Subgroups — C₃₁×#18012 #21474771

Order 7936.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3108	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2969	C₂	1	1	yes	maximal, characteristic
3	Q₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1593	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1587	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1583	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1495	C₂	1	1	yes	maximal, characteristic
14	C₃₁×#1479	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#529	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#483	C₄	1	1	yes	characteristic
19	C₃₁×#480	C₄	1	1	yes	characteristic
20	C₃₁×#479	C₄	1	1	yes	characteristic
21	C₃₁×#479	C₄	1	1	yes	characteristic
22	C₃₁×#464	C₄	1	1	yes	characteristic
23	C₃₁×#464	C₄	1	1	yes	characteristic
24	C₃₁×#451	C₄	1	1	yes	characteristic
25	C₃₁×#448	C₄	1	1	yes	characteristic
26	C₃₁×#446	—	2	1	no	—
27	C₃₁×#446	—	2	1	no	—
28	C₃₁×#446	C₂²	1	1	yes	characteristic
29	C₃₁×#446	C₂²	1	1	yes	characteristic
30	C₃₁×#446	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#436	—	2	1	no	—
39	C₃₁×#436	C₂²	1	1	yes	characteristic
40	C₃₁×#435	—	2	1	no	—
41	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#433	—	2	1	no	—
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₃₁×#431	C₂²	1	1	yes	characteristic
45	C₃₁×#431	C₂²	1	1	yes	characteristic
46	C₃₁×#423	—	2	1	no	—
47	C₃₁×#423	C₂²	1	1	yes	characteristic
48	C₃₁×#405	—	2	1	no	—
49	C₄×M₁₆×C₃₁	—	2	1	no	—
50	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
51	C₃₁×#335	C₂²	1	1	yes	characteristic
52	C₃₁×#335	C₂²	1	1	yes	characteristic
53	C₃₁×#334	C₂²	1	1	yes	characteristic
54	C₃₁×#334	C₂²	1	1	yes	characteristic
55	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
61	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₃₁×#322	C₂²	1	1	yes	characteristic
64	C₃₁×#322	C₂²	1	1	yes	characteristic
65	C₃₁×#322	C₂²	1	1	yes	characteristic
66	C₈²×C₃₁	C₂²	1	1	yes	characteristic
67	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂×M₁₆×C₃₁	—	2	1	no	—
73	C₂×M₁₆×C₃₁	—	2	1	no	—
74	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
75	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	—	2	1	no	—
77	C₃₁×#126	—	2	1	no	—
78	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
79	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
80	C₃₁×#123	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
82	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₃₁	—	2	1	no	—
84	C₄×Q₈×C₃₁	—	2	1	no	—
85	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₃₁	—	2	1	no	—
91	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#117	—	2	1	no	—
93	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	—	2	1	no	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
98	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
99	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
100	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
101	C₃₁×C₄⋊C₈	—	2	1	no	—
102	C₃₁×C₄⋊C₈	—	2	1	no	—
103	C₃₁×C₄⋊C₈	—	2	1	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	—	2	1	no	—
107	C₃₁×C₄⋊C₈	—	2	1	no	—
108	C₃₁×C₄⋊C₈	—	2	1	no	—
109	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
110	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
116	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
117	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
118	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
119	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
120	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
121	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
122	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
123	C₃₁×C₂²⋊C₈	—	2	1	no	—
124	C₃₁×C₂²⋊C₈	—	2	1	no	—
125	C₃₁×C₂²⋊C₈	—	2	1	no	—
126	C₃₁×C₂²⋊C₈	—	2	1	no	—
127	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
128	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
129	C₃₁×#97	—	2	1	no	—
130	C₃₁×#97	—	2	1	no	—
131	C₃₁×#97	—	2	1	no	—
132	C₃₁×#97	D₄	1	1	yes	characteristic
133	C₃₁×#97	D₄	1	1	yes	characteristic
134	C₄×C₈×C₃₁	—	2	1	no	—
135	C₄×C₈×C₃₁	—	2	1	no	—
136	C₄×C₈×C₃₁	—	2	1	no	—
137	C₄×C₈×C₃₁	—	2	1	no	—
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×Q₈×C₃₁	—	2	1	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	—	4	1	no	—
147	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	2	1	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
158	Q₁₆×C₃₁	—	2	2	no	—
159	Q₁₆×C₃₁	—	4	1	no	—
160	SD₁₆×C₃₁	—	2	2	no	—
161	SD₁₆×C₃₁	—	4	1	no	—
162	M₁₆×C₃₁	—	2	2	no	—
163	M₁₆×C₃₁	—	4	1	no	—
164	M₁₆×C₃₁	—	4	2	no	—
165	M₁₆×C₃₁	—	4	2	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	1	no	—
172	C₂×C₈×C₃₁	—	4	2	no	—
173	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	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	1	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
194	C₂²⋊C₄×C₃₁	—	2	1	no	—
195	C₂²⋊C₄×C₃₁	—	2	1	no	—
196	C₂²⋊C₄×C₃₁	—	2	1	no	—
197	C₂²⋊C₄×C₃₁	—	2	1	no	—
198	C₂²⋊C₄×C₃₁	—	2	1	no	—
199	C₄²×C₃₁	—	2	1	no	—
200	C₄²×C₃₁	—	2	1	no	—
201	C₄²×C₃₁	—	2	1	no	—
202	C₄²×C₃₁	—	2	1	no	—
203	C₄²×C₃₁	—	2	1	no	—
204	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₃₁	#41	1	1	yes	characteristic
207	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
208	#18012	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₃₁	—	2	1	no	—
210	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
211	Q₈×C₃₁	—	2	1	no	—
212	Q₈×C₃₁	—	2	1	no	—
213	Q₈×C₃₁	—	2	1	no	—
214	Q₈×C₃₁	—	2	1	no	—
215	Q₈×C₃₁	—	2	1	no	—
216	Q₈×C₃₁	—	4	2	no	—
217	Q₈×C₃₁	#131	1	2	yes	—
218	D₄×C₃₁	—	2	1	no	—
219	D₄×C₃₁	—	2	1	no	—
220	D₄×C₃₁	—	8	1	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	1	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	4	1	no	—
233	C₂×C₄×C₃₁	—	4	2	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	8	1	no	—
237	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
243	C₈×C₃₁	—	2	1	no	—
244	C₈×C₃₁	—	2	1	no	—
245	C₈×C₃₁	—	2	1	no	—
246	C₈×C₃₁	—	2	1	no	—
247	C₈×C₃₁	—	2	1	no	—
248	C₈×C₃₁	—	2	1	no	—
249	C₈×C₃₁	—	2	1	no	—
250	C₈×C₃₁	—	4	1	no	—
251	C₈×C₃₁	—	4	1	no	—
252	C₈×C₃₁	—	4	1	no	—
253	C₈×C₃₁	—	4	1	no	—
254	C₈×C₃₁	—	4	1	no	—
255	C₈×C₃₁	—	4	1	no	—
256	C₈×C₃₁	#137	1	2	yes	—
257	#3108	C₂×C₃₁	1	1	yes	characteristic
258	#2969	C₂×C₃₁	1	1	yes	characteristic
259	Q₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1593	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1590	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1587	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#1585	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1584	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1583	C₂×C₃₁	1	1	yes	characteristic, complemented
266	SD₁₆⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1495	C₂×C₃₁	1	1	yes	characteristic
270	#1479	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂²×C₃₁	—	2	1	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	4	1	no	—
276	C₂²×C₃₁	—	4	1	no	—
277	C₂²×C₃₁	—	4	1	no	—
278	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
279	C₄×C₃₁	—	2	1	no	—
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	2	1	no	—
283	C₄×C₃₁	—	2	1	no	—
284	C₄×C₃₁	—	4	1	no	—
285	C₄×C₃₁	—	4	1	no	—
286	C₄×C₃₁	—	4	1	no	—
287	C₄×C₃₁	—	4	1	no	—
288	C₄×C₃₁	—	4	1	no	—
289	C₄×C₃₁	—	4	1	no	—
290	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
291	C₄×C₃₁	#568	1	1	yes	characteristic
292	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
293	C₄×C₃₁	#575	1	1	yes	characteristic
294	#529	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
296	#483	C₄×C₃₁	1	1	yes	characteristic
297	#480	C₄×C₃₁	1	1	yes	characteristic
298	#479	C₄×C₃₁	1	1	yes	characteristic
299	#479	C₄×C₃₁	1	1	yes	characteristic
300	#464	C₄×C₃₁	1	1	yes	characteristic
301	#464	C₄×C₃₁	1	1	yes	characteristic
302	#451	C₄×C₃₁	1	1	yes	characteristic
303	#448	C₄×C₃₁	1	1	yes	characteristic
304	#446	—	2	1	no	—
305	#446	—	2	1	no	—
306	#446	C₂²×C₃₁	1	1	yes	characteristic
307	#446	C₂²×C₃₁	1	1	yes	characteristic
308	#446	C₂²×C₃₁	1	1	yes	characteristic
309	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
311	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
312	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
313	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
314	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
315	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
316	#436	—	2	1	no	—
317	#436	C₂²×C₃₁	1	1	yes	characteristic
318	#435	—	2	1	no	—
319	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
320	#433	—	2	1	no	—
321	#431	C₂²×C₃₁	1	1	yes	characteristic
322	#431	C₂²×C₃₁	1	1	yes	characteristic
323	#431	C₂²×C₃₁	1	1	yes	characteristic
324	#423	—	2	1	no	—
325	#423	C₂²×C₃₁	1	1	yes	characteristic
326	#405	—	2	1	no	—
327	C₄×M₁₆	—	2	1	no	—
328	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
329	#335	C₂²×C₃₁	1	1	yes	characteristic
330	#335	C₂²×C₃₁	1	1	yes	characteristic
331	#334	C₂²×C₃₁	1	1	yes	characteristic
332	#334	C₂²×C₃₁	1	1	yes	characteristic
333	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
334	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
335	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
336	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
337	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
339	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
340	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
341	#322	C₂²×C₃₁	1	1	yes	characteristic
342	#322	C₂²×C₃₁	1	1	yes	characteristic
343	#322	C₂²×C₃₁	1	1	yes	characteristic
344	C₈²	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×C₃₁	—	4	1	no	—
346	C₂×C₃₁	—	8	1	no	—
347	C₂×C₃₁	#2933	1	1	yes	characteristic
348	C₂×C₃₁	C₄×#137	1	1	yes	characteristic
349	C₂×C₃₁	#2957	1	1	yes	characteristic
350	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
353	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
354	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₂×M₁₆	—	2	1	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
358	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₈	—	2	1	no	—
360	#126	—	2	1	no	—
361	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
362	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
363	#123	—	2	1	no	—
364	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
365	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
366	C₄×Q₈	—	2	1	no	—
367	C₄×Q₈	—	2	1	no	—
368	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
369	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
370	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
372	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
375	#117	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄²	—	2	1	no	—
379	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
380	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
381	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
382	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
383	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	—	2	1	no	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
393	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
394	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
398	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
399	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
400	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
401	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
402	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
403	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
404	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
405	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
406	C₂²⋊C₈	—	2	1	no	—
407	C₂²⋊C₈	—	2	1	no	—
408	C₂²⋊C₈	—	2	1	no	—
409	C₂²⋊C₈	—	2	1	no	—
410	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
411	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
412	#97	—	2	1	no	—
413	#97	—	2	1	no	—
414	#97	—	2	1	no	—
415	#97	D₄×C₃₁	1	1	yes	characteristic
416	#97	D₄×C₃₁	1	1	yes	characteristic
417	C₄×C₈	—	2	1	no	—
418	C₄×C₈	—	2	1	no	—
419	C₄×C₈	—	2	1	no	—
420	C₄×C₈	—	2	1	no	—
421	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
426	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
427	C₃₁	#18012	1	1	yes	characteristic, direct product factor, Sylow
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	4	1	no	—
431	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
433	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
442	Q₁₆	—	2	2	no	—
443	Q₁₆	—	4	1	no	—
444	SD₁₆	—	2	2	no	—
445	SD₁₆	—	4	1	no	—
446	M₁₆	—	2	2	no	—
447	M₁₆	—	4	1	no	—
448	M₁₆	—	4	2	no	—
449	M₁₆	—	4	2	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	4	2	no	—
457	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
458	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
460	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
461	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
475	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
476	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
477	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
478	C₂²⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	2	1	no	—
481	C₂²⋊C₄	—	2	1	no	—
482	C₂²⋊C₄	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
489	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
490	C₄²	#41×C₃₁	1	1	yes	characteristic
491	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
492	C₂³	—	2	1	no	—
493	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	2	1	no	—
498	Q₈	—	2	1	no	—
499	Q₈	—	4	2	no	—
500	Q₈	C₃₁×#131	1	2	yes	—
501	D₄	—	2	1	no	—
502	D₄	—	2	1	no	—
503	D₄	—	8	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	8	1	no	—
520	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
521	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
526	C₈	—	2	1	no	—
527	C₈	—	2	1	no	—
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	2	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	1	no	—
539	C₈	C₃₁×#137	1	2	yes	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
547	C₄	—	2	1	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	4	1	no	—
558	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
559	C₄	C₃₁×#568	1	1	yes	characteristic
560	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
561	C₄	C₃₁×#575	1	1	yes	characteristic
562	C₂	—	4	1	no	—
563	C₂	—	8	1	no	—
564	C₂	C₃₁×#2933	1	1	yes	characteristic
565	C₂	C₄×C₃₁×#137	1	1	yes	characteristic
566	C₂	C₃₁×#2957	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×#121	C₂×C₄
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
3. Q₈×M₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1593	C₂	semidirect
5. C₃₁×#1590	C₂	semidirect
6. C₃₁×#1587	C₂	semidirect
7. C₃₁×#1585	C₂	semidirect
8. C₃₁×#1584	C₂	semidirect
9. C₃₁×#1583	C₂	semidirect
10. C₃₁×SD₁₆⋊C₈	C₂	semidirect
11. C₈×Q₁₆×C₃₁	C₂	semidirect
12. C₃₁×#1574	C₂	semidirect
14. C₃₁×#1479	C₂	semidirect
15. C₃₁×#1453	C₂	semidirect
208. #18012	C₃₁	direct (N × H)
259. Q₈×M₁₆	C₂×C₃₁	semidirect
260. #1593	C₂×C₃₁	semidirect
261. #1590	C₂×C₃₁	semidirect
262. #1587	C₂×C₃₁	semidirect
263. #1585	C₂×C₃₁	semidirect
264. #1584	C₂×C₃₁	semidirect
265. #1583	C₂×C₃₁	semidirect
266. SD₁₆⋊C₈	C₂×C₃₁	semidirect
267. C₈×Q₁₆	C₂×C₃₁	semidirect
268. #1574	C₂×C₃₁	semidirect
270. #1479	C₂×C₃₁	semidirect
271. #1453	C₂×C₃₁	semidirect
427. C₃₁	#18012	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×#3108	2
C₃₁×#2969	2
Q₈×M₁₆×C₃₁	2
C₃₁×#1593	2
C₃₁×#1590	2
C₃₁×#1587	2
C₃₁×#1585	2
C₃₁×#1584	2
C₃₁×#1583	2
C₃₁×SD₁₆⋊C₈	2
C₈×Q₁₆×C₃₁	2
C₃₁×#1574	2
C₃₁×#1495	2
C₃₁×#1479	2
C₃₁×#1453	2
#18012	31

Subgroups — C31×#18012 #21474771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18012 #21474771