Subgroups — C₃₁×#60720 #21517479

Order 7936.

1846 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3181	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3178	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3172	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3059	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3057	C₂	1	2	yes	maximal
12	C₃₁×#2950	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2949	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
17	C₂×C₃₁×#131	C₂²	1	2	yes	—
18	C₃₁×#559	—	2	2	no	—
19	C₃₁×#557	—	2	2	no	—
20	C₃₁×#556	—	2	2	no	—
21	C₃₁×#556	—	2	2	no	—
22	C₃₁×#552	—	2	4	no	—
23	C₃₁×#551	—	2	4	no	—
24	C₃₁×#542	C₂²	1	4	yes	—
25	C₃₁×#541	C₂²	1	4	yes	complemented
26	C₃₁×#536	C₂²	1	2	yes	—
27	C₃₁×#536	C₂²	1	1	yes	characteristic
28	C₃₁×#535	C₂²	1	1	yes	characteristic
29	C₃₁×#533	C₂²	1	1	yes	characteristic
30	C₃₁×#533	C₂²	1	1	yes	characteristic
31	C₃₁×#533	C₂²	1	4	yes	complemented
32	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
33	C₃₁×#529	C₂²	1	4	yes	—
34	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
35	C₃₁×#519	C₂²	1	2	yes	—
36	C₃₁×#519	C₂²	1	1	yes	characteristic
37	C₃₁×#518	C₂²	1	1	yes	characteristic
38	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#501	C₂²	1	2	yes	complemented
40	C₃₁×#501	C₂²	1	4	yes	complemented
41	C₃₁×#501	C₂²	1	2	yes	complemented
42	C₃₁×#497	C₂²	1	2	yes	—
43	C₃₁×#497	C₂²	1	4	yes	—
44	C₃₁×#497	C₂²	1	2	yes	—
45	C₃₁×#491	C₂²	1	4	yes	complemented
46	C₃₁×#489	C₂²	1	4	yes	—
47	C₃₁×#488	C₂²	1	4	yes	—
48	C₃₁×#487	C₂²	1	4	yes	complemented
49	C₃₁×#470	C₂²	1	8	yes	complemented
50	C₃₁×#468	C₂²	1	8	yes	—
51	C₃₁×#467	C₂²	1	8	yes	complemented
52	C₃₁×#465	C₂²	1	8	yes	—
53	C₃₁×#441	C₂²	1	8	yes	complemented
54	C₃₁×#440	C₂²	1	8	yes	—
55	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
56	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
57	C₃₁×#427	C₂²	1	2	yes	—
58	C₃₁×#427	C₂²	1	2	yes	—
59	C₃₁×#419	C₂²	1	2	yes	—
60	C₃₁×#419	C₂²	1	2	yes	—
61	C₃₁×#419	C₂²	1	2	yes	—
62	C₃₁×#418	C₂²	1	2	yes	—
63	C₃₁×#405	C₂²	1	2	yes	—
64	C₃₁×#405	C₂²	1	4	yes	complemented
65	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
66	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
67	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
68	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
69	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
72	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×Q₁₆×C₃₁	—	2	8	no	—
74	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
75	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₃₁	—	2	8	no	—
77	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
78	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
79	C₃₁×#131	—	2	8	no	—
80	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
81	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
82	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
83	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
84	C₃₁×C₄⋊Q₈	—	2	1	no	—
85	C₃₁×C₄⋊Q₈	—	2	2	no	—
86	C₃₁×C₄⋊Q₈	—	2	2	no	—
87	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
88	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
89	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
90	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
91	C₃₁×#127	—	2	1	no	—
92	C₃₁×#126	—	2	4	no	—
93	C₃₁×#126	—	2	8	no	—
94	C₃₁×#126	—	2	4	no	—
95	C₃₁×#125	—	2	8	no	—
96	C₃₁×#125	—	2	4	no	—
97	C₃₁×#125	C₂³	1	2	yes	—
98	C₃₁×#125	C₂³	1	2	yes	—
99	C₃₁×#124	—	2	1	no	—
100	C₃₁×#124	—	2	2	no	—
101	C₃₁×#124	—	2	1	no	—
102	C₃₁×#124	—	2	4	no	—
103	C₃₁×#124	—	2	2	no	—
104	C₃₁×#124	C₂³	1	2	yes	—
105	C₃₁×#124	C₂³	1	2	yes	—
106	C₃₁×#123	—	2	2	no	—
107	C₃₁×#123	—	2	4	no	—
108	C₃₁×#123	—	2	4	no	—
109	C₃₁×#123	—	2	4	no	—
110	C₃₁×C₂²⋊Q₈	—	2	2	no	—
111	C₃₁×C₂²⋊Q₈	—	2	2	no	—
112	C₃₁×C₂²⋊Q₈	—	2	4	no	—
113	C₃₁×C₂²⋊Q₈	—	2	4	no	—
114	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
115	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
116	C₃₁×#121	C₂³	1	4	yes	—
117	C₃₁×#120	—	2	2	no	—
118	C₄×Q₈×C₃₁	—	2	4	no	—
119	C₄×Q₈×C₃₁	—	2	2	no	—
120	C₄×Q₈×C₃₁	—	2	2	no	—
121	C₄×Q₈×C₃₁	—	2	2	no	—
122	C₄×Q₈×C₃₁	—	2	4	no	—
123	C₄×Q₈×C₃₁	D₄	1	2	yes	—
124	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
125	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
126	C₄×D₄×C₃₁	—	2	4	no	—
127	C₄×D₄×C₃₁	—	2	2	no	—
128	C₄×D₄×C₃₁	D₄	1	4	yes	—
129	C₄×D₄×C₃₁	D₄	1	2	yes	—
130	C₄×D₄×C₃₁	C₂³	1	4	yes	—
131	C₃₁×#117	—	2	4	no	—
132	C₃₁×#117	—	2	2	no	—
133	C₃₁×#117	—	2	4	no	—
134	C₃₁×#117	—	2	4	no	—
135	C₃₁×#117	C₂³	1	2	yes	—
136	C₃₁×#117	C₂³	1	1	yes	characteristic
137	C₃₁×#117	C₂³	1	1	yes	characteristic
138	C₃₁×#117	C₂³	1	1	yes	characteristic
139	C₃₁×#117	C₂³	1	2	yes	—
140	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
143	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
145	C₃₁×#107	C₂³	1	4	yes	—
146	C₃₁×#107	C₂³	1	4	yes	—
147	C₃₁×#106	C₂³	1	4	yes	—
148	C₃₁×#106	C₂³	1	4	yes	—
149	C₃₁×#103	C₂³	1	8	yes	—
150	C₃₁×#103	C₂³	1	8	yes	—
151	C₃₁×#103	C₂³	1	8	yes	—
152	C₃₁×#102	C₂³	1	8	yes	—
153	C₃₁×#97	C₂³	1	1	yes	characteristic
154	C₃₁×#97	C₂³	1	1	yes	characteristic
155	C₃₁×#97	C₂³	1	2	yes	—
156	C₃₁×#97	C₂³	1	2	yes	—
157	C₃₁×#97	C₂³	1	4	yes	—
158	C₄×C₈×C₃₁	C₂³	1	2	yes	—
159	C₄×C₈×C₃₁	C₂³	1	4	yes	—
160	C₂⁴×C₃₁	—	2	1	no	—
161	#41×C₃₁	—	2	4	no	—
162	C₂×Q₈×C₃₁	—	2	2	no	—
163	C₂×Q₈×C₃₁	—	2	1	no	—
164	C₂×Q₈×C₃₁	—	2	2	no	—
165	C₂×Q₈×C₃₁	—	2	2	no	—
166	C₂×Q₈×C₃₁	—	2	4	no	—
167	C₂×Q₈×C₃₁	—	2	4	no	—
168	C₂×Q₈×C₃₁	—	2	4	no	—
169	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
171	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
172	C₂×D₄×C₃₁	—	2	4	no	—
173	C₂×D₄×C₃₁	—	2	1	no	—
174	C₂×D₄×C₃₁	—	2	4	no	—
175	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₃₁	—	2	1	no	—
179	C₂²×C₄×C₃₁	—	2	2	no	—
180	C₂²×C₄×C₃₁	—	2	4	no	—
181	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	Q₁₆×C₃₁	—	2	8	no	—
186	Q₁₆×C₃₁	—	2	8	no	—
187	SD₁₆×C₃₁	—	2	8	no	—
188	SD₁₆×C₃₁	—	2	8	no	—
189	M₁₆×C₃₁	—	2	4	no	—
190	M₁₆×C₃₁	—	2	8	no	—
191	C₂×C₈×C₃₁	—	2	8	no	—
192	C₂×C₈×C₃₁	#41	1	8	yes	—
193	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
194	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
195	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	2	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	—	2	4	no	—
201	C₄⋊C₄×C₃₁	—	2	8	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	2	no	—
204	C₄⋊C₄×C₃₁	—	2	2	no	—
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	8	no	—
207	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
208	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
209	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
210	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
211	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
212	C₂²⋊C₄×C₃₁	—	2	1	no	—
213	C₂²⋊C₄×C₃₁	—	2	1	no	—
214	C₂²⋊C₄×C₃₁	—	2	2	no	—
215	C₂²⋊C₄×C₃₁	—	2	8	no	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
221	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
222	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
223	C₄²×C₃₁	—	2	4	no	—
224	C₄²×C₃₁	—	2	8	no	—
225	C₄²×C₃₁	—	2	1	no	—
226	C₄²×C₃₁	—	2	1	no	—
227	C₄²×C₃₁	—	2	2	no	—
228	C₄²×C₃₁	C₂×D₄	1	4	yes	—
229	C₄²×C₃₁	C₂×D₄	1	2	yes	—
230	C₄²×C₃₁	C₂⁴	1	2	yes	—
231	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
232	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
233	#60720	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂³×C₃₁	—	2	2	no	—
235	C₂³×C₃₁	—	4	2	no	—
236	C₂³×C₃₁	—	8	2	no	—
237	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	Q₈×C₃₁	—	2	2	no	—
240	Q₈×C₃₁	—	2	4	no	—
241	Q₈×C₃₁	—	2	4	no	—
242	Q₈×C₃₁	—	2	4	no	—
243	Q₈×C₃₁	—	4	4	no	—
244	Q₈×C₃₁	—	4	2	no	—
245	Q₈×C₃₁	—	4	4	no	—
246	Q₈×C₃₁	—	8	2	no	—
247	D₄×C₃₁	—	2	4	no	—
248	D₄×C₃₁	—	2	2	no	—
249	D₄×C₃₁	—	2	4	no	—
250	D₄×C₃₁	—	4	2	no	—
251	D₄×C₃₁	—	8	4	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	4	4	no	—
259	C₂×C₄×C₃₁	—	4	2	no	—
260	C₂×C₄×C₃₁	—	4	2	no	—
261	C₂×C₄×C₃₁	—	4	4	no	—
262	C₂×C₄×C₃₁	—	8	4	no	—
263	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
269	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
270	C₈×C₃₁	—	2	4	no	—
271	C₈×C₃₁	C₂×#41	1	4	yes	—
272	C₈×C₃₁	C₂×#41	1	4	yes	—
273	#3510	C₂×C₃₁	1	1	yes	characteristic
274	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#3181	C₂×C₃₁	1	4	yes	complemented
276	#3178	C₂×C₃₁	1	8	yes	complemented
277	#3172	C₂×C₃₁	1	4	yes	complemented
278	#3162	C₂×C₃₁	1	1	yes	characteristic, complemented
279	C₂×#497	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#3140	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#3139	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#3059	C₂×C₃₁	1	2	yes	complemented
283	#3057	C₂×C₃₁	1	2	yes	—
284	#2950	C₂×C₃₁	1	2	yes	complemented
285	#2949	C₂×C₃₁	1	2	yes	complemented
286	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	4	no	—
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	4	4	no	—
291	C₂²×C₃₁	—	4	2	no	—
292	C₂²×C₃₁	—	16	2	no	—
293	C₂²×C₃₁	#532	1	2	yes	—
294	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
295	C₄×C₃₁	—	2	2	no	—
296	C₄×C₃₁	—	2	4	no	—
297	C₄×C₃₁	—	2	2	no	—
298	C₄×C₃₁	—	2	2	no	—
299	C₄×C₃₁	—	4	2	no	—
300	C₄×C₃₁	—	4	2	no	—
301	C₄×C₃₁	—	4	4	no	—
302	C₄×C₃₁	—	4	2	no	—
303	C₄×C₃₁	—	4	4	no	—
304	C₄×C₃₁	#532	1	2	yes	—
305	C₄×C₃₁	C₂²×#41	1	2	yes	—
306	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
307	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
308	C₂×#131	C₂²×C₃₁	1	2	yes	—
309	#559	—	2	2	no	—
310	#557	—	2	2	no	—
311	#556	—	2	2	no	—
312	#556	—	2	2	no	—
313	#552	—	2	4	no	—
314	#551	—	2	4	no	—
315	#542	C₂²×C₃₁	1	4	yes	—
316	#541	C₂²×C₃₁	1	4	yes	complemented
317	#536	C₂²×C₃₁	1	2	yes	—
318	#536	C₂²×C₃₁	1	1	yes	characteristic
319	#535	C₂²×C₃₁	1	1	yes	characteristic
320	#533	C₂²×C₃₁	1	1	yes	characteristic
321	#533	C₂²×C₃₁	1	1	yes	characteristic
322	#533	C₂²×C₃₁	1	4	yes	complemented
323	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
324	#529	C₂²×C₃₁	1	4	yes	—
325	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
326	#519	C₂²×C₃₁	1	2	yes	—
327	#519	C₂²×C₃₁	1	1	yes	characteristic
328	#518	C₂²×C₃₁	1	1	yes	characteristic
329	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
330	#501	C₂²×C₃₁	1	2	yes	complemented
331	#501	C₂²×C₃₁	1	4	yes	complemented
332	#501	C₂²×C₃₁	1	2	yes	complemented
333	#497	C₂²×C₃₁	1	2	yes	—
334	#497	C₂²×C₃₁	1	4	yes	—
335	#497	C₂²×C₃₁	1	2	yes	—
336	#491	C₂²×C₃₁	1	4	yes	complemented
337	#489	C₂²×C₃₁	1	4	yes	—
338	#488	C₂²×C₃₁	1	4	yes	—
339	#487	C₂²×C₃₁	1	4	yes	complemented
340	#470	C₂²×C₃₁	1	8	yes	complemented
341	#468	C₂²×C₃₁	1	8	yes	—
342	#467	C₂²×C₃₁	1	8	yes	complemented
343	#465	C₂²×C₃₁	1	8	yes	—
344	#441	C₂²×C₃₁	1	8	yes	complemented
345	#440	C₂²×C₃₁	1	8	yes	—
346	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
347	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
348	#427	C₂²×C₃₁	1	2	yes	—
349	#427	C₂²×C₃₁	1	2	yes	—
350	#419	C₂²×C₃₁	1	2	yes	—
351	#419	C₂²×C₃₁	1	2	yes	—
352	#419	C₂²×C₃₁	1	2	yes	—
353	#418	C₂²×C₃₁	1	2	yes	—
354	#405	C₂²×C₃₁	1	2	yes	—
355	#405	C₂²×C₃₁	1	4	yes	complemented
356	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
357	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
358	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
360	C₂×C₃₁	—	2	2	no	—
361	C₂×C₃₁	—	4	2	no	—
362	C₂×C₃₁	—	8	2	no	—
363	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
364	C₂×C₃₁	#3586	1	2	yes	—
365	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
366	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
368	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂×Q₁₆	—	2	8	no	—
370	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
371	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
372	C₂×SD₁₆	—	2	8	no	—
373	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
374	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
375	#131	—	2	8	no	—
376	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
377	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
378	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
379	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
380	C₄⋊Q₈	—	2	1	no	—
381	C₄⋊Q₈	—	2	2	no	—
382	C₄⋊Q₈	—	2	2	no	—
383	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
384	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
385	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
386	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
387	#127	—	2	1	no	—
388	#126	—	2	4	no	—
389	#126	—	2	8	no	—
390	#126	—	2	4	no	—
391	#125	—	2	8	no	—
392	#125	—	2	4	no	—
393	#125	C₂³×C₃₁	1	2	yes	—
394	#125	C₂³×C₃₁	1	2	yes	—
395	#124	—	2	1	no	—
396	#124	—	2	2	no	—
397	#124	—	2	1	no	—
398	#124	—	2	4	no	—
399	#124	—	2	2	no	—
400	#124	C₂³×C₃₁	1	2	yes	—
401	#124	C₂³×C₃₁	1	2	yes	—
402	#123	—	2	2	no	—
403	#123	—	2	4	no	—
404	#123	—	2	4	no	—
405	#123	—	2	4	no	—
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	4	no	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
411	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
412	#121	C₂³×C₃₁	1	4	yes	—
413	#120	—	2	2	no	—
414	C₄×Q₈	—	2	4	no	—
415	C₄×Q₈	—	2	2	no	—
416	C₄×Q₈	—	2	2	no	—
417	C₄×Q₈	—	2	2	no	—
418	C₄×Q₈	—	2	4	no	—
419	C₄×Q₈	D₄×C₃₁	1	2	yes	—
420	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
421	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
422	C₄×D₄	—	2	4	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	D₄×C₃₁	1	4	yes	—
425	C₄×D₄	D₄×C₃₁	1	2	yes	—
426	C₄×D₄	C₂³×C₃₁	1	4	yes	—
427	#117	—	2	4	no	—
428	#117	—	2	2	no	—
429	#117	—	2	4	no	—
430	#117	—	2	4	no	—
431	#117	C₂³×C₃₁	1	1	yes	characteristic
432	#117	C₂³×C₃₁	1	1	yes	characteristic
433	#117	C₂³×C₃₁	1	2	yes	—
434	#117	C₂³×C₃₁	1	1	yes	characteristic
435	#117	C₂³×C₃₁	1	2	yes	—
436	C₂×C₄⋊C₄	—	2	2	no	—
437	C₂×C₂²⋊C₄	—	2	2	no	—
438	C₂×C₂²⋊C₄	—	2	2	no	—
439	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
441	#107	C₂³×C₃₁	1	4	yes	—
442	#107	C₂³×C₃₁	1	4	yes	—
443	#106	C₂³×C₃₁	1	4	yes	—
444	#106	C₂³×C₃₁	1	4	yes	—
445	#103	C₂³×C₃₁	1	8	yes	—
446	#103	C₂³×C₃₁	1	8	yes	—
447	#103	C₂³×C₃₁	1	8	yes	—
448	#102	C₂³×C₃₁	1	8	yes	—
449	#97	C₂³×C₃₁	1	1	yes	characteristic
450	#97	C₂³×C₃₁	1	1	yes	characteristic
451	#97	C₂³×C₃₁	1	2	yes	—
452	#97	C₂³×C₃₁	1	2	yes	—
453	#97	C₂³×C₃₁	1	4	yes	—
454	C₄×C₈	C₂³×C₃₁	1	2	yes	—
455	C₄×C₈	C₂³×C₃₁	1	4	yes	—
456	C₃₁	#60720	1	1	yes	characteristic, direct product factor, Sylow
457	C₂⁴	—	2	1	no	—
458	#41	—	2	4	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	2	no	—
462	C₂×Q₈	—	2	2	no	—
463	C₂×Q₈	—	2	4	no	—
464	C₂×Q₈	—	2	4	no	—
465	C₂×Q₈	—	2	4	no	—
466	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
467	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
468	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
473	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
481	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
482	Q₁₆	—	2	8	no	—
483	Q₁₆	—	2	8	no	—
484	SD₁₆	—	2	8	no	—
485	SD₁₆	—	2	8	no	—
486	M₁₆	—	2	4	no	—
487	M₁₆	—	2	8	no	—
488	C₂×C₈	—	2	8	no	—
489	C₂×C₈	#41×C₃₁	1	8	yes	—
490	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
491	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
492	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
505	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
506	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
507	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
508	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
509	C₂²⋊C₄	—	2	1	no	—
510	C₂²⋊C₄	—	2	1	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	8	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
519	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
520	C₄²	—	2	4	no	—
521	C₄²	—	2	8	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	1	no	—
524	C₄²	—	2	2	no	—
525	C₄²	C₂×D₄×C₃₁	1	4	yes	—
526	C₄²	C₂×D₄×C₃₁	1	2	yes	—
527	C₄²	C₂⁴×C₃₁	1	2	yes	—
528	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
529	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
530	C₂³	—	2	2	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	8	2	no	—
533	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
534	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
535	Q₈	—	2	2	no	—
536	Q₈	—	2	4	no	—
537	Q₈	—	2	4	no	—
538	Q₈	—	2	4	no	—
539	Q₈	—	4	4	no	—
540	Q₈	—	4	2	no	—
541	Q₈	—	4	4	no	—
542	Q₈	—	8	2	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	2	no	—
545	D₄	—	2	4	no	—
546	D₄	—	4	2	no	—
547	D₄	—	8	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	8	4	no	—
559	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
560	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
561	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
562	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
563	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
564	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
565	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
566	C₈	—	2	4	no	—
567	C₈	C₂×#41×C₃₁	1	4	yes	—
568	C₈	C₂×#41×C₃₁	1	4	yes	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	16	2	no	—
575	C₂²	C₃₁×#532	1	2	yes	—
576	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	4	no	—
580	C₄	—	2	2	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	4	no	—
586	C₄	C₃₁×#532	1	2	yes	—
587	C₄	C₂²×#41×C₃₁	1	2	yes	—
588	C₂	—	2	2	no	—
589	C₂	—	4	2	no	—
590	C₂	—	8	2	no	—
591	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
592	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×#535	C₂²
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
2. C₃₁×#3509	C₂	semidirect
3. C₃₁×#3181	C₂	semidirect
4. C₃₁×#3178	C₂	semidirect
5. C₃₁×#3172	C₂	semidirect
6. C₃₁×#3162	C₂	semidirect
7. C₂×C₃₁×#497	C₂	semidirect
8. C₃₁×#3140	C₂	semidirect
9. C₃₁×#3139	C₂	semidirect
10. C₃₁×#3059	C₂	semidirect
12. C₃₁×#2950	C₂	semidirect
13. C₃₁×#2949	C₂	semidirect
14. C₃₁×#2875	C₂	semidirect
25. C₃₁×#541	C₂²	semidirect
31. C₃₁×#533	C₂²	semidirect
39. C₃₁×#501	C₂²	semidirect
40. C₃₁×#501	C₂²	semidirect
41. C₃₁×#501	C₂²	semidirect
45. C₃₁×#491	C₂²	semidirect
48. C₃₁×#487	C₂²	semidirect
49. C₃₁×#470	C₂²	semidirect
51. C₃₁×#467	C₂²	semidirect
53. C₃₁×#441	C₂²	semidirect
55. C₄×Q₁₆×C₃₁	C₂²	semidirect
64. C₃₁×#405	C₂²	semidirect
66. C₄×M₁₆×C₃₁	C₂²	semidirect
68. C₂×C₃₁×#97	C₂²	semidirect
233. #60720	C₃₁	direct (N × H)
274. #3509	C₂×C₃₁	semidirect
275. #3181	C₂×C₃₁	semidirect
276. #3178	C₂×C₃₁	semidirect
277. #3172	C₂×C₃₁	semidirect
278. #3162	C₂×C₃₁	semidirect
279. C₂×#497	C₂×C₃₁	semidirect
280. #3140	C₂×C₃₁	semidirect
281. #3139	C₂×C₃₁	semidirect
282. #3059	C₂×C₃₁	semidirect
284. #2950	C₂×C₃₁	semidirect
285. #2949	C₂×C₃₁	semidirect
286. #2875	C₂×C₃₁	semidirect
316. #541	C₂²×C₃₁	semidirect
322. #533	C₂²×C₃₁	semidirect
330. #501	C₂²×C₃₁	semidirect
331. #501	C₂²×C₃₁	semidirect
332. #501	C₂²×C₃₁	semidirect
336. #491	C₂²×C₃₁	semidirect
339. #487	C₂²×C₃₁	semidirect
340. #470	C₂²×C₃₁	semidirect
342. #467	C₂²×C₃₁	semidirect
344. #441	C₂²×C₃₁	semidirect
346. C₄×Q₁₆	C₂²×C₃₁	semidirect
355. #405	C₂²×C₃₁	semidirect
357. C₄×M₁₆	C₂²×C₃₁	semidirect
359. C₂×#97	C₂²×C₃₁	semidirect
456. C₃₁	#60720	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×#3510	2
C₃₁×#3509	2
C₃₁×#3181	2
C₃₁×#3178	2
C₃₁×#3172	2
C₃₁×#3162	2
C₂×C₃₁×#497	2
C₃₁×#3140	2
C₃₁×#3139	2
C₃₁×#3059	2
C₃₁×#3057	2
C₃₁×#2950	2
C₃₁×#2949	2
C₃₁×#2875	2
#60720	31

Subgroups — C31×#60720 #21517479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60720 #21517479