Subgroups — C₃₁×#17629 #21474388

Order 7936.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3506	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2973	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2353	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1516	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1515	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1514	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1513	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1506	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1505	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1504	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1501	C₂	1	2	yes	maximal, complemented
12	C₃₁×#559	—	2	2	no	—
13	C₃₁×#556	C₄	1	2	yes	complemented
14	C₃₁×#555	C₄	1	2	yes	complemented
15	C₃₁×#553	—	2	2	no	—
16	C₃₁×#550	C₄	1	2	yes	complemented
17	C₃₁×#547	C₄	1	2	yes	complemented
18	C₃₁×#538	—	2	1	no	—
19	C₃₁×#538	—	2	1	no	—
20	C₃₁×#535	—	2	1	no	—
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#525	C₂²	1	2	yes	—
24	C₃₁×#525	C₂²	1	2	yes	—
25	C₃₁×#520	—	2	1	no	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#435	—	2	2	no	—
28	C₃₁×#435	—	2	2	no	—
29	C₃₁×#432	C₂²	1	1	yes	characteristic
30	C₃₁×#432	C₂²	1	1	yes	characteristic
31	C₃₁×#407	C₂²	1	2	yes	—
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₃₁×#388	—	2	1	no	—
34	C₃₁×#387	—	2	1	no	—
35	C₃₁×#386	—	2	1	no	—
36	C₃₁×#383	C₂²	1	1	yes	characteristic
37	C₃₁×#381	C₂²	1	1	yes	characteristic
38	C₃₁×#380	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#376	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
43	C₃₁×#328	C₂²	1	2	yes	—
44	C₃₁×#328	C₂²	1	2	yes	—
45	C₃₁×#328	C₂²	1	2	yes	—
46	C₃₁×#328	C₂²	1	2	yes	—
47	C₃₁×#327	C₂²	1	2	yes	—
48	C₃₁×#327	C₂²	1	2	yes	—
49	C₃₁×#327	C₂²	1	2	yes	—
50	C₃₁×#327	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	1	no	—
52	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×#41×C₃₁	C₂×C₄	1	2	yes	complemented
54	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	—	2	1	no	—
56	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
61	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
63	C₃₁×#127	—	2	2	no	—
64	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
65	C₃₁×#126	—	2	2	no	—
66	C₃₁×#126	—	2	2	no	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#124	—	2	2	no	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	—	2	2	no	—
71	C₃₁×#124	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
78	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
79	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
80	C₃₁×#121	—	2	2	no	—
81	C₃₁×#121	—	2	2	no	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	2	no	—
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	C₂×C₄	1	2	yes	—
86	C₃₁×#121	C₂×C₄	1	2	yes	complemented
87	C₃₁×#121	C₂³	1	2	yes	—
88	C₃₁×#120	—	2	2	no	—
89	C₃₁×#120	—	2	2	no	—
90	C₄×Q₈×C₃₁	—	2	2	no	—
91	C₄×Q₈×C₃₁	—	2	1	no	—
92	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	—	2	2	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	2	no	—
97	C₄×D₄×C₃₁	—	2	2	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
100	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
101	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
102	C₄×D₄×C₃₁	D₄	1	2	yes	—
103	C₄×D₄×C₃₁	D₄	1	2	yes	—
104	C₃₁×#117	—	2	1	no	—
105	C₃₁×#117	—	2	2	no	—
106	C₃₁×#117	C₂×C₄	1	2	yes	—
107	C₃₁×#117	C₂×C₄	1	2	yes	complemented
108	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
110	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	—	2	1	no	—
117	C₂×C₄²×C₃₁	—	2	1	no	—
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₃₁×C₄⋊C₈	—	2	1	no	—
120	C₃₁×C₄⋊C₈	—	2	1	no	—
121	C₃₁×C₂²⋊C₈	—	2	2	no	—
122	C₃₁×C₂²⋊C₈	—	2	2	no	—
123	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
124	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
125	C₃₁×#97	—	2	1	no	—
126	C₃₁×#97	—	2	1	no	—
127	C₃₁×#95	—	2	1	no	—
128	C₃₁×#95	—	2	1	no	—
129	C₃₁×#95	—	2	1	no	—
130	C₃₁×#95	—	2	1	no	—
131	C₃₁×#95	—	2	1	no	—
132	C₃₁×#95	C₂³	1	1	yes	characteristic
133	C₃₁×#95	C₂³	1	1	yes	characteristic
134	C₃₁×#95	C₂³	1	1	yes	characteristic
135	C₃₁×#95	C₂³	1	1	yes	characteristic
136	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
138	#41×C₃₁	—	4	2	no	—
139	#41×C₃₁	—	4	2	no	—
140	#41×C₃₁	—	8	2	no	—
141	#41×C₃₁	—	8	2	no	—
142	C₂×Q₈×C₃₁	—	2	1	no	—
143	C₂×Q₈×C₃₁	—	2	1	no	—
144	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
145	C₂×D₄×C₃₁	—	2	2	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	2	2	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	—	4	1	no	—
153	C₂×D₄×C₃₁	—	4	1	no	—
154	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
155	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
156	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
157	C₂²×C₄×C₃₁	—	2	1	no	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	1	no	—
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	4	2	no	—
165	C₂²×C₄×C₃₁	—	4	2	no	—
166	C₂²×C₄×C₃₁	—	8	1	no	—
167	C₂²×C₄×C₃₁	—	8	1	no	—
168	C₂²×C₄×C₃₁	—	8	1	no	—
169	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	M₁₆×C₃₁	—	8	2	no	—
179	C₂×C₈×C₃₁	—	2	2	no	—
180	C₂×C₈×C₃₁	—	2	2	no	—
181	C₂×C₈×C₃₁	—	8	2	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	1	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	—	8	1	no	—
191	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
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	C₂²⋊C₄×C₃₁	—	2	2	no	—
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	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	4	2	no	—
205	C₂²⋊C₄×C₃₁	—	4	2	no	—
206	C₂²⋊C₄×C₃₁	—	8	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	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
211	C₄²×C₃₁	—	2	2	no	—
212	C₄²×C₃₁	—	2	2	no	—
213	C₄²×C₃₁	—	2	2	no	—
214	C₄²×C₃₁	—	4	2	no	—
215	C₄²×C₃₁	—	8	1	no	—
216	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
218	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
220	#17629	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂³×C₃₁	—	2	1	no	—
222	C₂³×C₃₁	—	2	1	no	—
223	C₂³×C₃₁	—	2	1	no	—
224	C₂³×C₃₁	—	2	1	no	—
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	4	1	no	—
227	C₂³×C₃₁	—	4	1	no	—
228	C₂³×C₃₁	—	4	1	no	—
229	C₂³×C₃₁	—	4	1	no	—
230	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
231	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
232	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	Q₈×C₃₁	—	4	2	no	—
234	Q₈×C₃₁	—	4	2	no	—
235	Q₈×C₃₁	—	8	1	no	—
236	D₄×C₃₁	—	4	2	no	—
237	D₄×C₃₁	—	4	2	no	—
238	D₄×C₃₁	—	4	2	no	—
239	D₄×C₃₁	—	4	2	no	—
240	D₄×C₃₁	—	4	2	no	—
241	D₄×C₃₁	—	4	2	no	—
242	D₄×C₃₁	—	8	1	no	—
243	D₄×C₃₁	—	8	2	no	—
244	D₄×C₃₁	—	8	2	no	—
245	D₄×C₃₁	—	8	2	no	—
246	D₄×C₃₁	—	8	2	no	—
247	C₂×C₄×C₃₁	—	2	1	no	—
248	C₂×C₄×C₃₁	—	2	1	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	4	1	no	—
254	C₂×C₄×C₃₁	—	4	1	no	—
255	C₂×C₄×C₃₁	—	4	1	no	—
256	C₂×C₄×C₃₁	—	4	2	no	—
257	C₂×C₄×C₃₁	—	4	2	no	—
258	C₂×C₄×C₃₁	—	4	1	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	—	4	1	no	—
263	C₂×C₄×C₃₁	—	4	1	no	—
264	C₂×C₄×C₃₁	—	4	1	no	—
265	C₂×C₄×C₃₁	—	4	1	no	—
266	C₂×C₄×C₃₁	—	4	1	no	—
267	C₂×C₄×C₃₁	—	4	1	no	—
268	C₂×C₄×C₃₁	—	4	1	no	—
269	C₂×C₄×C₃₁	—	4	1	no	—
270	C₂×C₄×C₃₁	—	4	1	no	—
271	C₂×C₄×C₃₁	—	4	1	no	—
272	C₂×C₄×C₃₁	—	4	1	no	—
273	C₂×C₄×C₃₁	—	8	2	no	—
274	C₂×C₄×C₃₁	—	8	2	no	—
275	C₂×C₄×C₃₁	—	8	4	no	—
276	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
278	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
279	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
283	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
284	C₈×C₃₁	—	4	2	no	—
285	C₈×C₃₁	—	4	2	no	—
286	#3506	C₂×C₃₁	1	1	yes	characteristic
287	#2973	C₂×C₃₁	1	1	yes	characteristic, complemented
288	#2353	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#1516	C₂×C₃₁	1	2	yes	complemented
290	#1515	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#1514	C₂×C₃₁	1	1	yes	characteristic, complemented
292	#1513	C₂×C₃₁	1	2	yes	complemented
293	#1506	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#1505	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1504	C₂×C₃₁	1	2	yes	complemented
296	#1501	C₂×C₃₁	1	2	yes	complemented
297	C₂²×C₃₁	—	2	1	no	—
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	2	no	—
303	C₂²×C₃₁	—	2	2	no	—
304	C₂²×C₃₁	—	2	2	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	—	4	2	no	—
307	C₂²×C₃₁	—	4	2	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	—	4	2	no	—
310	C₂²×C₃₁	—	8	1	no	—
311	C₂²×C₃₁	—	8	1	no	—
312	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
313	C₂²×C₃₁	#575	1	1	yes	characteristic
314	C₂²×C₃₁	#575	1	1	yes	characteristic
315	C₄×C₃₁	—	2	1	no	—
316	C₄×C₃₁	—	2	1	no	—
317	C₄×C₃₁	—	2	2	no	—
318	C₄×C₃₁	—	2	2	no	—
319	C₄×C₃₁	—	2	1	no	—
320	C₄×C₃₁	—	2	1	no	—
321	C₄×C₃₁	—	2	1	no	—
322	C₄×C₃₁	—	2	1	no	—
323	C₄×C₃₁	—	4	2	no	—
324	C₄×C₃₁	—	4	2	no	—
325	C₄×C₃₁	—	4	2	no	—
326	C₄×C₃₁	—	4	2	no	—
327	C₄×C₃₁	—	8	1	no	—
328	C₄×C₃₁	—	8	1	no	—
329	C₄×C₃₁	—	8	1	no	—
330	#559	—	2	2	no	—
331	#556	C₄×C₃₁	1	2	yes	complemented
332	#555	C₄×C₃₁	1	2	yes	complemented
333	#553	—	2	2	no	—
334	#550	C₄×C₃₁	1	2	yes	complemented
335	#547	C₄×C₃₁	1	2	yes	complemented
336	#538	—	2	1	no	—
337	#538	—	2	1	no	—
338	#535	—	2	1	no	—
339	#533	C₂²×C₃₁	1	1	yes	characteristic
340	#532	C₂²×C₃₁	1	1	yes	characteristic
341	#525	C₂²×C₃₁	1	2	yes	—
342	#525	C₂²×C₃₁	1	2	yes	—
343	#520	—	2	1	no	—
344	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
345	#435	—	2	2	no	—
346	#435	—	2	2	no	—
347	#432	C₂²×C₃₁	1	1	yes	characteristic
348	#432	C₂²×C₃₁	1	1	yes	characteristic
349	#407	C₂²×C₃₁	1	2	yes	—
350	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
351	#388	—	2	1	no	—
352	#387	—	2	1	no	—
353	#386	—	2	1	no	—
354	#383	C₂²×C₃₁	1	1	yes	characteristic
355	#381	C₂²×C₃₁	1	1	yes	characteristic
356	#380	C₂²×C₃₁	1	1	yes	characteristic
357	C₄×C₂²⋊C₄	—	2	1	no	—
358	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
359	#376	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
361	#328	C₂²×C₃₁	1	2	yes	—
362	#328	C₂²×C₃₁	1	2	yes	—
363	#328	C₂²×C₃₁	1	2	yes	—
364	#328	C₂²×C₃₁	1	2	yes	—
365	#327	C₂²×C₃₁	1	2	yes	—
366	#327	C₂²×C₃₁	1	2	yes	—
367	#327	C₂²×C₃₁	1	2	yes	—
368	#327	C₂²×C₃₁	1	2	yes	—
369	C₂×C₃₁	—	2	1	no	—
370	C₂×C₃₁	—	2	1	no	—
371	C₂×C₃₁	—	4	2	no	—
372	C₂×C₃₁	—	4	2	no	—
373	C₂×C₃₁	—	8	2	no	—
374	C₂×C₃₁	#2883	1	1	yes	characteristic
375	C₂×C₃₁	#2896	1	1	yes	characteristic
376	C₂×C₃₁	#2901	1	1	yes	characteristic
377	C₂×#41	—	2	1	no	—
378	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
379	C₂×#41	C₂×C₄×C₃₁	1	2	yes	complemented
380	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
381	C₂³×C₄	—	2	1	no	—
382	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
384	C₂×M₁₆	—	2	2	no	—
385	C₂²×C₈	—	2	2	no	—
386	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
387	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
388	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
389	#127	—	2	2	no	—
390	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
391	#126	—	2	2	no	—
392	#126	—	2	2	no	—
393	#125	—	2	1	no	—
394	#124	—	2	2	no	—
395	#124	—	2	2	no	—
396	#124	—	2	2	no	—
397	#124	—	2	1	no	—
398	#123	—	2	2	no	—
399	#123	—	2	2	no	—
400	#123	—	2	2	no	—
401	#123	—	2	2	no	—
402	C₂²⋊Q₈	—	2	1	no	—
403	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
404	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
405	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
406	#121	—	2	2	no	—
407	#121	—	2	2	no	—
408	#121	—	2	1	no	—
409	#121	—	2	2	no	—
410	#121	—	2	2	no	—
411	#121	C₂×C₄×C₃₁	1	2	yes	—
412	#121	C₂×C₄×C₃₁	1	2	yes	complemented
413	#121	C₂³×C₃₁	1	2	yes	—
414	#120	—	2	2	no	—
415	#120	—	2	2	no	—
416	C₄×Q₈	—	2	2	no	—
417	C₄×Q₈	—	2	1	no	—
418	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	—	2	2	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	—	2	1	no	—
425	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
426	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
427	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
428	C₄×D₄	D₄×C₃₁	1	2	yes	—
429	C₄×D₄	D₄×C₃₁	1	2	yes	—
430	#117	—	2	1	no	—
431	#117	—	2	2	no	—
432	#117	C₂×C₄×C₃₁	1	2	yes	—
433	#117	C₂×C₄×C₃₁	1	2	yes	complemented
434	C₂×C₄⋊C₄	—	2	2	no	—
435	C₂×C₄⋊C₄	—	2	1	no	—
436	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
442	C₂×C₄²	—	2	1	no	—
443	C₂×C₄²	—	2	1	no	—
444	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
445	C₄⋊C₈	—	2	1	no	—
446	C₄⋊C₈	—	2	1	no	—
447	C₂²⋊C₈	—	2	2	no	—
448	C₂²⋊C₈	—	2	2	no	—
449	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
450	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
451	#97	—	2	1	no	—
452	#97	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	#95	—	2	1	no	—
458	#95	C₂³×C₃₁	1	1	yes	characteristic
459	#95	C₂³×C₃₁	1	1	yes	characteristic
460	#95	C₂³×C₃₁	1	1	yes	characteristic
461	#95	C₂³×C₃₁	1	1	yes	characteristic
462	C₃₁	#17629	1	1	yes	characteristic, direct product factor, Sylow
463	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
465	#41	—	4	2	no	—
466	#41	—	4	2	no	—
467	#41	—	8	2	no	—
468	#41	—	8	2	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	2	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	1	no	—
480	C₂×D₄	—	4	1	no	—
481	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
482	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
483	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	4	2	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	—	8	1	no	—
494	C₂²×C₄	—	8	1	no	—
495	C₂²×C₄	—	8	1	no	—
496	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
497	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
498	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
501	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
503	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
505	M₁₆	—	8	2	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	2	2	no	—
508	C₂×C₈	—	8	2	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	8	1	no	—
518	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
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	2	yes	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	2	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	2	no	—
531	C₂²⋊C₄	—	4	2	no	—
532	C₂²⋊C₄	—	4	2	no	—
533	C₂²⋊C₄	—	8	1	no	—
534	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
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	2	no	—
539	C₄²	—	2	2	no	—
540	C₄²	—	2	2	no	—
541	C₄²	—	4	2	no	—
542	C₄²	—	8	1	no	—
543	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
544	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
545	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
546	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
547	C₂³	—	2	1	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	4	1	no	—
553	C₂³	—	4	1	no	—
554	C₂³	—	4	1	no	—
555	C₂³	—	4	1	no	—
556	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
557	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
558	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
559	Q₈	—	4	2	no	—
560	Q₈	—	4	2	no	—
561	Q₈	—	8	1	no	—
562	D₄	—	4	2	no	—
563	D₄	—	4	2	no	—
564	D₄	—	4	2	no	—
565	D₄	—	4	2	no	—
566	D₄	—	4	2	no	—
567	D₄	—	4	2	no	—
568	D₄	—	8	1	no	—
569	D₄	—	8	2	no	—
570	D₄	—	8	2	no	—
571	D₄	—	8	2	no	—
572	D₄	—	8	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	8	2	no	—
600	C₂×C₄	—	8	2	no	—
601	C₂×C₄	—	8	4	no	—
602	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
603	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
604	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
605	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
606	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
607	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
608	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
610	C₈	—	4	2	no	—
611	C₈	—	4	2	no	—
612	C₂²	—	2	1	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	2	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	2	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	8	1	no	—
626	C₂²	—	8	1	no	—
627	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
628	C₂²	C₃₁×#575	1	1	yes	characteristic
629	C₂²	C₃₁×#575	1	1	yes	characteristic
630	C₄	—	2	1	no	—
631	C₄	—	2	1	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	2	1	no	—
637	C₄	—	2	1	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	2	no	—
641	C₄	—	4	2	no	—
642	C₄	—	8	1	no	—
643	C₄	—	8	1	no	—
644	C₄	—	8	1	no	—
645	C₂	—	2	1	no	—
646	C₂	—	2	1	no	—
647	C₂	—	4	2	no	—
648	C₂	—	4	2	no	—
649	C₂	—	8	2	no	—
650	C₂	C₃₁×#2883	1	1	yes	characteristic
651	C₂	C₃₁×#2896	1	1	yes	characteristic
652	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	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3506	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2973	C₂	semidirect
3. C₃₁×#2353	C₂	semidirect
4. C₃₁×#1516	C₂	semidirect
5. C₃₁×#1515	C₂	semidirect
6. C₃₁×#1514	C₂	semidirect
7. C₃₁×#1513	C₂	semidirect
8. C₃₁×#1506	C₂	semidirect
9. C₃₁×#1505	C₂	semidirect
10. C₃₁×#1504	C₂	semidirect
11. C₃₁×#1501	C₂	semidirect
13. C₃₁×#556	C₄	semidirect
14. C₃₁×#555	C₄	semidirect
16. C₃₁×#550	C₄	semidirect
17. C₃₁×#547	C₄	semidirect
53. C₂×#41×C₃₁	C₂×C₄	semidirect
60. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
61. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
62. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
64. C₃₁×#127	C₂×C₄	semidirect
78. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
86. C₃₁×#121	C₂×C₄	semidirect
92. C₄×Q₈×C₃₁	C₂×C₄	semidirect
101. C₄×D₄×C₃₁	C₂×C₄	semidirect
107. C₃₁×#117	C₂×C₄	semidirect
220. #17629	C₃₁	direct (N × H)
287. #2973	C₂×C₃₁	semidirect
288. #2353	C₂×C₃₁	semidirect
289. #1516	C₂×C₃₁	semidirect
290. #1515	C₂×C₃₁	semidirect
291. #1514	C₂×C₃₁	semidirect
292. #1513	C₂×C₃₁	semidirect
293. #1506	C₂×C₃₁	semidirect
294. #1505	C₂×C₃₁	semidirect
295. #1504	C₂×C₃₁	semidirect
296. #1501	C₂×C₃₁	semidirect
331. #556	C₄×C₃₁	semidirect
332. #555	C₄×C₃₁	semidirect
334. #550	C₄×C₃₁	semidirect
335. #547	C₄×C₃₁	semidirect
379. C₂×#41	C₂×C₄×C₃₁	semidirect
386. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
387. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
388. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
390. #127	C₂×C₄×C₃₁	semidirect
404. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
412. #121	C₂×C₄×C₃₁	semidirect
418. C₄×Q₈	C₂×C₄×C₃₁	semidirect
427. C₄×D₄	C₂×C₄×C₃₁	semidirect
433. #117	C₂×C₄×C₃₁	semidirect
462. C₃₁	#17629	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×#3506	2
C₃₁×#2973	2
C₃₁×#2353	2
C₃₁×#1516	2
C₃₁×#1515	2
C₃₁×#1514	2
C₃₁×#1513	2
C₃₁×#1506	2
C₃₁×#1505	2
C₃₁×#1504	2
C₃₁×#1501	2
#17629	31

Subgroups — C31×#17629 #21474388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17629 #21474388