Subgroups — C₃₁×#60719 #21517478

Order 7936.

1966 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3180	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3179	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3171	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3058	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3057	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2949	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2948	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₃₁×#131	C₂²	1	2	yes	—
18	C₃₁×#559	—	2	2	no	—
19	C₃₁×#559	—	2	2	no	—
20	C₃₁×#556	—	2	2	no	—
21	C₃₁×#551	—	2	4	no	—
22	C₃₁×#550	—	2	2	no	—
23	C₃₁×#546	—	2	4	no	—
24	C₃₁×#542	C₂²	1	4	yes	complemented
25	C₃₁×#540	C₂²	1	4	yes	complemented
26	C₃₁×#536	C₂²	1	1	yes	characteristic
27	C₃₁×#535	C₂²	1	2	yes	—
28	C₃₁×#535	C₂²	1	1	yes	characteristic
29	C₃₁×#533	C₂²	1	1	yes	characteristic
30	C₃₁×#533	C₂²	1	1	yes	characteristic
31	C₃₁×#532	C₂²	1	4	yes	complemented
32	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
33	C₃₁×#529	C₂²	1	4	yes	complemented
34	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
35	C₃₁×#519	C₂²	1	1	yes	characteristic
36	C₃₁×#518	C₂²	1	1	yes	characteristic
37	C₃₁×#518	C₂²	1	2	yes	—
38	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#501	C₂²	1	2	yes	complemented
40	C₃₁×#501	C₂²	1	4	yes	complemented
41	C₃₁×#501	C₂²	1	2	yes	complemented
42	C₃₁×#496	C₂²	1	2	yes	—
43	C₃₁×#496	C₂²	1	4	yes	—
44	C₃₁×#496	C₂²	1	2	yes	—
45	C₃₁×#490	C₂²	1	4	yes	complemented
46	C₃₁×#489	C₂²	1	4	yes	complemented
47	C₃₁×#488	C₂²	1	4	yes	complemented
48	C₃₁×#486	C₂²	1	4	yes	complemented
49	C₃₁×#469	C₂²	1	8	yes	complemented
50	C₃₁×#468	C₂²	1	8	yes	complemented
51	C₃₁×#466	C₂²	1	8	yes	complemented
52	C₃₁×#465	C₂²	1	8	yes	complemented
53	C₃₁×#442	C₂²	1	8	yes	complemented
54	C₃₁×#440	C₂²	1	8	yes	complemented
55	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
56	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
57	C₃₁×#427	C₂²	1	2	yes	—
58	C₃₁×#427	C₂²	1	2	yes	—
59	C₃₁×#419	C₂²	1	2	yes	—
60	C₃₁×#418	C₂²	1	2	yes	—
61	C₃₁×#418	C₂²	1	2	yes	—
62	C₃₁×#418	C₂²	1	2	yes	—
63	C₃₁×#405	C₂²	1	2	yes	—
64	C₃₁×#405	C₂²	1	4	yes	complemented
65	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
66	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
67	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
68	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
69	C₂×#41×C₃₁	—	2	4	no	—
70	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₃₁	—	2	2	no	—
73	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
75	C₂×SD₁₆×C₃₁	—	2	8	no	—
76	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
78	C₂×D₈×C₃₁	—	2	8	no	—
79	C₂×D₈×C₃₁	C₂³	1	4	yes	—
80	C₂×D₈×C₃₁	C₂³	1	4	yes	—
81	C₃₁×#131	—	2	8	no	—
82	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
83	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
84	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
85	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
86	C₃₁×C₄⋊Q₈	—	2	1	no	—
87	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
88	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
89	C₃₁×#127	—	2	2	no	—
90	C₃₁×#127	—	2	1	no	—
91	C₃₁×#127	—	2	2	no	—
92	C₃₁×#127	C₂³	1	2	yes	—
93	C₃₁×#127	C₂³	1	2	yes	—
94	C₃₁×#126	—	2	4	no	—
95	C₃₁×#126	—	2	4	no	—
96	C₃₁×#125	C₂³	1	2	yes	—
97	C₃₁×#125	C₂³	1	2	yes	—
98	C₃₁×#124	—	2	1	no	—
99	C₃₁×#124	—	2	2	no	—
100	C₃₁×#124	—	2	1	no	—
101	C₃₁×#124	—	2	4	no	—
102	C₃₁×#124	—	2	2	no	—
103	C₃₁×#124	—	2	4	no	—
104	C₃₁×#124	C₂³	1	2	yes	—
105	C₃₁×#124	C₂³	1	2	yes	—
106	C₃₁×#123	—	2	2	no	—
107	C₃₁×#123	—	2	4	no	—
108	C₃₁×#123	—	2	8	no	—
109	C₃₁×#123	—	2	4	no	—
110	C₃₁×C₂²⋊Q₈	—	2	2	no	—
111	C₃₁×C₂²⋊Q₈	—	2	2	no	—
112	C₃₁×C₂²⋊Q₈	—	2	4	no	—
113	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
114	C₃₁×#121	—	2	4	no	—
115	C₃₁×#121	—	2	8	no	—
116	C₃₁×#121	C₂³	1	8	yes	—
117	C₃₁×#121	C₂³	1	4	yes	—
118	C₃₁×#120	—	2	4	no	—
119	C₃₁×#120	—	2	2	no	—
120	C₄×Q₈×C₃₁	—	2	2	no	—
121	C₄×Q₈×C₃₁	—	2	2	no	—
122	C₄×Q₈×C₃₁	—	2	2	no	—
123	C₄×Q₈×C₃₁	D₄	1	2	yes	—
124	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
125	C₄×D₄×C₃₁	—	2	4	no	—
126	C₄×D₄×C₃₁	—	2	4	no	—
127	C₄×D₄×C₃₁	—	2	2	no	—
128	C₄×D₄×C₃₁	—	2	4	no	—
129	C₄×D₄×C₃₁	D₄	1	4	yes	—
130	C₄×D₄×C₃₁	D₄	1	2	yes	—
131	C₄×D₄×C₃₁	C₂³	1	4	yes	—
132	C₄×D₄×C₃₁	C₂³	1	8	yes	—
133	C₃₁×#117	—	2	2	no	—
134	C₃₁×#117	—	2	4	no	—
135	C₃₁×#117	C₂³	1	1	yes	characteristic
136	C₃₁×#117	C₂³	1	1	yes	characteristic
137	C₃₁×#117	C₂³	1	2	yes	—
138	C₃₁×#117	C₂³	1	1	yes	characteristic
139	C₃₁×#117	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
141	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
142	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
143	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
145	C₃₁×#107	C₂³	1	4	yes	—
146	C₃₁×#107	C₂³	1	4	yes	—
147	C₃₁×#106	C₂³	1	4	yes	—
148	C₃₁×#106	C₂³	1	4	yes	—
149	C₃₁×#103	C₂³	1	8	yes	—
150	C₃₁×#102	C₂³	1	8	yes	—
151	C₃₁×#102	C₂³	1	8	yes	—
152	C₃₁×#102	C₂³	1	8	yes	—
153	C₃₁×#97	C₂³	1	1	yes	characteristic
154	C₃₁×#97	C₂³	1	1	yes	characteristic
155	C₃₁×#97	C₂³	1	2	yes	—
156	C₃₁×#97	C₂³	1	2	yes	—
157	C₃₁×#97	C₂³	1	4	yes	—
158	C₄×C₈×C₃₁	C₂³	1	2	yes	—
159	C₄×C₈×C₃₁	C₂³	1	4	yes	—
160	C₂⁴×C₃₁	—	2	1	no	—
161	C₂⁴×C₃₁	—	2	2	no	—
162	#41×C₃₁	—	2	4	no	—
163	#41×C₃₁	—	8	8	no	—
164	C₂×Q₈×C₃₁	—	2	1	no	—
165	C₂×Q₈×C₃₁	—	2	2	no	—
166	C₂×Q₈×C₃₁	—	2	2	no	—
167	C₂×Q₈×C₃₁	—	2	4	no	—
168	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
170	C₂×D₄×C₃₁	—	2	4	no	—
171	C₂×D₄×C₃₁	—	2	8	no	—
172	C₂×D₄×C₃₁	—	2	1	no	—
173	C₂×D₄×C₃₁	—	2	2	no	—
174	C₂×D₄×C₃₁	—	2	4	no	—
175	C₂×D₄×C₃₁	—	2	4	no	—
176	C₂×D₄×C₃₁	—	2	4	no	—
177	C₂×D₄×C₃₁	—	2	4	no	—
178	C₂×D₄×C₃₁	—	2	4	no	—
179	C₂×D₄×C₃₁	—	8	4	no	—
180	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
183	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
184	C₂²×C₄×C₃₁	—	2	1	no	—
185	C₂²×C₄×C₃₁	—	2	4	no	—
186	C₂²×C₄×C₃₁	—	2	8	no	—
187	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
191	SD₁₆×C₃₁	—	2	8	no	—
192	SD₁₆×C₃₁	—	2	8	no	—
193	D₈×C₃₁	—	2	8	no	—
194	D₈×C₃₁	—	2	8	no	—
195	M₁₆×C₃₁	—	2	4	no	—
196	M₁₆×C₃₁	—	2	8	no	—
197	C₂×C₈×C₃₁	—	2	8	no	—
198	C₂×C₈×C₃₁	#41	1	8	yes	complemented
199	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
200	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
201	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₃₁	—	2	2	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	—	2	4	no	—
205	C₄⋊C₄×C₃₁	—	2	2	no	—
206	C₄⋊C₄×C₃₁	—	2	2	no	—
207	C₄⋊C₄×C₃₁	—	2	4	no	—
208	C₄⋊C₄×C₃₁	—	2	2	no	—
209	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
210	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
211	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
212	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
213	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
214	C₂²⋊C₄×C₃₁	—	2	2	no	—
215	C₂²⋊C₄×C₃₁	—	2	1	no	—
216	C₂²⋊C₄×C₃₁	—	2	1	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	2	8	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
221	C₂²⋊C₄×C₃₁	—	2	4	no	—
222	C₂²⋊C₄×C₃₁	—	2	4	no	—
223	C₂²⋊C₄×C₃₁	—	2	8	no	—
224	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
225	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
226	C₄²×C₃₁	—	2	4	no	—
227	C₄²×C₃₁	—	2	2	no	—
228	C₄²×C₃₁	—	2	1	no	—
229	C₄²×C₃₁	—	2	1	no	—
230	C₄²×C₃₁	C₂×D₄	1	4	yes	—
231	C₄²×C₃₁	C₂×D₄	1	2	yes	—
232	C₄²×C₃₁	C₂⁴	1	2	yes	—
233	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
234	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
235	#60719	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂³×C₃₁	—	2	2	no	—
237	C₂³×C₃₁	—	2	4	no	—
238	C₂³×C₃₁	—	4	2	no	—
239	C₂³×C₃₁	—	4	4	no	—
240	C₂³×C₃₁	—	8	4	no	—
241	C₂³×C₃₁	—	8	2	no	—
242	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
243	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	Q₈×C₃₁	—	2	2	no	—
245	Q₈×C₃₁	—	2	4	no	—
246	Q₈×C₃₁	—	4	4	no	—
247	Q₈×C₃₁	—	4	2	no	—
248	Q₈×C₃₁	—	8	2	no	—
249	D₄×C₃₁	—	2	4	no	—
250	D₄×C₃₁	—	2	2	no	—
251	D₄×C₃₁	—	2	4	no	—
252	D₄×C₃₁	—	2	4	no	—
253	D₄×C₃₁	—	2	4	no	—
254	D₄×C₃₁	—	4	8	no	—
255	D₄×C₃₁	—	4	2	no	—
256	D₄×C₃₁	—	4	4	no	—
257	D₄×C₃₁	—	8	4	no	—
258	D₄×C₃₁	—	8	4	no	—
259	D₄×C₃₁	—	8	8	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	4	2	no	—
266	C₂×C₄×C₃₁	—	4	4	no	—
267	C₂×C₄×C₃₁	—	4	2	no	—
268	C₂×C₄×C₃₁	—	8	4	no	—
269	C₂×C₄×C₃₁	—	8	8	no	—
270	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
271	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
276	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
277	C₈×C₃₁	—	2	4	no	—
278	C₈×C₃₁	C₂×#41	1	4	yes	—
279	C₈×C₃₁	C₂×#41	1	4	yes	—
280	#3510	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#3508	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#3180	C₂×C₃₁	1	4	yes	complemented
283	#3179	C₂×C₃₁	1	8	yes	complemented
284	#3171	C₂×C₃₁	1	4	yes	complemented
285	#3162	C₂×C₃₁	1	1	yes	characteristic, complemented
286	C₂×#496	C₂×C₃₁	1	1	yes	characteristic, complemented
287	#3140	C₂×C₃₁	1	1	yes	characteristic, complemented
288	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#3058	C₂×C₃₁	1	2	yes	complemented
290	#3057	C₂×C₃₁	1	2	yes	complemented
291	#2949	C₂×C₃₁	1	2	yes	complemented
292	#2948	C₂×C₃₁	1	2	yes	complemented
293	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
294	C₂²×C₃₁	—	2	2	no	—
295	C₂²×C₃₁	—	2	4	no	—
296	C₂²×C₃₁	—	2	2	no	—
297	C₂²×C₃₁	—	4	8	no	—
298	C₂²×C₃₁	—	4	4	no	—
299	C₂²×C₃₁	—	4	2	no	—
300	C₂²×C₃₁	—	4	4	no	—
301	C₂²×C₃₁	—	16	4	no	—
302	C₂²×C₃₁	—	16	2	no	—
303	C₂²×C₃₁	#532	1	2	yes	—
304	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
305	C₄×C₃₁	—	2	2	no	—
306	C₄×C₃₁	—	2	4	no	—
307	C₄×C₃₁	—	2	2	no	—
308	C₄×C₃₁	—	2	2	no	—
309	C₄×C₃₁	—	4	4	no	—
310	C₄×C₃₁	—	4	2	no	—
311	C₄×C₃₁	—	4	2	no	—
312	C₄×C₃₁	—	4	2	no	—
313	C₄×C₃₁	#532	1	2	yes	—
314	C₄×C₃₁	C₂²×#41	1	2	yes	—
315	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
316	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
317	C₂×#131	C₂²×C₃₁	1	2	yes	—
318	#559	—	2	2	no	—
319	#559	—	2	2	no	—
320	#556	—	2	2	no	—
321	#551	—	2	4	no	—
322	#550	—	2	2	no	—
323	#546	—	2	4	no	—
324	#542	C₂²×C₃₁	1	4	yes	complemented
325	#540	C₂²×C₃₁	1	4	yes	complemented
326	#536	C₂²×C₃₁	1	1	yes	characteristic
327	#535	C₂²×C₃₁	1	2	yes	—
328	#535	C₂²×C₃₁	1	1	yes	characteristic
329	#533	C₂²×C₃₁	1	1	yes	characteristic
330	#533	C₂²×C₃₁	1	1	yes	characteristic
331	#532	C₂²×C₃₁	1	4	yes	complemented
332	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
333	#529	C₂²×C₃₁	1	4	yes	complemented
334	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
335	#519	C₂²×C₃₁	1	1	yes	characteristic
336	#518	C₂²×C₃₁	1	1	yes	characteristic
337	#518	C₂²×C₃₁	1	2	yes	—
338	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
339	#501	C₂²×C₃₁	1	2	yes	complemented
340	#501	C₂²×C₃₁	1	4	yes	complemented
341	#501	C₂²×C₃₁	1	2	yes	complemented
342	#496	C₂²×C₃₁	1	2	yes	—
343	#496	C₂²×C₃₁	1	4	yes	—
344	#496	C₂²×C₃₁	1	2	yes	—
345	#490	C₂²×C₃₁	1	4	yes	complemented
346	#489	C₂²×C₃₁	1	4	yes	complemented
347	#488	C₂²×C₃₁	1	4	yes	complemented
348	#486	C₂²×C₃₁	1	4	yes	complemented
349	#469	C₂²×C₃₁	1	8	yes	complemented
350	#468	C₂²×C₃₁	1	8	yes	complemented
351	#466	C₂²×C₃₁	1	8	yes	complemented
352	#465	C₂²×C₃₁	1	8	yes	complemented
353	#442	C₂²×C₃₁	1	8	yes	complemented
354	#440	C₂²×C₃₁	1	8	yes	complemented
355	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
356	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
357	#427	C₂²×C₃₁	1	2	yes	—
358	#427	C₂²×C₃₁	1	2	yes	—
359	#419	C₂²×C₃₁	1	2	yes	—
360	#418	C₂²×C₃₁	1	2	yes	—
361	#418	C₂²×C₃₁	1	2	yes	—
362	#418	C₂²×C₃₁	1	2	yes	—
363	#405	C₂²×C₃₁	1	2	yes	—
364	#405	C₂²×C₃₁	1	4	yes	complemented
365	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
366	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
367	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
369	C₂×C₃₁	—	2	2	no	—
370	C₂×C₃₁	—	4	2	no	—
371	C₂×C₃₁	—	8	4	no	—
372	C₂×C₃₁	—	8	2	no	—
373	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
374	C₂×C₃₁	#3585	1	2	yes	—
375	C₂×#41	—	2	4	no	—
376	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
377	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₂²×D₄	—	2	2	no	—
379	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
380	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
381	C₂×SD₁₆	—	2	8	no	—
382	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
383	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
384	C₂×D₈	—	2	8	no	—
385	C₂×D₈	C₂³×C₃₁	1	4	yes	—
386	C₂×D₈	C₂³×C₃₁	1	4	yes	—
387	#131	—	2	8	no	—
388	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
389	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
390	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
391	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
392	C₄⋊Q₈	—	2	1	no	—
393	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
394	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
395	#127	—	2	2	no	—
396	#127	—	2	1	no	—
397	#127	—	2	2	no	—
398	#127	C₂³×C₃₁	1	2	yes	—
399	#127	C₂³×C₃₁	1	2	yes	—
400	#126	—	2	4	no	—
401	#126	—	2	4	no	—
402	#125	C₂³×C₃₁	1	2	yes	—
403	#125	C₂³×C₃₁	1	2	yes	—
404	#124	—	2	1	no	—
405	#124	—	2	2	no	—
406	#124	—	2	1	no	—
407	#124	—	2	4	no	—
408	#124	—	2	2	no	—
409	#124	—	2	4	no	—
410	#124	C₂³×C₃₁	1	2	yes	—
411	#124	C₂³×C₃₁	1	2	yes	—
412	#123	—	2	2	no	—
413	#123	—	2	4	no	—
414	#123	—	2	8	no	—
415	#123	—	2	4	no	—
416	C₂²⋊Q₈	—	2	2	no	—
417	C₂²⋊Q₈	—	2	2	no	—
418	C₂²⋊Q₈	—	2	4	no	—
419	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
420	#121	—	2	4	no	—
421	#121	—	2	8	no	—
422	#121	C₂³×C₃₁	1	4	yes	—
423	#121	C₂³×C₃₁	1	8	yes	—
424	#120	—	2	4	no	—
425	#120	—	2	2	no	—
426	C₄×Q₈	—	2	2	no	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×Q₈	—	2	2	no	—
429	C₄×Q₈	D₄×C₃₁	1	2	yes	—
430	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
431	C₄×D₄	—	2	4	no	—
432	C₄×D₄	—	2	4	no	—
433	C₄×D₄	—	2	2	no	—
434	C₄×D₄	—	2	4	no	—
435	C₄×D₄	D₄×C₃₁	1	4	yes	—
436	C₄×D₄	D₄×C₃₁	1	2	yes	—
437	C₄×D₄	C₂³×C₃₁	1	4	yes	—
438	C₄×D₄	C₂³×C₃₁	1	8	yes	—
439	#117	—	2	2	no	—
440	#117	—	2	4	no	—
441	#117	C₂³×C₃₁	1	2	yes	—
442	#117	C₂³×C₃₁	1	1	yes	characteristic
443	#117	C₂³×C₃₁	1	1	yes	characteristic
444	#117	C₂³×C₃₁	1	1	yes	characteristic
445	#117	C₂³×C₃₁	1	2	yes	—
446	C₂×C₂²⋊C₄	—	2	2	no	—
447	C₂×C₂²⋊C₄	—	2	4	no	—
448	C₂×C₂²⋊C₄	—	2	2	no	—
449	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
451	#107	C₂³×C₃₁	1	4	yes	—
452	#107	C₂³×C₃₁	1	4	yes	—
453	#106	C₂³×C₃₁	1	4	yes	—
454	#106	C₂³×C₃₁	1	4	yes	—
455	#103	C₂³×C₃₁	1	8	yes	—
456	#102	C₂³×C₃₁	1	8	yes	—
457	#102	C₂³×C₃₁	1	8	yes	—
458	#102	C₂³×C₃₁	1	8	yes	—
459	#97	C₂³×C₃₁	1	1	yes	characteristic
460	#97	C₂³×C₃₁	1	1	yes	characteristic
461	#97	C₂³×C₃₁	1	2	yes	—
462	#97	C₂³×C₃₁	1	2	yes	—
463	#97	C₂³×C₃₁	1	4	yes	—
464	C₄×C₈	C₂³×C₃₁	1	2	yes	—
465	C₄×C₈	C₂³×C₃₁	1	4	yes	—
466	C₃₁	#60719	1	1	yes	characteristic, direct product factor, Sylow
467	C₂⁴	—	2	1	no	—
468	C₂⁴	—	2	2	no	—
469	#41	—	2	4	no	—
470	#41	—	8	8	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	2	2	no	—
474	C₂×Q₈	—	2	4	no	—
475	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	8	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	8	4	no	—
487	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
488	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
490	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	8	no	—
494	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
495	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
498	SD₁₆	—	2	8	no	—
499	SD₁₆	—	2	8	no	—
500	D₈	—	2	8	no	—
501	D₈	—	2	8	no	—
502	M₁₆	—	2	4	no	—
503	M₁₆	—	2	8	no	—
504	C₂×C₈	—	2	8	no	—
505	C₂×C₈	#41×C₃₁	1	8	yes	complemented
506	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
507	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
508	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
517	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
518	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
519	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
520	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	1	no	—
523	C₂²⋊C₄	—	2	1	no	—
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₂²⋊C₄	—	2	8	no	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	8	no	—
531	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
532	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
533	C₄²	—	2	4	no	—
534	C₄²	—	2	2	no	—
535	C₄²	—	2	1	no	—
536	C₄²	—	2	1	no	—
537	C₄²	C₂×D₄×C₃₁	1	4	yes	—
538	C₄²	C₂×D₄×C₃₁	1	2	yes	—
539	C₄²	C₂⁴×C₃₁	1	2	yes	—
540	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
541	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
542	C₂³	—	2	2	no	—
543	C₂³	—	2	4	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	4	4	no	—
546	C₂³	—	8	4	no	—
547	C₂³	—	8	2	no	—
548	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
549	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
550	Q₈	—	2	2	no	—
551	Q₈	—	2	4	no	—
552	Q₈	—	4	4	no	—
553	Q₈	—	4	2	no	—
554	Q₈	—	8	2	no	—
555	D₄	—	2	4	no	—
556	D₄	—	2	2	no	—
557	D₄	—	2	4	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	4	no	—
560	D₄	—	4	8	no	—
561	D₄	—	4	2	no	—
562	D₄	—	4	4	no	—
563	D₄	—	8	4	no	—
564	D₄	—	8	4	no	—
565	D₄	—	8	8	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	8	4	no	—
575	C₂×C₄	—	8	8	no	—
576	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
577	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
581	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
582	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
583	C₈	—	2	4	no	—
584	C₈	C₂×#41×C₃₁	1	4	yes	—
585	C₈	C₂×#41×C₃₁	1	4	yes	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	8	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	16	4	no	—
594	C₂²	—	16	2	no	—
595	C₂²	C₃₁×#532	1	2	yes	—
596	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	4	no	—
600	C₄	—	2	2	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	2	no	—
604	C₄	—	4	2	no	—
605	C₄	C₃₁×#532	1	2	yes	—
606	C₄	C₂²×#41×C₃₁	1	2	yes	—
607	C₂	—	2	2	no	—
608	C₂	—	4	2	no	—
609	C₂	—	8	4	no	—
610	C₂	—	8	2	no	—
611	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
612	C₂	C₃₁×#3585	1	2	yes	—

	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×#3510	C₂	semidirect
2. C₃₁×#3508	C₂	semidirect
3. C₃₁×#3180	C₂	semidirect
4. C₃₁×#3179	C₂	semidirect
5. C₃₁×#3171	C₂	semidirect
6. C₃₁×#3162	C₂	semidirect
7. C₂×C₃₁×#496	C₂	semidirect
8. C₃₁×#3140	C₂	semidirect
9. C₃₁×#3138	C₂	semidirect
10. C₃₁×#3058	C₂	semidirect
11. C₃₁×#3057	C₂	semidirect
12. C₃₁×#2949	C₂	semidirect
13. C₃₁×#2948	C₂	semidirect
14. C₃₁×#2875	C₂	semidirect
24. C₃₁×#542	C₂²	semidirect
25. C₃₁×#540	C₂²	semidirect
31. C₃₁×#532	C₂²	semidirect
33. C₃₁×#529	C₂²	semidirect
39. C₃₁×#501	C₂²	semidirect
40. C₃₁×#501	C₂²	semidirect
41. C₃₁×#501	C₂²	semidirect
45. C₃₁×#490	C₂²	semidirect
46. C₃₁×#489	C₂²	semidirect
47. C₃₁×#488	C₂²	semidirect
48. C₃₁×#486	C₂²	semidirect
49. C₃₁×#469	C₂²	semidirect
50. C₃₁×#468	C₂²	semidirect
51. C₃₁×#466	C₂²	semidirect
52. C₃₁×#465	C₂²	semidirect
53. C₃₁×#442	C₂²	semidirect
54. C₃₁×#440	C₂²	semidirect
55. C₄×SD₁₆×C₃₁	C₂²	semidirect
56. C₄×D₈×C₃₁	C₂²	semidirect
64. C₃₁×#405	C₂²	semidirect
66. C₄×M₁₆×C₃₁	C₂²	semidirect
68. C₂×C₃₁×#97	C₂²	semidirect
198. C₂×C₈×C₃₁	#41	semidirect
235. #60719	C₃₁	direct (N × H)
280. #3510	C₂×C₃₁	semidirect
281. #3508	C₂×C₃₁	semidirect
282. #3180	C₂×C₃₁	semidirect
283. #3179	C₂×C₃₁	semidirect
284. #3171	C₂×C₃₁	semidirect
285. #3162	C₂×C₃₁	semidirect
286. C₂×#496	C₂×C₃₁	semidirect
287. #3140	C₂×C₃₁	semidirect
288. #3138	C₂×C₃₁	semidirect
289. #3058	C₂×C₃₁	semidirect
290. #3057	C₂×C₃₁	semidirect
291. #2949	C₂×C₃₁	semidirect
292. #2948	C₂×C₃₁	semidirect
293. #2875	C₂×C₃₁	semidirect
324. #542	C₂²×C₃₁	semidirect
325. #540	C₂²×C₃₁	semidirect
331. #532	C₂²×C₃₁	semidirect
333. #529	C₂²×C₃₁	semidirect
339. #501	C₂²×C₃₁	semidirect
340. #501	C₂²×C₃₁	semidirect
341. #501	C₂²×C₃₁	semidirect
345. #490	C₂²×C₃₁	semidirect
346. #489	C₂²×C₃₁	semidirect
347. #488	C₂²×C₃₁	semidirect
348. #486	C₂²×C₃₁	semidirect
349. #469	C₂²×C₃₁	semidirect
350. #468	C₂²×C₃₁	semidirect
351. #466	C₂²×C₃₁	semidirect
352. #465	C₂²×C₃₁	semidirect
353. #442	C₂²×C₃₁	semidirect
354. #440	C₂²×C₃₁	semidirect
355. C₄×SD₁₆	C₂²×C₃₁	semidirect
356. C₄×D₈	C₂²×C₃₁	semidirect
364. #405	C₂²×C₃₁	semidirect
366. C₄×M₁₆	C₂²×C₃₁	semidirect
368. C₂×#97	C₂²×C₃₁	semidirect
466. C₃₁	#60719	direct (N × H)
505. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3510	2
C₃₁×#3508	2
C₃₁×#3180	2
C₃₁×#3179	2
C₃₁×#3171	2
C₃₁×#3162	2
C₂×C₃₁×#496	2
C₃₁×#3140	2
C₃₁×#3138	2
C₃₁×#3058	2
C₃₁×#3057	2
C₃₁×#2949	2
C₃₁×#2948	2
C₃₁×#2875	2
#60719	31

Subgroups — C31×#60719 #21517478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60719 #21517478