Subgroups — C₃₁×#17910 #21474669

Order 7936.

720 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3105	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1586	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1585	C₂	1	2	yes	maximal, complemented
6	C₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1573	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1490	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1483	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₃₁	—	2	1	no	—
13	C₃₁×#529	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#484	C₄	1	1	yes	characteristic
16	C₃₁×#481	C₄	1	1	yes	characteristic
17	C₃₁×#478	C₄	1	2	yes	—
18	C₃₁×#463	C₄	1	2	yes	—
19	C₃₁×#450	C₄	1	1	yes	characteristic
20	C₃₁×#447	C₄	1	1	yes	characteristic
21	C₃₁×#446	C₂²	1	1	yes	characteristic, complemented
22	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
24	C₄×D₈×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#436	C₂²	1	1	yes	characteristic, complemented
26	C₃₁×#435	—	2	1	no	—
27	C₈×D₄×C₃₁	—	2	1	no	—
28	C₈×D₄×C₃₁	—	2	1	no	—
29	C₈×D₄×C₃₁	—	2	1	no	—
30	C₈×D₄×C₃₁	C₂²	1	2	yes	—
31	C₈×D₄×C₃₁	C₂²	1	2	yes	—
32	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#433	—	2	1	no	—
34	C₃₁×#432	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
38	C₃₁×#405	—	2	1	no	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	2	yes	complemented
42	C₃₁×#334	C₂²	1	2	yes	complemented
43	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
44	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
45	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
46	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
47	C₃₁×#322	C₂²	1	2	yes	complemented
48	C₈²×C₃₁	C₂²	1	2	yes	complemented
49	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	4	4	no	—
60	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₃₁×#126	—	2	1	no	—
63	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
64	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
65	C₃₁×#123	—	2	1	no	—
66	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
67	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₃₁	—	2	1	no	—
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	2	1	no	—
72	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
73	C₄×D₄×C₃₁	C₂³	1	2	yes	—
74	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#117	—	2	1	no	—
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×#106	C₂×C₄	1	2	yes	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	1	no	—
84	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
85	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×#103	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	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	2	no	—
94	C₃₁×C₂²⋊C₈	—	2	2	no	—
95	C₃₁×C₂²⋊C₈	—	2	1	no	—
96	C₃₁×C₂²⋊C₈	—	2	1	no	—
97	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
98	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
99	C₃₁×#97	—	2	1	no	—
100	C₃₁×#97	—	2	1	no	—
101	C₃₁×#97	—	2	1	no	—
102	C₄×C₈×C₃₁	—	2	1	no	—
103	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
104	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
105	C₄×C₈×C₃₁	C₂³	1	2	yes	—
106	C₄×C₈×C₃₁	C₂³	1	2	yes	—
107	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
109	C₂⁴×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×D₄×C₃₁	—	2	4	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	4	2	no	—
121	C₂²×C₄×C₃₁	—	4	2	no	—
122	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
125	SD₁₆×C₃₁	—	2	4	no	—
126	D₈×C₃₁	—	2	4	no	—
127	M₁₆×C₃₁	—	2	4	no	—
128	M₁₆×C₃₁	—	4	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	4	1	no	—
135	C₂×C₈×C₃₁	—	4	4	no	—
136	C₂×C₈×C₃₁	—	4	2	no	—
137	C₂×C₈×C₃₁	—	4	2	no	—
138	C₂×C₈×C₃₁	—	4	4	no	—
139	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	1	no	—
158	C₂²⋊C₄×C₃₁	—	2	2	no	—
159	C₂²⋊C₄×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₃₁	#41	1	1	yes	characteristic
164	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
165	#17910	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₃₁	—	2	4	no	—
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
172	Q₈×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	2	4	no	—
174	D₄×C₃₁	—	2	2	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	#131	1	4	yes	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
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	4	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₈×C₃₁	—	2	4	no	—
199	C₈×C₃₁	—	2	2	no	—
200	C₈×C₃₁	—	2	2	no	—
201	C₈×C₃₁	—	2	1	no	—
202	C₈×C₃₁	—	2	1	no	—
203	C₈×C₃₁	—	2	1	no	—
204	C₈×C₃₁	—	4	1	no	—
205	C₈×C₃₁	—	4	2	no	—
206	C₈×C₃₁	—	4	1	no	—
207	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
208	#3105	C₂×C₃₁	1	1	yes	characteristic
209	#2966	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1586	C₂×C₃₁	1	2	yes	complemented
212	#1585	C₂×C₃₁	1	2	yes	complemented
213	C₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
214	C₈×D₈	C₂×C₃₁	1	2	yes	complemented
215	#1573	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1490	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1483	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂²×C₃₁	—	2	1	no	—
220	C₂²×C₃₁	—	2	1	no	—
221	C₂²×C₃₁	—	2	4	no	—
222	C₂²×C₃₁	—	4	8	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	—	4	1	no	—
226	C₂²×C₃₁	—	4	1	no	—
227	C₂²×C₃₁	—	4	1	no	—
228	C₂²×C₃₁	#443	1	2	yes	—
229	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
230	C₄×C₃₁	—	2	1	no	—
231	C₄×C₃₁	—	2	1	no	—
232	C₄×C₃₁	—	2	1	no	—
233	C₄×C₃₁	—	2	1	no	—
234	C₄×C₃₁	—	4	1	no	—
235	C₄×C₃₁	—	4	1	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	4	1	no	—
238	C₄×C₃₁	C₂×#131	1	2	yes	—
239	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₄×C₃₁	#575	1	1	yes	characteristic
241	C₂³×C₈	—	2	1	no	—
242	#529	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
244	#484	C₄×C₃₁	1	1	yes	characteristic
245	#481	C₄×C₃₁	1	1	yes	characteristic
246	#478	C₄×C₃₁	1	2	yes	—
247	#463	C₄×C₃₁	1	2	yes	—
248	#450	C₄×C₃₁	1	1	yes	characteristic
249	#447	C₄×C₃₁	1	1	yes	characteristic
250	#446	C₂²×C₃₁	1	1	yes	characteristic, complemented
251	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
252	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
253	C₄×D₈	C₂²×C₃₁	1	2	yes	—
254	#436	C₂²×C₃₁	1	1	yes	characteristic, complemented
255	#435	—	2	1	no	—
256	C₈×D₄	—	2	1	no	—
257	C₈×D₄	—	2	1	no	—
258	C₈×D₄	—	2	1	no	—
259	C₈×D₄	C₂²×C₃₁	1	2	yes	—
260	C₈×D₄	C₂²×C₃₁	1	2	yes	—
261	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
262	#433	—	2	1	no	—
263	#432	C₂²×C₃₁	1	1	yes	characteristic
264	#431	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₂²⋊C₈	—	2	1	no	—
267	#405	—	2	1	no	—
268	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
270	#335	C₂²×C₃₁	1	2	yes	complemented
271	#334	C₂²×C₃₁	1	2	yes	complemented
272	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
273	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
274	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
275	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
276	#322	C₂²×C₃₁	1	2	yes	complemented
277	C₈²	C₂²×C₃₁	1	2	yes	complemented
278	C₂×C₃₁	—	2	2	no	—
279	C₂×C₃₁	—	4	4	no	—
280	C₂×C₃₁	—	8	1	no	—
281	C₂×C₃₁	#2929	1	1	yes	characteristic
282	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
284	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₂³×C₄	—	2	1	no	—
286	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
287	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
288	C₂×M₁₆	—	2	1	no	—
289	C₂×M₁₆	—	2	1	no	—
290	C₂²×C₈	—	2	1	no	—
291	C₂²×C₈	—	2	2	no	—
292	C₂²×C₈	—	2	2	no	—
293	C₂²×C₈	—	2	1	no	—
294	C₂²×C₈	—	4	4	no	—
295	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
297	#126	—	2	1	no	—
298	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
299	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
300	#123	—	2	1	no	—
301	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
302	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
303	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₄×D₄	—	2	1	no	—
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
308	C₄×D₄	C₂³×C₃₁	1	2	yes	—
309	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
310	#117	—	2	1	no	—
311	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
312	C₂×C₂²⋊C₄	—	2	1	no	—
313	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
314	#107	C₂×C₄×C₃₁	1	2	yes	—
315	#106	C₂×C₄×C₃₁	1	2	yes	—
316	C₄⋊C₈	—	2	1	no	—
317	C₄⋊C₈	—	2	1	no	—
318	C₄⋊C₈	—	2	1	no	—
319	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
320	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
323	#103	C₂×C₄×C₃₁	1	2	yes	—
324	#102	C₂×C₄×C₃₁	1	2	yes	—
325	#102	C₂×C₄×C₃₁	1	2	yes	—
326	#102	C₂×C₄×C₃₁	1	2	yes	—
327	C₂²⋊C₈	—	2	1	no	—
328	C₂²⋊C₈	—	2	2	no	—
329	C₂²⋊C₈	—	2	2	no	—
330	C₂²⋊C₈	—	2	1	no	—
331	C₂²⋊C₈	—	2	1	no	—
332	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
333	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
334	#97	—	2	1	no	—
335	#97	—	2	1	no	—
336	#97	—	2	1	no	—
337	C₄×C₈	—	2	1	no	—
338	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
339	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
340	C₄×C₈	C₂³×C₃₁	1	2	yes	—
341	C₄×C₈	C₂³×C₃₁	1	2	yes	—
342	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₃₁	#17910	1	1	yes	characteristic, direct product factor, Sylow
345	C₂⁴	—	2	1	no	—
346	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
347	C₂×D₄	—	2	4	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
361	SD₁₆	—	2	4	no	—
362	D₈	—	2	4	no	—
363	M₁₆	—	2	4	no	—
364	M₁₆	—	4	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	4	1	no	—
371	C₂×C₈	—	4	4	no	—
372	C₂×C₈	—	4	2	no	—
373	C₂×C₈	—	4	2	no	—
374	C₂×C₈	—	4	4	no	—
375	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
376	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
377	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
387	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
388	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
389	C₂²⋊C₄	—	2	1	no	—
390	C₂²⋊C₄	—	2	2	no	—
391	C₂²⋊C₄	—	2	1	no	—
392	C₂²⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
398	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₄²	#41×C₃₁	1	1	yes	characteristic
400	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
401	C₂³	—	2	4	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
407	Q₈	—	2	2	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	2	no	—
411	D₄	—	4	2	no	—
412	D₄	C₃₁×#131	1	4	yes	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₈	—	2	4	no	—
434	C₈	—	2	2	no	—
435	C₈	—	2	2	no	—
436	C₈	—	2	1	no	—
437	C₈	—	2	1	no	—
438	C₈	—	2	1	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	2	no	—
441	C₈	—	4	1	no	—
442	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	4	8	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	1	no	—
451	C₂²	—	4	1	no	—
452	C₂²	C₃₁×#443	1	2	yes	—
453	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	2	1	no	—
457	C₄	—	2	1	no	—
458	C₄	—	4	1	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	1	no	—
461	C₄	—	4	1	no	—
462	C₄	C₂×C₃₁×#131	1	2	yes	—
463	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
464	C₄	C₃₁×#575	1	1	yes	characteristic
465	C₂	—	2	2	no	—
466	C₂	—	4	4	no	—
467	C₂	—	8	1	no	—
468	C₂	C₃₁×#2929	1	1	yes	characteristic
469	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
470	C₂	C₂×C₃₁×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#447	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2966	C₂	semidirect
3. C₂×C₈×D₄×C₃₁	C₂	semidirect
4. C₃₁×#1586	C₂	semidirect
5. C₃₁×#1585	C₂	semidirect
6. C₈×SD₁₆×C₃₁	C₂	semidirect
7. C₈×D₈×C₃₁	C₂	semidirect
8. C₃₁×#1573	C₂	semidirect
9. C₃₁×#1490	C₂	semidirect
10. C₃₁×#1483	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
21. C₃₁×#446	C₂²	semidirect
22. C₈×Q₈×C₃₁	C₂²	semidirect
25. C₃₁×#436	C₂²	semidirect
32. C₈×D₄×C₃₁	C₂²	semidirect
41. C₃₁×#335	C₂²	semidirect
42. C₃₁×#334	C₂²	semidirect
43. C₃₁×Q₈⋊C₈	C₂²	semidirect
46. C₃₁×D₄⋊C₈	C₂²	semidirect
47. C₃₁×#322	C₂²	semidirect
48. C₈²×C₃₁	C₂²	semidirect
165. #17910	C₃₁	direct (N × H)
209. #2966	C₂×C₃₁	semidirect
210. C₂×C₈×D₄	C₂×C₃₁	semidirect
211. #1586	C₂×C₃₁	semidirect
212. #1585	C₂×C₃₁	semidirect
213. C₈×SD₁₆	C₂×C₃₁	semidirect
214. C₈×D₈	C₂×C₃₁	semidirect
215. #1573	C₂×C₃₁	semidirect
216. #1490	C₂×C₃₁	semidirect
217. #1483	C₂×C₃₁	semidirect
218. C₈×M₁₆	C₂×C₃₁	semidirect
250. #446	C₂²×C₃₁	semidirect
251. C₈×Q₈	C₂²×C₃₁	semidirect
254. #436	C₂²×C₃₁	semidirect
261. C₈×D₄	C₂²×C₃₁	semidirect
270. #335	C₂²×C₃₁	semidirect
271. #334	C₂²×C₃₁	semidirect
272. Q₈⋊C₈	C₂²×C₃₁	semidirect
275. D₄⋊C₈	C₂²×C₃₁	semidirect
276. #322	C₂²×C₃₁	semidirect
277. C₈²	C₂²×C₃₁	semidirect
344. C₃₁	#17910	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×#3105	2
C₃₁×#2966	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#1586	2
C₃₁×#1585	2
C₈×SD₁₆×C₃₁	2
C₈×D₈×C₃₁	2
C₃₁×#1573	2
C₃₁×#1490	2
C₃₁×#1483	2
C₈×M₁₆×C₃₁	2
#17910	31

Subgroups — C31×#17910 #21474669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17910 #21474669