Subgroups — C₃₁×#17687 #21474446

Order 7936.

896 conjugacy classes of subgroups.

The subgroup lattice has 600 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	1	yes	maximal, characteristic
2	C₃₁×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2970	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1535	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1532	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1529	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1528	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1528	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1525	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1524	C₂	1	2	yes	maximal, complemented
12	C₃₁×#553	—	2	2	no	—
13	C₃₁×#553	—	2	2	no	—
14	C₃₁×#553	—	2	1	no	—
15	C₃₁×#553	—	2	1	no	—
16	C₃₁×#552	—	2	1	no	—
17	C₃₁×#552	—	2	1	no	—
18	C₃₁×#541	C₄	1	2	yes	—
19	C₃₁×#540	C₄	1	2	yes	—
20	C₃₁×#537	C₄	1	1	yes	characteristic
21	C₃₁×#537	C₄	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#529	C₂²	1	2	yes	—
24	C₃₁×#529	C₂²	1	2	yes	—
25	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
26	C₃₁×#520	C₄	1	1	yes	characteristic
27	C₃₁×#520	C₄	1	1	yes	characteristic
28	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#446	—	2	1	no	—
30	C₈×Q₈×C₃₁	—	2	1	no	—
31	C₃₁×#436	—	2	2	no	—
32	C₃₁×#436	—	2	1	no	—
33	C₃₁×#435	—	2	2	no	—
34	C₈×D₄×C₃₁	—	2	1	no	—
35	C₃₁×#432	C₂²	1	1	yes	characteristic
36	C₃₁×#432	C₂²	1	2	yes	—
37	C₃₁×#432	C₂²	1	1	yes	characteristic
38	C₃₁×#432	C₂²	1	2	yes	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
41	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#97	C₂²	1	2	yes	—
43	C₃₁×#330	C₂²	1	2	yes	—
44	C₃₁×#330	C₂²	1	2	yes	—
45	C₃₁×#330	C₂²	1	2	yes	—
46	C₃₁×#330	C₂²	1	2	yes	—
47	C₃₁×#329	C₂²	1	1	yes	characteristic
48	C₃₁×#329	C₂²	1	2	yes	—
49	C₃₁×#329	C₂²	1	1	yes	characteristic
50	C₃₁×#329	C₂²	1	1	yes	characteristic
51	C₃₁×#329	C₂²	1	1	yes	characteristic
52	C₃₁×#329	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
61	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
62	C₃₁×#126	—	2	2	no	—
63	C₃₁×#126	—	2	2	no	—
64	C₃₁×#126	—	2	2	no	—
65	C₃₁×#126	—	2	2	no	—
66	C₃₁×#125	—	2	2	no	—
67	C₃₁×#125	—	2	2	no	—
68	C₃₁×#125	—	2	1	no	—
69	C₃₁×#125	—	2	1	no	—
70	C₃₁×#125	C₂×C₄	1	2	yes	—
71	C₃₁×#125	C₂³	1	2	yes	—
72	C₃₁×#124	C₂³	1	1	yes	characteristic
73	C₃₁×#124	C₂³	1	1	yes	characteristic
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×#123	—	2	2	no	—
76	C₃₁×#123	—	2	2	no	—
77	C₃₁×#123	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	1	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
81	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	1	no	—
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	—	2	2	no	—
86	C₃₁×#121	—	2	1	no	—
87	C₃₁×#121	—	2	1	no	—
88	C₃₁×#121	—	2	2	no	—
89	C₃₁×#121	—	2	2	no	—
90	C₃₁×#121	C₂×C₄	1	2	yes	—
91	C₃₁×#121	C₂×C₄	1	2	yes	—
92	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
93	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
94	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
96	C₄×D₄×C₃₁	—	2	2	no	—
97	C₄×D₄×C₃₁	—	2	2	no	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	—	2	2	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
103	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
104	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
105	C₄×D₄×C₃₁	D₄	1	2	yes	—
106	C₄×D₄×C₃₁	D₄	1	2	yes	—
107	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
108	C₃₁×#117	—	2	1	no	—
109	C₃₁×#117	—	2	1	no	—
110	C₃₁×#117	—	2	2	no	—
111	C₃₁×#117	—	2	2	no	—
112	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
113	C₃₁×#117	C₂×C₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
117	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊C₈	—	2	1	no	—
123	C₃₁×C₄⋊C₈	—	2	2	no	—
124	C₃₁×C₄⋊C₈	—	2	2	no	—
125	C₃₁×C₄⋊C₈	—	2	1	no	—
126	C₃₁×C₂²⋊C₈	—	2	2	no	—
127	C₃₁×C₂²⋊C₈	—	2	1	no	—
128	C₃₁×C₂²⋊C₈	—	2	1	no	—
129	C₃₁×C₂²⋊C₈	—	2	2	no	—
130	C₃₁×#97	—	2	2	no	—
131	C₃₁×#97	C₂³	1	1	yes	characteristic
132	C₃₁×#97	C₂³	1	2	yes	—
133	C₃₁×#97	C₂³	1	1	yes	characteristic
134	C₃₁×#97	C₂³	1	1	yes	characteristic
135	C₃₁×#97	C₂³	1	2	yes	—
136	C₃₁×#97	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₃₁	—	2	1	no	—
138	C₄×C₈×C₃₁	—	2	2	no	—
139	C₄×C₈×C₃₁	—	2	1	no	—
140	#41×C₃₁	—	4	2	no	—
141	#41×C₃₁	—	4	2	no	—
142	#41×C₃₁	—	4	2	no	—
143	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
144	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
146	C₂×D₄×C₃₁	—	2	1	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	2	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	—	2	1	no	—
152	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
154	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	—	2	1	no	—
157	C₂²×C₄×C₃₁	—	2	1	no	—
158	C₂²×C₄×C₃₁	—	2	1	no	—
159	C₂²×C₄×C₃₁	—	2	1	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
168	M₁₆×C₃₁	—	4	2	no	—
169	M₁₆×C₃₁	—	4	4	no	—
170	M₁₆×C₃₁	—	4	2	no	—
171	C₂×C₈×C₃₁	—	2	1	no	—
172	C₂×C₈×C₃₁	—	2	1	no	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	2	no	—
175	C₂×C₈×C₃₁	—	2	1	no	—
176	C₂×C₈×C₃₁	—	2	2	no	—
177	C₂×C₈×C₃₁	—	4	4	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	2	2	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	2	yes	—
195	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
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	1	no	—
203	C₂²⋊C₄×C₃₁	—	2	1	no	—
204	C₂²⋊C₄×C₃₁	—	2	1	no	—
205	C₂²⋊C₄×C₃₁	—	2	1	no	—
206	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
207	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
208	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
209	C₄²×C₃₁	—	2	2	no	—
210	C₄²×C₃₁	—	2	2	no	—
211	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
212	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂×D₄	1	2	yes	—
214	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂×D₄	1	2	yes	—
217	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
218	#17687	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₃₁	—	2	1	no	—
220	C₂³×C₃₁	—	2	1	no	—
221	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	Q₈×C₃₁	—	2	2	no	—
224	Q₈×C₃₁	—	4	1	no	—
225	Q₈×C₃₁	—	4	1	no	—
226	D₄×C₃₁	—	2	2	no	—
227	D₄×C₃₁	—	4	2	no	—
228	D₄×C₃₁	—	4	1	no	—
229	D₄×C₃₁	—	4	1	no	—
230	D₄×C₃₁	—	4	2	no	—
231	D₄×C₃₁	—	4	2	no	—
232	D₄×C₃₁	—	8	1	no	—
233	D₄×C₃₁	—	8	1	no	—
234	D₄×C₃₁	—	8	2	no	—
235	D₄×C₃₁	—	8	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	4	2	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	2	no	—
248	C₂×C₄×C₃₁	—	8	2	no	—
249	C₂×C₄×C₃₁	—	8	2	no	—
250	C₂×C₄×C₃₁	—	8	1	no	—
251	C₂×C₄×C₃₁	—	8	1	no	—
252	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
261	C₈×C₃₁	—	2	2	no	—
262	C₈×C₃₁	—	2	2	no	—
263	C₈×C₃₁	—	4	2	no	—
264	C₈×C₃₁	—	4	1	no	—
265	C₈×C₃₁	—	4	2	no	—
266	C₈×C₃₁	—	4	1	no	—
267	#3512	C₂×C₃₁	1	1	yes	characteristic
268	#2971	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#2970	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1536	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1535	C₂×C₃₁	1	2	yes	complemented
272	#1532	C₂×C₃₁	1	2	yes	complemented
273	#1529	C₂×C₃₁	1	2	yes	complemented
274	#1528	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1528	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1525	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1524	C₂×C₃₁	1	2	yes	complemented
278	C₂²×C₃₁	—	2	1	no	—
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	—	2	1	no	—
282	C₂²×C₃₁	—	2	2	no	—
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	4	1	no	—
285	C₂²×C₃₁	—	4	1	no	—
286	C₂²×C₃₁	—	4	1	no	—
287	C₂²×C₃₁	—	4	1	no	—
288	C₂²×C₃₁	—	4	1	no	—
289	C₂²×C₃₁	—	4	1	no	—
290	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	2	2	no	—
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	2	1	no	—
298	C₄×C₃₁	—	2	1	no	—
299	C₄×C₃₁	—	4	1	no	—
300	C₄×C₃₁	—	4	1	no	—
301	C₄×C₃₁	—	4	2	no	—
302	C₄×C₃₁	—	4	2	no	—
303	C₄×C₃₁	#413	1	2	yes	—
304	#553	—	2	2	no	—
305	#553	—	2	2	no	—
306	#553	—	2	1	no	—
307	#553	—	2	1	no	—
308	#552	—	2	1	no	—
309	#552	—	2	1	no	—
310	#541	C₄×C₃₁	1	2	yes	—
311	#540	C₄×C₃₁	1	2	yes	—
312	#537	C₄×C₃₁	1	1	yes	characteristic
313	#537	C₄×C₃₁	1	1	yes	characteristic
314	#532	C₂²×C₃₁	1	1	yes	characteristic
315	#529	C₂²×C₃₁	1	2	yes	—
316	#529	C₂²×C₃₁	1	2	yes	—
317	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
318	#520	C₄×C₃₁	1	1	yes	characteristic
319	#520	C₄×C₃₁	1	1	yes	characteristic
320	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
321	#446	—	2	1	no	—
322	C₈×Q₈	—	2	1	no	—
323	#436	—	2	2	no	—
324	#436	—	2	1	no	—
325	#435	—	2	2	no	—
326	C₈×D₄	—	2	1	no	—
327	#432	C₂²×C₃₁	1	1	yes	characteristic
328	#432	C₂²×C₃₁	1	1	yes	characteristic
329	#432	C₂²×C₃₁	1	2	yes	—
330	#432	C₂²×C₃₁	1	2	yes	—
331	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
332	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
333	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×#97	C₂²×C₃₁	1	2	yes	—
335	#330	C₂²×C₃₁	1	2	yes	—
336	#330	C₂²×C₃₁	1	2	yes	—
337	#330	C₂²×C₃₁	1	2	yes	—
338	#330	C₂²×C₃₁	1	2	yes	—
339	#329	C₂²×C₃₁	1	2	yes	—
340	#329	C₂²×C₃₁	1	2	yes	—
341	#329	C₂²×C₃₁	1	1	yes	characteristic
342	#329	C₂²×C₃₁	1	1	yes	characteristic
343	#329	C₂²×C₃₁	1	1	yes	characteristic
344	#329	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×C₃₁	—	4	1	no	—
346	C₂×C₃₁	—	4	2	no	—
347	C₂×C₃₁	—	8	1	no	—
348	C₂×C₃₁	—	8	1	no	—
349	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
350	C₂×C₃₁	#2901	1	1	yes	characteristic
351	C₂×C₃₁	#2901	1	1	yes	characteristic
352	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
353	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
354	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₂×M₁₆	—	2	1	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	2	no	—
358	C₂²×C₈	—	2	2	no	—
359	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
360	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
361	#126	—	2	2	no	—
362	#126	—	2	2	no	—
363	#126	—	2	2	no	—
364	#126	—	2	2	no	—
365	#125	—	2	2	no	—
366	#125	—	2	2	no	—
367	#125	—	2	1	no	—
368	#125	—	2	1	no	—
369	#125	C₂×C₄×C₃₁	1	2	yes	—
370	#125	C₂³×C₃₁	1	2	yes	—
371	#124	C₂³×C₃₁	1	1	yes	characteristic
372	#124	C₂³×C₃₁	1	1	yes	characteristic
373	#123	—	2	2	no	—
374	#123	—	2	2	no	—
375	#123	—	2	2	no	—
376	#123	—	2	2	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
380	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
381	#121	—	2	1	no	—
382	#121	—	2	1	no	—
383	#121	—	2	2	no	—
384	#121	—	2	2	no	—
385	#121	—	2	1	no	—
386	#121	—	2	1	no	—
387	#121	—	2	2	no	—
388	#121	—	2	2	no	—
389	#121	C₂×C₄×C₃₁	1	2	yes	—
390	#121	C₂×C₄×C₃₁	1	2	yes	—
391	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
392	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
393	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
394	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	1	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
402	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
403	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
404	C₄×D₄	D₄×C₃₁	1	2	yes	—
405	C₄×D₄	D₄×C₃₁	1	2	yes	—
406	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
407	#117	—	2	1	no	—
408	#117	—	2	1	no	—
409	#117	—	2	2	no	—
410	#117	—	2	2	no	—
411	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
412	#117	C₂×C₄×C₃₁	1	2	yes	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
421	C₄⋊C₈	—	2	1	no	—
422	C₄⋊C₈	—	2	2	no	—
423	C₄⋊C₈	—	2	2	no	—
424	C₄⋊C₈	—	2	1	no	—
425	C₂²⋊C₈	—	2	2	no	—
426	C₂²⋊C₈	—	2	1	no	—
427	C₂²⋊C₈	—	2	1	no	—
428	C₂²⋊C₈	—	2	2	no	—
429	#97	—	2	2	no	—
430	#97	C₂³×C₃₁	1	1	yes	characteristic
431	#97	C₂³×C₃₁	1	2	yes	—
432	#97	C₂³×C₃₁	1	1	yes	characteristic
433	#97	C₂³×C₃₁	1	2	yes	—
434	#97	C₂³×C₃₁	1	1	yes	characteristic
435	#97	C₂³×C₃₁	1	1	yes	characteristic
436	C₄×C₈	—	2	1	no	—
437	C₄×C₈	—	2	2	no	—
438	C₄×C₈	—	2	1	no	—
439	C₃₁	#17687	1	1	yes	characteristic, direct product factor, Sylow
440	#41	—	4	2	no	—
441	#41	—	4	2	no	—
442	#41	—	4	2	no	—
443	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
445	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
453	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
454	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
455	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
464	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
468	M₁₆	—	4	2	no	—
469	M₁₆	—	4	4	no	—
470	M₁₆	—	4	2	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	4	4	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
491	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
492	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
493	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
494	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
495	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	2	1	no	—
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
507	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
508	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
509	C₄²	—	2	2	no	—
510	C₄²	—	2	2	no	—
511	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
512	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₃₁	1	2	yes	—
514	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
515	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
516	C₄²	C₂×D₄×C₃₁	1	2	yes	—
517	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
521	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	Q₈	—	2	2	no	—
523	Q₈	—	4	1	no	—
524	Q₈	—	4	1	no	—
525	D₄	—	2	2	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	1	no	—
528	D₄	—	4	1	no	—
529	D₄	—	4	2	no	—
530	D₄	—	4	2	no	—
531	D₄	—	8	1	no	—
532	D₄	—	8	1	no	—
533	D₄	—	8	2	no	—
534	D₄	—	8	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	8	2	no	—
548	C₂×C₄	—	8	2	no	—
549	C₂×C₄	—	8	1	no	—
550	C₂×C₄	—	8	1	no	—
551	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
552	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
553	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
554	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
556	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
557	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
560	C₈	—	2	2	no	—
561	C₈	—	2	2	no	—
562	C₈	—	4	2	no	—
563	C₈	—	4	1	no	—
564	C₈	—	4	2	no	—
565	C₈	—	4	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
579	C₄	—	2	1	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	1	no	—
583	C₄	—	2	2	no	—
584	C₄	—	2	2	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	2	no	—
591	C₄	C₃₁×#413	1	2	yes	—
592	C₂	—	4	1	no	—
593	C₂	—	4	2	no	—
594	C₂	—	8	1	no	—
595	C₂	—	8	1	no	—
596	C₂	C₂×C₃₁×#413	1	1	yes	characteristic
597	C₂	C₃₁×#2901	1	1	yes	characteristic
598	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×#3512	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×#2971	C₂	semidirect
3. C₃₁×#2970	C₂	semidirect
4. C₃₁×#1536	C₂	semidirect
5. C₃₁×#1535	C₂	semidirect
6. C₃₁×#1532	C₂	semidirect
7. C₃₁×#1529	C₂	semidirect
8. C₃₁×#1528	C₂	semidirect
9. C₃₁×#1528	C₂	semidirect
10. C₃₁×#1525	C₂	semidirect
11. C₃₁×#1524	C₂	semidirect
218. #17687	C₃₁	direct (N × H)
268. #2971	C₂×C₃₁	semidirect
269. #2970	C₂×C₃₁	semidirect
270. #1536	C₂×C₃₁	semidirect
271. #1535	C₂×C₃₁	semidirect
272. #1532	C₂×C₃₁	semidirect
273. #1529	C₂×C₃₁	semidirect
274. #1528	C₂×C₃₁	semidirect
275. #1528	C₂×C₃₁	semidirect
276. #1525	C₂×C₃₁	semidirect
277. #1524	C₂×C₃₁	semidirect
439. C₃₁	#17687	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×#2971	2
C₃₁×#2970	2
C₃₁×#1536	2
C₃₁×#1535	2
C₃₁×#1532	2
C₃₁×#1529	2
C₃₁×#1528	2
C₃₁×#1525	2
C₃₁×#1524	2
#17687	31

Subgroups — C31×#17687 #21474446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17687 #21474446