Subgroups — C₃₁×#17653 #21474412

Order 7936.

964 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3515	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2970	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1532	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1531	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1529	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1528	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1526	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1524	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1524	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#330	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#557	C₄	1	1	yes	characteristic
14	C₃₁×#554	C₄	1	1	yes	characteristic
15	C₃₁×#554	C₄	1	1	yes	characteristic
16	C₃₁×#553	C₄	1	2	yes	—
17	C₃₁×#552	—	2	2	no	—
18	C₃₁×#552	—	2	1	no	—
19	C₃₁×#552	C₄	1	2	yes	—
20	C₃₁×#547	—	2	1	no	—
21	C₃₁×#547	C₄	1	1	yes	characteristic
22	C₃₁×#541	—	2	1	no	—
23	C₃₁×#539	—	2	2	no	—
24	C₃₁×#533	C₂²	1	1	yes	characteristic
25	C₃₁×#529	C₂²	1	1	yes	characteristic
26	C₃₁×#529	C₂²	1	2	yes	—
27	C₂×C₃₁×#125	C₂²	1	2	yes	—
28	C₃₁×#520	—	2	1	no	—
29	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#446	—	2	1	no	—
31	C₃₁×#435	—	2	2	no	—
32	C₈×D₄×C₃₁	—	2	1	no	—
33	C₃₁×#432	C₂²	1	2	yes	—
34	C₃₁×#432	C₂²	1	1	yes	characteristic
35	C₃₁×#405	—	2	1	no	—
36	C₃₁×#405	—	2	2	no	—
37	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#97	C₂²	1	2	yes	—
41	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#97	C₂²	1	2	yes	—
43	C₃₁×#330	C₂²	1	1	yes	characteristic
44	C₃₁×#330	C₂²	1	2	yes	—
45	C₃₁×#330	C₂²	1	2	yes	—
46	C₃₁×#330	C₂²	1	2	yes	—
47	C₃₁×#330	C₂²	1	1	yes	characteristic
48	C₃₁×#330	C₂²	1	2	yes	—
49	C₃₁×#330	C₂²	1	2	yes	—
50	C₃₁×#329	C₂²	1	1	yes	characteristic
51	C₃₁×#329	C₂²	1	2	yes	—
52	C₃₁×#329	C₂²	1	1	yes	characteristic
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
55	C₃₁×#131	—	2	8	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
65	C₃₁×#126	—	2	2	no	—
66	C₃₁×#126	—	2	2	no	—
67	C₃₁×#126	—	2	2	no	—
68	C₃₁×#126	—	2	1	no	—
69	C₃₁×#125	—	2	2	no	—
70	C₃₁×#125	—	2	2	no	—
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×#125	—	2	2	no	—
73	C₃₁×#125	—	2	2	no	—
74	C₃₁×#125	C₂×C₄	1	2	yes	—
75	C₃₁×#125	C₂×C₄	1	2	yes	—
76	C₃₁×#125	C₂³	1	1	yes	characteristic
77	C₃₁×#125	C₂³	1	2	yes	—
78	C₃₁×#124	C₂³	1	1	yes	characteristic
79	C₃₁×#123	—	2	2	no	—
80	C₃₁×#123	—	2	1	no	—
81	C₃₁×#123	—	2	2	no	—
82	C₃₁×#123	—	2	2	no	—
83	C₃₁×#123	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	—	2	1	no	—
85	C₃₁×C₂²⋊Q₈	—	2	1	no	—
86	C₃₁×C₂²⋊Q₈	—	2	1	no	—
87	C₃₁×C₂²⋊Q₈	—	2	2	no	—
88	C₃₁×C₂²⋊Q₈	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	—	2	2	no	—
90	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
91	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
92	C₃₁×#121	—	2	2	no	—
93	C₃₁×#121	C₂×C₄	1	2	yes	—
94	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	no	—
97	C₄×Q₈×C₃₁	—	2	1	no	—
98	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
99	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
100	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
101	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	—	2	2	no	—
106	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
108	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
109	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
110	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
111	C₄×D₄×C₃₁	D₄	1	2	yes	—
112	C₃₁×#117	—	2	2	no	—
113	C₃₁×#117	—	2	1	no	—
114	C₃₁×#117	—	2	2	no	—
115	C₃₁×#117	C₂×C₄	1	2	yes	—
116	C₃₁×#117	D₄	1	1	yes	characteristic
117	C₃₁×#117	D₄	1	2	yes	—
118	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₃₁×C₄⋊C₈	—	2	1	no	—
126	C₃₁×C₄⋊C₈	—	2	2	no	—
127	C₃₁×C₂²⋊C₈	—	2	2	no	—
128	C₃₁×C₂²⋊C₈	—	2	1	no	—
129	C₃₁×#97	—	2	2	no	—
130	C₃₁×#97	—	2	1	no	—
131	C₃₁×#97	—	2	2	no	—
132	C₃₁×#97	C₂³	1	1	yes	characteristic
133	C₃₁×#97	C₂³	1	1	yes	characteristic
134	C₃₁×#97	C₂³	1	2	yes	—
135	C₃₁×#97	C₂³	1	1	yes	characteristic
136	C₃₁×#97	C₂³	1	2	yes	—
137	C₃₁×#97	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	—	2	2	no	—
139	C₄×C₈×C₃₁	—	2	1	no	—
140	C₄×C₈×C₃₁	—	2	1	no	—
141	#41×C₃₁	—	2	4	no	—
142	#41×C₃₁	—	4	4	no	—
143	C₂×Q₈×C₃₁	—	2	1	no	—
144	C₂×Q₈×C₃₁	—	2	1	no	—
145	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₂×D₄×C₃₁	—	2	2	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
150	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	—	2	2	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	M₁₆×C₃₁	—	2	8	no	—
163	M₁₆×C₃₁	—	2	4	no	—
164	M₁₆×C₃₁	—	4	2	no	—
165	C₂×C₈×C₃₁	—	2	8	no	—
166	C₂×C₈×C₃₁	—	2	4	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	1	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	—	4	4	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
193	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
194	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
197	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₂²⋊C₄×C₃₁	—	2	1	no	—
201	C₂²⋊C₄×C₃₁	—	2	1	no	—
202	C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	1	no	—
205	C₂²⋊C₄×C₃₁	—	2	1	no	—
206	C₂²⋊C₄×C₃₁	—	2	1	no	—
207	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
208	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
209	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
210	C₄²×C₃₁	—	2	2	no	—
211	C₄²×C₃₁	—	2	2	no	—
212	C₄²×C₃₁	—	2	1	no	—
213	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
214	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂×D₄	1	2	yes	—
216	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₃₁	C₂×D₄	1	2	yes	—
219	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
220	#17653	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂³×C₃₁	—	2	1	no	—
222	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
224	Q₈×C₃₁	—	2	2	no	—
225	Q₈×C₃₁	—	4	1	no	—
226	Q₈×C₃₁	—	4	2	no	—
227	Q₈×C₃₁	—	8	1	no	—
228	D₄×C₃₁	—	2	4	no	—
229	D₄×C₃₁	—	4	1	no	—
230	D₄×C₃₁	—	4	4	no	—
231	D₄×C₃₁	—	8	1	no	—
232	D₄×C₃₁	#104	1	4	yes	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	4	2	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	4	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	8	2	no	—
247	C₂×C₄×C₃₁	—	8	2	no	—
248	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
250	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
257	C₈×C₃₁	—	2	2	no	—
258	C₈×C₃₁	—	2	4	no	—
259	C₈×C₃₁	—	2	2	no	—
260	C₈×C₃₁	—	2	2	no	—
261	C₈×C₃₁	—	4	2	no	—
262	C₈×C₃₁	—	4	1	no	—
263	#3515	C₂×C₃₁	1	1	yes	characteristic
264	#2970	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#1532	C₂×C₃₁	1	2	yes	complemented
267	#1531	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1529	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1528	C₂×C₃₁	1	2	yes	complemented
270	#1526	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1524	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#1524	C₂×C₃₁	1	2	yes	complemented
273	C₂×#330	C₂×C₃₁	1	2	yes	complemented
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	1	no	—
277	C₂²×C₃₁	—	2	1	no	—
278	C₂²×C₃₁	—	2	1	no	—
279	C₂²×C₃₁	—	4	1	no	—
280	C₂²×C₃₁	—	4	1	no	—
281	C₂²×C₃₁	—	4	1	no	—
282	C₂²×C₃₁	C₂×#104	1	4	yes	—
283	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
284	C₄×C₃₁	—	2	1	no	—
285	C₄×C₃₁	—	2	1	no	—
286	C₄×C₃₁	—	2	2	no	—
287	C₄×C₃₁	—	2	1	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	1	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	4	1	no	—
293	C₄×C₃₁	—	4	1	no	—
294	C₄×C₃₁	—	4	2	no	—
295	C₄×C₃₁	—	4	1	no	—
296	C₄×C₃₁	—	4	2	no	—
297	C₄×C₃₁	C₂×#104	1	2	yes	—
298	C₂×#131	—	2	1	no	—
299	#557	C₄×C₃₁	1	1	yes	characteristic
300	#554	C₄×C₃₁	1	1	yes	characteristic
301	#554	C₄×C₃₁	1	1	yes	characteristic
302	#553	C₄×C₃₁	1	2	yes	—
303	#552	—	2	2	no	—
304	#552	—	2	1	no	—
305	#552	C₄×C₃₁	1	2	yes	—
306	#547	—	2	1	no	—
307	#547	C₄×C₃₁	1	1	yes	characteristic
308	#541	—	2	1	no	—
309	#539	—	2	2	no	—
310	#533	C₂²×C₃₁	1	1	yes	characteristic
311	#529	C₂²×C₃₁	1	1	yes	characteristic
312	#529	C₂²×C₃₁	1	2	yes	—
313	C₂×#125	C₂²×C₃₁	1	2	yes	—
314	#520	—	2	1	no	—
315	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
316	#446	—	2	1	no	—
317	#435	—	2	2	no	—
318	C₈×D₄	—	2	1	no	—
319	#432	C₂²×C₃₁	1	2	yes	—
320	#432	C₂²×C₃₁	1	1	yes	characteristic
321	#405	—	2	1	no	—
322	#405	—	2	2	no	—
323	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
324	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
325	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×#97	C₂²×C₃₁	1	2	yes	—
327	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×#97	C₂²×C₃₁	1	2	yes	—
329	#330	C₂²×C₃₁	1	1	yes	characteristic
330	#330	C₂²×C₃₁	1	2	yes	—
331	#330	C₂²×C₃₁	1	2	yes	—
332	#330	C₂²×C₃₁	1	2	yes	—
333	#330	C₂²×C₃₁	1	1	yes	characteristic
334	#330	C₂²×C₃₁	1	2	yes	—
335	#330	C₂²×C₃₁	1	2	yes	—
336	#329	C₂²×C₃₁	1	1	yes	characteristic
337	#329	C₂²×C₃₁	1	2	yes	—
338	#329	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×C₃₁	—	2	4	no	—
340	C₂×C₃₁	—	4	1	no	—
341	C₂×C₃₁	—	8	1	no	—
342	C₂×C₃₁	#2889	1	1	yes	characteristic
343	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
344	C₂×C₃₁	#2901	1	1	yes	characteristic
345	C₂×#41	—	2	1	no	—
346	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
347	#131	—	2	8	no	—
348	C₂×M₁₆	—	2	2	no	—
349	C₂×M₁₆	—	2	1	no	—
350	C₂×M₁₆	—	2	1	no	—
351	C₂²×C₈	—	2	2	no	—
352	C₂²×C₈	—	2	1	no	—
353	C₂²×C₈	—	2	2	no	—
354	C₄⋊Q₈	—	2	1	no	—
355	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
357	#126	—	2	2	no	—
358	#126	—	2	2	no	—
359	#126	—	2	2	no	—
360	#126	—	2	1	no	—
361	#125	—	2	2	no	—
362	#125	—	2	2	no	—
363	#125	—	2	1	no	—
364	#125	—	2	2	no	—
365	#125	—	2	2	no	—
366	#125	C₂×C₄×C₃₁	1	2	yes	—
367	#125	C₂×C₄×C₃₁	1	2	yes	—
368	#125	C₂³×C₃₁	1	1	yes	characteristic
369	#125	C₂³×C₃₁	1	2	yes	—
370	#124	C₂³×C₃₁	1	1	yes	characteristic
371	#123	—	2	2	no	—
372	#123	—	2	1	no	—
373	#123	—	2	2	no	—
374	#123	—	2	2	no	—
375	#123	—	2	2	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
383	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
384	#121	—	2	2	no	—
385	#121	C₂×C₄×C₃₁	1	2	yes	—
386	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
391	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
392	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
393	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
394	C₄×D₄	—	2	1	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
399	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
400	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
401	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
402	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
403	C₄×D₄	D₄×C₃₁	1	2	yes	—
404	#117	—	2	2	no	—
405	#117	—	2	1	no	—
406	#117	—	2	2	no	—
407	#117	C₂×C₄×C₃₁	1	2	yes	—
408	#117	D₄×C₃₁	1	1	yes	characteristic
409	#117	D₄×C₃₁	1	2	yes	—
410	C₂×C₄⋊C₄	—	2	2	no	—
411	C₂×C₄⋊C₄	—	2	2	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
414	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
415	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
416	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
417	C₄⋊C₈	—	2	1	no	—
418	C₄⋊C₈	—	2	2	no	—
419	C₂²⋊C₈	—	2	2	no	—
420	C₂²⋊C₈	—	2	1	no	—
421	#97	—	2	2	no	—
422	#97	—	2	1	no	—
423	#97	—	2	2	no	—
424	#97	C₂³×C₃₁	1	1	yes	characteristic
425	#97	C₂³×C₃₁	1	1	yes	characteristic
426	#97	C₂³×C₃₁	1	1	yes	characteristic
427	#97	C₂³×C₃₁	1	2	yes	—
428	#97	C₂³×C₃₁	1	2	yes	—
429	#97	C₂³×C₃₁	1	1	yes	characteristic
430	C₄×C₈	—	2	2	no	—
431	C₄×C₈	—	2	1	no	—
432	C₄×C₈	—	2	1	no	—
433	C₃₁	#17653	1	1	yes	characteristic, direct product factor, Sylow
434	#41	—	2	4	no	—
435	#41	—	4	4	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
439	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
443	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
450	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	M₁₆	—	2	8	no	—
456	M₁₆	—	2	4	no	—
457	M₁₆	—	4	2	no	—
458	C₂×C₈	—	2	8	no	—
459	C₂×C₈	—	2	4	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	2	no	—
466	C₂×C₈	—	4	4	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
484	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
485	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
486	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
487	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
488	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
489	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
490	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	2	1	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	1	no	—
498	C₂²⋊C₄	—	2	1	no	—
499	C₂²⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
501	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
502	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	1	no	—
506	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
507	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
508	C₄²	C₂×D₄×C₃₁	1	2	yes	—
509	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
510	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₃₁	1	2	yes	—
512	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
513	C₂³	—	2	1	no	—
514	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
515	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
516	Q₈	—	2	2	no	—
517	Q₈	—	4	1	no	—
518	Q₈	—	4	2	no	—
519	Q₈	—	8	1	no	—
520	D₄	—	2	4	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	4	no	—
523	D₄	—	8	1	no	—
524	D₄	C₃₁×#104	1	4	yes	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	4	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
541	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
542	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
543	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
544	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
545	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
546	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₈	—	2	2	no	—
550	C₈	—	2	4	no	—
551	C₈	—	2	2	no	—
552	C₈	—	2	2	no	—
553	C₈	—	4	2	no	—
554	C₈	—	4	1	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	1	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	—	4	1	no	—
563	C₂²	C₂×C₃₁×#104	1	4	yes	—
564	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
565	C₄	—	2	1	no	—
566	C₄	—	2	1	no	—
567	C₄	—	2	2	no	—
568	C₄	—	2	1	no	—
569	C₄	—	2	2	no	—
570	C₄	—	2	1	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	1	no	—
573	C₄	—	4	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	1	no	—
577	C₄	—	4	2	no	—
578	C₄	C₂×C₃₁×#104	1	2	yes	—
579	C₂	—	2	4	no	—
580	C₂	—	4	1	no	—
581	C₂	—	8	1	no	—
582	C₂	C₃₁×#2889	1	1	yes	characteristic
583	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic
584	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3515	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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×#2970	C₂	semidirect
3. C₃₁×#2875	C₂	semidirect
4. C₃₁×#1532	C₂	semidirect
5. C₃₁×#1531	C₂	semidirect
6. C₃₁×#1529	C₂	semidirect
7. C₃₁×#1528	C₂	semidirect
8. C₃₁×#1526	C₂	semidirect
9. C₃₁×#1524	C₂	semidirect
10. C₃₁×#1524	C₂	semidirect
11. C₂×C₃₁×#330	C₂	semidirect
220. #17653	C₃₁	direct (N × H)
264. #2970	C₂×C₃₁	semidirect
265. #2875	C₂×C₃₁	semidirect
266. #1532	C₂×C₃₁	semidirect
267. #1531	C₂×C₃₁	semidirect
268. #1529	C₂×C₃₁	semidirect
269. #1528	C₂×C₃₁	semidirect
270. #1526	C₂×C₃₁	semidirect
271. #1524	C₂×C₃₁	semidirect
272. #1524	C₂×C₃₁	semidirect
273. C₂×#330	C₂×C₃₁	semidirect
433. C₃₁	#17653	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×#3515	2
C₃₁×#2970	2
C₃₁×#2875	2
C₃₁×#1532	2
C₃₁×#1531	2
C₃₁×#1529	2
C₃₁×#1528	2
C₃₁×#1526	2
C₃₁×#1524	2
C₂×C₃₁×#330	2
#17653	31

Subgroups — C31×#17653 #21474412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17653 #21474412