Subgroups — C₃₁×#17630 #21474389

Order 7936.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 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×#2975	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2349	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1517	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1513	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1504	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1502	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#559	—	2	2	no	—
11	C₃₁×#556	C₄	1	2	yes	complemented
12	C₃₁×#555	C₄	1	2	yes	complemented
13	C₃₁×#553	—	2	2	no	—
14	C₃₁×#550	C₄	1	2	yes	complemented
15	C₃₁×#547	C₄	1	2	yes	complemented
16	C₃₁×#538	—	2	2	no	—
17	C₃₁×#535	—	2	1	no	—
18	C₃₁×#533	C₂²	1	1	yes	characteristic
19	C₃₁×#532	C₂²	1	1	yes	characteristic
20	C₃₁×#525	C₂²	1	4	yes	—
21	C₃₁×#520	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	—	2	1	no	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#436	—	2	2	no	—
25	C₃₁×#436	—	2	2	no	—
26	C₃₁×#431	C₂²	1	1	yes	characteristic
27	C₃₁×#431	C₂²	1	1	yes	characteristic
28	C₃₁×#407	C₂²	1	4	yes	—
29	C₃₁×#389	—	2	1	no	—
30	C₃₁×#385	—	2	2	no	—
31	C₃₁×#382	C₂²	1	1	yes	characteristic
32	C₃₁×#380	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#95	C₂²	1	2	yes	—
35	C₃₁×#328	C₂²	1	2	yes	—
36	C₃₁×#328	C₂²	1	2	yes	—
37	C₃₁×#328	C₂²	1	4	yes	—
38	C₃₁×#327	C₂²	1	2	yes	—
39	C₃₁×#327	C₂²	1	2	yes	—
40	C₃₁×#327	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	—	2	1	no	—
42	C₂×#41×C₃₁	C₂×C₄	1	4	yes	complemented
43	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	2	no	—
45	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
48	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
49	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
50	C₃₁×#127	—	2	2	no	—
51	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
52	C₃₁×#126	—	2	4	no	—
53	C₃₁×#125	—	2	1	no	—
54	C₃₁×#124	—	2	2	no	—
55	C₃₁×#124	—	2	4	no	—
56	C₃₁×#124	—	2	1	no	—
57	C₃₁×#123	—	2	4	no	—
58	C₃₁×#123	—	2	4	no	—
59	C₃₁×C₂²⋊Q₈	—	2	1	no	—
60	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	complemented
61	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
62	C₃₁×#121	—	2	2	no	—
63	C₃₁×#121	—	2	2	no	—
64	C₃₁×#121	—	2	1	no	—
65	C₃₁×#121	—	2	4	no	—
66	C₃₁×#121	C₂×C₄	1	4	yes	complemented
67	C₃₁×#121	C₂³	1	2	yes	—
68	C₃₁×#120	—	2	4	no	—
69	C₄×Q₈×C₃₁	—	2	2	no	—
70	C₄×Q₈×C₃₁	—	2	1	no	—
71	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
72	C₄×D₄×C₃₁	—	2	4	no	—
73	C₄×D₄×C₃₁	—	2	4	no	—
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	4	4	no	—
77	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
78	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
79	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
80	C₄×D₄×C₃₁	D₄	1	2	yes	—
81	C₄×D₄×C₃₁	D₄	1	2	yes	—
82	C₃₁×#117	—	2	1	no	—
83	C₃₁×#117	—	2	2	no	—
84	C₃₁×#117	C₂×C₄	1	4	yes	complemented
85	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
92	C₂×C₄²×C₃₁	—	2	1	no	—
93	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	—	2	1	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₂²⋊C₈	—	2	4	no	—
97	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
98	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
99	C₄×C₈×C₃₁	—	2	1	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₃₁×#95	—	2	2	no	—
102	C₃₁×#95	—	2	1	no	—
103	C₃₁×#95	C₂³	1	1	yes	characteristic
104	C₃₁×#95	C₂³	1	2	yes	—
105	C₃₁×#95	C₂³	1	1	yes	characteristic
106	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
107	#41×C₃₁	—	4	2	no	—
108	#41×C₃₁	—	4	2	no	—
109	#41×C₃₁	—	8	4	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	4	no	—
115	C₂×D₄×C₃₁	—	2	4	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
121	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
122	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	4	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	4	4	no	—
129	C₂²×C₄×C₃₁	—	4	4	no	—
130	C₂²×C₄×C₃₁	—	8	2	no	—
131	C₂²×C₄×C₃₁	—	8	2	no	—
132	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
138	M₁₆×C₃₁	—	8	4	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	4	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	4	2	no	—
149	C₄⋊C₄×C₃₁	—	4	2	no	—
150	C₄⋊C₄×C₃₁	—	8	2	no	—
151	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
152	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
153	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
154	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
155	C₂²⋊C₄×C₃₁	—	2	4	no	—
156	C₂²⋊C₄×C₃₁	—	2	4	no	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₃₁	—	2	2	no	—
161	C₂²⋊C₄×C₃₁	—	4	4	no	—
162	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
163	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
164	C₄²×C₃₁	—	2	4	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	4	2	no	—
167	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
171	#17630	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
178	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₃₁	—	4	2	no	—
180	Q₈×C₃₁	—	4	2	no	—
181	Q₈×C₃₁	—	8	1	no	—
182	D₄×C₃₁	—	2	4	no	—
183	D₄×C₃₁	—	2	4	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	8	1	no	—
189	D₄×C₃₁	—	8	4	no	—
190	D₄×C₃₁	—	8	4	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	4	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	4	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	8	4	no	—
210	C₂×C₄×C₃₁	—	8	8	no	—
211	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
212	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
217	C₂×C₄×C₃₁	#137	1	2	yes	—
218	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₈×C₃₁	—	4	2	no	—
221	C₈×C₃₁	—	4	2	no	—
222	#3506	C₂×C₃₁	1	1	yes	characteristic
223	#2975	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2349	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1518	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1517	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1513	C₂×C₃₁	1	4	yes	complemented
228	#1504	C₂×C₃₁	1	4	yes	complemented
229	#1503	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1502	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	1	no	—
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	2	4	no	—
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	2	4	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
242	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
243	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
244	C₄×C₃₁	—	2	1	no	—
245	C₄×C₃₁	—	2	1	no	—
246	C₄×C₃₁	—	2	4	no	—
247	C₄×C₃₁	—	2	1	no	—
248	C₄×C₃₁	—	2	1	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	8	1	no	—
254	C₄×C₃₁	—	8	2	no	—
255	C₄×C₃₁	#417	1	2	yes	—
256	C₄×C₃₁	#417	1	2	yes	—
257	#559	—	2	2	no	—
258	#556	C₄×C₃₁	1	2	yes	complemented
259	#555	C₄×C₃₁	1	2	yes	complemented
260	#553	—	2	2	no	—
261	#550	C₄×C₃₁	1	2	yes	complemented
262	#547	C₄×C₃₁	1	2	yes	complemented
263	#538	—	2	2	no	—
264	#535	—	2	1	no	—
265	#533	C₂²×C₃₁	1	1	yes	characteristic
266	#532	C₂²×C₃₁	1	1	yes	characteristic
267	#525	C₂²×C₃₁	1	4	yes	—
268	#520	—	2	1	no	—
269	C₂×C₄×D₄	—	2	1	no	—
270	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
271	#436	—	2	2	no	—
272	#436	—	2	2	no	—
273	#431	C₂²×C₃₁	1	1	yes	characteristic
274	#431	C₂²×C₃₁	1	1	yes	characteristic
275	#407	C₂²×C₃₁	1	4	yes	—
276	#389	—	2	1	no	—
277	#385	—	2	2	no	—
278	#382	C₂²×C₃₁	1	1	yes	characteristic
279	#380	C₂²×C₃₁	1	2	yes	—
280	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×#95	C₂²×C₃₁	1	2	yes	—
282	#328	C₂²×C₃₁	1	2	yes	—
283	#328	C₂²×C₃₁	1	2	yes	—
284	#328	C₂²×C₃₁	1	4	yes	—
285	#327	C₂²×C₃₁	1	2	yes	—
286	#327	C₂²×C₃₁	1	2	yes	—
287	#327	C₂²×C₃₁	1	4	yes	—
288	C₂×C₃₁	—	2	1	no	—
289	C₂×C₃₁	—	2	1	no	—
290	C₂×C₃₁	—	4	4	no	—
291	C₂×C₃₁	—	8	2	no	—
292	C₂×C₃₁	#2883	1	1	yes	characteristic
293	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
294	C₂×C₃₁	#2901	1	1	yes	characteristic
295	C₂×#41	—	2	1	no	—
296	C₂×#41	C₂×C₄×C₃₁	1	4	yes	complemented
297	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
298	C₂³×C₄	—	2	2	no	—
299	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
300	C₂×M₁₆	—	2	4	no	—
301	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
302	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
303	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
304	#127	—	2	2	no	—
305	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
306	#126	—	2	4	no	—
307	#125	—	2	1	no	—
308	#124	—	2	2	no	—
309	#124	—	2	4	no	—
310	#124	—	2	1	no	—
311	#123	—	2	4	no	—
312	#123	—	2	4	no	—
313	C₂²⋊Q₈	—	2	1	no	—
314	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
315	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
316	#121	—	2	2	no	—
317	#121	—	2	2	no	—
318	#121	—	2	1	no	—
319	#121	—	2	4	no	—
320	#121	C₂×C₄×C₃₁	1	4	yes	complemented
321	#121	C₂³×C₃₁	1	2	yes	—
322	#120	—	2	4	no	—
323	C₄×Q₈	—	2	2	no	—
324	C₄×Q₈	—	2	1	no	—
325	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
326	C₄×D₄	—	2	4	no	—
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	—	2	2	no	—
329	C₄×D₄	—	2	1	no	—
330	C₄×D₄	—	4	4	no	—
331	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
332	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
333	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
334	C₄×D₄	D₄×C₃₁	1	2	yes	—
335	C₄×D₄	D₄×C₃₁	1	2	yes	—
336	#117	—	2	1	no	—
337	#117	—	2	2	no	—
338	#117	C₂×C₄×C₃₁	1	4	yes	complemented
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
348	C₄⋊C₈	—	2	1	no	—
349	C₄⋊C₈	—	2	1	no	—
350	C₂²⋊C₈	—	2	4	no	—
351	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
352	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	—	2	1	no	—
355	#95	—	2	2	no	—
356	#95	—	2	1	no	—
357	#95	C₂³×C₃₁	1	1	yes	characteristic
358	#95	C₂³×C₃₁	1	2	yes	—
359	#95	C₂³×C₃₁	1	1	yes	characteristic
360	C₃₁	#17630	1	1	yes	characteristic, direct product factor, Sylow
361	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
362	#41	—	4	2	no	—
363	#41	—	4	2	no	—
364	#41	—	8	4	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
376	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
377	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	—	8	2	no	—
387	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
388	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
391	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
393	M₁₆	—	8	4	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	8	2	no	—
406	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
407	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
408	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
409	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	4	4	no	—
417	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
418	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
419	C₄²	—	2	4	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	4	2	no	—
422	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
423	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
432	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	Q₈	—	4	2	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	8	1	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	2	no	—
441	D₄	—	4	2	no	—
442	D₄	—	8	1	no	—
443	D₄	—	8	4	no	—
444	D₄	—	8	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	8	4	no	—
464	C₂×C₄	—	8	8	no	—
465	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
471	C₂×C₄	C₃₁×#137	1	2	yes	—
472	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
474	C₈	—	4	2	no	—
475	C₈	—	4	2	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	4	4	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
487	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
488	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	4	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	2	no	—
499	C₄	—	8	1	no	—
500	C₄	C₃₁×#417	1	2	yes	—
501	C₄	C₃₁×#417	1	2	yes	—
502	C₂	—	2	1	no	—
503	C₂	—	2	1	no	—
504	C₂	—	4	4	no	—
505	C₂	—	8	2	no	—
506	C₂	C₃₁×#2883	1	1	yes	characteristic
507	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
508	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×#2975	C₂	semidirect
3. C₃₁×#2349	C₂	semidirect
4. C₃₁×#1518	C₂	semidirect
5. C₃₁×#1517	C₂	semidirect
6. C₃₁×#1513	C₂	semidirect
7. C₃₁×#1504	C₂	semidirect
8. C₃₁×#1503	C₂	semidirect
9. C₃₁×#1502	C₂	semidirect
11. C₃₁×#556	C₄	semidirect
12. C₃₁×#555	C₄	semidirect
14. C₃₁×#550	C₄	semidirect
15. C₃₁×#547	C₄	semidirect
42. C₂×#41×C₃₁	C₂×C₄	semidirect
47. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
48. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
49. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
51. C₃₁×#127	C₂×C₄	semidirect
60. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
66. C₃₁×#121	C₂×C₄	semidirect
71. C₄×Q₈×C₃₁	C₂×C₄	semidirect
79. C₄×D₄×C₃₁	C₂×C₄	semidirect
84. C₃₁×#117	C₂×C₄	semidirect
171. #17630	C₃₁	direct (N × H)
223. #2975	C₂×C₃₁	semidirect
224. #2349	C₂×C₃₁	semidirect
225. #1518	C₂×C₃₁	semidirect
226. #1517	C₂×C₃₁	semidirect
227. #1513	C₂×C₃₁	semidirect
228. #1504	C₂×C₃₁	semidirect
229. #1503	C₂×C₃₁	semidirect
230. #1502	C₂×C₃₁	semidirect
258. #556	C₄×C₃₁	semidirect
259. #555	C₄×C₃₁	semidirect
261. #550	C₄×C₃₁	semidirect
262. #547	C₄×C₃₁	semidirect
296. C₂×#41	C₂×C₄×C₃₁	semidirect
301. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
302. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
303. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
305. #127	C₂×C₄×C₃₁	semidirect
314. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
320. #121	C₂×C₄×C₃₁	semidirect
325. C₄×Q₈	C₂×C₄×C₃₁	semidirect
333. C₄×D₄	C₂×C₄×C₃₁	semidirect
338. #117	C₂×C₄×C₃₁	semidirect
360. C₃₁	#17630	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×#2975	2
C₃₁×#2349	2
C₃₁×#1518	2
C₃₁×#1517	2
C₃₁×#1513	2
C₃₁×#1504	2
C₃₁×#1503	2
C₃₁×#1502	2
#17630	31

Subgroups — C31×#17630 #21474389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17630 #21474389