Subgroups — C₃₁×#60883 #21517642

Order 7936.

1742 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3241	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3235	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3234	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3096	C₂	1	4	yes	maximal, complemented
11	C₃₁×#3081	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#559	—	2	2	no	—
14	C₃₁×#557	—	2	2	no	—
15	C₃₁×#554	—	2	4	no	—
16	C₃₁×#554	—	2	2	no	—
17	C₃₁×#550	—	2	2	no	—
18	C₃₁×#546	—	2	4	no	—
19	C₃₁×#544	C₂²	1	2	yes	—
20	C₃₁×#544	C₂²	1	4	yes	complemented
21	C₃₁×#540	C₂²	1	4	yes	—
22	C₃₁×#535	C₂²	1	2	yes	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#533	C₂²	1	1	yes	characteristic
25	C₃₁×#533	C₂²	1	2	yes	complemented
26	C₃₁×#532	C₂²	1	4	yes	—
27	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#125	C₂²	1	2	yes	complemented
30	C₃₁×#520	C₂²	1	2	yes	—
31	C₃₁×#518	C₂²	1	2	yes	—
32	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#501	C₂²	1	2	yes	complemented
34	C₃₁×#501	C₂²	1	2	yes	complemented
35	C₃₁×#501	C₂²	1	4	yes	complemented
36	C₃₁×#501	C₂²	1	4	yes	complemented
37	C₃₁×#499	C₂²	1	4	yes	complemented
38	C₃₁×#490	C₂²	1	4	yes	—
39	C₃₁×#490	C₂²	1	4	yes	—
40	C₃₁×#490	C₂²	1	2	yes	—
41	C₃₁×#490	C₂²	1	2	yes	—
42	C₃₁×#486	C₂²	1	4	yes	—
43	C₃₁×#482	C₂²	1	8	yes	complemented
44	C₃₁×#482	C₂²	1	8	yes	complemented
45	C₃₁×#481	C₂²	1	8	yes	complemented
46	C₃₁×#480	C₂²	1	8	yes	complemented
47	C₃₁×#478	C₂²	1	8	yes	complemented
48	C₃₁×#478	C₂²	1	8	yes	complemented
49	C₃₁×#476	C₂²	1	8	yes	complemented
50	C₃₁×#474	C₂²	1	8	yes	complemented
51	C₃₁×#433	C₂²	1	4	yes	complemented
52	C₃₁×#432	C₂²	1	4	yes	complemented
53	C₃₁×#428	C₂²	1	4	yes	—
54	C₃₁×#427	C₂²	1	4	yes	—
55	C₃₁×#424	C₂²	1	2	yes	—
56	C₃₁×#418	C₂²	1	4	yes	—
57	C₂×C₃₁×#102	C₂²	1	4	yes	—
58	C₃₁×#408	C₂²	1	2	yes	—
59	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
60	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
61	C₂×#41×C₃₁	—	2	4	no	—
62	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₃₁	—	2	2	no	—
64	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
67	C₃₁×C₄⋊Q₈	—	2	2	no	—
68	C₃₁×C₄⋊Q₈	—	2	4	no	—
69	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
71	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
72	C₃₁×#127	—	2	2	no	—
73	C₃₁×#127	—	2	2	no	—
74	C₃₁×#127	C₂³	1	2	yes	—
75	C₃₁×#127	C₂³	1	2	yes	—
76	C₃₁×#125	—	2	2	no	—
77	C₃₁×#125	—	2	4	no	—
78	C₃₁×#125	—	2	2	no	—
79	C₃₁×#125	—	2	4	no	—
80	C₃₁×#125	C₂³	1	4	yes	—
81	C₃₁×#125	C₂³	1	2	yes	—
82	C₃₁×#125	C₂³	1	2	yes	—
83	C₃₁×#125	C₂³	1	1	yes	characteristic
84	C₃₁×#125	C₂³	1	1	yes	characteristic
85	C₃₁×#125	C₂³	1	2	yes	—
86	C₃₁×#124	—	2	2	no	—
87	C₃₁×#124	—	2	4	no	—
88	C₃₁×#123	—	2	8	no	—
89	C₃₁×C₂²⋊Q₈	—	2	2	no	—
90	C₃₁×C₂²⋊Q₈	—	2	2	no	—
91	C₃₁×C₂²⋊Q₈	—	2	4	no	—
92	C₃₁×C₂²⋊Q₈	—	2	4	no	—
93	C₃₁×C₂²⋊Q₈	—	2	4	no	—
94	C₃₁×C₂²⋊Q₈	—	2	4	no	—
95	C₃₁×#121	—	2	4	no	—
96	C₃₁×#121	—	2	8	no	—
97	C₃₁×#121	C₂³	1	8	yes	—
98	C₃₁×#120	—	2	4	no	—
99	C₄×Q₈×C₃₁	—	2	2	no	—
100	C₄×Q₈×C₃₁	—	2	2	no	—
101	C₄×Q₈×C₃₁	—	2	2	no	—
102	C₄×Q₈×C₃₁	—	2	4	no	—
103	C₄×Q₈×C₃₁	D₄	1	2	yes	—
104	C₄×D₄×C₃₁	—	2	4	no	—
105	C₄×D₄×C₃₁	—	2	2	no	—
106	C₄×D₄×C₃₁	—	2	4	no	—
107	C₄×D₄×C₃₁	—	2	4	no	—
108	C₄×D₄×C₃₁	D₄	1	4	yes	—
109	C₄×D₄×C₃₁	D₄	1	2	yes	—
110	C₄×D₄×C₃₁	C₂³	1	8	yes	—
111	C₃₁×#117	C₂³	1	2	yes	—
112	C₃₁×#117	C₂³	1	2	yes	—
113	C₃₁×#117	C₂³	1	4	yes	—
114	C₃₁×#117	C₂³	1	1	yes	characteristic
115	C₃₁×#117	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
123	C₃₁×#107	C₂³	1	4	yes	—
124	C₃₁×#107	C₂³	1	4	yes	—
125	C₃₁×#107	C₂³	1	8	yes	—
126	C₃₁×#106	C₂³	1	4	yes	—
127	C₃₁×#106	C₂³	1	4	yes	—
128	C₃₁×#106	C₂³	1	8	yes	—
129	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
130	C₃₁×#102	C₂³	1	8	yes	—
131	C₃₁×#102	C₂³	1	8	yes	—
132	C₃₁×#102	C₂³	1	8	yes	—
133	C₃₁×#102	C₂³	1	8	yes	—
134	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
135	C₃₁×#97	C₂³	1	2	yes	—
136	C₃₁×#97	C₂³	1	2	yes	—
137	C₃₁×#97	C₂³	1	2	yes	—
138	C₄×C₈×C₃₁	C₂³	1	2	yes	—
139	C₂⁴×C₃₁	—	2	2	no	—
140	#41×C₃₁	—	2	4	no	—
141	#41×C₃₁	—	8	8	no	—
142	C₂×Q₈×C₃₁	—	2	2	no	—
143	C₂×Q₈×C₃₁	—	2	4	no	—
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂×D₄×C₃₁	—	2	8	no	—
147	C₂×D₄×C₃₁	—	2	2	no	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	8	no	—
151	C₂×D₄×C₃₁	—	8	4	no	—
152	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	M₁₆×C₃₁	—	2	4	no	—
163	C₂×C₈×C₃₁	—	2	4	no	—
164	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
165	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
166	C₄⋊C₄×C₃₁	—	2	4	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	4	no	—
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	4	no	—
176	C₄⋊C₄×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
180	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
181	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
182	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
183	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	4	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₂²⋊C₄×C₃₁	—	2	8	no	—
189	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
190	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
191	C₄²×C₃₁	—	2	2	no	—
192	C₄²×C₃₁	—	2	4	no	—
193	C₄²×C₃₁	—	2	2	no	—
194	C₄²×C₃₁	C₂×D₄	1	4	yes	—
195	C₄²×C₃₁	C₂×D₄	1	2	yes	—
196	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
197	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
198	C₄²×C₃₁	C₂⁴	1	2	yes	—
199	#60883	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	4	4	no	—
202	C₂³×C₃₁	—	8	4	no	—
203	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
204	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	2	no	—
206	Q₈×C₃₁	—	4	4	no	—
207	Q₈×C₃₁	—	8	4	no	—
208	Q₈×C₃₁	—	8	2	no	—
209	D₄×C₃₁	—	2	4	no	—
210	D₄×C₃₁	—	2	2	no	—
211	D₄×C₃₁	—	2	8	no	—
212	D₄×C₃₁	—	4	8	no	—
213	D₄×C₃₁	—	4	4	no	—
214	D₄×C₃₁	—	8	4	no	—
215	D₄×C₃₁	—	8	8	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	4	4	no	—
223	C₂×C₄×C₃₁	—	8	8	no	—
224	C₂×C₄×C₃₁	—	8	2	no	—
225	C₂×C₄×C₃₁	—	8	4	no	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
229	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
232	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
233	C₈×C₃₁	—	2	4	no	—
234	C₈×C₃₁	—	2	4	no	—
235	#3508	C₂×C₃₁	1	1	yes	characteristic
236	#3504	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#3241	C₂×C₃₁	1	8	yes	complemented
238	#3235	C₂×C₃₁	1	4	yes	complemented
239	#3234	C₂×C₃₁	1	4	yes	complemented
240	#3166	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#3162	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂×#490	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#3096	C₂×C₃₁	1	4	yes	complemented
245	#3081	C₂×C₃₁	1	4	yes	complemented
246	#2923	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂²×C₃₁	—	2	2	no	—
248	C₂²×C₃₁	—	2	4	no	—
249	C₂²×C₃₁	—	2	2	no	—
250	C₂²×C₃₁	—	4	8	no	—
251	C₂²×C₃₁	—	4	4	no	—
252	C₂²×C₃₁	—	16	4	no	—
253	C₂²×C₃₁	#536	1	2	yes	—
254	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	2	4	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	2	2	no	—
259	C₄×C₃₁	—	4	4	no	—
260	C₄×C₃₁	—	4	4	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	#536	1	2	yes	—
264	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
265	#559	—	2	2	no	—
266	#557	—	2	2	no	—
267	#554	—	2	4	no	—
268	#554	—	2	2	no	—
269	#550	—	2	2	no	—
270	#546	—	2	4	no	—
271	#544	C₂²×C₃₁	1	2	yes	—
272	#544	C₂²×C₃₁	1	4	yes	complemented
273	#540	C₂²×C₃₁	1	4	yes	—
274	#535	C₂²×C₃₁	1	2	yes	—
275	#533	C₂²×C₃₁	1	1	yes	characteristic
276	#533	C₂²×C₃₁	1	1	yes	characteristic
277	#533	C₂²×C₃₁	1	2	yes	complemented
278	#532	C₂²×C₃₁	1	4	yes	—
279	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×#125	C₂²×C₃₁	1	2	yes	complemented
282	#520	C₂²×C₃₁	1	2	yes	—
283	#518	C₂²×C₃₁	1	2	yes	—
284	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
285	#501	C₂²×C₃₁	1	2	yes	complemented
286	#501	C₂²×C₃₁	1	2	yes	complemented
287	#501	C₂²×C₃₁	1	4	yes	complemented
288	#501	C₂²×C₃₁	1	4	yes	complemented
289	#499	C₂²×C₃₁	1	4	yes	complemented
290	#490	C₂²×C₃₁	1	4	yes	—
291	#490	C₂²×C₃₁	1	4	yes	—
292	#490	C₂²×C₃₁	1	2	yes	—
293	#490	C₂²×C₃₁	1	2	yes	—
294	#486	C₂²×C₃₁	1	4	yes	—
295	#482	C₂²×C₃₁	1	8	yes	complemented
296	#482	C₂²×C₃₁	1	8	yes	complemented
297	#481	C₂²×C₃₁	1	8	yes	complemented
298	#480	C₂²×C₃₁	1	8	yes	complemented
299	#478	C₂²×C₃₁	1	8	yes	complemented
300	#478	C₂²×C₃₁	1	8	yes	complemented
301	#476	C₂²×C₃₁	1	8	yes	complemented
302	#474	C₂²×C₃₁	1	8	yes	complemented
303	#433	C₂²×C₃₁	1	4	yes	complemented
304	#432	C₂²×C₃₁	1	4	yes	complemented
305	#428	C₂²×C₃₁	1	4	yes	—
306	#427	C₂²×C₃₁	1	4	yes	—
307	#424	C₂²×C₃₁	1	2	yes	—
308	#418	C₂²×C₃₁	1	4	yes	—
309	C₂×#102	C₂²×C₃₁	1	4	yes	—
310	#408	C₂²×C₃₁	1	2	yes	—
311	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×C₃₁	—	2	2	no	—
314	C₂×C₃₁	—	4	2	no	—
315	C₂×C₃₁	—	8	4	no	—
316	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
317	C₂×C₃₁	#3585	1	2	yes	—
318	C₂×#41	—	2	4	no	—
319	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
320	C₂²×D₄	—	2	2	no	—
321	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
322	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
323	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	—	2	4	no	—
326	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
327	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
328	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
329	#127	—	2	2	no	—
330	#127	—	2	2	no	—
331	#127	C₂³×C₃₁	1	2	yes	—
332	#127	C₂³×C₃₁	1	2	yes	—
333	#125	—	2	2	no	—
334	#125	—	2	4	no	—
335	#125	—	2	2	no	—
336	#125	—	2	4	no	—
337	#125	C₂³×C₃₁	1	2	yes	—
338	#125	C₂³×C₃₁	1	2	yes	—
339	#125	C₂³×C₃₁	1	1	yes	characteristic
340	#125	C₂³×C₃₁	1	4	yes	—
341	#125	C₂³×C₃₁	1	1	yes	characteristic
342	#125	C₂³×C₃₁	1	2	yes	—
343	#124	—	2	2	no	—
344	#124	—	2	4	no	—
345	#123	—	2	8	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	4	no	—
349	C₂²⋊Q₈	—	2	4	no	—
350	C₂²⋊Q₈	—	2	4	no	—
351	C₂²⋊Q₈	—	2	4	no	—
352	#121	—	2	4	no	—
353	#121	—	2	8	no	—
354	#121	C₂³×C₃₁	1	8	yes	—
355	#120	—	2	4	no	—
356	C₄×Q₈	—	2	2	no	—
357	C₄×Q₈	—	2	2	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	—	2	4	no	—
360	C₄×Q₈	D₄×C₃₁	1	2	yes	—
361	C₄×D₄	—	2	4	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	4	no	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	D₄×C₃₁	1	4	yes	—
366	C₄×D₄	D₄×C₃₁	1	2	yes	—
367	C₄×D₄	C₂³×C₃₁	1	8	yes	—
368	#117	C₂³×C₃₁	1	2	yes	—
369	#117	C₂³×C₃₁	1	2	yes	—
370	#117	C₂³×C₃₁	1	4	yes	—
371	#117	C₂³×C₃₁	1	1	yes	characteristic
372	#117	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	—	2	2	no	—
374	C₂×C₄⋊C₄	—	2	4	no	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
377	C₂×C₂²⋊C₄	—	2	4	no	—
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
380	#107	C₂³×C₃₁	1	4	yes	—
381	#107	C₂³×C₃₁	1	4	yes	—
382	#107	C₂³×C₃₁	1	8	yes	—
383	#106	C₂³×C₃₁	1	4	yes	—
384	#106	C₂³×C₃₁	1	4	yes	—
385	#106	C₂³×C₃₁	1	8	yes	—
386	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
387	#102	C₂³×C₃₁	1	8	yes	—
388	#102	C₂³×C₃₁	1	8	yes	—
389	#102	C₂³×C₃₁	1	8	yes	—
390	#102	C₂³×C₃₁	1	8	yes	—
391	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
392	#97	C₂³×C₃₁	1	2	yes	—
393	#97	C₂³×C₃₁	1	2	yes	—
394	#97	C₂³×C₃₁	1	2	yes	—
395	C₄×C₈	C₂³×C₃₁	1	2	yes	—
396	C₃₁	#60883	1	1	yes	characteristic, direct product factor, Sylow
397	C₂⁴	—	2	2	no	—
398	#41	—	2	4	no	—
399	#41	—	8	8	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	4	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂×D₄	—	2	8	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	8	4	no	—
410	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
411	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
412	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
417	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
418	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
419	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
420	M₁₆	—	2	4	no	—
421	C₂×C₈	—	2	4	no	—
422	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
423	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	4	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	4	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
437	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
438	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
439	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
440	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
441	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	2	no	—
445	C₂²⋊C₄	—	2	8	no	—
446	C₂²⋊C₄	—	2	8	no	—
447	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
448	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
449	C₄²	—	2	2	no	—
450	C₄²	—	2	4	no	—
451	C₄²	—	2	2	no	—
452	C₄²	C₂×D₄×C₃₁	1	4	yes	—
453	C₄²	C₂×D₄×C₃₁	1	2	yes	—
454	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
455	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
456	C₄²	C₂⁴×C₃₁	1	2	yes	—
457	C₂³	—	2	4	no	—
458	C₂³	—	4	4	no	—
459	C₂³	—	8	4	no	—
460	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
461	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	Q₈	—	2	2	no	—
463	Q₈	—	4	4	no	—
464	Q₈	—	8	4	no	—
465	Q₈	—	8	2	no	—
466	D₄	—	2	4	no	—
467	D₄	—	2	2	no	—
468	D₄	—	2	8	no	—
469	D₄	—	4	8	no	—
470	D₄	—	4	4	no	—
471	D₄	—	8	4	no	—
472	D₄	—	8	8	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	8	8	no	—
481	C₂×C₄	—	8	2	no	—
482	C₂×C₄	—	8	4	no	—
483	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
486	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
487	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
489	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
490	C₈	—	2	4	no	—
491	C₈	—	2	4	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	4	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	8	no	—
497	C₂²	—	16	4	no	—
498	C₂²	C₃₁×#536	1	2	yes	—
499	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	4	no	—
503	C₄	—	2	2	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	4	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	C₃₁×#536	1	2	yes	—
509	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
510	C₂	—	2	2	no	—
511	C₂	—	4	2	no	—
512	C₂	—	8	4	no	—
513	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
514	C₂	C₃₁×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3508	C₂
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
2. C₃₁×#3504	C₂	semidirect
3. C₃₁×#3241	C₂	semidirect
4. C₃₁×#3235	C₂	semidirect
5. C₃₁×#3234	C₂	semidirect
6. C₃₁×#3166	C₂	semidirect
7. C₃₁×#3162	C₂	semidirect
8. C₃₁×#3138	C₂	semidirect
9. C₂×C₃₁×#490	C₂	semidirect
10. C₃₁×#3096	C₂	semidirect
11. C₃₁×#3081	C₂	semidirect
12. C₃₁×#2923	C₂	semidirect
20. C₃₁×#544	C₂²	semidirect
25. C₃₁×#533	C₂²	semidirect
29. C₂×C₃₁×#125	C₂²	semidirect
33. C₃₁×#501	C₂²	semidirect
34. C₃₁×#501	C₂²	semidirect
35. C₃₁×#501	C₂²	semidirect
36. C₃₁×#501	C₂²	semidirect
37. C₃₁×#499	C₂²	semidirect
43. C₃₁×#482	C₂²	semidirect
44. C₃₁×#482	C₂²	semidirect
45. C₃₁×#481	C₂²	semidirect
46. C₃₁×#480	C₂²	semidirect
47. C₃₁×#478	C₂²	semidirect
48. C₃₁×#478	C₂²	semidirect
49. C₃₁×#476	C₂²	semidirect
50. C₃₁×#474	C₂²	semidirect
51. C₃₁×#433	C₂²	semidirect
52. C₃₁×#432	C₂²	semidirect
199. #60883	C₃₁	direct (N × H)
236. #3504	C₂×C₃₁	semidirect
237. #3241	C₂×C₃₁	semidirect
238. #3235	C₂×C₃₁	semidirect
239. #3234	C₂×C₃₁	semidirect
240. #3166	C₂×C₃₁	semidirect
241. #3162	C₂×C₃₁	semidirect
242. #3138	C₂×C₃₁	semidirect
243. C₂×#490	C₂×C₃₁	semidirect
244. #3096	C₂×C₃₁	semidirect
245. #3081	C₂×C₃₁	semidirect
246. #2923	C₂×C₃₁	semidirect
272. #544	C₂²×C₃₁	semidirect
277. #533	C₂²×C₃₁	semidirect
281. C₂×#125	C₂²×C₃₁	semidirect
285. #501	C₂²×C₃₁	semidirect
286. #501	C₂²×C₃₁	semidirect
287. #501	C₂²×C₃₁	semidirect
288. #501	C₂²×C₃₁	semidirect
289. #499	C₂²×C₃₁	semidirect
295. #482	C₂²×C₃₁	semidirect
296. #482	C₂²×C₃₁	semidirect
297. #481	C₂²×C₃₁	semidirect
298. #480	C₂²×C₃₁	semidirect
299. #478	C₂²×C₃₁	semidirect
300. #478	C₂²×C₃₁	semidirect
301. #476	C₂²×C₃₁	semidirect
302. #474	C₂²×C₃₁	semidirect
303. #433	C₂²×C₃₁	semidirect
304. #432	C₂²×C₃₁	semidirect
396. C₃₁	#60883	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×#3508	2
C₃₁×#3504	2
C₃₁×#3241	2
C₃₁×#3235	2
C₃₁×#3234	2
C₃₁×#3166	2
C₃₁×#3162	2
C₃₁×#3138	2
C₂×C₃₁×#490	2
C₃₁×#3096	2
C₃₁×#3081	2
C₃₁×#2923	2
#60883	31

Subgroups — C31×#60883 #21517642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60883 #21517642