Subgroups — C₃₁×#62595 #21519354

Order 7936.

1966 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3517	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3359	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3332	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3330	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3239	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3214	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3171	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3141	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2971	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
13	C₃₁×#557	C₂²	1	1	yes	characteristic
14	C₃₁×#553	—	2	3	no	—
15	C₃₁×#550	—	2	1	no	—
16	C₃₁×#550	C₂²	1	3	yes	—
17	C₃₁×#550	C₂²	1	1	yes	characteristic
18	D₄×Q₈×C₃₁	—	2	1	no	—
19	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#547	—	2	3	no	—
21	C₃₁×#547	C₂²	1	3	yes	—
22	C₃₁×#547	C₂²	1	6	yes	—
23	C₃₁×#540	—	2	3	no	—
24	C₃₁×#536	—	2	3	no	—
25	C₃₁×#532	C₂²	1	3	yes	—
26	C₃₁×#532	C₂²	1	3	yes	—
27	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
28	C₃₁×#529	C₂²	1	3	yes	—
29	C₃₁×#520	—	2	1	no	—
30	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#501	C₂²	1	3	yes	—
32	C₃₁×#496	C₂²	1	3	yes	—
33	C₃₁×#490	C₂²	1	1	yes	characteristic
34	C₃₁×#489	C₂²	1	3	yes	—
35	C₃₁×#489	C₂²	1	6	yes	—
36	C₃₁×#487	C₂²	1	6	yes	—
37	C₃₁×#486	C₂²	1	6	yes	—
38	C₃₁×#484	C₂²	1	6	yes	—
39	C₃₁×#481	C₂²	1	6	yes	—
40	C₃₁×#475	C₂²	1	6	yes	—
41	C₃₁×#474	C₂²	1	6	yes	—
42	C₃₁×#469	C₂²	1	6	yes	—
43	C₃₁×#468	C₂²	1	6	yes	—
44	C₃₁×#461	C₂²	1	6	yes	—
45	C₃₁×#459	C₂²	1	6	yes	—
46	C₃₁×#449	C₂²	1	12	yes	—
47	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
48	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	—
49	C₄×D₈×C₃₁	C₂²	1	6	yes	—
50	C₃₁×#436	C₂²	1	6	yes	—
51	C₃₁×#432	C₂²	1	3	yes	—
52	C₃₁×#417	C₂²	1	12	yes	—
53	C₄×M₁₆×C₃₁	C₂²	1	3	yes	—
54	C₃₁×2₋¹⁺⁴	—	4	8	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	—	2	6	no	—
57	C₂×#41×C₃₁	—	2	3	no	—
58	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
59	C₂×#41×C₃₁	C₂³	1	6	yes	—
60	C₂²×Q₈×C₃₁	—	2	2	no	—
61	C₂²×Q₈×C₃₁	—	2	3	no	—
62	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
63	C₂×D₈×C₃₁	C₂³	1	6	yes	—
64	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
65	C₃₁×C₄⋊Q₈	—	2	3	no	—
66	C₃₁×C₄⋊Q₈	—	2	3	no	—
67	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
70	C₃₁×#127	—	2	3	no	—
71	C₃₁×#127	C₂³	1	3	yes	—
72	C₃₁×#127	C₂³	1	1	yes	characteristic
73	C₃₁×#126	—	2	3	no	—
74	C₃₁×#125	—	2	3	no	—
75	C₃₁×#125	C₂³	1	1	yes	characteristic
76	C₃₁×#124	—	2	3	no	—
77	C₃₁×#124	C₂³	1	3	yes	—
78	C₃₁×#123	—	2	3	no	—
79	C₃₁×#123	—	2	12	no	—
80	C₃₁×#123	—	2	6	no	—
81	C₃₁×C₂²⋊Q₈	—	2	6	no	—
82	C₃₁×C₂²⋊Q₈	—	2	6	no	—
83	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
84	C₃₁×#121	—	2	3	no	—
85	C₃₁×#121	—	2	3	no	—
86	C₃₁×#121	—	2	3	no	—
87	C₃₁×#121	—	2	3	no	—
88	C₃₁×#121	—	2	3	no	—
89	C₃₁×#121	—	2	6	no	—
90	C₃₁×#121	C₂³	1	6	yes	—
91	C₃₁×#121	C₂³	1	3	yes	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
95	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₃₁	—	2	3	no	—
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	—	2	3	no	—
99	C₄×D₄×C₃₁	—	2	3	no	—
100	C₄×D₄×C₃₁	—	2	3	no	—
101	C₄×D₄×C₃₁	D₄	1	3	yes	—
102	C₄×D₄×C₃₁	C₂³	1	3	yes	—
103	C₄×D₄×C₃₁	C₂³	1	6	yes	—
104	C₄×D₄×C₃₁	C₂³	1	3	yes	—
105	C₃₁×#117	—	2	3	no	—
106	C₃₁×#117	D₄	1	3	yes	—
107	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
108	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
111	C₃₁×#107	C₂³	1	6	yes	—
112	C₃₁×#106	C₂³	1	6	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
114	C₃₁×#103	C₂³	1	6	yes	—
115	C₃₁×#103	C₂³	1	12	yes	—
116	C₃₁×#102	C₂³	1	6	yes	—
117	C₃₁×#102	C₂³	1	12	yes	—
118	C₃₁×#102	C₂³	1	2	yes	—
119	C₃₁×#102	C₂³	1	2	yes	—
120	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
121	C₃₁×#97	C₂³	1	1	yes	characteristic
122	C₃₁×#97	C₂³	1	3	yes	—
123	C₄×C₈×C₃₁	C₂³	1	6	yes	—
124	#41×C₃₁	—	2	4	no	—
125	#41×C₃₁	—	4	12	no	—
126	#41×C₃₁	—	4	12	no	—
127	#41×C₃₁	—	4	24	no	—
128	#41×C₃₁	—	4	4	no	—
129	C₂×Q₈×C₃₁	—	2	6	no	—
130	C₂×Q₈×C₃₁	—	2	3	no	—
131	C₂×Q₈×C₃₁	—	4	12	no	—
132	C₂×Q₈×C₃₁	—	4	8	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×Q₈×C₃₁	C₂⁴	1	3	yes	—
135	C₂×D₄×C₃₁	—	2	6	no	—
136	C₂×D₄×C₃₁	—	2	3	no	—
137	C₂×D₄×C₃₁	—	2	6	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	3	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	3	no	—
142	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
143	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
144	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
145	C₂²×C₄×C₃₁	—	2	6	no	—
146	C₂²×C₄×C₃₁	—	2	3	no	—
147	C₂²×C₄×C₃₁	—	2	6	no	—
148	C₂²×C₄×C₃₁	—	2	6	no	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
152	SD₁₆×C₃₁	—	4	6	no	—
153	D₈×C₃₁	—	4	6	no	—
154	M₁₆×C₃₁	—	4	6	no	—
155	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
156	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
157	C₄⋊C₄×C₃₁	—	2	3	no	—
158	C₄⋊C₄×C₃₁	—	2	3	no	—
159	C₄⋊C₄×C₃₁	—	2	3	no	—
160	C₄⋊C₄×C₃₁	—	2	6	no	—
161	C₄⋊C₄×C₃₁	—	2	6	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	3	no	—
165	C₄⋊C₄×C₃₁	—	2	3	no	—
166	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
167	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
169	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂²⋊C₄×C₃₁	—	2	3	no	—
172	C₂²⋊C₄×C₃₁	—	2	3	no	—
173	C₂²⋊C₄×C₃₁	—	2	3	no	—
174	C₂²⋊C₄×C₃₁	—	2	3	no	—
175	C₂²⋊C₄×C₃₁	—	2	6	no	—
176	C₂²⋊C₄×C₃₁	—	2	3	no	—
177	C₂²⋊C₄×C₃₁	—	2	6	no	—
178	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
179	C₄²×C₃₁	—	2	3	no	—
180	C₄²×C₃₁	C₂×D₄	1	3	yes	—
181	C₄²×C₃₁	C₂×D₄	1	3	yes	—
182	C₄²×C₃₁	C₂⁴	1	3	yes	—
183	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
184	#62595	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	3	no	—
187	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
188	Q₈×C₃₁	—	2	6	no	—
189	Q₈×C₃₁	—	4	6	no	—
190	Q₈×C₃₁	—	4	12	no	—
191	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
192	D₄×C₃₁	—	2	6	no	—
193	D₄×C₃₁	—	2	2	no	—
194	D₄×C₃₁	—	2	6	no	—
195	D₄×C₃₁	—	2	2	no	—
196	D₄×C₃₁	—	4	2	no	—
197	D₄×C₃₁	—	4	2	no	—
198	D₄×C₃₁	—	4	12	no	—
199	D₄×C₃₁	—	4	6	no	—
200	D₄×C₃₁	—	8	3	no	—
201	D₄×C₃₁	—	8	3	no	—
202	D₄×C₃₁	—	8	6	no	—
203	C₂×C₄×C₃₁	—	2	6	no	—
204	C₂×C₄×C₃₁	—	2	6	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	3	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	6	no	—
209	C₂×C₄×C₃₁	—	4	3	no	—
210	C₂×C₄×C₃₁	—	4	12	no	—
211	C₂×C₄×C₃₁	—	4	12	no	—
212	C₂×C₄×C₃₁	—	4	6	no	—
213	C₂×C₄×C₃₁	—	8	6	no	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
219	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	2	6	no	—
222	#3517	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#3499	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#3359	C₂×C₃₁	1	2	yes	complemented
225	#3358	C₂×C₃₁	1	2	yes	complemented
226	#3332	C₂×C₃₁	1	6	yes	complemented
227	#3330	C₂×C₃₁	1	6	yes	complemented
228	#3239	C₂×C₃₁	1	3	yes	complemented
229	#3214	C₂×C₃₁	1	3	yes	complemented
230	#3171	C₂×C₃₁	1	3	yes	complemented
231	#3141	C₂×C₃₁	1	3	yes	complemented
232	#2971	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂²×C₃₁	—	2	3	no	—
234	C₂²×C₃₁	—	2	3	no	—
235	C₂²×C₃₁	—	2	3	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	3	no	—
238	C₂²×C₃₁	—	4	3	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	4	3	no	—
242	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₃₁	—	2	1	no	—
244	C₄×C₃₁	—	2	3	no	—
245	C₄×C₃₁	—	2	3	no	—
246	C₄×C₃₁	—	2	1	no	—
247	C₄×C₃₁	—	4	1	no	—
248	C₄×C₃₁	—	4	3	no	—
249	C₄×C₃₁	—	4	1	no	—
250	C₄×C₃₁	—	4	6	no	—
251	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
252	C₄×C₃₁	#585	1	2	yes	—
253	C₂×2₋¹⁺⁴	—	2	1	no	—
254	#557	C₂²×C₃₁	1	1	yes	characteristic
255	#553	—	2	3	no	—
256	#550	—	2	1	no	—
257	#550	C₂²×C₃₁	1	3	yes	—
258	#550	C₂²×C₃₁	1	1	yes	characteristic
259	D₄×Q₈	—	2	1	no	—
260	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
261	#547	—	2	3	no	—
262	#547	C₂²×C₃₁	1	3	yes	—
263	#547	C₂²×C₃₁	1	6	yes	—
264	#540	—	2	3	no	—
265	#536	—	2	3	no	—
266	#532	C₂²×C₃₁	1	3	yes	—
267	#532	C₂²×C₃₁	1	3	yes	—
268	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
269	#529	C₂²×C₃₁	1	3	yes	—
270	#520	—	2	1	no	—
271	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
272	#501	C₂²×C₃₁	1	3	yes	—
273	#496	C₂²×C₃₁	1	3	yes	—
274	#490	C₂²×C₃₁	1	1	yes	characteristic
275	#489	C₂²×C₃₁	1	3	yes	—
276	#489	C₂²×C₃₁	1	6	yes	—
277	#487	C₂²×C₃₁	1	6	yes	—
278	#486	C₂²×C₃₁	1	6	yes	—
279	#484	C₂²×C₃₁	1	6	yes	—
280	#481	C₂²×C₃₁	1	6	yes	—
281	#475	C₂²×C₃₁	1	6	yes	—
282	#474	C₂²×C₃₁	1	6	yes	—
283	#469	C₂²×C₃₁	1	6	yes	—
284	#468	C₂²×C₃₁	1	6	yes	—
285	#461	C₂²×C₃₁	1	6	yes	—
286	#459	C₂²×C₃₁	1	6	yes	—
287	#449	C₂²×C₃₁	1	12	yes	—
288	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
289	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	—
290	C₄×D₈	C₂²×C₃₁	1	6	yes	—
291	#436	C₂²×C₃₁	1	6	yes	—
292	#432	C₂²×C₃₁	1	3	yes	—
293	#417	C₂²×C₃₁	1	12	yes	—
294	C₄×M₁₆	C₂²×C₃₁	1	3	yes	—
295	C₂×C₃₁	—	4	3	no	—
296	C₂×C₃₁	—	8	2	no	—
297	C₂×C₃₁	—	8	3	no	—
298	C₂×C₃₁	#3500	1	1	yes	characteristic
299	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂×C₃₁	#3586	1	1	yes	characteristic
301	2₋¹⁺⁴	—	4	8	no	—
302	C₂×#41	—	2	1	no	—
303	C₂×#41	—	2	6	no	—
304	C₂×#41	—	2	3	no	—
305	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
306	C₂×#41	C₂³×C₃₁	1	6	yes	—
307	C₂²×Q₈	—	2	2	no	—
308	C₂²×Q₈	—	2	3	no	—
309	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
310	C₂×D₈	C₂³×C₃₁	1	6	yes	—
311	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
312	C₄⋊Q₈	—	2	3	no	—
313	C₄⋊Q₈	—	2	3	no	—
314	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
315	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
316	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
317	#127	—	2	3	no	—
318	#127	C₂³×C₃₁	1	1	yes	characteristic
319	#127	C₂³×C₃₁	1	3	yes	—
320	#126	—	2	3	no	—
321	#125	—	2	3	no	—
322	#125	C₂³×C₃₁	1	1	yes	characteristic
323	#124	—	2	3	no	—
324	#124	C₂³×C₃₁	1	3	yes	—
325	#123	—	2	3	no	—
326	#123	—	2	12	no	—
327	#123	—	2	6	no	—
328	C₂²⋊Q₈	—	2	6	no	—
329	C₂²⋊Q₈	—	2	6	no	—
330	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
331	#121	—	2	3	no	—
332	#121	—	2	3	no	—
333	#121	—	2	3	no	—
334	#121	—	2	3	no	—
335	#121	—	2	3	no	—
336	#121	—	2	6	no	—
337	#121	C₂³×C₃₁	1	3	yes	—
338	#121	C₂³×C₃₁	1	6	yes	—
339	C₄×Q₈	—	2	1	no	—
340	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
341	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
342	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₄×D₄	—	2	3	no	—
344	C₄×D₄	—	2	3	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	—	2	3	no	—
347	C₄×D₄	—	2	3	no	—
348	C₄×D₄	D₄×C₃₁	1	3	yes	—
349	C₄×D₄	C₂³×C₃₁	1	3	yes	—
350	C₄×D₄	C₂³×C₃₁	1	6	yes	—
351	C₄×D₄	C₂³×C₃₁	1	3	yes	—
352	#117	—	2	3	no	—
353	#117	D₄×C₃₁	1	3	yes	—
354	C₂×C₄⋊C₄	—	2	3	no	—
355	C₂×C₄⋊C₄	—	2	3	no	—
356	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
357	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
358	#107	C₂³×C₃₁	1	6	yes	—
359	#106	C₂³×C₃₁	1	6	yes	—
360	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
361	#103	C₂³×C₃₁	1	6	yes	—
362	#103	C₂³×C₃₁	1	12	yes	—
363	#102	C₂³×C₃₁	1	6	yes	—
364	#102	C₂³×C₃₁	1	12	yes	—
365	#102	C₂³×C₃₁	1	2	yes	—
366	#102	C₂³×C₃₁	1	2	yes	—
367	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
368	#97	C₂³×C₃₁	1	1	yes	characteristic
369	#97	C₂³×C₃₁	1	3	yes	—
370	C₄×C₈	C₂³×C₃₁	1	6	yes	—
371	C₃₁	#62595	1	1	yes	characteristic, direct product factor, Sylow
372	#41	—	2	4	no	—
373	#41	—	4	12	no	—
374	#41	—	4	12	no	—
375	#41	—	4	24	no	—
376	#41	—	4	4	no	—
377	C₂×Q₈	—	2	6	no	—
378	C₂×Q₈	—	2	3	no	—
379	C₂×Q₈	—	4	12	no	—
380	C₂×Q₈	—	4	8	no	—
381	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂×Q₈	C₂⁴×C₃₁	1	3	yes	—
383	C₂×D₄	—	2	6	no	—
384	C₂×D₄	—	2	3	no	—
385	C₂×D₄	—	2	6	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	3	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	3	no	—
390	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
391	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
392	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
393	C₂²×C₄	—	2	6	no	—
394	C₂²×C₄	—	2	3	no	—
395	C₂²×C₄	—	2	6	no	—
396	C₂²×C₄	—	2	6	no	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
399	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
400	SD₁₆	—	4	6	no	—
401	D₈	—	4	6	no	—
402	M₁₆	—	4	6	no	—
403	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
404	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
405	C₄⋊C₄	—	2	3	no	—
406	C₄⋊C₄	—	2	3	no	—
407	C₄⋊C₄	—	2	3	no	—
408	C₄⋊C₄	—	2	6	no	—
409	C₄⋊C₄	—	2	6	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	3	no	—
413	C₄⋊C₄	—	2	3	no	—
414	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
415	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
416	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
417	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
418	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
419	C₂²⋊C₄	—	2	3	no	—
420	C₂²⋊C₄	—	2	3	no	—
421	C₂²⋊C₄	—	2	3	no	—
422	C₂²⋊C₄	—	2	3	no	—
423	C₂²⋊C₄	—	2	6	no	—
424	C₂²⋊C₄	—	2	3	no	—
425	C₂²⋊C₄	—	2	6	no	—
426	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
427	C₄²	—	2	3	no	—
428	C₄²	C₂×D₄×C₃₁	1	3	yes	—
429	C₄²	C₂×D₄×C₃₁	1	3	yes	—
430	C₄²	C₂⁴×C₃₁	1	3	yes	—
431	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
432	C₂³	—	2	2	no	—
433	C₂³	—	2	3	no	—
434	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
435	Q₈	—	2	6	no	—
436	Q₈	—	4	6	no	—
437	Q₈	—	4	12	no	—
438	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
439	D₄	—	2	6	no	—
440	D₄	—	2	2	no	—
441	D₄	—	2	6	no	—
442	D₄	—	2	2	no	—
443	D₄	—	4	2	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	12	no	—
446	D₄	—	4	6	no	—
447	D₄	—	8	3	no	—
448	D₄	—	8	3	no	—
449	D₄	—	8	6	no	—
450	C₂×C₄	—	2	6	no	—
451	C₂×C₄	—	2	6	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	3	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	6	no	—
456	C₂×C₄	—	4	3	no	—
457	C₂×C₄	—	4	12	no	—
458	C₂×C₄	—	4	12	no	—
459	C₂×C₄	—	4	6	no	—
460	C₂×C₄	—	8	6	no	—
461	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
464	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
465	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
466	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
467	C₈	—	2	2	no	—
468	C₈	—	2	6	no	—
469	C₂²	—	2	3	no	—
470	C₂²	—	2	3	no	—
471	C₂²	—	2	3	no	—
472	C₂²	—	4	3	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	3	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	3	no	—
478	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
479	C₄	—	2	3	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	3	no	—
483	C₄	—	4	3	no	—
484	C₄	—	4	6	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	1	no	—
487	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
488	C₄	C₃₁×#585	1	2	yes	—
489	C₂	—	4	3	no	—
490	C₂	—	8	3	no	—
491	C₂	—	8	2	no	—
492	C₂	C₃₁×#3500	1	1	yes	characteristic
493	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
494	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3517	C₂	semidirect
2. C₃₁×#3499	C₂	semidirect
3. C₃₁×#3359	C₂	semidirect
4. C₃₁×#3358	C₂	semidirect
5. C₃₁×#3332	C₂	semidirect
6. C₃₁×#3330	C₂	semidirect
7. C₃₁×#3239	C₂	semidirect
8. C₃₁×#3214	C₂	semidirect
9. C₃₁×#3171	C₂	semidirect
10. C₃₁×#3141	C₂	semidirect
11. C₃₁×#2971	C₂	semidirect
184. #62595	C₃₁	direct (N × H)
222. #3517	C₂×C₃₁	semidirect
223. #3499	C₂×C₃₁	semidirect
224. #3359	C₂×C₃₁	semidirect
225. #3358	C₂×C₃₁	semidirect
226. #3332	C₂×C₃₁	semidirect
227. #3330	C₂×C₃₁	semidirect
228. #3239	C₂×C₃₁	semidirect
229. #3214	C₂×C₃₁	semidirect
230. #3171	C₂×C₃₁	semidirect
231. #3141	C₂×C₃₁	semidirect
232. #2971	C₂×C₃₁	semidirect
371. C₃₁	#62595	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×#3517	2
C₃₁×#3499	2
C₃₁×#3359	2
C₃₁×#3358	2
C₃₁×#3332	2
C₃₁×#3330	2
C₃₁×#3239	2
C₃₁×#3214	2
C₃₁×#3171	2
C₃₁×#3141	2
C₃₁×#2971	2
#62595	31

Subgroups — C31×#62595 #21519354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62595 #21519354