Subgroups — C₃₁×#18534 #21475293

Order 7936.

1168 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3112	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3109	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1637	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1627	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1556	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1491	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1490	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₃₁	—	2	1	no	—
11	D₄×Q₈×C₃₁	—	2	2	no	—
12	D₄×Q₈×C₃₁	—	2	2	no	—
13	D₄×Q₈×C₃₁	—	2	2	no	—
14	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#489	—	2	2	no	—
24	C₃₁×#487	C₂²	1	2	yes	complemented
25	C₃₁×#486	C₂²	1	2	yes	complemented
26	C₃₁×#481	—	2	1	no	—
27	C₃₁×#480	—	2	1	no	—
28	C₃₁×#477	C₂²	1	2	yes	complemented
29	C₃₁×#476	C₂²	1	2	yes	—
30	C₃₁×#460	C₂²	1	2	yes	complemented
31	C₃₁×#459	C₂²	1	2	yes	—
32	C₃₁×#448	—	2	1	no	—
33	C₃₁×#447	—	2	1	no	—
34	C₃₁×#442	—	2	2	no	—
35	C₃₁×#440	—	2	2	no	—
36	C₃₁×#432	C₂²	1	1	yes	characteristic
37	C₃₁×#432	C₂²	1	1	yes	characteristic
38	C₃₁×#417	—	2	2	no	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#332	C₂²	1	4	yes	complemented
41	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₃₁×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	—	2	2	no	—
45	C₂²×Q₈×C₃₁	—	2	4	no	—
46	C₂²×Q₈×C₃₁	—	2	4	no	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	4	8	no	—
49	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	2	no	—
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	4	no	—
57	C₃₁×C₄⋊Q₈	—	2	4	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₃₁×#127	C₂³	1	1	yes	characteristic
64	C₃₁×#124	—	2	1	no	—
65	C₃₁×C₂²⋊Q₈	—	2	4	no	—
66	C₃₁×C₂²⋊Q₈	—	2	8	no	—
67	C₃₁×C₂²⋊Q₈	—	2	4	no	—
68	C₃₁×C₂²⋊Q₈	—	2	8	no	—
69	C₃₁×#121	—	2	1	no	—
70	C₃₁×#121	—	2	1	no	—
71	C₄×Q₈×C₃₁	—	2	1	no	—
72	C₄×Q₈×C₃₁	—	2	1	no	—
73	C₄×Q₈×C₃₁	D₄	1	2	yes	—
74	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
75	C₄×D₄×C₃₁	—	2	4	no	—
76	C₄×D₄×C₃₁	—	2	4	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	D₄	1	2	yes	—
82	C₄×D₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
87	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	—	2	1	no	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#107	—	2	2	no	—
94	C₃₁×#106	—	2	2	no	—
95	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
96	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
97	C₃₁×#103	—	2	2	no	—
98	C₃₁×#103	—	2	4	no	—
99	C₃₁×#102	—	2	2	no	—
100	C₃₁×#102	—	2	4	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	2	no	—
103	C₃₁×C₂²⋊C₈	—	2	1	no	—
104	C₃₁×C₂²⋊C₈	—	2	1	no	—
105	C₃₁×#97	—	2	1	no	—
106	C₃₁×#97	—	2	1	no	—
107	C₃₁×#97	—	2	1	no	—
108	C₄×C₈×C₃₁	C₂³	1	2	yes	—
109	C₂⁴×C₃₁	—	2	1	no	—
110	#41×C₃₁	—	4	4	no	—
111	C₂×Q₈×C₃₁	—	2	4	no	—
112	C₂×Q₈×C₃₁	—	2	8	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×Q₈×C₃₁	—	4	8	no	—
118	C₂×Q₈×C₃₁	—	4	4	no	—
119	C₂×Q₈×C₃₁	—	4	16	no	—
120	C₂×Q₈×C₃₁	—	4	8	no	—
121	C₂×Q₈×C₃₁	—	4	8	no	—
122	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	—	2	1	no	—
126	C₂×D₄×C₃₁	—	2	1	no	—
127	C₂×D₄×C₃₁	—	2	1	no	—
128	C₂×D₄×C₃₁	—	4	1	no	—
129	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	8	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	4	8	no	—
138	C₂²×C₄×C₃₁	—	4	4	no	—
139	C₂²×C₄×C₃₁	—	4	1	no	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	SD₁₆×C₃₁	—	4	4	no	—
144	D₈×C₃₁	—	4	4	no	—
145	M₁₆×C₃₁	—	4	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	8	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
160	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	2	no	—
164	C₂²⋊C₄×C₃₁	—	2	2	no	—
165	C₂²⋊C₄×C₃₁	—	4	1	no	—
166	C₂²⋊C₄×C₃₁	—	4	1	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
173	#18534	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	4	no	—
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	1	no	—
179	C₂³×C₃₁	#120	1	1	yes	characteristic
180	Q₈×C₃₁	—	2	4	no	—
181	Q₈×C₃₁	—	2	2	no	—
182	Q₈×C₃₁	—	4	16	no	—
183	Q₈×C₃₁	—	4	4	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	4	4	no	—
186	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
187	D₄×C₃₁	—	2	4	no	—
188	D₄×C₃₁	—	2	2	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	8	2	no	—
193	D₄×C₃₁	#137	1	4	yes	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	4	16	no	—
203	C₂×C₄×C₃₁	—	4	4	no	—
204	C₂×C₄×C₃₁	—	4	4	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	4	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	8	2	no	—
210	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
211	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
212	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₈×C₃₁	—	4	2	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	—	4	2	no	—
219	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#3141	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#3112	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#3109	C₂×C₃₁	1	1	yes	characteristic
223	#1637	C₂×C₃₁	1	4	yes	complemented
224	#1627	C₂×C₃₁	1	4	yes	complemented
225	#1556	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1491	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1490	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂²×C₃₁	—	2	1	no	—
229	C₂²×C₃₁	—	2	1	no	—
230	C₂²×C₃₁	—	2	4	no	—
231	C₂²×C₃₁	—	4	8	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	8	1	no	—
235	C₂²×C₃₁	—	8	1	no	—
236	C₂²×C₃₁	—	8	1	no	—
237	C₂²×C₃₁	#454	1	2	yes	—
238	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
239	C₄×C₃₁	—	2	4	no	—
240	C₄×C₃₁	—	2	1	no	—
241	C₄×C₃₁	—	2	1	no	—
242	C₄×C₃₁	—	2	1	no	—
243	C₄×C₃₁	—	2	1	no	—
244	C₄×C₃₁	—	4	4	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
249	C₄×C₃₁	C₂×#137	1	2	yes	—
250	C₂³×Q₈	—	2	1	no	—
251	D₄×Q₈	—	2	2	no	—
252	D₄×Q₈	—	2	2	no	—
253	D₄×Q₈	—	2	2	no	—
254	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
255	#532	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊Q₈	—	2	1	no	—
257	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₂²⋊Q₈	—	2	1	no	—
259	C₂×C₂²⋊Q₈	—	2	2	no	—
260	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄×D₄	—	2	1	no	—
262	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
263	#489	—	2	2	no	—
264	#487	C₂²×C₃₁	1	2	yes	complemented
265	#486	C₂²×C₃₁	1	2	yes	complemented
266	#481	—	2	1	no	—
267	#480	—	2	1	no	—
268	#477	C₂²×C₃₁	1	2	yes	complemented
269	#476	C₂²×C₃₁	1	2	yes	—
270	#460	C₂²×C₃₁	1	2	yes	complemented
271	#459	C₂²×C₃₁	1	2	yes	—
272	#448	—	2	1	no	—
273	#447	—	2	1	no	—
274	#442	—	2	2	no	—
275	#440	—	2	2	no	—
276	#432	C₂²×C₃₁	1	1	yes	characteristic
277	#432	C₂²×C₃₁	1	1	yes	characteristic
278	#417	—	2	2	no	—
279	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
280	#332	C₂²×C₃₁	1	4	yes	complemented
281	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	complemented
282	D₄⋊C₈	C₂²×C₃₁	1	4	yes	—
283	C₂×C₃₁	—	2	2	no	—
284	C₂×C₃₁	—	4	4	no	—
285	C₂×C₃₁	—	16	1	no	—
286	C₂×C₃₁	#3003	1	1	yes	characteristic
287	C₂×C₃₁	#3004	1	1	yes	characteristic
288	C₂×C₃₁	C₂×#454	1	1	yes	characteristic
289	C₂×#41	—	2	1	no	—
290	C₂²×Q₈	—	2	2	no	—
291	C₂²×Q₈	—	2	4	no	—
292	C₂²×Q₈	—	2	4	no	—
293	C₂²×Q₈	—	2	1	no	—
294	C₂²×Q₈	—	4	8	no	—
295	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
296	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
297	C₂³×C₄	—	2	2	no	—
298	C₂³×C₄	—	2	1	no	—
299	C₂×SD₁₆	—	2	2	no	—
300	C₂×D₈	—	2	2	no	—
301	C₂×M₁₆	—	2	1	no	—
302	C₄⋊Q₈	—	2	4	no	—
303	C₄⋊Q₈	—	2	4	no	—
304	C₄⋊Q₈	—	2	1	no	—
305	C₄⋊Q₈	—	2	1	no	—
306	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
307	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
309	#127	C₂³×C₃₁	1	1	yes	characteristic
310	#124	—	2	1	no	—
311	C₂²⋊Q₈	—	2	4	no	—
312	C₂²⋊Q₈	—	2	8	no	—
313	C₂²⋊Q₈	—	2	4	no	—
314	C₂²⋊Q₈	—	2	8	no	—
315	#121	—	2	1	no	—
316	#121	—	2	1	no	—
317	C₄×Q₈	—	2	1	no	—
318	C₄×Q₈	—	2	1	no	—
319	C₄×Q₈	D₄×C₃₁	1	2	yes	—
320	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
321	C₄×D₄	—	2	4	no	—
322	C₄×D₄	—	2	4	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	—	2	1	no	—
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	D₄×C₃₁	1	2	yes	—
328	C₄×D₄	C₂³×C₃₁	1	2	yes	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	1	no	—
337	C₂×C₄²	—	2	1	no	—
338	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
339	#107	—	2	2	no	—
340	#106	—	2	2	no	—
341	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
342	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
343	#103	—	2	2	no	—
344	#103	—	2	4	no	—
345	#102	—	2	2	no	—
346	#102	—	2	4	no	—
347	#102	—	2	2	no	—
348	#102	—	2	2	no	—
349	C₂²⋊C₈	—	2	1	no	—
350	C₂²⋊C₈	—	2	1	no	—
351	#97	—	2	1	no	—
352	#97	—	2	1	no	—
353	#97	—	2	1	no	—
354	C₄×C₈	C₂³×C₃₁	1	2	yes	—
355	C₃₁	#18534	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	—	2	1	no	—
357	#41	—	4	4	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	8	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	4	8	no	—
365	C₂×Q₈	—	4	4	no	—
366	C₂×Q₈	—	4	16	no	—
367	C₂×Q₈	—	4	8	no	—
368	C₂×Q₈	—	4	8	no	—
369	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	4	1	no	—
376	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	8	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	4	1	no	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	SD₁₆	—	4	4	no	—
391	D₈	—	4	4	no	—
392	M₁₆	—	4	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₄⋊C₄	—	2	8	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	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	—	2	2	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
407	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
408	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
409	C₂²⋊C₄	—	2	8	no	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	4	1	no	—
413	C₂²⋊C₄	—	4	1	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	2	2	no	—
416	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
418	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
420	C₂³	—	2	4	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	1	no	—
425	C₂³	C₃₁×#120	1	1	yes	characteristic
426	Q₈	—	2	4	no	—
427	Q₈	—	2	2	no	—
428	Q₈	—	4	16	no	—
429	Q₈	—	4	4	no	—
430	Q₈	—	4	2	no	—
431	Q₈	—	4	4	no	—
432	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
433	D₄	—	2	4	no	—
434	D₄	—	2	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	8	2	no	—
439	D₄	C₃₁×#137	1	4	yes	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	4	16	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	8	2	no	—
456	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
457	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
458	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	C₈	—	4	2	no	—
463	C₈	—	4	2	no	—
464	C₈	—	4	2	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	8	1	no	—
472	C₂²	—	8	1	no	—
473	C₂²	—	8	1	no	—
474	C₂²	C₃₁×#454	1	2	yes	—
475	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
476	C₄	—	2	4	no	—
477	C₄	—	2	1	no	—
478	C₄	—	2	1	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	4	4	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	8	1	no	—
485	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
486	C₄	C₂×C₃₁×#137	1	2	yes	—
487	C₂	—	2	2	no	—
488	C₂	—	4	4	no	—
489	C₂	—	16	1	no	—
490	C₂	C₃₁×#3003	1	1	yes	characteristic
491	C₂	C₃₁×#3004	1	1	yes	characteristic
492	C₂	C₂×C₃₁×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3109	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Q₈×C₃₁	C₂	semidirect
2. C₃₁×#3141	C₂	semidirect
3. C₃₁×#3112	C₂	semidirect
5. C₃₁×#1637	C₂	semidirect
6. C₃₁×#1627	C₂	semidirect
7. C₃₁×#1556	C₂	semidirect
8. C₃₁×#1491	C₂	semidirect
9. C₃₁×#1490	C₂	semidirect
24. C₃₁×#487	C₂²	semidirect
25. C₃₁×#486	C₂²	semidirect
28. C₃₁×#477	C₂²	semidirect
30. C₃₁×#460	C₂²	semidirect
40. C₃₁×#332	C₂²	semidirect
41. C₃₁×Q₈⋊C₈	C₂²	semidirect
173. #18534	C₃₁	direct (N × H)
219. C₂×D₄×Q₈	C₂×C₃₁	semidirect
220. #3141	C₂×C₃₁	semidirect
221. #3112	C₂×C₃₁	semidirect
223. #1637	C₂×C₃₁	semidirect
224. #1627	C₂×C₃₁	semidirect
225. #1556	C₂×C₃₁	semidirect
226. #1491	C₂×C₃₁	semidirect
227. #1490	C₂×C₃₁	semidirect
264. #487	C₂²×C₃₁	semidirect
265. #486	C₂²×C₃₁	semidirect
268. #477	C₂²×C₃₁	semidirect
270. #460	C₂²×C₃₁	semidirect
280. #332	C₂²×C₃₁	semidirect
281. Q₈⋊C₈	C₂²×C₃₁	semidirect
355. C₃₁	#18534	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄×Q₈×C₃₁	2
C₃₁×#3141	2
C₃₁×#3112	2
C₃₁×#3109	2
C₃₁×#1637	2
C₃₁×#1627	2
C₃₁×#1556	2
C₃₁×#1491	2
C₃₁×#1490	2
#18534	31

Subgroups — C31×#18534 #21475293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18534 #21475293