Subgroups — C₃₁×#60964 #21517723

Order 7936.

1844 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3256	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3232	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3211	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3195	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3193	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3192	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#3183	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3117	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3114	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#529	—	2	2	no	—
13	C₃₁×#529	—	2	2	no	—
14	C₃₁×#529	—	2	2	no	—
15	C₃₁×#529	—	2	2	no	—
16	C₃₁×#529	C₂²	1	4	yes	—
17	C₃₁×#529	C₂²	1	4	yes	—
18	C₃₁×#529	C₂²	1	4	yes	complemented
19	C₃₁×#529	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#126	—	2	2	no	—
21	C₂×C₃₁×#125	C₂²	1	2	yes	complemented
22	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
23	C₂×C₃₁×#123	—	2	2	no	—
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#121	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
27	C₂×C₄×D₄×C₃₁	—	2	2	no	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
29	C₂×C₃₁×#117	—	2	2	no	—
30	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#499	C₂²	1	4	yes	complemented
32	C₃₁×#495	C₂²	1	4	yes	complemented
33	C₃₁×#488	C₂²	1	8	yes	complemented
34	C₃₁×#484	C₂²	1	4	yes	complemented
35	C₃₁×#483	C₂²	1	4	yes	complemented
36	C₃₁×#481	C₂²	1	4	yes	—
37	C₃₁×#480	C₂²	1	4	yes	—
38	C₃₁×#479	C₂²	1	8	yes	complemented
39	C₃₁×#478	C₂²	1	8	yes	complemented
40	C₃₁×#470	C₂²	1	4	yes	complemented
41	C₃₁×#469	C₂²	1	4	yes	—
42	C₃₁×#468	C₂²	1	8	yes	complemented
43	C₃₁×#464	C₂²	1	8	yes	complemented
44	C₃₁×#463	C₂²	1	8	yes	complemented
45	C₃₁×#451	C₂²	1	4	yes	complemented
46	C₃₁×#450	C₂²	1	4	yes	complemented
47	C₃₁×#448	C₂²	1	4	yes	—
48	C₃₁×#447	C₂²	1	4	yes	—
49	C₃₁×#442	C₂²	1	4	yes	complemented
50	C₃₁×#441	C₂²	1	4	yes	complemented
51	C₃₁×#440	C₂²	1	8	yes	complemented
52	C₃₁×#432	C₂²	1	4	yes	complemented
53	C₃₁×#431	C₂²	1	4	yes	complemented
54	C₃₁×#423	C₂²	1	2	yes	complemented
55	C₃₁×#407	C₂²	1	1	yes	characteristic
56	C₃₁×#407	C₂²	1	1	yes	characteristic
57	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
58	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×D₄×C₃₁	—	2	2	no	—
60	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	—	2	2	no	—
62	C₂³×C₄×C₃₁	D₄	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
65	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
66	C₂×D₈×C₃₁	C₂³	1	4	yes	—
67	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
68	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
69	C₃₁×#126	—	2	4	no	—
70	C₃₁×#126	—	2	4	no	—
71	C₃₁×#126	—	2	4	no	—
72	C₃₁×#126	—	2	4	no	—
73	C₃₁×#125	—	2	2	no	—
74	C₃₁×#125	—	2	2	no	—
75	C₃₁×#125	C₂³	1	4	yes	—
76	C₃₁×#125	C₂³	1	4	yes	—
77	C₃₁×#124	—	2	2	no	—
78	C₃₁×#124	—	2	2	no	—
79	C₃₁×#124	C₂³	1	4	yes	—
80	C₃₁×#124	C₂³	1	4	yes	—
81	C₃₁×#123	—	2	4	no	—
82	C₃₁×#123	—	2	4	no	—
83	C₃₁×#123	—	2	4	no	—
84	C₃₁×#123	—	2	4	no	—
85	C₃₁×C₂²⋊Q₈	—	2	2	no	—
86	C₃₁×C₂²⋊Q₈	—	2	2	no	—
87	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
88	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
89	C₃₁×#121	—	2	2	no	—
90	C₃₁×#121	—	2	2	no	—
91	C₃₁×#121	C₂³	1	4	yes	—
92	C₃₁×#121	C₂³	1	4	yes	—
93	C₄×Q₈×C₃₁	—	2	2	no	—
94	C₄×Q₈×C₃₁	—	2	2	no	—
95	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
96	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
97	C₄×D₄×C₃₁	—	2	4	no	—
98	C₄×D₄×C₃₁	—	2	4	no	—
99	C₄×D₄×C₃₁	—	2	4	no	—
100	C₄×D₄×C₃₁	—	2	4	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	C₂³	1	4	yes	—
104	C₄×D₄×C₃₁	C₂³	1	4	yes	—
105	C₃₁×#117	—	2	4	no	—
106	C₃₁×#117	—	2	4	no	—
107	C₃₁×#117	—	2	4	no	—
108	C₃₁×#117	—	2	4	no	—
109	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄²×C₃₁	—	2	2	no	—
121	C₂×C₄²×C₃₁	D₄	1	2	yes	—
122	C₂×C₄²×C₃₁	D₄	1	2	yes	—
123	C₂×C₄²×C₃₁	D₄	1	2	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
128	C₃₁×#107	C₂³	1	8	yes	—
129	C₃₁×#106	C₂³	1	4	yes	—
130	C₃₁×#106	C₂³	1	4	yes	—
131	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
132	C₃₁×#103	C₂³	1	8	yes	—
133	C₃₁×#103	C₂³	1	8	yes	—
134	C₃₁×#102	C₂³	1	8	yes	—
135	C₃₁×#102	C₂³	1	8	yes	—
136	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
137	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
138	C₃₁×#97	C₂³	1	4	yes	—
139	C₄×C₈×C₃₁	C₂³	1	4	yes	—
140	C₂⁴×C₃₁	—	2	2	no	—
141	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×Q₈×C₃₁	—	2	2	no	—
143	C₂×Q₈×C₃₁	—	4	4	no	—
144	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	2	4	no	—
147	C₂×D₄×C₃₁	—	2	2	no	—
148	C₂×D₄×C₃₁	—	4	4	no	—
149	C₂×D₄×C₃₁	—	4	8	no	—
150	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	2	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	4	2	no	—
157	C₂²×C₄×C₃₁	—	4	4	no	—
158	C₂²×C₄×C₃₁	—	4	8	no	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	Q₁₆×C₃₁	—	4	4	no	—
170	SD₁₆×C₃₁	—	4	8	no	—
171	D₈×C₃₁	—	4	4	no	—
172	M₁₆×C₃₁	—	4	2	no	—
173	M₁₆×C₃₁	—	4	2	no	—
174	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
175	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
176	C₄⋊C₄×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	4	no	—
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
188	C₂²⋊C₄×C₃₁	—	2	4	no	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	—	2	4	no	—
192	C₂²⋊C₄×C₃₁	—	2	4	no	—
193	C₂²⋊C₄×C₃₁	—	2	4	no	—
194	C₂²⋊C₄×C₃₁	—	2	4	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	4	no	—
198	C₄²×C₃₁	—	2	4	no	—
199	C₄²×C₃₁	—	2	4	no	—
200	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₃₁	C₂×D₄	1	2	yes	—
202	C₄²×C₃₁	C₂×D₄	1	2	yes	—
203	C₄²×C₃₁	C₂×D₄	1	2	yes	—
204	C₄²×C₃₁	C₂×D₄	1	2	yes	—
205	C₄²×C₃₁	C₂×D₄	1	2	yes	—
206	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
208	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
209	C₄²×C₃₁	C₂⁴	1	2	yes	—
210	#60964	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂³×C₃₁	—	2	4	no	—
212	C₂³×C₃₁	—	2	4	no	—
213	C₂³×C₃₁	—	4	1	no	—
214	C₂³×C₃₁	—	4	4	no	—
215	C₂³×C₃₁	—	4	4	no	—
216	C₂³×C₃₁	—	4	4	no	—
217	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
219	Q₈×C₃₁	—	2	8	no	—
220	Q₈×C₃₁	—	4	4	no	—
221	D₄×C₃₁	—	2	8	no	—
222	D₄×C₃₁	—	4	4	no	—
223	D₄×C₃₁	—	4	8	no	—
224	D₄×C₃₁	—	4	8	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	4	2	no	—
233	C₂×C₄×C₃₁	—	4	1	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	4	2	no	—
236	C₂×C₄×C₃₁	—	4	2	no	—
237	C₂×C₄×C₃₁	—	4	2	no	—
238	C₂×C₄×C₃₁	—	4	8	no	—
239	C₂×C₄×C₃₁	—	4	8	no	—
240	C₂×C₄×C₃₁	—	4	4	no	—
241	C₂×C₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	—	4	4	no	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
250	C₈×C₃₁	—	2	4	no	—
251	C₈×C₃₁	—	2	4	no	—
252	C₂×#529	C₂×C₃₁	1	2	yes	complemented
253	#3256	C₂×C₃₁	1	4	yes	complemented
254	#3232	C₂×C₃₁	1	4	yes	complemented
255	#3211	C₂×C₃₁	1	4	yes	complemented
256	#3195	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3193	C₂×C₃₁	1	2	yes	complemented
258	#3192	C₂×C₃₁	1	1	yes	characteristic
259	#3183	C₂×C₃₁	1	4	yes	complemented
260	#3117	C₂×C₃₁	1	4	yes	complemented
261	#3114	C₂×C₃₁	1	4	yes	complemented
262	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	2	2	no	—
265	C₂²×C₃₁	—	2	1	no	—
266	C₂²×C₃₁	—	2	2	no	—
267	C₂²×C₃₁	—	4	4	no	—
268	C₂²×C₃₁	—	4	4	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	4	no	—
271	C₂²×C₃₁	—	8	8	no	—
272	C₂²×C₃₁	#575	1	4	yes	—
273	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	2	no	—
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	2	2	no	—
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	4	4	no	—
281	C₄×C₃₁	—	4	4	no	—
282	C₄×C₃₁	—	4	4	no	—
283	C₄×C₃₁	#585	1	2	yes	—
284	#529	—	2	2	no	—
285	#529	—	2	2	no	—
286	#529	—	2	2	no	—
287	#529	—	2	2	no	—
288	#529	C₂²×C₃₁	1	4	yes	—
289	#529	C₂²×C₃₁	1	4	yes	—
290	#529	C₂²×C₃₁	1	4	yes	complemented
291	#529	C₂²×C₃₁	1	4	yes	complemented
292	C₂×#126	—	2	2	no	—
293	C₂×#125	C₂²×C₃₁	1	2	yes	complemented
294	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
295	C₂×#123	—	2	2	no	—
296	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
297	C₂×#121	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
299	C₂×C₄×D₄	—	2	2	no	—
300	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
301	C₂×#117	—	2	2	no	—
302	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
303	#499	C₂²×C₃₁	1	4	yes	complemented
304	#495	C₂²×C₃₁	1	4	yes	complemented
305	#488	C₂²×C₃₁	1	8	yes	complemented
306	#484	C₂²×C₃₁	1	4	yes	complemented
307	#483	C₂²×C₃₁	1	4	yes	complemented
308	#481	C₂²×C₃₁	1	4	yes	—
309	#480	C₂²×C₃₁	1	4	yes	—
310	#479	C₂²×C₃₁	1	8	yes	complemented
311	#478	C₂²×C₃₁	1	8	yes	complemented
312	#470	C₂²×C₃₁	1	4	yes	complemented
313	#469	C₂²×C₃₁	1	4	yes	—
314	#468	C₂²×C₃₁	1	8	yes	complemented
315	#464	C₂²×C₃₁	1	8	yes	complemented
316	#463	C₂²×C₃₁	1	8	yes	complemented
317	#451	C₂²×C₃₁	1	4	yes	complemented
318	#450	C₂²×C₃₁	1	4	yes	complemented
319	#448	C₂²×C₃₁	1	4	yes	—
320	#447	C₂²×C₃₁	1	4	yes	—
321	#442	C₂²×C₃₁	1	4	yes	complemented
322	#441	C₂²×C₃₁	1	4	yes	complemented
323	#440	C₂²×C₃₁	1	8	yes	complemented
324	#432	C₂²×C₃₁	1	4	yes	complemented
325	#431	C₂²×C₃₁	1	4	yes	complemented
326	#423	C₂²×C₃₁	1	2	yes	complemented
327	#407	C₂²×C₃₁	1	1	yes	characteristic
328	#407	C₂²×C₃₁	1	1	yes	characteristic
329	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
330	C₂×C₃₁	—	2	4	no	—
331	C₂×C₃₁	—	4	1	no	—
332	C₂×C₃₁	—	8	4	no	—
333	C₂×C₃₁	#3449	1	1	yes	characteristic
334	C₂×C₃₁	C₂×#575	1	2	yes	—
335	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
336	C₂²×D₄	—	2	2	no	—
337	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
338	C₂³×C₄	—	2	2	no	—
339	C₂³×C₄	D₄×C₃₁	1	2	yes	—
340	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
342	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
343	C₂×D₈	C₂³×C₃₁	1	4	yes	—
344	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
345	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
346	#126	—	2	4	no	—
347	#126	—	2	4	no	—
348	#126	—	2	4	no	—
349	#126	—	2	4	no	—
350	#125	—	2	2	no	—
351	#125	—	2	2	no	—
352	#125	C₂³×C₃₁	1	4	yes	—
353	#125	C₂³×C₃₁	1	4	yes	—
354	#124	—	2	2	no	—
355	#124	—	2	2	no	—
356	#124	C₂³×C₃₁	1	4	yes	—
357	#124	C₂³×C₃₁	1	4	yes	—
358	#123	—	2	4	no	—
359	#123	—	2	4	no	—
360	#123	—	2	4	no	—
361	#123	—	2	4	no	—
362	C₂²⋊Q₈	—	2	2	no	—
363	C₂²⋊Q₈	—	2	2	no	—
364	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
365	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
366	#121	—	2	2	no	—
367	#121	—	2	2	no	—
368	#121	C₂³×C₃₁	1	4	yes	—
369	#121	C₂³×C₃₁	1	4	yes	—
370	C₄×Q₈	—	2	2	no	—
371	C₄×Q₈	—	2	2	no	—
372	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
373	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	—	2	4	no	—
376	C₄×D₄	—	2	4	no	—
377	C₄×D₄	—	2	4	no	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	—	2	2	no	—
380	C₄×D₄	C₂³×C₃₁	1	4	yes	—
381	C₄×D₄	C₂³×C₃₁	1	4	yes	—
382	#117	—	2	4	no	—
383	#117	—	2	4	no	—
384	#117	—	2	4	no	—
385	#117	—	2	4	no	—
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄⋊C₄	—	2	2	no	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
392	C₂×C₂²⋊C₄	—	2	2	no	—
393	C₂×C₂²⋊C₄	—	2	2	no	—
394	C₂×C₂²⋊C₄	—	2	2	no	—
395	C₂×C₂²⋊C₄	—	2	2	no	—
396	C₂×C₂²⋊C₄	—	2	2	no	—
397	C₂×C₄²	—	2	2	no	—
398	C₂×C₄²	D₄×C₃₁	1	2	yes	—
399	C₂×C₄²	D₄×C₃₁	1	2	yes	—
400	C₂×C₄²	D₄×C₃₁	1	2	yes	—
401	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
402	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
403	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
405	#107	C₂³×C₃₁	1	8	yes	—
406	#106	C₂³×C₃₁	1	4	yes	—
407	#106	C₂³×C₃₁	1	4	yes	—
408	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
409	#103	C₂³×C₃₁	1	8	yes	—
410	#103	C₂³×C₃₁	1	8	yes	—
411	#102	C₂³×C₃₁	1	8	yes	—
412	#102	C₂³×C₃₁	1	8	yes	—
413	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
414	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
415	#97	C₂³×C₃₁	1	4	yes	—
416	C₄×C₈	C₂³×C₃₁	1	4	yes	—
417	C₃₁	#60964	1	1	yes	characteristic, direct product factor, Sylow
418	C₂⁴	—	2	2	no	—
419	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	4	4	no	—
422	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	4	4	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	4	2	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	8	no	—
437	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
438	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
439	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
440	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
442	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
443	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
444	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
445	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
447	Q₁₆	—	4	4	no	—
448	SD₁₆	—	4	8	no	—
449	D₈	—	4	4	no	—
450	M₁₆	—	4	2	no	—
451	M₁₆	—	4	2	no	—
452	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
453	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
465	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₄²	—	2	4	no	—
477	C₄²	—	2	4	no	—
478	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₄²	C₂×D₄×C₃₁	1	2	yes	—
480	C₄²	C₂×D₄×C₃₁	1	2	yes	—
481	C₄²	C₂×D₄×C₃₁	1	2	yes	—
482	C₄²	C₂×D₄×C₃₁	1	2	yes	—
483	C₄²	C₂×D₄×C₃₁	1	2	yes	—
484	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
486	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
487	C₄²	C₂⁴×C₃₁	1	2	yes	—
488	C₂³	—	2	4	no	—
489	C₂³	—	2	4	no	—
490	C₂³	—	4	1	no	—
491	C₂³	—	4	4	no	—
492	C₂³	—	4	4	no	—
493	C₂³	—	4	4	no	—
494	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
495	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
496	Q₈	—	2	8	no	—
497	Q₈	—	4	4	no	—
498	D₄	—	2	8	no	—
499	D₄	—	4	4	no	—
500	D₄	—	4	8	no	—
501	D₄	—	4	8	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	8	no	—
516	C₂×C₄	—	4	8	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
521	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
524	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
525	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
526	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
527	C₈	—	2	4	no	—
528	C₈	—	2	4	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	4	4	no	—
534	C₂²	—	4	4	no	—
535	C₂²	—	4	4	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	8	8	no	—
538	C₂²	C₃₁×#575	1	4	yes	—
539	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
540	C₄	—	2	2	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	4	4	no	—
547	C₄	—	4	4	no	—
548	C₄	—	4	4	no	—
549	C₄	C₃₁×#585	1	2	yes	—
550	C₂	—	2	4	no	—
551	C₂	—	4	1	no	—
552	C₂	—	8	4	no	—
553	C₂	C₃₁×#3449	1	1	yes	characteristic
554	C₂	C₂×C₃₁×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3192	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#529	C₂	semidirect
2. C₃₁×#3256	C₂	semidirect
3. C₃₁×#3232	C₂	semidirect
4. C₃₁×#3211	C₂	semidirect
5. C₃₁×#3195	C₂	semidirect
6. C₃₁×#3193	C₂	semidirect
8. C₃₁×#3183	C₂	semidirect
9. C₃₁×#3117	C₂	semidirect
10. C₃₁×#3114	C₂	semidirect
11. C₃₁×#2920	C₂	semidirect
18. C₃₁×#529	C₂²	semidirect
19. C₃₁×#529	C₂²	semidirect
21. C₂×C₃₁×#125	C₂²	semidirect
22. C₂×C₃₁×#124	C₂²	semidirect
26. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
28. C₂×C₄×D₄×C₃₁	C₂²	semidirect
31. C₃₁×#499	C₂²	semidirect
32. C₃₁×#495	C₂²	semidirect
33. C₃₁×#488	C₂²	semidirect
34. C₃₁×#484	C₂²	semidirect
35. C₃₁×#483	C₂²	semidirect
38. C₃₁×#479	C₂²	semidirect
39. C₃₁×#478	C₂²	semidirect
40. C₃₁×#470	C₂²	semidirect
42. C₃₁×#468	C₂²	semidirect
43. C₃₁×#464	C₂²	semidirect
44. C₃₁×#463	C₂²	semidirect
45. C₃₁×#451	C₂²	semidirect
46. C₃₁×#450	C₂²	semidirect
49. C₃₁×#442	C₂²	semidirect
50. C₃₁×#441	C₂²	semidirect
51. C₃₁×#440	C₂²	semidirect
52. C₃₁×#432	C₂²	semidirect
53. C₃₁×#431	C₂²	semidirect
54. C₃₁×#423	C₂²	semidirect
57. C₄×M₁₆×C₃₁	C₂²	semidirect
210. #60964	C₃₁	direct (N × H)
252. C₂×#529	C₂×C₃₁	semidirect
253. #3256	C₂×C₃₁	semidirect
254. #3232	C₂×C₃₁	semidirect
255. #3211	C₂×C₃₁	semidirect
256. #3195	C₂×C₃₁	semidirect
257. #3193	C₂×C₃₁	semidirect
259. #3183	C₂×C₃₁	semidirect
260. #3117	C₂×C₃₁	semidirect
261. #3114	C₂×C₃₁	semidirect
262. #2920	C₂×C₃₁	semidirect
290. #529	C₂²×C₃₁	semidirect
291. #529	C₂²×C₃₁	semidirect
293. C₂×#125	C₂²×C₃₁	semidirect
294. C₂×#124	C₂²×C₃₁	semidirect
298. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
300. C₂×C₄×D₄	C₂²×C₃₁	semidirect
303. #499	C₂²×C₃₁	semidirect
304. #495	C₂²×C₃₁	semidirect
305. #488	C₂²×C₃₁	semidirect
306. #484	C₂²×C₃₁	semidirect
307. #483	C₂²×C₃₁	semidirect
310. #479	C₂²×C₃₁	semidirect
311. #478	C₂²×C₃₁	semidirect
312. #470	C₂²×C₃₁	semidirect
314. #468	C₂²×C₃₁	semidirect
315. #464	C₂²×C₃₁	semidirect
316. #463	C₂²×C₃₁	semidirect
317. #451	C₂²×C₃₁	semidirect
318. #450	C₂²×C₃₁	semidirect
321. #442	C₂²×C₃₁	semidirect
322. #441	C₂²×C₃₁	semidirect
323. #440	C₂²×C₃₁	semidirect
324. #432	C₂²×C₃₁	semidirect
325. #431	C₂²×C₃₁	semidirect
326. #423	C₂²×C₃₁	semidirect
329. C₄×M₁₆	C₂²×C₃₁	semidirect
417. C₃₁	#60964	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#529	2
C₃₁×#3256	2
C₃₁×#3232	2
C₃₁×#3211	2
C₃₁×#3195	2
C₃₁×#3193	2
C₃₁×#3192	2
C₃₁×#3183	2
C₃₁×#3117	2
C₃₁×#3114	2
C₃₁×#2920	2
#60964	31

Subgroups — C31×#60964 #21517723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60964 #21517723