	2	2	no	—
594	C₄⋊C₄	—	2	4	no	—
595	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
596	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
597	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
598	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
599	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
600	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
601	C₂²⋊C₄	—	2	4	no	—
602	C₂²⋊C₄	—	2	4	no	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	4	no	—
608	C₂²⋊C₄	—	2	4	no	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	—	2	4	no	—
611	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
612	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
613	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
614	C₄²	—	2	4	no	—
615	C₄²	—	2	4	no	—
616	C₄²	—	2	4	no	—
617	C₄²	—	2	2	no	—
618	C₄²	C₂×D₄×C₃₁	1	2	yes	—
619	C₄²	C₂×D₄×C₃₁	1	2	yes	—
620	C₄²	C₂⁴×C₃₁	1	4	yes	—
621	C₂³	—	2	2	no	—
622	C₂³	—	2	2	no	—
623	C₂³	—	4	2	no	—
624	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
625	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
626	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
627	Q₈	—	2	4	no	—
628	Q₈	—	2	2	no	—
629	Q₈	—	2	4	no	—
630	Q₈	—	2	2	no	—
631	Q₈	—	4	4	no	—
632	Q₈	—	8	4	no	—
633	D₄	—	2	2	no	—
634	D₄	—	2	2	no	—
635	D₄	—	4	4	no	—
636	D₄	—	4	4	no	—
637	D₄	—	8	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	4	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	4	no	—
650	C₂×C₄	—	4	4	no	—
651	C₂×C₄	—	4	4	no	—
652	C₂×C₄	—	4	4	no	—
653	C₂×C₄	—	8	4	no	—
654	C₂×C₄	—	8	2	no	—
655	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
656	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
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	C₃₁×H(5,𝔽₂)	1	2	yes	—
661	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
662	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
663	C₈	—	2	1	no	—
664	C₈	—	2	1	no	—
665	C₈	—	2	1	no	—
666	C₈	—	2	1	no	—
667	C₈	—	2	4	no	—
668	C₂²	—	2	2	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	2	1	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	2	2	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	8	2	no	—
677	C₂²	C₃₁×#537	1	2	yes	—
678	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
679	C₄	—	2	1	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	4	no	—
682	C₄	—	2	4	no	—
683	C₄	—	4	2	no	—
684	C₄	—	4	4	no	—
685	C₄	—	4	4	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	2	no	—
688	C₄	C₃₁×#537	1	2	yes	—
689	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
690	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
691	C₂	—	2	2	no	—
692	C₂	—	4	1	no	—
693	C₂	—	4	1	no	—
694	C₂	—	8	2	no	—
695	C₂	C₂×C₃₁×#537	1	1	yes	characteristic
696	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3054	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3495	C₂	semidirect
2. C₃₁×#3263	C₂	semidirect
3. C₃₁×#3261	C₂	semidirect
4. C₃₁×#3258	C₂	semidirect
5. C₃₁×#3256	C₂	semidirect
6. C₃₁×#3255	C₂	semidirect
7. C₃₁×#3252	C₂	semidirect
8. C₃₁×#3084	C₂	semidirect
9. C₃₁×#3082	C₂	semidirect
10. C₃₁×#3056	C₂	semidirect
12. C₂×C₃₁×#470	C₂	semidirect
13. C₃₁×#3046	C₂	semidirect
14. C₃₁×#3044	C₂	semidirect
15. C₃₁×#2905	C₂	semidirect
38. C₃₁×#479	C₂²	semidirect
39. C₃₁×#479	C₂²	semidirect
40. C₃₁×#479	C₂²	semidirect
41. C₃₁×#478	C₂²	semidirect
42. C₃₁×#477	C₂²	semidirect
43. C₃₁×#476	C₂²	semidirect
44. C₃₁×#475	C₂²	semidirect
45. C₃₁×#475	C₂²	semidirect
60. C₃₁×#464	C₂²	semidirect
61. C₃₁×#464	C₂²	semidirect
62. C₃₁×#464	C₂²	semidirect
63. C₃₁×#463	C₂²	semidirect
64. C₃₁×#462	C₂²	semidirect
65. C₃₁×#462	C₂²	semidirect
66. C₃₁×#462	C₂²	semidirect
67. C₃₁×#461	C₂²	semidirect
272. #60939	C₃₁	direct (N × H)
320. #3495	C₂×C₃₁	semidirect
321. #3263	C₂×C₃₁	semidirect
322. #3261	C₂×C₃₁	semidirect
323. #3258	C₂×C₃₁	semidirect
324. #3256	C₂×C₃₁	semidirect
325. #3255	C₂×C₃₁	semidirect
326. #3252	C₂×C₃₁	semidirect
327. #3084	C₂×C₃₁	semidirect
328. #3082	C₂×C₃₁	semidirect
329. #3056	C₂×C₃₁	semidirect
331. C₂×#470	C₂×C₃₁	semidirect
332. #3046	C₂×C₃₁	semidirect
333. #3044	C₂×C₃₁	semidirect
334. #2905	C₂×C₃₁	semidirect
380. #479	C₂²×C₃₁	semidirect
381. #479	C₂²×C₃₁	semidirect
382. #479	C₂²×C₃₁	semidirect
383. #478	C₂²×C₃₁	semidirect
384. #477	C₂²×C₃₁	semidirect
385. #476	C₂²×C₃₁	semidirect
386. #475	C₂²×C₃₁	semidirect
387. #475	C₂²×C₃₁	semidirect
402. #464	C₂²×C₃₁	semidirect
403. #464	C₂²×C₃₁	semidirect
404. #464	C₂²×C₃₁	semidirect
405. #463	C₂²×C₃₁	semidirect
406. #462	C₂²×C₃₁	semidirect
407. #462	C₂²×C₃₁	semidirect
408. #462	C₂²×C₃₁	semidirect
409. #461	C₂²×C₃₁	semidirect
534. C₃₁	#60939	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×#3495	2
C₃₁×#3263	2
C₃₁×#3261	2
C₃₁×#3258	2
C₃₁×#3256	2
C₃₁×#3255	2
C₃₁×#3252	2
C₃₁×#3084	2
C₃₁×#3082	2
C₃₁×#3056	2
C₃₁×#3054	2
C₂×C₃₁×#470	2
C₃₁×#3046	2
C₃₁×#3044	2
C₃₁×#2905	2
#60939	31

Subgroups — C31×#60939 #21517698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60939 #21517698