Subgroups — C₃₁×#18032 #21474791

Order 7936.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3104	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2986	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2984	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1593	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1587	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1587	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1584	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1572	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#1497	C₂	1	1	yes	maximal, characteristic
14	C₃₁×#1479	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#1454	C₂	1	1	yes	maximal, characteristic
16	C₃₁×#533	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#484	C₄	1	1	yes	characteristic
19	C₃₁×#484	C₄	1	1	yes	characteristic
20	C₃₁×#477	C₄	1	1	yes	characteristic
21	C₃₁×#477	C₄	1	1	yes	characteristic
22	C₃₁×#462	C₄	1	1	yes	characteristic
23	C₃₁×#462	C₄	1	1	yes	characteristic
24	C₃₁×#451	C₄	1	1	yes	characteristic
25	C₃₁×#451	C₄	1	1	yes	characteristic
26	C₃₁×#446	—	2	1	no	—
27	C₃₁×#446	—	2	1	no	—
28	C₃₁×#446	—	2	1	no	—
29	C₃₁×#446	C₂²	1	1	yes	characteristic
30	C₃₁×#446	C₂²	1	1	yes	characteristic
31	C₃₁×#446	C₂²	1	1	yes	characteristic
32	C₃₁×#446	C₂²	1	1	yes	characteristic
33	C₈×Q₈×C₃₁	—	2	1	no	—
34	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#433	—	2	1	no	—
43	C₃₁×#433	—	2	1	no	—
44	C₃₁×#432	—	2	1	no	—
45	C₃₁×#432	C₂²	1	1	yes	characteristic
46	C₃₁×#431	—	2	1	no	—
47	C₃₁×#431	C₂²	1	1	yes	characteristic
48	C₃₁×#431	C₂²	1	1	yes	characteristic
49	C₃₁×#431	C₂²	1	1	yes	characteristic
50	C₃₁×#431	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×Q₈⋊C₈	C₂²	1	1	yes	characteristic
62	C₃₁×Q₈⋊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×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
73	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
74	C₃₁×#125	—	2	1	no	—
75	C₃₁×C₂²⋊Q₈	—	2	1	no	—
76	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
77	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	—	2	1	no	—
81	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#117	—	2	1	no	—
88	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
93	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
94	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
95	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×C₄⋊C₈	—	2	1	no	—
99	C₃₁×C₄⋊C₈	—	2	1	no	—
100	C₃₁×C₄⋊C₈	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
106	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
107	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
109	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
114	C₃₁×#103	C₂×C₄	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×C₂²⋊C₈	—	2	1	no	—
122	C₃₁×C₂²⋊C₈	—	2	1	no	—
123	C₃₁×C₂²⋊C₈	—	2	1	no	—
124	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
125	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
126	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
127	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
128	C₃₁×#97	—	2	1	no	—
129	C₃₁×#97	—	2	1	no	—
130	C₃₁×#97	—	2	1	no	—
131	C₃₁×#97	—	2	1	no	—
132	C₃₁×#97	—	2	1	no	—
133	C₄×C₈×C₃₁	—	2	1	no	—
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	1	no	—
146	C₂×Q₈×C₃₁	—	2	2	no	—
147	C₂×Q₈×C₃₁	—	4	1	no	—
148	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₂×Q₈×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	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
157	Q₁₆×C₃₁	—	2	2	no	—
158	Q₁₆×C₃₁	—	2	2	no	—
159	Q₁₆×C₃₁	—	4	1	no	—
160	Q₁₆×C₃₁	—	4	1	no	—
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
191	C₂²⋊C₄×C₃₁	—	2	1	no	—
192	C₂²⋊C₄×C₃₁	—	2	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	—	2	1	no	—
196	C₄²×C₃₁	—	2	1	no	—
197	C₄²×C₃₁	—	2	1	no	—
198	C₄²×C₃₁	—	2	1	no	—
199	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
200	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₃₁	#41	1	1	yes	characteristic
202	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
203	#18032	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	1	no	—
206	Q₈×C₃₁	—	2	1	no	—
207	Q₈×C₃₁	—	2	1	no	—
208	Q₈×C₃₁	—	2	1	no	—
209	Q₈×C₃₁	—	2	1	no	—
210	Q₈×C₃₁	—	2	1	no	—
211	Q₈×C₃₁	—	2	1	no	—
212	Q₈×C₃₁	—	4	2	no	—
213	Q₈×C₃₁	—	4	2	no	—
214	Q₈×C₃₁	#131	1	2	yes	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	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	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
236	C₈×C₃₁	—	2	1	no	—
237	C₈×C₃₁	—	2	1	no	—
238	C₈×C₃₁	—	2	1	no	—
239	C₈×C₃₁	—	2	1	no	—
240	C₈×C₃₁	—	2	1	no	—
241	C₈×C₃₁	—	2	1	no	—
242	C₈×C₃₁	—	2	1	no	—
243	C₈×C₃₁	—	2	1	no	—
244	C₈×C₃₁	—	4	1	no	—
245	C₈×C₃₁	—	4	1	no	—
246	C₈×C₃₁	—	4	1	no	—
247	C₈×C₃₁	—	4	1	no	—
248	C₈×C₃₁	—	4	1	no	—
249	C₈×C₃₁	—	4	1	no	—
250	#3104	C₂×C₃₁	1	1	yes	characteristic
251	#2986	C₂×C₃₁	1	1	yes	characteristic
252	#2984	C₂×C₃₁	1	1	yes	characteristic
253	#1593	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1587	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1587	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1584	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1584	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#1579	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1572	C₂×C₃₁	1	1	yes	characteristic
262	#1497	C₂×C₃₁	1	1	yes	characteristic
263	#1479	C₂×C₃₁	1	1	yes	characteristic
264	#1454	C₂×C₃₁	1	1	yes	characteristic
265	C₂²×C₃₁	—	2	1	no	—
266	C₂²×C₃₁	—	2	1	no	—
267	C₂²×C₃₁	—	2	1	no	—
268	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
269	C₄×C₃₁	—	2	1	no	—
270	C₄×C₃₁	—	2	2	no	—
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	4	1	no	—
275	C₄×C₃₁	—	4	1	no	—
276	C₄×C₃₁	—	4	1	no	—
277	C₄×C₃₁	—	4	1	no	—
278	C₄×C₃₁	—	4	1	no	—
279	C₄×C₃₁	—	4	1	no	—
280	C₄×C₃₁	—	4	1	no	—
281	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
282	C₄×C₃₁	#568	1	1	yes	characteristic
283	C₄×C₃₁	#578	1	1	yes	characteristic
284	C₄×C₃₁	#578	1	1	yes	characteristic
285	#533	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
287	#484	C₄×C₃₁	1	1	yes	characteristic
288	#484	C₄×C₃₁	1	1	yes	characteristic
289	#477	C₄×C₃₁	1	1	yes	characteristic
290	#477	C₄×C₃₁	1	1	yes	characteristic
291	#462	C₄×C₃₁	1	1	yes	characteristic
292	#462	C₄×C₃₁	1	1	yes	characteristic
293	#451	C₄×C₃₁	1	1	yes	characteristic
294	#451	C₄×C₃₁	1	1	yes	characteristic
295	#446	—	2	1	no	—
296	#446	—	2	1	no	—
297	#446	—	2	1	no	—
298	#446	C₂²×C₃₁	1	1	yes	characteristic
299	#446	C₂²×C₃₁	1	1	yes	characteristic
300	#446	C₂²×C₃₁	1	1	yes	characteristic
301	#446	C₂²×C₃₁	1	1	yes	characteristic
302	C₈×Q₈	—	2	1	no	—
303	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
304	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
305	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
306	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
307	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
308	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
309	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
310	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
311	#433	—	2	1	no	—
312	#433	—	2	1	no	—
313	#432	—	2	1	no	—
314	#432	C₂²×C₃₁	1	1	yes	characteristic
315	#431	—	2	1	no	—
316	#431	C₂²×C₃₁	1	1	yes	characteristic
317	#431	C₂²×C₃₁	1	1	yes	characteristic
318	#431	C₂²×C₃₁	1	1	yes	characteristic
319	#431	C₂²×C₃₁	1	1	yes	characteristic
320	#335	C₂²×C₃₁	1	1	yes	characteristic
321	#335	C₂²×C₃₁	1	1	yes	characteristic
322	#334	C₂²×C₃₁	1	1	yes	characteristic
323	#334	C₂²×C₃₁	1	1	yes	characteristic
324	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
325	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
326	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
327	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
328	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
329	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
330	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
331	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
332	#322	C₂²×C₃₁	1	1	yes	characteristic
333	#322	C₂²×C₃₁	1	1	yes	characteristic
334	#322	C₂²×C₃₁	1	1	yes	characteristic
335	C₈²	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×C₃₁	—	4	1	no	—
337	C₂×C₃₁	#2933	1	1	yes	characteristic
338	C₂×C₃₁	#2950	1	1	yes	characteristic
339	C₂×C₃₁	#2955	1	1	yes	characteristic
340	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
341	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
342	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
343	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
344	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
345	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
346	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
347	#125	—	2	1	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₄×Q₈	—	2	1	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
358	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
359	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
360	#117	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
365	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
366	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
367	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
368	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
369	C₄⋊C₈	—	2	1	no	—
370	C₄⋊C₈	—	2	1	no	—
371	C₄⋊C₈	—	2	1	no	—
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
379	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
381	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
382	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
384	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
385	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
386	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
387	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
388	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
389	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
390	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
391	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
392	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
393	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
394	C₂²⋊C₈	—	2	1	no	—
395	C₂²⋊C₈	—	2	1	no	—
396	C₂²⋊C₈	—	2	1	no	—
397	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
398	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
399	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
400	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
401	#97	—	2	1	no	—
402	#97	—	2	1	no	—
403	#97	—	2	1	no	—
404	#97	—	2	1	no	—
405	#97	—	2	1	no	—
406	C₄×C₈	—	2	1	no	—
407	C₄×C₈	—	2	1	no	—
408	C₄×C₈	—	2	1	no	—
409	C₄×C₈	—	2	1	no	—
410	C₄×C₈	—	2	1	no	—
411	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
412	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
415	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
416	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
417	C₃₁	#18032	1	1	yes	characteristic, direct product factor, Sylow
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	4	1	no	—
422	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
423	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
424	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
425	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
431	Q₁₆	—	2	2	no	—
432	Q₁₆	—	2	2	no	—
433	Q₁₆	—	4	1	no	—
434	Q₁₆	—	4	1	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
442	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
443	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
444	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
445	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
446	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
462	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
463	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
464	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄²	#41×C₃₁	1	1	yes	characteristic
476	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
477	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
478	Q₈	—	2	1	no	—
479	Q₈	—	2	1	no	—
480	Q₈	—	2	1	no	—
481	Q₈	—	2	1	no	—
482	Q₈	—	2	1	no	—
483	Q₈	—	2	1	no	—
484	Q₈	—	2	1	no	—
485	Q₈	—	4	2	no	—
486	Q₈	—	4	2	no	—
487	Q₈	C₃₁×#131	1	2	yes	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
506	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
509	C₈	—	2	1	no	—
510	C₈	—	2	1	no	—
511	C₈	—	2	1	no	—
512	C₈	—	2	1	no	—
513	C₈	—	2	1	no	—
514	C₈	—	2	1	no	—
515	C₈	—	2	1	no	—
516	C₈	—	2	1	no	—
517	C₈	—	4	1	no	—
518	C₈	—	4	1	no	—
519	C₈	—	4	1	no	—
520	C₈	—	4	1	no	—
521	C₈	—	4	1	no	—
522	C₈	—	4	1	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
527	C₄	—	2	1	no	—
528	C₄	—	2	2	no	—
529	C₄	—	2	1	no	—
530	C₄	—	2	1	no	—
531	C₄	—	2	1	no	—
532	C₄	—	4	1	no	—
533	C₄	—	4	1	no	—
534	C₄	—	4	1	no	—
535	C₄	—	4	1	no	—
536	C₄	—	4	1	no	—
537	C₄	—	4	1	no	—
538	C₄	—	4	1	no	—
539	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
540	C₄	C₃₁×#568	1	1	yes	characteristic
541	C₄	C₃₁×#578	1	1	yes	characteristic
542	C₄	C₃₁×#578	1	1	yes	characteristic
543	C₂	—	4	1	no	—
544	C₂	C₃₁×#2933	1	1	yes	characteristic
545	C₂	C₃₁×#2950	1	1	yes	characteristic
546	C₂	C₃₁×#2955	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	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1593	C₂	semidirect
5. C₃₁×#1591	C₂	semidirect
6. C₃₁×#1587	C₂	semidirect
7. C₃₁×#1587	C₂	semidirect
8. C₃₁×#1584	C₂	semidirect
9. C₃₁×#1584	C₂	semidirect
10. C₃₁×#1579	C₂	semidirect
11. C₈×Q₁₆×C₃₁	C₂	semidirect
203. #18032	C₃₁	direct (N × H)
253. #1593	C₂×C₃₁	semidirect
254. #1591	C₂×C₃₁	semidirect
255. #1587	C₂×C₃₁	semidirect
256. #1587	C₂×C₃₁	semidirect
257. #1584	C₂×C₃₁	semidirect
258. #1584	C₂×C₃₁	semidirect
259. #1579	C₂×C₃₁	semidirect
260. C₈×Q₁₆	C₂×C₃₁	semidirect
417. C₃₁	#18032	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×#3104	2
C₃₁×#2986	2
C₃₁×#2984	2
C₃₁×#1593	2
C₃₁×#1591	2
C₃₁×#1587	2
C₃₁×#1584	2
C₃₁×#1579	2
C₈×Q₁₆×C₃₁	2
C₃₁×#1572	2
C₃₁×#1497	2
C₃₁×#1479	2
C₃₁×#1454	2
#18032	31

Subgroups — C31×#18032 #21474791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18032 #21474791