Subgroups — C₃₁×#19644 #21476403

Order 7936.

754 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3210	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3207	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3182	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1719	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1717	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1711	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1708	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#533	C₂²	1	1	yes	characteristic
17	C₃₁×#532	C₂²	1	1	yes	characteristic
18	C₃₁×#532	C₂²	1	1	yes	characteristic
19	C₃₁×#532	C₂²	1	1	yes	characteristic
20	C₃₁×#500	C₂²	1	1	yes	characteristic
21	C₃₁×#500	C₂²	1	1	yes	characteristic
22	C₃₁×#498	C₂²	1	1	yes	characteristic
23	C₃₁×#494	C₂²	1	1	yes	characteristic
24	C₃₁×#493	C₂²	1	1	yes	characteristic
25	C₃₁×#493	C₂²	1	1	yes	characteristic
26	C₃₁×#493	C₂²	1	1	yes	characteristic
27	C₃₁×#492	C₂²	1	1	yes	characteristic
28	C₃₁×#492	C₂²	1	1	yes	characteristic
29	C₃₁×#492	C₂²	1	1	yes	characteristic
30	C₃₁×#488	—	2	2	no	—
31	C₃₁×#487	C₂²	1	1	yes	characteristic
32	C₃₁×#487	C₂²	1	1	yes	characteristic
33	C₃₁×#487	C₂²	1	1	yes	characteristic
34	C₃₁×#487	C₂²	1	1	yes	characteristic
35	C₃₁×#487	C₂²	1	1	yes	characteristic
36	C₃₁×#486	C₂²	1	1	yes	characteristic
37	C₃₁×#486	C₂²	1	1	yes	characteristic
38	C₃₁×#486	C₂²	1	1	yes	characteristic
39	C₃₁×#486	C₂²	1	1	yes	characteristic
40	C₃₁×#486	C₂²	1	1	yes	characteristic
41	C₃₁×#486	C₂²	1	1	yes	characteristic
42	C₃₁×#486	C₂²	1	1	yes	characteristic
43	C₃₁×#486	C₂²	1	1	yes	characteristic
44	C₃₁×#486	C₂²	1	1	yes	characteristic
45	C₃₁×#469	—	2	2	no	—
46	C₃₁×#468	—	2	2	no	—
47	C₃₁×#466	—	2	2	no	—
48	C₃₁×#464	—	2	2	no	—
49	C₃₁×#463	—	2	2	no	—
50	C₃₁×#452	—	2	2	no	—
51	C₃₁×#449	—	2	2	no	—
52	C₃₁×#442	—	2	2	no	—
53	C₃₁×#440	—	2	2	no	—
54	C₄×D₈×C₃₁	—	2	2	no	—
55	C₃₁×#431	C₂²	1	1	yes	characteristic
56	C₃₁×#416	—	2	2	no	—
57	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
59	C₃₁×#322	C₂²	1	1	yes	characteristic
60	C₃₁×#322	C₂²	1	1	yes	characteristic
61	C₈²×C₃₁	C₂²	1	1	yes	characteristic
62	C₈²×C₃₁	C₂²	1	1	yes	characteristic
63	C₂×#41×C₃₁	—	2	1	no	—
64	C₂×#41×C₃₁	—	2	1	no	—
65	C₂×#41×C₃₁	—	2	1	no	—
66	C₂×Q₁₆×C₃₁	—	2	2	no	—
67	C₂×SD₁₆×C₃₁	—	2	2	no	—
68	C₂×SD₁₆×C₃₁	—	2	2	no	—
69	C₂×SD₁₆×C₃₁	—	2	2	no	—
70	C₂×D₈×C₃₁	—	2	2	no	—
71	C₂×D₈×C₃₁	—	2	2	no	—
72	C₂×D₈×C₃₁	—	2	2	no	—
73	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
74	C₂²×C₈×C₃₁	D₄	1	2	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×#127	C₂³	1	1	yes	characteristic
81	C₃₁×#127	C₂³	1	1	yes	characteristic
82	C₃₁×#127	C₂³	1	1	yes	characteristic
83	C₃₁×#125	—	2	1	no	—
84	C₃₁×#124	—	2	1	no	—
85	C₃₁×#124	—	2	1	no	—
86	C₃₁×#124	—	2	1	no	—
87	C₃₁×C₂²⋊Q₈	—	2	1	no	—
88	C₃₁×C₂²⋊Q₈	—	2	1	no	—
89	C₃₁×#121	—	2	1	no	—
90	C₃₁×#121	—	2	1	no	—
91	C₃₁×#121	—	2	1	no	—
92	C₃₁×#121	—	2	1	no	—
93	C₃₁×#121	—	2	1	no	—
94	C₃₁×#121	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	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	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₃₁×#117	—	2	1	no	—
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₃₁×#107	—	2	2	no	—
106	C₃₁×#107	—	2	2	no	—
107	C₃₁×#106	—	2	2	no	—
108	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
109	C₃₁×#103	—	2	2	no	—
110	C₃₁×#103	—	2	2	no	—
111	C₃₁×#103	—	2	2	no	—
112	C₃₁×#103	—	2	2	no	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#102	—	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×#102	—	2	2	no	—
120	C₃₁×#102	—	2	2	no	—
121	C₃₁×#102	—	2	2	no	—
122	C₃₁×#102	—	2	2	no	—
123	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
124	C₃₁×#97	D₄	1	2	yes	—
125	C₄×C₈×C₃₁	D₄	1	2	yes	—
126	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
127	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
128	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
130	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
131	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
132	#41×C₃₁	—	4	4	no	—
133	#41×C₃₁	—	8	2	no	—
134	#41×C₃₁	—	8	2	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	2	1	no	—
139	C₂×Q₈×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	1	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	4	1	no	—
150	C₂×D₄×C₃₁	—	4	1	no	—
151	C₂×D₄×C₃₁	—	4	1	no	—
152	C₂²×C₄×C₃₁	—	4	1	no	—
153	C₂²×C₄×C₃₁	—	4	1	no	—
154	C₂²×C₄×C₃₁	—	4	1	no	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	Q₁₆×C₃₁	—	8	2	no	—
159	SD₁₆×C₃₁	—	4	4	no	—
160	SD₁₆×C₃₁	—	8	2	no	—
161	SD₁₆×C₃₁	—	8	2	no	—
162	D₈×C₃₁	—	4	4	no	—
163	D₈×C₃₁	—	4	4	no	—
164	D₈×C₃₁	—	8	2	no	—
165	M₁₆×C₃₁	—	2	4	no	—
166	C₂×C₈×C₃₁	—	4	2	no	—
167	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
169	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
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	—	4	1	no	—
184	C₄⋊C₄×C₃₁	—	4	1	no	—
185	C₂²⋊C₄×C₃₁	—	4	1	no	—
186	C₂²⋊C₄×C₃₁	—	4	1	no	—
187	C₂²⋊C₄×C₃₁	—	4	1	no	—
188	C₂²⋊C₄×C₃₁	—	4	1	no	—
189	C₂²⋊C₄×C₃₁	—	4	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	1	no	—
191	C₂²⋊C₄×C₃₁	—	4	1	no	—
192	C₄²×C₃₁	—	4	1	no	—
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
197	#19644	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₃₁	—	4	1	no	—
199	C₂³×C₃₁	—	4	1	no	—
200	C₂³×C₃₁	—	4	1	no	—
201	C₂³×C₃₁	#127	1	1	yes	characteristic
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	Q₈×C₃₁	—	4	2	no	—
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	2	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	—	4	2	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	8	2	no	—
217	D₄×C₃₁	—	8	2	no	—
218	D₄×C₃₁	—	8	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	8	2	no	—
228	C₂×C₄×C₃₁	—	8	2	no	—
229	C₂×C₄×C₃₁	—	8	2	no	—
230	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
231	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
232	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₈×C₃₁	—	2	2	no	—
237	C₈×C₃₁	—	2	2	no	—
238	C₈×C₃₁	—	2	2	no	—
239	C₈×C₃₁	—	2	2	no	—
240	C₈×C₃₁	—	2	2	no	—
241	C₈×C₃₁	#135	1	4	yes	—
242	#3210	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#3207	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#3184	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#3182	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1723	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1721	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1719	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1717	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1711	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1709	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1708	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂²×C₃₁	—	2	1	no	—
258	C₂²×C₃₁	—	2	1	no	—
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	8	1	no	—
261	C₂²×C₃₁	—	8	1	no	—
262	C₂²×C₃₁	—	8	1	no	—
263	C₂²×C₃₁	—	8	1	no	—
264	C₂²×C₃₁	—	8	1	no	—
265	C₂²×C₃₁	—	8	1	no	—
266	C₂²×C₃₁	—	8	1	no	—
267	C₂²×C₃₁	—	8	1	no	—
268	C₂²×C₃₁	—	8	1	no	—
269	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
270	C₄×C₃₁	—	2	1	no	—
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	8	1	no	—
274	C₄×C₃₁	—	8	1	no	—
275	C₄×C₃₁	—	8	1	no	—
276	C₄×C₃₁	—	8	1	no	—
277	C₄×C₃₁	—	8	1	no	—
278	C₄×C₃₁	C₂×#135	1	2	yes	—
279	C₄×C₃₁	#575	1	2	yes	—
280	C₄×C₃₁	#576	1	2	yes	—
281	#533	C₂²×C₃₁	1	1	yes	characteristic
282	#532	C₂²×C₃₁	1	1	yes	characteristic
283	#532	C₂²×C₃₁	1	1	yes	characteristic
284	#532	C₂²×C₃₁	1	1	yes	characteristic
285	#500	C₂²×C₃₁	1	1	yes	characteristic
286	#500	C₂²×C₃₁	1	1	yes	characteristic
287	#498	C₂²×C₃₁	1	1	yes	characteristic
288	#494	C₂²×C₃₁	1	1	yes	characteristic
289	#493	C₂²×C₃₁	1	1	yes	characteristic
290	#493	C₂²×C₃₁	1	1	yes	characteristic
291	#493	C₂²×C₃₁	1	1	yes	characteristic
292	#492	C₂²×C₃₁	1	1	yes	characteristic
293	#492	C₂²×C₃₁	1	1	yes	characteristic
294	#492	C₂²×C₃₁	1	1	yes	characteristic
295	#488	—	2	2	no	—
296	#487	C₂²×C₃₁	1	1	yes	characteristic
297	#487	C₂²×C₃₁	1	1	yes	characteristic
298	#487	C₂²×C₃₁	1	1	yes	characteristic
299	#487	C₂²×C₃₁	1	1	yes	characteristic
300	#487	C₂²×C₃₁	1	1	yes	characteristic
301	#486	C₂²×C₃₁	1	1	yes	characteristic
302	#486	C₂²×C₃₁	1	1	yes	characteristic
303	#486	C₂²×C₃₁	1	1	yes	characteristic
304	#486	C₂²×C₃₁	1	1	yes	characteristic
305	#486	C₂²×C₃₁	1	1	yes	characteristic
306	#486	C₂²×C₃₁	1	1	yes	characteristic
307	#486	C₂²×C₃₁	1	1	yes	characteristic
308	#486	C₂²×C₃₁	1	1	yes	characteristic
309	#486	C₂²×C₃₁	1	1	yes	characteristic
310	#469	—	2	2	no	—
311	#468	—	2	2	no	—
312	#466	—	2	2	no	—
313	#464	—	2	2	no	—
314	#463	—	2	2	no	—
315	#452	—	2	2	no	—
316	#449	—	2	2	no	—
317	#442	—	2	2	no	—
318	#440	—	2	2	no	—
319	C₄×D₈	—	2	2	no	—
320	#431	C₂²×C₃₁	1	1	yes	characteristic
321	#416	—	2	2	no	—
322	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
324	#322	C₂²×C₃₁	1	1	yes	characteristic
325	#322	C₂²×C₃₁	1	1	yes	characteristic
326	C₈²	C₂²×C₃₁	1	1	yes	characteristic
327	C₈²	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×C₃₁	—	4	1	no	—
329	C₂×C₃₁	—	16	1	no	—
330	C₂×C₃₁	—	16	1	no	—
331	C₂×C₃₁	—	16	1	no	—
332	C₂×C₃₁	#3147	1	1	yes	characteristic
333	C₂×C₃₁	#3153	1	1	yes	characteristic
334	C₂×C₃₁	#3155	1	1	yes	characteristic
335	C₂×#41	—	2	1	no	—
336	C₂×#41	—	2	1	no	—
337	C₂×#41	—	2	1	no	—
338	C₂×Q₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×D₈	—	2	2	no	—
343	C₂×D₈	—	2	2	no	—
344	C₂×D₈	—	2	2	no	—
345	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
346	C₂²×C₈	D₄×C₃₁	1	2	yes	—
347	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
349	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
350	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
352	#127	C₂³×C₃₁	1	1	yes	characteristic
353	#127	C₂³×C₃₁	1	1	yes	characteristic
354	#127	C₂³×C₃₁	1	1	yes	characteristic
355	#125	—	2	1	no	—
356	#124	—	2	1	no	—
357	#124	—	2	1	no	—
358	#124	—	2	1	no	—
359	C₂²⋊Q₈	—	2	1	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	#121	—	2	1	no	—
362	#121	—	2	1	no	—
363	#121	—	2	1	no	—
364	#121	—	2	1	no	—
365	#121	—	2	1	no	—
366	#121	—	2	1	no	—
367	C₄×Q₈	—	2	1	no	—
368	C₄×Q₈	—	2	1	no	—
369	C₄×D₄	—	2	1	no	—
370	C₄×D₄	—	2	1	no	—
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	—	2	1	no	—
375	#117	—	2	1	no	—
376	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
377	#107	—	2	2	no	—
378	#107	—	2	2	no	—
379	#106	—	2	2	no	—
380	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
381	#103	—	2	2	no	—
382	#103	—	2	2	no	—
383	#103	—	2	2	no	—
384	#103	—	2	2	no	—
385	#103	—	2	2	no	—
386	#102	—	2	2	no	—
387	#102	—	2	2	no	—
388	#102	—	2	2	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	—	2	2	no	—
395	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
396	#97	D₄×C₃₁	1	2	yes	—
397	C₄×C₈	D₄×C₃₁	1	2	yes	—
398	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
399	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
401	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
402	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
403	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
404	C₃₁	#19644	1	1	yes	characteristic, direct product factor, Sylow
405	#41	—	4	4	no	—
406	#41	—	8	2	no	—
407	#41	—	8	2	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	4	1	no	—
423	C₂×D₄	—	4	1	no	—
424	C₂×D₄	—	4	1	no	—
425	C₂²×C₄	—	4	1	no	—
426	C₂²×C₄	—	4	1	no	—
427	C₂²×C₄	—	4	1	no	—
428	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
431	Q₁₆	—	8	2	no	—
432	SD₁₆	—	4	4	no	—
433	SD₁₆	—	8	2	no	—
434	SD₁₆	—	8	2	no	—
435	D₈	—	4	4	no	—
436	D₈	—	4	4	no	—
437	D₈	—	8	2	no	—
438	M₁₆	—	2	4	no	—
439	C₂×C₈	—	4	2	no	—
440	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
441	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
442	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
443	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
444	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
445	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
457	C₄⋊C₄	—	4	1	no	—
458	C₂²⋊C₄	—	4	1	no	—
459	C₂²⋊C₄	—	4	1	no	—
460	C₂²⋊C₄	—	4	1	no	—
461	C₂²⋊C₄	—	4	1	no	—
462	C₂²⋊C₄	—	4	1	no	—
463	C₂²⋊C₄	—	4	1	no	—
464	C₂²⋊C₄	—	4	1	no	—
465	C₄²	—	4	1	no	—
466	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
467	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
469	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
470	C₂³	—	4	1	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	1	no	—
473	C₂³	C₃₁×#127	1	1	yes	characteristic
474	Q₈	—	4	2	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	2	no	—
477	Q₈	—	4	2	no	—
478	Q₈	—	4	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	2	no	—
483	D₄	—	4	2	no	—
484	D₄	—	4	2	no	—
485	D₄	—	4	2	no	—
486	D₄	—	4	2	no	—
487	D₄	—	4	2	no	—
488	D₄	—	8	2	no	—
489	D₄	—	8	2	no	—
490	D₄	—	8	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	8	2	no	—
500	C₂×C₄	—	8	2	no	—
501	C₂×C₄	—	8	2	no	—
502	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
503	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
504	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₈	—	2	2	no	—
509	C₈	—	2	2	no	—
510	C₈	—	2	2	no	—
511	C₈	—	2	2	no	—
512	C₈	—	2	2	no	—
513	C₈	C₃₁×#135	1	4	yes	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	8	1	no	—
518	C₂²	—	8	1	no	—
519	C₂²	—	8	1	no	—
520	C₂²	—	8	1	no	—
521	C₂²	—	8	1	no	—
522	C₂²	—	8	1	no	—
523	C₂²	—	8	1	no	—
524	C₂²	—	8	1	no	—
525	C₂²	—	8	1	no	—
526	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
527	C₄	—	2	1	no	—
528	C₄	—	2	1	no	—
529	C₄	—	2	1	no	—
530	C₄	—	8	1	no	—
531	C₄	—	8	1	no	—
532	C₄	—	8	1	no	—
533	C₄	—	8	1	no	—
534	C₄	—	8	1	no	—
535	C₄	C₂×C₃₁×#135	1	2	yes	—
536	C₄	C₃₁×#575	1	2	yes	—
537	C₄	C₃₁×#576	1	2	yes	—
538	C₂	—	4	1	no	—
539	C₂	—	16	1	no	—
540	C₂	—	16	1	no	—
541	C₂	—	16	1	no	—
542	C₂	C₃₁×#3147	1	1	yes	characteristic
543	C₂	C₃₁×#3153	1	1	yes	characteristic
544	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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×#3210	C₂	semidirect
2. C₃₁×#3207	C₂	semidirect
3. C₃₁×#3184	C₂	semidirect
4. C₃₁×#3182	C₂	semidirect
5. C₃₁×#3169	C₂	semidirect
6. C₃₁×#3131	C₂	semidirect
7. C₃₁×#1723	C₂	semidirect
8. C₃₁×#1721	C₂	semidirect
9. C₃₁×#1719	C₂	semidirect
10. C₃₁×#1717	C₂	semidirect
11. C₃₁×#1711	C₂	semidirect
12. C₃₁×#1709	C₂	semidirect
13. C₃₁×#1708	C₂	semidirect
14. C₃₁×#1706	C₂	semidirect
15. C₈×M₁₆×C₃₁	C₂	semidirect
197. #19644	C₃₁	direct (N × H)
242. #3210	C₂×C₃₁	semidirect
243. #3207	C₂×C₃₁	semidirect
244. #3184	C₂×C₃₁	semidirect
245. #3182	C₂×C₃₁	semidirect
246. #3169	C₂×C₃₁	semidirect
247. #3131	C₂×C₃₁	semidirect
248. #1723	C₂×C₃₁	semidirect
249. #1721	C₂×C₃₁	semidirect
250. #1719	C₂×C₃₁	semidirect
251. #1717	C₂×C₃₁	semidirect
252. #1711	C₂×C₃₁	semidirect
253. #1709	C₂×C₃₁	semidirect
254. #1708	C₂×C₃₁	semidirect
255. #1706	C₂×C₃₁	semidirect
256. C₈×M₁₆	C₂×C₃₁	semidirect
404. C₃₁	#19644	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×#3210	2
C₃₁×#3207	2
C₃₁×#3184	2
C₃₁×#3182	2
C₃₁×#3169	2
C₃₁×#3131	2
C₃₁×#1723	2
C₃₁×#1721	2
C₃₁×#1719	2
C₃₁×#1717	2
C₃₁×#1711	2
C₃₁×#1709	2
C₃₁×#1708	2
C₃₁×#1706	2
C₈×M₁₆×C₃₁	2
#19644	31

Subgroups — C31×#19644 #21476403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19644 #21476403