Subgroups — C₃₁×#17994 #21474753

Order 7936.

748 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2935	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1589	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1581	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1581	C₂	1	2	yes	maximal, complemented
7	C₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1571	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₂²×M₁₆×C₃₁	—	2	1	no	—
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#480	C₄	1	2	yes	—
17	C₃₁×#474	C₄	1	1	yes	characteristic
18	C₃₁×#474	C₄	1	1	yes	characteristic
19	C₃₁×#459	C₄	1	1	yes	characteristic
20	C₃₁×#459	C₄	1	1	yes	characteristic
21	C₃₁×#447	C₄	1	2	yes	—
22	C₄×D₈×C₃₁	C₂²	1	2	yes	—
23	C₄×D₈×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#436	C₂²	1	2	yes	—
25	C₃₁×#436	C₂²	1	2	yes	complemented
26	C₃₁×#435	—	2	2	no	—
27	C₃₁×#435	—	2	2	no	—
28	C₈×D₄×C₃₁	C₂²	1	2	yes	—
29	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#423	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃₁×#408	—	2	1	no	—
34	C₃₁×#407	—	2	1	no	—
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	2	yes	complemented
38	C₃₁×#335	C₂²	1	2	yes	complemented
39	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
45	C₃₁×#322	C₂²	1	2	yes	complemented
46	C₈²×C₃₁	C₂²	1	2	yes	complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
52	C₃₁×#131	—	4	4	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	4	4	no	—
57	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	—	2	2	no	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
64	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
65	C₃₁×#124	—	2	1	no	—
66	C₃₁×#121	—	2	1	no	—
67	C₃₁×#121	C₂×C₄	1	2	yes	—
68	C₄×D₄×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	—	2	1	no	—
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×D₄×C₃₁	C₂³	1	2	yes	—
75	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	C₂×C₄	1	2	yes	—
80	C₃₁×#107	C₂×C₄	1	2	yes	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
84	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
88	C₃₁×#102	C₂×C₄	1	2	yes	—
89	C₃₁×#102	C₂×C₄	1	2	yes	—
90	C₃₁×#102	C₂×C₄	1	2	yes	—
91	C₃₁×#102	C₂×C₄	1	2	yes	—
92	C₃₁×C₂²⋊C₈	—	2	2	no	—
93	C₃₁×C₂²⋊C₈	—	2	2	no	—
94	C₃₁×C₂²⋊C₈	—	2	2	no	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
97	C₃₁×#97	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	—	2	1	no	—
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	C₂⁴×C₃₁	—	2	1	no	—
104	#41×C₃₁	—	4	2	no	—
105	#41×C₃₁	—	4	2	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	4	1	no	—
113	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	4	2	no	—
123	C₂²×C₄×C₃₁	—	4	2	no	—
124	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
127	D₈×C₃₁	—	2	4	no	—
128	D₈×C₃₁	—	4	2	no	—
129	M₁₆×C₃₁	—	2	2	no	—
130	M₁₆×C₃₁	—	4	2	no	—
131	M₁₆×C₃₁	—	4	4	no	—
132	M₁₆×C₃₁	—	4	4	no	—
133	M₁₆×C₃₁	D₈	1	4	yes	complemented
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	4	2	no	—
140	C₂×C₈×C₃₁	—	4	2	no	—
141	C₂×C₈×C₃₁	—	4	4	no	—
142	C₂×C₈×C₃₁	—	4	4	no	—
143	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
145	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂²⋊C₄×C₃₁	—	2	1	no	—
153	C₂²⋊C₄×C₃₁	—	2	1	no	—
154	C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂²⋊C₄×C₃₁	—	2	1	no	—
156	C₂²⋊C₄×C₃₁	—	2	1	no	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₃₁	#41	1	1	yes	characteristic
161	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
162	#17994	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₃₁	—	2	1	no	—
164	C₂³×C₃₁	—	2	1	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	—	8	1	no	—
169	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
170	Q₈×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	2	2	no	—
172	D₄×C₃₁	—	2	1	no	—
173	D₄×C₃₁	—	2	1	no	—
174	D₄×C₃₁	—	2	1	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	4	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	8	1	no	—
180	D₄×C₃₁	#131	1	2	yes	complemented
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
200	C₂×C₄×C₃₁	#135	1	2	yes	—
201	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	2	2	no	—
207	C₈×C₃₁	—	2	2	no	—
208	C₈×C₃₁	—	4	1	no	—
209	C₈×C₃₁	—	4	1	no	—
210	C₈×C₃₁	—	4	1	no	—
211	C₈×C₃₁	—	4	2	no	—
212	C₈×C₃₁	C₂×D₈	1	4	yes	—
213	#3100	C₂×C₃₁	1	1	yes	characteristic
214	#2935	C₂×C₃₁	1	1	yes	characteristic, complemented
215	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1589	C₂×C₃₁	1	2	yes	complemented
217	#1581	C₂×C₃₁	1	2	yes	complemented
218	#1581	C₂×C₃₁	1	2	yes	complemented
219	C₈×D₈	C₂×C₃₁	1	2	yes	complemented
220	#1571	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1492	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂×D₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂²×C₃₁	—	2	1	no	—
225	C₂²×C₃₁	—	2	1	no	—
226	C₂²×C₃₁	—	2	2	no	—
227	C₂²×C₃₁	—	4	1	no	—
228	C₂²×C₃₁	—	4	1	no	—
229	C₂²×C₃₁	—	4	1	no	—
230	C₂²×C₃₁	—	4	2	no	—
231	C₂²×C₃₁	—	4	1	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	1	no	—
235	C₂²×C₃₁	—	8	4	no	—
236	C₂²×C₃₁	C₄×D₈	1	2	yes	—
237	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
238	C₄×C₃₁	—	2	1	no	—
239	C₄×C₃₁	—	2	1	no	—
240	C₄×C₃₁	—	2	1	no	—
241	C₄×C₃₁	—	4	1	no	—
242	C₄×C₃₁	—	4	1	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	C₄×D₈	1	2	yes	—
245	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
246	C₄×C₃₁	#568	1	1	yes	characteristic
247	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
248	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
249	C₂×#131	—	2	1	no	—
250	C₂²×M₁₆	—	2	1	no	—
251	#532	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
253	#480	C₄×C₃₁	1	2	yes	—
254	#474	C₄×C₃₁	1	1	yes	characteristic
255	#474	C₄×C₃₁	1	1	yes	characteristic
256	#459	C₄×C₃₁	1	1	yes	characteristic
257	#459	C₄×C₃₁	1	1	yes	characteristic
258	#447	C₄×C₃₁	1	2	yes	—
259	C₄×D₈	C₂²×C₃₁	1	2	yes	—
260	C₄×D₈	C₂²×C₃₁	1	2	yes	—
261	#436	C₂²×C₃₁	1	2	yes	—
262	#436	C₂²×C₃₁	1	2	yes	complemented
263	#435	—	2	2	no	—
264	#435	—	2	2	no	—
265	C₈×D₄	C₂²×C₃₁	1	2	yes	—
266	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
267	#431	C₂²×C₃₁	1	1	yes	characteristic
268	#423	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
270	#408	—	2	1	no	—
271	#407	—	2	1	no	—
272	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
274	#335	C₂²×C₃₁	1	2	yes	complemented
275	#335	C₂²×C₃₁	1	2	yes	complemented
276	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
277	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
278	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
279	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
280	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
281	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
282	#322	C₂²×C₃₁	1	2	yes	complemented
283	C₈²	C₂²×C₃₁	1	2	yes	complemented
284	C₂×C₃₁	—	2	2	no	—
285	C₂×C₃₁	—	4	2	no	—
286	C₂×C₃₁	—	8	2	no	—
287	C₂×C₃₁	—	8	1	no	—
288	C₂×C₃₁	#2933	1	1	yes	characteristic
289	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
290	C₂×C₃₁	#2956	1	1	yes	characteristic
291	C₂×#41	—	2	1	no	—
292	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
293	C₂³×C₄	—	2	1	no	—
294	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
295	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
296	#131	—	4	4	no	—
297	C₂×M₁₆	—	2	1	no	—
298	C₂×M₁₆	—	2	2	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	—	4	4	no	—
301	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
302	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
303	C₂²×C₈	—	2	1	no	—
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	—	2	2	no	—
306	C₂²×C₈	—	2	2	no	—
307	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
308	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
309	#124	—	2	1	no	—
310	#121	—	2	1	no	—
311	#121	C₂×C₄×C₃₁	1	2	yes	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	—	2	1	no	—
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
317	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₄×D₄	C₂³×C₃₁	1	2	yes	—
320	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
321	C₂×C₂²⋊C₄	—	2	1	no	—
322	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
323	#107	C₂×C₄×C₃₁	1	2	yes	—
324	#107	C₂×C₄×C₃₁	1	2	yes	—
325	C₄⋊C₈	—	2	1	no	—
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
328	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
329	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
332	#102	C₂×C₄×C₃₁	1	2	yes	—
333	#102	C₂×C₄×C₃₁	1	2	yes	—
334	#102	C₂×C₄×C₃₁	1	2	yes	—
335	#102	C₂×C₄×C₃₁	1	2	yes	—
336	C₂²⋊C₈	—	2	2	no	—
337	C₂²⋊C₈	—	2	2	no	—
338	C₂²⋊C₈	—	2	2	no	—
339	C₂²⋊C₈	—	2	2	no	—
340	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
341	#97	D₄×C₃₁	1	2	yes	—
342	C₄×C₈	—	2	1	no	—
343	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₄×C₈	C₂³×C₃₁	1	2	yes	—
346	C₄×C₈	C₂³×C₃₁	1	2	yes	—
347	C₃₁	#17994	1	1	yes	characteristic, direct product factor, Sylow
348	C₂⁴	—	2	1	no	—
349	#41	—	4	2	no	—
350	#41	—	4	2	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
359	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
360	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
372	D₈	—	2	4	no	—
373	D₈	—	4	2	no	—
374	M₁₆	—	2	2	no	—
375	M₁₆	—	4	2	no	—
376	M₁₆	—	4	4	no	—
377	M₁₆	—	4	4	no	—
378	M₁₆	D₈×C₃₁	1	4	yes	complemented
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	4	2	no	—
385	C₂×C₈	—	4	2	no	—
386	C₂×C₈	—	4	4	no	—
387	C₂×C₈	—	4	4	no	—
388	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
389	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
390	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
391	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
395	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
396	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
397	C₂²⋊C₄	—	2	1	no	—
398	C₂²⋊C₄	—	2	1	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	2	1	no	—
402	C₂²⋊C₄	—	2	2	no	—
403	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
404	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₄²	#41×C₃₁	1	1	yes	characteristic
406	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
407	C₂³	—	2	1	no	—
408	C₂³	—	2	1	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	8	1	no	—
413	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
414	Q₈	—	4	2	no	—
415	D₄	—	2	2	no	—
416	D₄	—	2	1	no	—
417	D₄	—	2	1	no	—
418	D₄	—	2	1	no	—
419	D₄	—	2	2	no	—
420	D₄	—	4	2	no	—
421	D₄	—	4	4	no	—
422	D₄	—	4	2	no	—
423	D₄	—	8	1	no	—
424	D₄	C₃₁×#131	1	2	yes	complemented
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
444	C₂×C₄	C₃₁×#135	1	2	yes	—
445	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
448	C₈	—	2	2	no	—
449	C₈	—	2	2	no	—
450	C₈	—	2	2	no	—
451	C₈	—	2	2	no	—
452	C₈	—	4	1	no	—
453	C₈	—	4	1	no	—
454	C₈	—	4	1	no	—
455	C₈	—	4	2	no	—
456	C₈	C₂×D₈×C₃₁	1	4	yes	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	4	1	no	—
461	C₂²	—	4	1	no	—
462	C₂²	—	4	1	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	8	4	no	—
469	C₂²	C₄×D₈×C₃₁	1	2	yes	—
470	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
471	C₄	—	2	1	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	1	no	—
476	C₄	—	4	2	no	—
477	C₄	C₄×D₈×C₃₁	1	2	yes	—
478	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
479	C₄	C₃₁×#568	1	1	yes	characteristic
480	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
481	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
482	C₂	—	2	2	no	—
483	C₂	—	4	2	no	—
484	C₂	—	8	1	no	—
485	C₂	—	8	2	no	—
486	C₂	C₃₁×#2933	1	1	yes	characteristic
487	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
488	C₂	C₃₁×#2956	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	C₂×C₄×D₄
prime radical	C₃₁×#3100	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2935	C₂	semidirect
3. D₄×M₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1589	C₂	semidirect
5. C₃₁×#1581	C₂	semidirect
6. C₃₁×#1581	C₂	semidirect
7. C₈×D₈×C₃₁	C₂	semidirect
8. C₃₁×#1571	C₂	semidirect
9. C₃₁×#1492	C₂	semidirect
10. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
25. C₃₁×#436	C₂²	semidirect
29. C₈×D₄×C₃₁	C₂²	semidirect
37. C₃₁×#335	C₂²	semidirect
38. C₃₁×#335	C₂²	semidirect
43. C₃₁×D₄⋊C₈	C₂²	semidirect
44. C₃₁×D₄⋊C₈	C₂²	semidirect
45. C₃₁×#322	C₂²	semidirect
46. C₈²×C₃₁	C₂²	semidirect
133. M₁₆×C₃₁	D₈	semidirect
162. #17994	C₃₁	direct (N × H)
180. D₄×C₃₁	#131	semidirect
214. #2935	C₂×C₃₁	semidirect
215. D₄×M₁₆	C₂×C₃₁	semidirect
216. #1589	C₂×C₃₁	semidirect
217. #1581	C₂×C₃₁	semidirect
218. #1581	C₂×C₃₁	semidirect
219. C₈×D₈	C₂×C₃₁	semidirect
220. #1571	C₂×C₃₁	semidirect
221. #1492	C₂×C₃₁	semidirect
222. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
223. C₈×M₁₆	C₂×C₃₁	semidirect
262. #436	C₂²×C₃₁	semidirect
266. C₈×D₄	C₂²×C₃₁	semidirect
274. #335	C₂²×C₃₁	semidirect
275. #335	C₂²×C₃₁	semidirect
280. D₄⋊C₈	C₂²×C₃₁	semidirect
281. D₄⋊C₈	C₂²×C₃₁	semidirect
282. #322	C₂²×C₃₁	semidirect
283. C₈²	C₂²×C₃₁	semidirect
347. C₃₁	#17994	direct (N × H)
378. M₁₆	D₈×C₃₁	semidirect
424. D₄	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3100	2
C₃₁×#2935	2
D₄×M₁₆×C₃₁	2
C₃₁×#1589	2
C₃₁×#1581	2
C₈×D₈×C₃₁	2
C₃₁×#1571	2
C₃₁×#1492	2
C₂×C₃₁×D₄⋊C₈	2
C₈×M₁₆×C₃₁	2
#17994	31

Subgroups — C31×#17994 #21474753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17994 #21474753