Subgroups — C₃₁×#17522 #21474281

Order 7936.

1008 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3481	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1495	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1486	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1484	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1482	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1478	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1476	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#559	C₄	1	1	yes	characteristic
13	C₃₁×#554	C₄	1	1	yes	characteristic
14	C₃₁×#554	C₄	1	1	yes	characteristic
15	C₃₁×#553	—	2	1	no	—
16	C₃₁×#553	C₄	1	2	yes	—
17	C₃₁×#551	—	2	2	no	—
18	C₃₁×#546	—	2	1	no	—
19	C₃₁×#546	C₄	1	2	yes	—
20	D₄²×C₃₁	C₄	1	1	yes	characteristic
21	C₃₁×#532	C₂²	1	1	yes	characteristic
22	C₃₁×#529	—	2	1	no	—
23	C₃₁×#529	C₂²	1	2	yes	—
24	C₃₁×#525	—	2	2	no	—
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₃₁	—	2	1	no	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₈×Q₈×C₃₁	—	2	1	no	—
30	C₃₁×#436	—	2	1	no	—
31	C₃₁×#436	—	2	2	no	—
32	C₃₁×#433	—	2	2	no	—
33	C₃₁×#431	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#431	C₂²	1	2	yes	complemented
35	C₃₁×#431	C₂²	1	2	yes	complemented
36	C₃₁×#431	C₂²	1	2	yes	complemented
37	C₃₁×#424	—	2	1	no	—
38	C₃₁×#423	C₂²	1	2	yes	complemented
39	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#408	—	2	1	no	—
41	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
46	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
47	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
51	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
52	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×D₄×C₃₁	—	2	1	no	—
57	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂³×C₄×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	—	2	1	no	—
60	C₂×M₁₆×C₃₁	—	2	2	no	—
61	C₂×M₁₆×C₃₁	—	2	1	no	—
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
64	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
65	C₃₁×#126	—	2	2	no	—
66	C₃₁×#126	—	2	2	no	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#125	C₂×C₄	1	2	yes	—
69	C₃₁×#124	—	2	1	no	—
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#124	C₂×C₄	1	2	yes	—
72	C₃₁×#123	—	2	2	no	—
73	C₃₁×#123	—	2	2	no	—
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×#123	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	1	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
80	C₃₁×#121	—	2	2	no	—
81	C₃₁×#121	—	2	1	no	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	2	no	—
84	C₃₁×#121	C₂×C₄	1	2	yes	—
85	C₃₁×#121	C₂×C₄	1	2	yes	—
86	C₃₁×#120	—	2	2	no	—
87	C₃₁×#120	—	2	2	no	—
88	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
89	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
91	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×D₄×C₃₁	—	2	1	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	2	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	2	no	—
100	C₄×D₄×C₃₁	—	2	2	no	—
101	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
102	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
103	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
105	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
106	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
107	C₄×D₄×C₃₁	D₄	1	2	yes	—
108	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
109	C₄×D₄×C₃₁	C₂³	1	2	yes	—
110	C₃₁×#117	—	2	1	no	—
111	C₃₁×#117	—	2	2	no	—
112	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
113	C₃₁×#117	D₄	1	1	yes	characteristic
114	C₃₁×#117	D₄	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₄²×C₃₁	—	2	1	no	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
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	2	no	—
126	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
127	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
128	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
129	C₃₁×C₂²⋊C₈	—	2	2	no	—
130	C₃₁×C₂²⋊C₈	—	2	2	no	—
131	C₃₁×C₂²⋊C₈	—	2	2	no	—
132	C₃₁×C₂²⋊C₈	—	2	1	no	—
133	C₃₁×#97	—	2	1	no	—
134	C₄×C₈×C₃₁	—	2	1	no	—
135	C₄×C₈×C₃₁	—	2	1	no	—
136	C₄×C₈×C₃₁	—	2	2	no	—
137	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₃₁	C₂³	1	2	yes	—
140	C₂⁴×C₃₁	—	2	2	no	—
141	#41×C₃₁	—	2	2	no	—
142	#41×C₃₁	—	4	1	no	—
143	#41×C₃₁	—	4	2	no	—
144	#41×C₃₁	—	8	1	no	—
145	#41×C₃₁	—	8	1	no	—
146	C₂×Q₈×C₃₁	—	2	1	no	—
147	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	—	2	2	no	—
154	C₂×D₄×C₃₁	—	2	1	no	—
155	C₂×D₄×C₃₁	—	2	1	no	—
156	C₂×D₄×C₃₁	—	4	2	no	—
157	C₂×D₄×C₃₁	—	4	4	no	—
158	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
159	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
160	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
161	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	2	1	no	—
166	C₂²×C₄×C₃₁	—	2	2	no	—
167	C₂²×C₄×C₃₁	—	2	2	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	8	2	no	—
170	C₂²×C₄×C₃₁	—	8	2	no	—
171	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
177	M₁₆×C₃₁	—	4	2	no	—
178	M₁₆×C₃₁	—	8	2	no	—
179	M₁₆×C₃₁	—	8	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	—	2	2	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	2	1	no	—
185	C₂×C₈×C₃₁	—	2	2	no	—
186	C₂×C₈×C₃₁	—	4	2	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	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
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	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
200	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
201	C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂²⋊C₄×C₃₁	—	2	1	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	2	no	—
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	1	no	—
208	C₂²⋊C₄×C₃₁	—	2	2	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	1	no	—
211	C₂²⋊C₄×C₃₁	—	2	2	no	—
212	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
213	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
214	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
215	C₄²×C₃₁	—	2	1	no	—
216	C₄²×C₃₁	—	2	1	no	—
217	C₄²×C₃₁	—	2	1	no	—
218	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
219	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
220	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₃₁	C₂×D₄	1	2	yes	—
222	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₃₁	C₂×D₄	1	2	yes	—
224	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
225	#17522	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂³×C₃₁	—	2	1	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	4	4	no	—
229	C₂³×C₃₁	—	4	4	no	—
230	C₂³×C₃₁	—	8	2	no	—
231	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
232	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	Q₈×C₃₁	—	2	2	no	—
234	Q₈×C₃₁	—	2	1	no	—
235	Q₈×C₃₁	—	2	1	no	—
236	Q₈×C₃₁	—	8	1	no	—
237	D₄×C₃₁	—	2	2	no	—
238	D₄×C₃₁	—	2	1	no	—
239	D₄×C₃₁	—	2	2	no	—
240	D₄×C₃₁	—	2	2	no	—
241	D₄×C₃₁	—	4	2	no	—
242	D₄×C₃₁	—	4	4	no	—
243	D₄×C₃₁	—	4	2	no	—
244	D₄×C₃₁	—	4	4	no	—
245	D₄×C₃₁	—	4	2	no	—
246	D₄×C₃₁	—	8	1	no	—
247	D₄×C₃₁	—	8	2	no	—
248	D₄×C₃₁	#131	1	2	yes	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	1	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	1	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	4	2	no	—
263	C₂×C₄×C₃₁	—	4	2	no	—
264	C₂×C₄×C₃₁	—	4	1	no	—
265	C₂×C₄×C₃₁	—	4	2	no	—
266	C₂×C₄×C₃₁	—	4	1	no	—
267	C₂×C₄×C₃₁	—	4	2	no	—
268	C₂×C₄×C₃₁	—	4	2	no	—
269	C₂×C₄×C₃₁	—	8	2	no	—
270	C₂×C₄×C₃₁	—	8	2	no	—
271	C₂×C₄×C₃₁	—	8	2	no	—
272	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
274	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
275	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
281	C₈×C₃₁	—	4	1	no	—
282	C₈×C₃₁	—	4	1	no	—
283	C₈×C₃₁	—	4	2	no	—
284	C₈×C₃₁	—	4	1	no	—
285	C₈×C₃₁	—	4	2	no	—
286	C₈×C₃₁	—	4	1	no	—
287	#3481	C₂×C₃₁	1	1	yes	characteristic
288	#2991	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#1495	C₂×C₃₁	1	2	yes	complemented
291	#1492	C₂×C₃₁	1	1	yes	characteristic, complemented
292	#1489	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#1486	C₂×C₃₁	1	2	yes	complemented
294	#1484	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1482	C₂×C₃₁	1	2	yes	complemented
296	#1478	C₂×C₃₁	1	2	yes	complemented
297	#1476	C₂×C₃₁	1	1	yes	characteristic, complemented
298	C₂²×C₃₁	—	2	1	no	—
299	C₂²×C₃₁	—	2	2	no	—
300	C₂²×C₃₁	—	2	2	no	—
301	C₂²×C₃₁	—	2	2	no	—
302	C₂²×C₃₁	—	2	1	no	—
303	C₂²×C₃₁	—	2	1	no	—
304	C₂²×C₃₁	—	2	2	no	—
305	C₂²×C₃₁	—	4	1	no	—
306	C₂²×C₃₁	—	4	1	no	—
307	C₂²×C₃₁	—	4	2	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	—	4	2	no	—
310	C₂²×C₃₁	—	8	8	no	—
311	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
312	C₄×C₃₁	—	2	1	no	—
313	C₄×C₃₁	—	2	2	no	—
314	C₄×C₃₁	—	2	2	no	—
315	C₄×C₃₁	—	2	1	no	—
316	C₄×C₃₁	—	2	1	no	—
317	C₄×C₃₁	—	2	2	no	—
318	C₄×C₃₁	—	4	1	no	—
319	C₄×C₃₁	—	4	1	no	—
320	C₄×C₃₁	—	4	2	no	—
321	C₄×C₃₁	—	4	1	no	—
322	C₄×C₃₁	#421	1	2	yes	—
323	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
324	C₄×C₃₁	#568	1	1	yes	characteristic
325	C₄×C₃₁	#576	1	1	yes	characteristic
326	C₄×C₃₁	#577	1	1	yes	characteristic
327	#559	C₄×C₃₁	1	1	yes	characteristic
328	#554	C₄×C₃₁	1	1	yes	characteristic
329	#554	C₄×C₃₁	1	1	yes	characteristic
330	#553	—	2	1	no	—
331	#553	C₄×C₃₁	1	2	yes	—
332	#551	—	2	2	no	—
333	#546	—	2	1	no	—
334	#546	C₄×C₃₁	1	2	yes	—
335	D₄²	C₄×C₃₁	1	1	yes	characteristic
336	#532	C₂²×C₃₁	1	1	yes	characteristic
337	#529	—	2	1	no	—
338	#529	C₂²×C₃₁	1	2	yes	—
339	#525	—	2	2	no	—
340	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
341	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₄×D₄	—	2	1	no	—
343	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
344	C₈×Q₈	—	2	1	no	—
345	#436	—	2	1	no	—
346	#436	—	2	2	no	—
347	#433	—	2	2	no	—
348	#431	C₂²×C₃₁	1	1	yes	characteristic, complemented
349	#431	C₂²×C₃₁	1	2	yes	complemented
350	#431	C₂²×C₃₁	1	2	yes	complemented
351	#431	C₂²×C₃₁	1	2	yes	complemented
352	#424	—	2	1	no	—
353	#423	C₂²×C₃₁	1	2	yes	complemented
354	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
355	#408	—	2	1	no	—
356	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
357	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
358	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
359	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
360	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
361	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
362	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
363	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
364	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
365	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
366	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
367	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
368	C₂×C₃₁	—	4	1	no	—
369	C₂×C₃₁	—	4	2	no	—
370	C₂×C₃₁	—	4	2	no	—
371	C₂×C₃₁	—	8	2	no	—
372	C₂×C₃₁	#2880	1	1	yes	characteristic
373	C₂×C₃₁	#2897	1	1	yes	characteristic
374	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
375	C₂×#41	—	2	1	no	—
376	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
377	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
378	C₂²×D₄	—	2	1	no	—
379	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
380	C₂³×C₄	—	2	2	no	—
381	C₂×M₁₆	—	2	1	no	—
382	C₂×M₁₆	—	2	2	no	—
383	C₂×M₁₆	—	2	1	no	—
384	C₂²×C₈	—	2	1	no	—
385	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
386	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
387	#126	—	2	2	no	—
388	#126	—	2	2	no	—
389	#125	—	2	1	no	—
390	#125	C₂×C₄×C₃₁	1	2	yes	—
391	#124	—	2	1	no	—
392	#124	—	2	1	no	—
393	#124	C₂×C₄×C₃₁	1	2	yes	—
394	#123	—	2	2	no	—
395	#123	—	2	2	no	—
396	#123	—	2	2	no	—
397	#123	—	2	2	no	—
398	C₂²⋊Q₈	—	2	1	no	—
399	C₂²⋊Q₈	—	2	2	no	—
400	C₂²⋊Q₈	—	2	2	no	—
401	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
402	#121	—	2	2	no	—
403	#121	—	2	1	no	—
404	#121	—	2	1	no	—
405	#121	—	2	2	no	—
406	#121	C₂×C₄×C₃₁	1	2	yes	—
407	#121	C₂×C₄×C₃₁	1	2	yes	—
408	#120	—	2	2	no	—
409	#120	—	2	2	no	—
410	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
411	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
412	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
413	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	C₄×D₄	—	2	1	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	1	no	—
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	—	2	2	no	—
423	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
424	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
425	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
426	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
427	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
428	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
429	C₄×D₄	D₄×C₃₁	1	2	yes	—
430	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
431	C₄×D₄	C₂³×C₃₁	1	2	yes	—
432	#117	—	2	1	no	—
433	#117	—	2	2	no	—
434	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
435	#117	D₄×C₃₁	1	2	yes	—
436	#117	D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
439	C₂×C₂²⋊C₄	—	2	2	no	—
440	C₂×C₂²⋊C₄	—	2	2	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
443	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
444	C₄⋊C₈	—	2	1	no	—
445	C₄⋊C₈	—	2	2	no	—
446	C₄⋊C₈	—	2	2	no	—
447	C₄⋊C₈	—	2	2	no	—
448	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
449	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
450	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
451	C₂²⋊C₈	—	2	2	no	—
452	C₂²⋊C₈	—	2	2	no	—
453	C₂²⋊C₈	—	2	2	no	—
454	C₂²⋊C₈	—	2	1	no	—
455	#97	—	2	1	no	—
456	C₄×C₈	—	2	1	no	—
457	C₄×C₈	—	2	1	no	—
458	C₄×C₈	—	2	2	no	—
459	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
460	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
461	C₄×C₈	C₂³×C₃₁	1	2	yes	—
462	C₃₁	#17522	1	1	yes	characteristic, direct product factor, Sylow
463	C₂⁴	—	2	2	no	—
464	#41	—	2	2	no	—
465	#41	—	4	1	no	—
466	#41	—	4	2	no	—
467	#41	—	8	1	no	—
468	#41	—	8	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
471	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	2	2	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	4	2	no	—
480	C₂×D₄	—	4	4	no	—
481	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
482	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
483	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
484	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	8	2	no	—
493	C₂²×C₄	—	8	2	no	—
494	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
495	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
497	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
500	M₁₆	—	4	2	no	—
501	M₁₆	—	8	2	no	—
502	M₁₆	—	8	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	2	no	—
508	C₂×C₈	—	2	2	no	—
509	C₂×C₈	—	4	2	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
518	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
519	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
520	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
521	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
522	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
523	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	1	no	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	2	no	—
528	C₂²⋊C₄	—	2	2	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	1	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	1	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
536	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
537	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
542	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
543	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
545	C₄²	C₂×D₄×C₃₁	1	2	yes	—
546	C₄²	C₂×D₄×C₃₁	1	2	yes	—
547	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
548	C₂³	—	2	1	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	4	4	no	—
551	C₂³	—	4	4	no	—
552	C₂³	—	8	2	no	—
553	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
554	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
555	Q₈	—	2	2	no	—
556	Q₈	—	2	1	no	—
557	Q₈	—	2	1	no	—
558	Q₈	—	8	1	no	—
559	D₄	—	2	2	no	—
560	D₄	—	2	2	no	—
561	D₄	—	2	1	no	—
562	D₄	—	2	2	no	—
563	D₄	—	4	2	no	—
564	D₄	—	4	4	no	—
565	D₄	—	4	2	no	—
566	D₄	—	4	4	no	—
567	D₄	—	4	2	no	—
568	D₄	—	8	1	no	—
569	D₄	—	8	2	no	—
570	D₄	C₃₁×#131	1	2	yes	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	8	2	no	—
592	C₂×C₄	—	8	2	no	—
593	C₂×C₄	—	8	2	no	—
594	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
595	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
596	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
597	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
598	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
599	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
600	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
601	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
602	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
603	C₈	—	4	1	no	—
604	C₈	—	4	1	no	—
605	C₈	—	4	2	no	—
606	C₈	—	4	1	no	—
607	C₈	—	4	2	no	—
608	C₈	—	4	1	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	4	1	no	—
617	C₂²	—	4	1	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	8	8	no	—
622	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
623	C₄	—	2	1	no	—
624	C₄	—	2	2	no	—
625	C₄	—	2	2	no	—
626	C₄	—	2	1	no	—
627	C₄	—	2	1	no	—
628	C₄	—	2	2	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	1	no	—
631	C₄	—	4	2	no	—
632	C₄	—	4	1	no	—
633	C₄	C₃₁×#421	1	2	yes	—
634	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
635	C₄	C₃₁×#568	1	1	yes	characteristic
636	C₄	C₃₁×#576	1	1	yes	characteristic
637	C₄	C₃₁×#577	1	1	yes	characteristic
638	C₂	—	4	2	no	—
639	C₂	—	4	1	no	—
640	C₂	—	4	2	no	—
641	C₂	—	8	2	no	—
642	C₂	C₃₁×#2880	1	1	yes	characteristic
643	C₂	C₃₁×#2897	1	1	yes	characteristic
644	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3481	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2991	C₂	semidirect
3. C₃₁×#2922	C₂	semidirect
4. C₃₁×#1495	C₂	semidirect
5. C₃₁×#1492	C₂	semidirect
6. C₃₁×#1489	C₂	semidirect
7. C₃₁×#1486	C₂	semidirect
8. C₃₁×#1484	C₂	semidirect
9. C₃₁×#1482	C₂	semidirect
10. C₃₁×#1478	C₂	semidirect
11. C₃₁×#1476	C₂	semidirect
33. C₃₁×#431	C₂²	semidirect
34. C₃₁×#431	C₂²	semidirect
35. C₃₁×#431	C₂²	semidirect
36. C₃₁×#431	C₂²	semidirect
38. C₃₁×#423	C₂²	semidirect
39. C₃₁×#423	C₂²	semidirect
41. C₄×M₁₆×C₃₁	C₂²	semidirect
42. C₂×C₄×C₈×C₃₁	C₂²	semidirect
43. C₃₁×Q₈⋊C₈	C₂²	semidirect
44. C₃₁×Q₈⋊C₈	C₂²	semidirect
45. C₃₁×Q₈⋊C₈	C₂²	semidirect
49. C₃₁×D₄⋊C₈	C₂²	semidirect
50. C₃₁×D₄⋊C₈	C₂²	semidirect
51. C₃₁×D₄⋊C₈	C₂²	semidirect
52. C₃₁×D₄⋊C₈	C₂²	semidirect
225. #17522	C₃₁	direct (N × H)
288. #2991	C₂×C₃₁	semidirect
289. #2922	C₂×C₃₁	semidirect
290. #1495	C₂×C₃₁	semidirect
291. #1492	C₂×C₃₁	semidirect
292. #1489	C₂×C₃₁	semidirect
293. #1486	C₂×C₃₁	semidirect
294. #1484	C₂×C₃₁	semidirect
295. #1482	C₂×C₃₁	semidirect
296. #1478	C₂×C₃₁	semidirect
297. #1476	C₂×C₃₁	semidirect
348. #431	C₂²×C₃₁	semidirect
349. #431	C₂²×C₃₁	semidirect
350. #431	C₂²×C₃₁	semidirect
351. #431	C₂²×C₃₁	semidirect
353. #423	C₂²×C₃₁	semidirect
354. #423	C₂²×C₃₁	semidirect
356. C₄×M₁₆	C₂²×C₃₁	semidirect
357. C₂×C₄×C₈	C₂²×C₃₁	semidirect
358. Q₈⋊C₈	C₂²×C₃₁	semidirect
359. Q₈⋊C₈	C₂²×C₃₁	semidirect
360. Q₈⋊C₈	C₂²×C₃₁	semidirect
364. D₄⋊C₈	C₂²×C₃₁	semidirect
365. D₄⋊C₈	C₂²×C₃₁	semidirect
366. D₄⋊C₈	C₂²×C₃₁	semidirect
367. D₄⋊C₈	C₂²×C₃₁	semidirect
462. C₃₁	#17522	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×#3481	2
C₃₁×#2991	2
C₃₁×#2922	2
C₃₁×#1495	2
C₃₁×#1492	2
C₃₁×#1489	2
C₃₁×#1486	2
C₃₁×#1484	2
C₃₁×#1482	2
C₃₁×#1478	2
C₃₁×#1476	2
#17522	31

Subgroups — C31×#17522 #21474281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17522 #21474281