Subgroups — C₃₁×#18602 #21475361

Order 7936.

900 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3216	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3184	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1653	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1640	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1626	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1620	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1553	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#559	C₂²	1	2	yes	—
13	C₃₁×#557	C₂²	1	2	yes	—
14	C₃₁×#556	—	2	1	no	—
15	C₃₁×#556	—	2	2	no	—
16	C₃₁×#554	—	2	1	no	—
17	C₃₁×#552	—	2	2	no	—
18	C₃₁×#539	—	2	2	no	—
19	C₃₁×#536	—	2	1	no	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#532	C₂²	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#529	—	2	2	no	—
24	C₃₁×#520	—	2	1	no	—
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#498	C₂²	1	1	yes	characteristic
27	C₃₁×#497	—	2	1	no	—
28	C₃₁×#493	C₂²	1	1	yes	characteristic
29	C₃₁×#489	—	2	1	no	—
30	C₃₁×#486	C₂²	1	1	yes	characteristic
31	C₃₁×#486	C₂²	1	1	yes	characteristic
32	C₃₁×#481	—	2	1	no	—
33	C₃₁×#475	C₂²	1	2	yes	—
34	C₃₁×#469	—	2	2	no	—
35	C₃₁×#462	C₂²	1	2	yes	—
36	C₃₁×#460	C₂²	1	2	yes	—
37	C₃₁×#460	C₂²	1	2	yes	—
38	C₃₁×#452	—	2	2	no	—
39	C₃₁×#448	—	2	1	no	—
40	C₃₁×#442	—	2	2	no	—
41	C₄×SD₁₆×C₃₁	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	1	no	—
52	C₂×#41×C₃₁	—	2	1	no	—
53	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×#127	C₂³	1	1	yes	characteristic
65	C₃₁×#127	C₂³	1	1	yes	characteristic
66	C₃₁×#126	—	2	4	no	—
67	C₃₁×#126	—	2	4	no	—
68	C₃₁×#125	—	2	2	no	—
69	C₃₁×#125	—	2	4	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#124	—	2	2	no	—
73	C₃₁×#124	—	2	1	no	—
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#123	—	2	4	no	—
76	C₃₁×#123	—	2	4	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	—	2	4	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	1	no	—
84	C₃₁×#121	—	2	1	no	—
85	C₃₁×#121	D₄	1	2	yes	—
86	C₄×Q₈×C₃₁	—	2	2	no	—
87	C₄×Q₈×C₃₁	—	2	2	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	D₄	1	2	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
94	C₄×D₄×C₃₁	—	2	4	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	2	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	D₄	1	2	yes	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	—	2	4	no	—
103	C₃₁×#117	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₃₁	—	2	1	no	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₃₁×#106	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#102	—	2	2	no	—
116	C₃₁×#102	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×C₂²⋊C₈	—	2	1	no	—
120	C₃₁×C₂²⋊C₈	—	2	1	no	—
121	C₃₁×#97	—	2	1	no	—
122	C₃₁×#97	—	2	1	no	—
123	C₄×C₈×C₃₁	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	#41×C₃₁	—	4	4	no	—
127	#41×C₃₁	—	8	2	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	4	2	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	4	1	no	—
142	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	4	1	no	—
148	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
151	Q₁₆×C₃₁	—	8	2	no	—
152	SD₁₆×C₃₁	—	4	4	no	—
153	SD₁₆×C₃₁	—	8	2	no	—
154	D₈×C₃₁	—	4	4	no	—
155	M₁₆×C₃₁	—	4	2	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	4	no	—
162	C₄⋊C₄×C₃₁	—	2	4	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₃₁	—	2	2	no	—
181	C₂²⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	4	1	no	—
184	C₂²⋊C₄×C₃₁	—	4	1	no	—
185	C₄²×C₃₁	—	2	4	no	—
186	C₄²×C₃₁	—	2	2	no	—
187	C₄²×C₃₁	—	2	2	no	—
188	C₄²×C₃₁	—	2	2	no	—
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
193	#18602	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	4	1	no	—
196	C₂³×C₃₁	#120	1	1	yes	characteristic
197	Q₈×C₃₁	—	2	2	no	—
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	2	2	no	—
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	Q₈×C₃₁	—	8	1	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	4	no	—
208	D₄×C₃₁	—	4	2	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	4	2	no	—
212	D₄×C₃₁	—	8	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	4	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	8	2	no	—
225	C₂×C₄×C₃₁	—	8	1	no	—
226	C₂×C₄×C₃₁	—	8	4	no	—
227	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
228	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
229	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₈×C₃₁	—	2	2	no	—
234	C₈×C₃₁	—	4	1	no	—
235	C₈×C₃₁	—	4	2	no	—
236	C₈×C₃₁	—	4	2	no	—
237	#3507	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#3216	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#3184	C₂×C₃₁	1	1	yes	characteristic
240	#3103	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1653	C₂×C₃₁	1	2	yes	complemented
242	#1640	C₂×C₃₁	1	2	yes	complemented
243	#1626	C₂×C₃₁	1	2	yes	complemented
244	#1620	C₂×C₃₁	1	2	yes	complemented
245	#1553	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1491	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1485	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂²×C₃₁	—	2	1	no	—
249	C₂²×C₃₁	—	2	1	no	—
250	C₂²×C₃₁	—	2	1	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	8	1	no	—
255	C₂²×C₃₁	—	8	1	no	—
256	C₂²×C₃₁	—	8	1	no	—
257	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	2	1	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	4	2	no	—
267	C₄×C₃₁	—	8	1	no	—
268	C₄×C₃₁	#575	1	2	yes	—
269	C₄×C₃₁	#577	1	2	yes	—
270	#559	C₂²×C₃₁	1	2	yes	—
271	#557	C₂²×C₃₁	1	2	yes	—
272	#556	—	2	1	no	—
273	#556	—	2	2	no	—
274	#554	—	2	1	no	—
275	#552	—	2	2	no	—
276	#539	—	2	2	no	—
277	#536	—	2	1	no	—
278	#533	C₂²×C₃₁	1	1	yes	characteristic
279	#532	C₂²×C₃₁	1	1	yes	characteristic
280	#532	C₂²×C₃₁	1	1	yes	characteristic
281	#529	—	2	2	no	—
282	#520	—	2	1	no	—
283	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
284	#498	C₂²×C₃₁	1	1	yes	characteristic
285	#497	—	2	1	no	—
286	#493	C₂²×C₃₁	1	1	yes	characteristic
287	#489	—	2	1	no	—
288	#486	C₂²×C₃₁	1	1	yes	characteristic
289	#486	C₂²×C₃₁	1	1	yes	characteristic
290	#481	—	2	1	no	—
291	#475	C₂²×C₃₁	1	2	yes	—
292	#469	—	2	2	no	—
293	#462	C₂²×C₃₁	1	2	yes	—
294	#460	C₂²×C₃₁	1	2	yes	—
295	#460	C₂²×C₃₁	1	2	yes	—
296	#452	—	2	2	no	—
297	#448	—	2	1	no	—
298	#442	—	2	2	no	—
299	C₄×SD₁₆	—	2	2	no	—
300	#432	C₂²×C₃₁	1	1	yes	characteristic
301	#431	C₂²×C₃₁	1	1	yes	characteristic
302	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
303	#332	C₂²×C₃₁	1	2	yes	—
304	#331	C₂²×C₃₁	1	2	yes	—
305	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
306	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
307	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
308	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
309	C₂×C₃₁	—	4	1	no	—
310	C₂×C₃₁	—	8	2	no	—
311	C₂×C₃₁	—	16	1	no	—
312	C₂×C₃₁	#3005	1	1	yes	characteristic
313	C₂×C₃₁	#3010	1	1	yes	characteristic
314	C₂×C₃₁	#3018	1	1	yes	characteristic
315	C₂×#41	—	2	1	no	—
316	C₂×#41	—	2	1	no	—
317	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
318	C₂×Q₁₆	—	2	2	no	—
319	C₂×SD₁₆	—	2	2	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×D₈	—	2	2	no	—
322	C₂×M₁₆	—	2	1	no	—
323	C₄⋊Q₈	—	2	2	no	—
324	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
325	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
326	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
327	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
328	#127	C₂³×C₃₁	1	1	yes	characteristic
329	#127	C₂³×C₃₁	1	1	yes	characteristic
330	#126	—	2	4	no	—
331	#126	—	2	4	no	—
332	#125	—	2	2	no	—
333	#125	—	2	4	no	—
334	#125	—	2	1	no	—
335	#124	—	2	2	no	—
336	#124	—	2	2	no	—
337	#124	—	2	1	no	—
338	#124	—	2	1	no	—
339	#123	—	2	4	no	—
340	#123	—	2	4	no	—
341	C₂²⋊Q₈	—	2	2	no	—
342	C₂²⋊Q₈	—	2	2	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	—	2	1	no	—
345	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
346	#121	—	2	1	no	—
347	#121	—	2	1	no	—
348	#121	—	2	1	no	—
349	#121	D₄×C₃₁	1	2	yes	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	2	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	D₄×C₃₁	1	2	yes	—
356	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
357	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
358	C₄×D₄	—	2	4	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	1	no	—
362	C₄×D₄	—	2	1	no	—
363	C₄×D₄	—	2	1	no	—
364	C₄×D₄	D₄×C₃₁	1	2	yes	—
365	#117	—	2	2	no	—
366	#117	—	2	4	no	—
367	#117	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄²	—	2	1	no	—
372	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
373	#106	—	2	2	no	—
374	#106	—	2	2	no	—
375	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
376	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
377	#103	—	2	2	no	—
378	#103	—	2	2	no	—
379	#102	—	2	2	no	—
380	#102	—	2	2	no	—
381	#102	—	2	2	no	—
382	#102	—	2	2	no	—
383	C₂²⋊C₈	—	2	1	no	—
384	C₂²⋊C₈	—	2	1	no	—
385	#97	—	2	1	no	—
386	#97	—	2	1	no	—
387	C₄×C₈	—	2	1	no	—
388	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
389	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
390	C₃₁	#18602	1	1	yes	characteristic, direct product factor, Sylow
391	#41	—	4	4	no	—
392	#41	—	8	2	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	4	2	no	—
398	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
399	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	4	1	no	—
407	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	4	1	no	—
413	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
414	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
415	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
416	Q₁₆	—	8	2	no	—
417	SD₁₆	—	4	4	no	—
418	SD₁₆	—	8	2	no	—
419	D₈	—	4	4	no	—
420	M₁₆	—	4	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	4	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
440	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
441	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
442	C₂²⋊C₄	—	2	4	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	4	1	no	—
449	C₂²⋊C₄	—	4	1	no	—
450	C₄²	—	2	4	no	—
451	C₄²	—	2	2	no	—
452	C₄²	—	2	2	no	—
453	C₄²	—	2	2	no	—
454	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
458	C₂³	—	2	2	no	—
459	C₂³	—	4	1	no	—
460	C₂³	C₃₁×#120	1	1	yes	characteristic
461	Q₈	—	2	2	no	—
462	Q₈	—	2	2	no	—
463	Q₈	—	2	2	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	4	2	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	8	1	no	—
469	D₄	—	4	2	no	—
470	D₄	—	4	2	no	—
471	D₄	—	4	4	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	8	2	no	—
489	C₂×C₄	—	8	1	no	—
490	C₂×C₄	—	8	4	no	—
491	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
492	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
493	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
494	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	4	1	no	—
499	C₈	—	4	2	no	—
500	C₈	—	4	2	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	8	1	no	—
508	C₂²	—	8	1	no	—
509	C₂²	—	8	1	no	—
510	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
511	C₄	—	2	1	no	—
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	2	1	no	—
515	C₄	—	4	2	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	8	1	no	—
521	C₄	C₃₁×#575	1	2	yes	—
522	C₄	C₃₁×#577	1	2	yes	—
523	C₂	—	4	1	no	—
524	C₂	—	8	2	no	—
525	C₂	—	16	1	no	—
526	C₂	C₃₁×#3005	1	1	yes	characteristic
527	C₂	C₃₁×#3010	1	1	yes	characteristic
528	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3184	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. C₃₁×#3507	C₂	semidirect
2. C₃₁×#3216	C₂	semidirect
4. C₃₁×#3103	C₂	semidirect
5. C₃₁×#1653	C₂	semidirect
6. C₃₁×#1640	C₂	semidirect
7. C₃₁×#1626	C₂	semidirect
8. C₃₁×#1620	C₂	semidirect
9. C₃₁×#1553	C₂	semidirect
10. C₃₁×#1491	C₂	semidirect
11. C₃₁×#1485	C₂	semidirect
193. #18602	C₃₁	direct (N × H)
237. #3507	C₂×C₃₁	semidirect
238. #3216	C₂×C₃₁	semidirect
240. #3103	C₂×C₃₁	semidirect
241. #1653	C₂×C₃₁	semidirect
242. #1640	C₂×C₃₁	semidirect
243. #1626	C₂×C₃₁	semidirect
244. #1620	C₂×C₃₁	semidirect
245. #1553	C₂×C₃₁	semidirect
246. #1491	C₂×C₃₁	semidirect
247. #1485	C₂×C₃₁	semidirect
390. C₃₁	#18602	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×#3507	2
C₃₁×#3216	2
C₃₁×#3184	2
C₃₁×#3103	2
C₃₁×#1653	2
C₃₁×#1640	2
C₃₁×#1626	2
C₃₁×#1620	2
C₃₁×#1553	2
C₃₁×#1491	2
C₃₁×#1485	2
#18602	31

Subgroups — C31×#18602 #21475361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18602 #21475361