Subgroups — C₃₁×#60861 #21517620

Order 7936.

1790 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3236	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3234	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3232	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3166	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3142	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3092	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3080	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#533	C₂²	1	2	yes	—
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	2	yes	—
15	C₃₁×#529	—	2	2	no	—
16	C₃₁×#529	—	2	2	no	—
17	C₃₁×#529	C₂²	1	4	yes	—
18	C₂×C₃₁×#125	—	2	2	no	—
19	C₂×C₃₁×#125	—	2	2	no	—
20	C₂×C₃₁×#125	—	2	2	no	—
21	C₂×C₃₁×#125	C₂²	1	2	yes	—
22	C₂×C₃₁×#125	C₂²	1	2	yes	—
23	C₂×C₃₁×#125	C₂²	1	4	yes	—
24	C₂×C₃₁×#125	C₂²	1	4	yes	—
25	C₃₁×#525	—	2	2	no	—
26	C₃₁×#525	C₂²	1	2	yes	—
27	C₄×#41×C₃₁	—	2	2	no	—
28	C₄×#41×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#501	C₂²	1	8	yes	complemented
33	C₃₁×#499	C₂²	1	8	yes	complemented
34	C₃₁×#491	C₂²	1	4	yes	complemented
35	C₃₁×#490	C₂²	1	4	yes	complemented
36	C₃₁×#488	C₂²	1	8	yes	complemented
37	C₃₁×#484	C₂²	1	8	yes	—
38	C₃₁×#483	C₂²	1	8	yes	—
39	C₃₁×#481	C₂²	1	8	yes	—
40	C₃₁×#480	C₂²	1	8	yes	—
41	C₃₁×#479	C₂²	1	16	yes	complemented
42	C₃₁×#478	C₂²	1	16	yes	complemented
43	C₃₁×#432	C₂²	1	4	yes	—
44	C₃₁×#431	C₂²	1	4	yes	—
45	C₃₁×#428	C₂²	1	8	yes	—
46	C₃₁×#423	C₂²	1	2	yes	—
47	C₃₁×#417	C₂²	1	8	yes	—
48	C₃₁×#407	C₂²	1	2	yes	—
49	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	—	2	2	no	—
51	C₂×#41×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	—	2	2	no	—
53	C₂³×C₄×C₃₁	D₄	1	2	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
56	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×#127	C₂³	1	1	yes	characteristic
59	C₃₁×#126	—	2	4	no	—
60	C₃₁×#125	—	2	2	no	—
61	C₃₁×#125	—	2	2	no	—
62	C₃₁×#125	—	2	4	no	—
63	C₃₁×#125	—	2	4	no	—
64	C₃₁×#125	—	2	4	no	—
65	C₃₁×#125	—	2	4	no	—
66	C₃₁×#125	—	2	4	no	—
67	C₃₁×#125	—	2	2	no	—
68	C₃₁×#125	C₂³	1	8	yes	—
69	C₃₁×#125	C₂³	1	4	yes	—
70	C₃₁×#125	C₂³	1	4	yes	—
71	C₃₁×#125	C₂³	1	2	yes	—
72	C₃₁×#124	—	2	2	no	—
73	C₃₁×#124	C₂³	1	2	yes	—
74	C₃₁×#123	—	2	4	no	—
75	C₃₁×#123	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
79	C₃₁×#121	—	2	2	no	—
80	C₃₁×#121	—	2	2	no	—
81	C₃₁×#121	C₂³	1	4	yes	—
82	C₃₁×#120	—	2	2	no	—
83	C₄×Q₈×C₃₁	—	2	2	no	—
84	C₄×Q₈×C₃₁	—	2	2	no	—
85	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
86	C₄×D₄×C₃₁	—	2	4	no	—
87	C₄×D₄×C₃₁	—	2	4	no	—
88	C₄×D₄×C₃₁	—	2	4	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	C₂³	1	4	yes	—
92	C₃₁×#117	—	2	2	no	—
93	C₃₁×#117	—	2	4	no	—
94	C₃₁×#117	—	2	2	no	—
95	C₃₁×#117	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
108	C₂×C₄²×C₃₁	—	2	2	no	—
109	C₂×C₄²×C₃₁	D₄	1	2	yes	—
110	C₂×C₄²×C₃₁	D₄	1	2	yes	—
111	C₂×C₄²×C₃₁	D₄	1	2	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
116	C₃₁×#107	C₂³	1	16	yes	—
117	C₃₁×#106	C₂³	1	16	yes	—
118	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
119	C₃₁×#103	C₂³	1	16	yes	—
120	C₃₁×#102	C₂³	1	16	yes	—
121	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
122	C₃₁×#97	C₂³	1	4	yes	—
123	C₄×C₈×C₃₁	C₂³	1	4	yes	—
124	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
125	#41×C₃₁	—	2	8	no	—
126	#41×C₃₁	—	4	8	no	—
127	C₂×Q₈×C₃₁	—	2	2	no	—
128	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	4	no	—
131	C₂×D₄×C₃₁	—	2	2	no	—
132	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	4	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	4	4	no	—
140	C₂²×C₄×C₃₁	—	4	8	no	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
151	M₁₆×C₃₁	—	2	8	no	—
152	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
153	C₄⋊C₄×C₃₁	—	2	8	no	—
154	C₄⋊C₄×C₃₁	—	2	8	no	—
155	C₄⋊C₄×C₃₁	—	2	8	no	—
156	C₄⋊C₄×C₃₁	—	2	8	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	4	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₄⋊C₄×C₃₁	—	2	4	no	—
163	C₄⋊C₄×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
166	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²⋊C₄×C₃₁	—	2	4	no	—
168	C₂²⋊C₄×C₃₁	—	2	2	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₄²×C₃₁	—	2	4	no	—
173	C₄²×C₃₁	—	2	2	no	—
174	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₃₁	C₂×D₄	1	2	yes	—
176	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂×D₄	1	2	yes	—
178	C₄²×C₃₁	C₂×D₄	1	2	yes	—
179	C₄²×C₃₁	C₂×D₄	1	2	yes	—
180	C₄²×C₃₁	C₂×D₄	1	2	yes	—
181	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
183	C₄²×C₃₁	C₂⁴	1	2	yes	—
184	#60861	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	4	no	—
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
189	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
190	Q₈×C₃₁	—	2	4	no	—
191	Q₈×C₃₁	—	4	4	no	—
192	D₄×C₃₁	—	2	4	no	—
193	D₄×C₃₁	—	4	4	no	—
194	D₄×C₃₁	—	4	4	no	—
195	D₄×C₃₁	—	4	8	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	8	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	8	no	—
209	C₂×C₄×C₃₁	—	4	8	no	—
210	C₂×C₄×C₃₁	—	4	8	no	—
211	C₂×C₄×C₃₁	—	4	8	no	—
212	C₂×C₄×C₃₁	—	4	4	no	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
220	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
221	C₈×C₃₁	—	2	8	no	—
222	#3444	C₂×C₃₁	1	1	yes	characteristic
223	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#3236	C₂×C₃₁	1	4	yes	complemented
225	#3234	C₂×C₃₁	1	4	yes	complemented
226	#3232	C₂×C₃₁	1	8	yes	complemented
227	#3166	C₂×C₃₁	1	2	yes	complemented
228	#3142	C₂×C₃₁	1	2	yes	complemented
229	#3092	C₂×C₃₁	1	4	yes	complemented
230	#3080	C₂×C₃₁	1	4	yes	complemented
231	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂²×C₃₁	—	2	8	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	2	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	#536	1	2	yes	—
239	C₂²×C₃₁	#575	1	4	yes	—
240	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	2	2	no	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	4	8	no	—
248	C₄×C₃₁	—	4	4	no	—
249	C₄×C₃₁	#536	1	2	yes	—
250	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
251	#533	C₂²×C₃₁	1	2	yes	—
252	#533	C₂²×C₃₁	1	1	yes	characteristic
253	#532	C₂²×C₃₁	1	1	yes	characteristic
254	#532	C₂²×C₃₁	1	2	yes	—
255	#529	—	2	2	no	—
256	#529	—	2	2	no	—
257	#529	C₂²×C₃₁	1	4	yes	—
258	C₂×#125	—	2	2	no	—
259	C₂×#125	—	2	2	no	—
260	C₂×#125	—	2	2	no	—
261	C₂×#125	C₂²×C₃₁	1	2	yes	—
262	C₂×#125	C₂²×C₃₁	1	2	yes	—
263	C₂×#125	C₂²×C₃₁	1	4	yes	—
264	C₂×#125	C₂²×C₃₁	1	4	yes	—
265	#525	—	2	2	no	—
266	#525	C₂²×C₃₁	1	2	yes	—
267	C₄×#41	—	2	2	no	—
268	C₄×#41	C₂²×C₃₁	1	2	yes	—
269	C₂²×C₄⋊C₄	—	2	2	no	—
270	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
271	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
272	#501	C₂²×C₃₁	1	8	yes	complemented
273	#499	C₂²×C₃₁	1	8	yes	complemented
274	#491	C₂²×C₃₁	1	4	yes	complemented
275	#490	C₂²×C₃₁	1	4	yes	complemented
276	#488	C₂²×C₃₁	1	8	yes	complemented
277	#484	C₂²×C₃₁	1	8	yes	—
278	#483	C₂²×C₃₁	1	8	yes	—
279	#481	C₂²×C₃₁	1	8	yes	—
280	#480	C₂²×C₃₁	1	8	yes	—
281	#479	C₂²×C₃₁	1	16	yes	complemented
282	#478	C₂²×C₃₁	1	16	yes	complemented
283	#432	C₂²×C₃₁	1	4	yes	—
284	#431	C₂²×C₃₁	1	4	yes	—
285	#428	C₂²×C₃₁	1	8	yes	—
286	#423	C₂²×C₃₁	1	2	yes	—
287	#417	C₂²×C₃₁	1	8	yes	—
288	#407	C₂²×C₃₁	1	2	yes	—
289	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
290	C₂×C₃₁	—	2	2	no	—
291	C₂×C₃₁	—	2	4	no	—
292	C₂×C₃₁	—	8	2	no	—
293	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
294	C₂×C₃₁	C₂×#575	1	2	yes	—
295	C₂×#41	—	2	2	no	—
296	C₂×#41	C₂³×C₃₁	1	2	yes	—
297	C₂³×C₄	—	2	2	no	—
298	C₂³×C₄	D₄×C₃₁	1	2	yes	—
299	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
301	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
303	#127	C₂³×C₃₁	1	1	yes	characteristic
304	#126	—	2	4	no	—
305	#125	—	2	2	no	—
306	#125	—	2	2	no	—
307	#125	—	2	4	no	—
308	#125	—	2	4	no	—
309	#125	—	2	4	no	—
310	#125	—	2	4	no	—
311	#125	—	2	4	no	—
312	#125	—	2	2	no	—
313	#125	C₂³×C₃₁	1	8	yes	—
314	#125	C₂³×C₃₁	1	4	yes	—
315	#125	C₂³×C₃₁	1	4	yes	—
316	#125	C₂³×C₃₁	1	2	yes	—
317	#124	—	2	2	no	—
318	#124	C₂³×C₃₁	1	2	yes	—
319	#123	—	2	4	no	—
320	#123	—	2	2	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
324	#121	—	2	2	no	—
325	#121	—	2	2	no	—
326	#121	C₂³×C₃₁	1	4	yes	—
327	#120	—	2	2	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
331	C₄×D₄	—	2	4	no	—
332	C₄×D₄	—	2	4	no	—
333	C₄×D₄	—	2	4	no	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	C₂³×C₃₁	1	4	yes	—
337	#117	—	2	2	no	—
338	#117	—	2	4	no	—
339	#117	—	2	2	no	—
340	#117	C₂³×C₃₁	1	2	yes	—
341	C₂×C₄⋊C₄	—	2	4	no	—
342	C₂×C₄⋊C₄	—	2	4	no	—
343	C₂×C₄⋊C₄	—	2	4	no	—
344	C₂×C₄⋊C₄	—	2	4	no	—
345	C₂×C₄⋊C₄	—	2	4	no	—
346	C₂×C₄⋊C₄	—	2	4	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
351	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂×C₄²	—	2	2	no	—
354	C₂×C₄²	D₄×C₃₁	1	2	yes	—
355	C₂×C₄²	D₄×C₃₁	1	2	yes	—
356	C₂×C₄²	D₄×C₃₁	1	2	yes	—
357	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
358	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
359	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
361	#107	C₂³×C₃₁	1	16	yes	—
362	#106	C₂³×C₃₁	1	16	yes	—
363	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
364	#103	C₂³×C₃₁	1	16	yes	—
365	#102	C₂³×C₃₁	1	16	yes	—
366	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
367	#97	C₂³×C₃₁	1	4	yes	—
368	C₄×C₈	C₂³×C₃₁	1	4	yes	—
369	C₃₁	#60861	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
371	#41	—	2	8	no	—
372	#41	—	4	8	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
391	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
393	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
395	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
397	M₁₆	—	2	8	no	—
398	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
399	C₄⋊C₄	—	2	8	no	—
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
412	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	4	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₄²	C₂×D₄×C₃₁	1	2	yes	—
422	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₄²	C₂×D₄×C₃₁	1	2	yes	—
424	C₄²	C₂×D₄×C₃₁	1	2	yes	—
425	C₄²	C₂×D₄×C₃₁	1	2	yes	—
426	C₄²	C₂×D₄×C₃₁	1	2	yes	—
427	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
428	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
429	C₄²	C₂⁴×C₃₁	1	2	yes	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	2	2	no	—
433	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
434	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
435	Q₈	—	2	4	no	—
436	Q₈	—	4	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	4	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	8	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	8	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	—	4	8	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
465	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
466	C₈	—	2	8	no	—
467	C₂²	—	2	8	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	4	no	—
473	C₂²	C₃₁×#536	1	2	yes	—
474	C₂²	C₃₁×#575	1	4	yes	—
475	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	8	no	—
483	C₄	—	4	4	no	—
484	C₄	C₃₁×#536	1	2	yes	—
485	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
486	C₂	—	2	2	no	—
487	C₂	—	2	4	no	—
488	C₂	—	8	2	no	—
489	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
490	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×#3444	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
2. C₂²×C₃₁×#125	C₂	semidirect
3. C₃₁×#3236	C₂	semidirect
4. C₃₁×#3234	C₂	semidirect
5. C₃₁×#3232	C₂	semidirect
6. C₃₁×#3166	C₂	semidirect
7. C₃₁×#3142	C₂	semidirect
8. C₃₁×#3092	C₂	semidirect
9. C₃₁×#3080	C₂	semidirect
10. C₃₁×#2920	C₂	semidirect
32. C₃₁×#501	C₂²	semidirect
33. C₃₁×#499	C₂²	semidirect
34. C₃₁×#491	C₂²	semidirect
35. C₃₁×#490	C₂²	semidirect
36. C₃₁×#488	C₂²	semidirect
41. C₃₁×#479	C₂²	semidirect
42. C₃₁×#478	C₂²	semidirect
184. #60861	C₃₁	direct (N × H)
223. C₂²×#125	C₂×C₃₁	semidirect
224. #3236	C₂×C₃₁	semidirect
225. #3234	C₂×C₃₁	semidirect
226. #3232	C₂×C₃₁	semidirect
227. #3166	C₂×C₃₁	semidirect
228. #3142	C₂×C₃₁	semidirect
229. #3092	C₂×C₃₁	semidirect
230. #3080	C₂×C₃₁	semidirect
231. #2920	C₂×C₃₁	semidirect
272. #501	C₂²×C₃₁	semidirect
273. #499	C₂²×C₃₁	semidirect
274. #491	C₂²×C₃₁	semidirect
275. #490	C₂²×C₃₁	semidirect
276. #488	C₂²×C₃₁	semidirect
281. #479	C₂²×C₃₁	semidirect
282. #478	C₂²×C₃₁	semidirect
369. C₃₁	#60861	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×#3444	2
C₂²×C₃₁×#125	2
C₃₁×#3236	2
C₃₁×#3234	2
C₃₁×#3232	2
C₃₁×#3166	2
C₃₁×#3142	2
C₃₁×#3092	2
C₃₁×#3080	2
C₃₁×#2920	2
#60861	31

Subgroups — C31×#60861 #21517620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60861 #21517620