Subgroups — C₃₁×#21785 #21478544

Order 7936.

1262 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3500	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2889	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2884	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1912	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1910	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1888	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1884	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1827	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1795	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
12	C₃₁×#568	C₂²	1	2	yes	complemented
13	C₃₁×#548	—	2	2	no	—
14	C₃₁×#546	—	2	2	no	—
15	C₃₁×#546	—	2	4	no	—
16	C₃₁×#536	C₄	1	2	yes	complemented
17	C₃₁×#536	C₂²	1	1	yes	characteristic
18	C₃₁×#535	C₄	1	2	yes	complemented
19	C₃₁×#535	C₂²	1	1	yes	characteristic
20	C₃₁×#532	C₄	1	2	yes	complemented
21	C₃₁×#532	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#124	C₄	1	2	yes	complemented
23	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
24	C₃₁×#519	C₂²	1	2	yes	complemented
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#421	C₂²	1	4	yes	—
27	C₃₁×#418	C₂²	1	4	yes	—
28	C₃₁×#413	—	2	2	no	—
29	C₃₁×#413	C₂²	1	2	yes	—
30	C₂×C₃₁×#101	—	2	2	no	—
31	C₂×C₃₁×#101	C₂²	1	2	yes	complemented
32	C₃₁×#410	—	2	4	no	—
33	C₃₁×#410	C₂²	1	2	yes	—
34	C₃₁×#410	C₂²	1	2	yes	complemented
35	C₃₁×#339	C₂²	1	4	yes	complemented
36	C₃₁×H(3,ℤ/4)	C₂²	1	4	yes	complemented
37	C₃₁×2₋¹⁺⁴	—	4	4	no	—
38	C₃₁×H(5,𝔽₂)	—	4	4	no	—
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×#41×C₃₁	D₄	1	2	yes	—
43	C₂×#41×C₃₁	D₄	1	4	yes	—
44	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
48	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
49	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	—	2	2	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#131	—	4	4	no	—
55	C₂×M₁₆×C₃₁	—	2	4	no	—
56	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
57	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
58	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
59	C₃₁×#127	C₂×C₄	1	2	yes	—
60	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
61	C₃₁×#124	—	2	4	no	—
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
64	C₃₁×#124	C₂×C₄	1	2	yes	—
65	C₃₁×#124	C₂×C₄	1	2	yes	—
66	C₃₁×#124	C₂×C₄	1	2	yes	—
67	C₃₁×#124	C₂×C₄	1	2	yes	—
68	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
69	C₃₁×#123	—	2	4	no	—
70	C₃₁×#123	—	2	2	no	—
71	C₃₁×C₂²⋊Q₈	—	2	4	no	—
72	C₃₁×C₂²⋊Q₈	—	2	2	no	—
73	C₃₁×#121	—	2	4	no	—
74	C₃₁×#121	—	2	2	no	—
75	C₃₁×#121	—	2	4	no	—
76	C₃₁×#121	—	2	2	no	—
77	C₃₁×#120	—	2	4	no	—
78	C₃₁×#120	—	2	2	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	—	4	4	no	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	—	2	4	no	—
84	C₄×D₄×C₃₁	—	2	2	no	—
85	C₃₁×#117	—	2	4	no	—
86	C₃₁×#117	C₂×C₄	1	2	yes	—
87	C₃₁×#117	C₂³	1	4	yes	—
88	C₃₁×#117	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#104	—	4	4	no	—
94	C₃₁×#102	—	4	4	no	—
95	C₃₁×#101	—	2	2	no	—
96	C₃₁×#101	—	2	4	no	—
97	C₃₁×#100	—	2	2	no	—
98	C₃₁×#99	—	2	4	no	—
99	C₃₁×#99	—	2	4	no	—
100	C₂⁴×C₃₁	—	2	2	no	—
101	C₂⁴×C₃₁	—	2	1	no	—
102	#41×C₃₁	—	2	4	no	—
103	#41×C₃₁	—	2	4	no	—
104	#41×C₃₁	—	2	4	no	—
105	#41×C₃₁	—	4	8	no	—
106	#41×C₃₁	—	4	4	no	—
107	#41×C₃₁	—	4	4	no	—
108	#41×C₃₁	—	4	4	no	—
109	#41×C₃₁	—	4	4	no	—
110	C₂×Q₈×C₃₁	—	2	4	no	—
111	C₂×Q₈×C₃₁	—	4	4	no	—
112	C₂×Q₈×C₃₁	—	4	4	no	—
113	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
114	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	—	2	4	no	—
126	C₂×D₄×C₃₁	—	4	4	no	—
127	C₂×D₄×C₃₁	—	4	8	no	—
128	C₂×D₄×C₃₁	—	4	4	no	—
129	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
130	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
132	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
141	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
142	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	M₁₆×C₃₁	—	2	4	no	—
147	M₁₆×C₃₁	—	2	4	no	—
148	M₁₆×C₃₁	—	4	4	no	—
149	C₂×C₈×C₃₁	—	2	4	no	—
150	C₂×C₈×C₃₁	—	4	4	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	4	4	no	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	2	2	no	—
159	C₂²⋊C₄×C₃₁	—	2	1	no	—
160	C₂²⋊C₄×C₃₁	—	2	4	no	—
161	C₂²⋊C₄×C₃₁	—	2	2	no	—
162	C₂²⋊C₄×C₃₁	—	2	4	no	—
163	C₂²⋊C₄×C₃₁	—	2	4	no	—
164	C₂²⋊C₄×C₃₁	—	2	4	no	—
165	C₂²⋊C₄×C₃₁	—	2	4	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	1	no	—
168	C₄²×C₃₁	—	2	4	no	—
169	C₄²×C₃₁	—	4	4	no	—
170	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
172	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
173	#21785	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	4	4	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	8	2	no	—
179	C₂³×C₃₁	—	8	4	no	—
180	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₃₁	—	2	4	no	—
184	Q₈×C₃₁	—	2	2	no	—
185	Q₈×C₃₁	—	2	4	no	—
186	Q₈×C₃₁	—	2	2	no	—
187	Q₈×C₃₁	—	2	2	no	—
188	Q₈×C₃₁	—	4	4	no	—
189	D₄×C₃₁	—	2	4	no	—
190	D₄×C₃₁	—	2	4	no	—
191	D₄×C₃₁	—	2	2	no	—
192	D₄×C₃₁	—	2	4	no	—
193	D₄×C₃₁	—	2	2	no	—
194	D₄×C₃₁	—	2	2	no	—
195	D₄×C₃₁	—	4	4	no	—
196	D₄×C₃₁	—	4	4	no	—
197	D₄×C₃₁	—	4	4	no	—
198	D₄×C₃₁	—	4	2	no	—
199	D₄×C₃₁	—	4	2	no	—
200	D₄×C₃₁	—	8	4	no	—
201	D₄×C₃₁	—	8	2	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	4	4	no	—
210	C₂×C₄×C₃₁	—	4	4	no	—
211	C₂×C₄×C₃₁	—	4	4	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	4	no	—
214	C₂×C₄×C₃₁	—	8	1	no	—
215	C₂×C₄×C₃₁	—	8	2	no	—
216	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
218	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
219	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
222	C₈×C₃₁	—	4	4	no	—
223	#3500	C₂×C₃₁	1	1	yes	characteristic
224	#2889	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#2884	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1912	C₂×C₃₁	1	2	yes	complemented
227	#1910	C₂×C₃₁	1	2	yes	complemented
228	#1888	C₂×C₃₁	1	2	yes	complemented
229	#1884	C₂×C₃₁	1	2	yes	complemented
230	#1827	C₂×C₃₁	1	2	yes	complemented
231	#1795	C₂×C₃₁	1	2	yes	complemented
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	4	1	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	1	no	—
241	C₂²×C₃₁	—	16	2	no	—
242	C₂²×C₃₁	—	16	4	no	—
243	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
244	C₂²×C₃₁	#534	1	2	yes	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	2	4	no	—
248	C₄×C₃₁	—	4	1	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	4	1	no	—
251	C₄×C₃₁	—	4	1	no	—
252	C₄×C₃₁	—	8	4	no	—
253	C₄×C₃₁	#518	1	2	yes	—
254	C₄×C₃₁	#534	1	2	yes	—
255	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
257	#568	C₂²×C₃₁	1	2	yes	complemented
258	#548	—	2	2	no	—
259	#546	—	2	2	no	—
260	#546	—	2	4	no	—
261	#536	C₄×C₃₁	1	2	yes	complemented
262	#536	C₂²×C₃₁	1	1	yes	characteristic
263	#535	C₄×C₃₁	1	2	yes	complemented
264	#535	C₂²×C₃₁	1	1	yes	characteristic
265	#532	C₄×C₃₁	1	2	yes	complemented
266	#532	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#124	C₄×C₃₁	1	2	yes	complemented
268	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
269	#519	C₂²×C₃₁	1	2	yes	complemented
270	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
271	#421	C₂²×C₃₁	1	4	yes	—
272	#418	C₂²×C₃₁	1	4	yes	—
273	#413	—	2	2	no	—
274	#413	C₂²×C₃₁	1	2	yes	—
275	C₂×#101	—	2	2	no	—
276	C₂×#101	C₂²×C₃₁	1	2	yes	complemented
277	#410	—	2	4	no	—
278	#410	C₂²×C₃₁	1	2	yes	—
279	#410	C₂²×C₃₁	1	2	yes	complemented
280	#339	C₂²×C₃₁	1	4	yes	complemented
281	H(3,ℤ/4)	C₂²×C₃₁	1	4	yes	complemented
282	C₂×C₃₁	—	2	1	no	—
283	C₂×C₃₁	—	2	2	no	—
284	C₂×C₃₁	—	4	2	no	—
285	C₂×C₃₁	—	4	2	no	—
286	C₂×C₃₁	—	8	2	no	—
287	C₂×C₃₁	—	8	1	no	—
288	C₂×C₃₁	#2308	1	1	yes	characteristic
289	2₋¹⁺⁴	—	4	4	no	—
290	H(5,𝔽₂)	—	4	4	no	—
291	C₂×#41	—	2	1	no	—
292	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
293	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
294	C₂×#41	D₄×C₃₁	1	2	yes	—
295	C₂×#41	D₄×C₃₁	1	4	yes	—
296	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
298	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
299	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
300	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
301	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₂²×D₄	—	2	2	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
305	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
306	#131	—	4	4	no	—
307	C₂×M₁₆	—	2	4	no	—
308	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
309	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
310	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
311	#127	C₂×C₄×C₃₁	1	2	yes	—
312	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
313	#124	—	2	4	no	—
314	#124	—	2	2	no	—
315	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
316	#124	C₂×C₄×C₃₁	1	2	yes	—
317	#124	C₂×C₄×C₃₁	1	2	yes	—
318	#124	C₂×C₄×C₃₁	1	2	yes	—
319	#124	C₂×C₄×C₃₁	1	2	yes	—
320	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
321	#123	—	2	4	no	—
322	#123	—	2	2	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	2	2	no	—
325	#121	—	2	4	no	—
326	#121	—	2	2	no	—
327	#121	—	2	4	no	—
328	#121	—	2	2	no	—
329	#120	—	2	4	no	—
330	#120	—	2	2	no	—
331	C₄×Q₈	—	2	1	no	—
332	C₄×Q₈	—	4	4	no	—
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	—	2	2	no	—
337	#117	—	2	4	no	—
338	#117	C₂×C₄×C₃₁	1	2	yes	—
339	#117	C₂³×C₃₁	1	4	yes	—
340	#117	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×C₂²⋊C₄	—	2	4	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
344	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
345	#104	—	4	4	no	—
346	#102	—	4	4	no	—
347	#101	—	2	2	no	—
348	#101	—	2	4	no	—
349	#100	—	2	2	no	—
350	#99	—	2	4	no	—
351	#99	—	2	4	no	—
352	C₃₁	#21785	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁴	—	2	2	no	—
354	C₂⁴	—	2	1	no	—
355	#41	—	2	4	no	—
356	#41	—	2	4	no	—
357	#41	—	2	4	no	—
358	#41	—	4	8	no	—
359	#41	—	4	4	no	—
360	#41	—	4	4	no	—
361	#41	—	4	4	no	—
362	#41	—	4	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	4	4	no	—
365	C₂×Q₈	—	4	4	no	—
366	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
367	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
368	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
369	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
370	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	4	4	no	—
380	C₂×D₄	—	4	8	no	—
381	C₂×D₄	—	4	4	no	—
382	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
383	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
384	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
385	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
386	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
394	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
395	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
399	M₁₆	—	2	4	no	—
400	M₁₆	—	2	4	no	—
401	M₁₆	—	4	4	no	—
402	C₂×C₈	—	2	4	no	—
403	C₂×C₈	—	4	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	4	4	no	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	1	no	—
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	C₂²⋊C₄	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	4	no	—
422	C₄²	—	4	4	no	—
423	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
424	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
425	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
426	C₂³	—	2	2	no	—
427	C₂³	—	2	1	no	—
428	C₂³	—	4	4	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	8	2	no	—
431	C₂³	—	8	4	no	—
432	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
433	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
434	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
435	Q₈	—	2	4	no	—
436	Q₈	—	2	2	no	—
437	Q₈	—	2	4	no	—
438	Q₈	—	2	2	no	—
439	Q₈	—	2	2	no	—
440	Q₈	—	4	4	no	—
441	D₄	—	2	4	no	—
442	D₄	—	2	4	no	—
443	D₄	—	2	2	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	4	4	no	—
448	D₄	—	4	4	no	—
449	D₄	—	4	4	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	2	no	—
452	D₄	—	8	4	no	—
453	D₄	—	8	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	8	1	no	—
467	C₂×C₄	—	8	2	no	—
468	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
469	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
470	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
471	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
474	C₈	—	4	4	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	4	1	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	16	2	no	—
485	C₂²	—	16	4	no	—
486	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
487	C₂²	C₃₁×#534	1	2	yes	—
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	4	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	2	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	1	no	—
495	C₄	—	8	4	no	—
496	C₄	C₃₁×#518	1	2	yes	—
497	C₄	C₃₁×#534	1	2	yes	—
498	C₂	—	2	1	no	—
499	C₂	—	2	2	no	—
500	C₂	—	4	2	no	—
501	C₂	—	4	2	no	—
502	C₂	—	8	2	no	—
503	C₂	—	8	1	no	—
504	C₂	C₃₁×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2308
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2889	C₂	semidirect
3. C₃₁×#2884	C₂	semidirect
4. C₃₁×#1912	C₂	semidirect
5. C₃₁×#1910	C₂	semidirect
6. C₃₁×#1888	C₂	semidirect
7. C₃₁×#1884	C₂	semidirect
8. C₃₁×#1827	C₂	semidirect
9. C₃₁×#1795	C₂	semidirect
12. C₃₁×#568	C₂²	semidirect
16. C₃₁×#536	C₄	semidirect
18. C₃₁×#535	C₄	semidirect
20. C₃₁×#532	C₄	semidirect
22. C₂×C₃₁×#124	C₄	semidirect
24. C₃₁×#519	C₂²	semidirect
31. C₂×C₃₁×#101	C₂²	semidirect
34. C₃₁×#410	C₂²	semidirect
35. C₃₁×#339	C₂²	semidirect
36. C₃₁×H(3,ℤ/4)	C₂²	semidirect
173. #21785	C₃₁	direct (N × H)
224. #2889	C₂×C₃₁	semidirect
225. #2884	C₂×C₃₁	semidirect
226. #1912	C₂×C₃₁	semidirect
227. #1910	C₂×C₃₁	semidirect
228. #1888	C₂×C₃₁	semidirect
229. #1884	C₂×C₃₁	semidirect
230. #1827	C₂×C₃₁	semidirect
231. #1795	C₂×C₃₁	semidirect
257. #568	C₂²×C₃₁	semidirect
261. #536	C₄×C₃₁	semidirect
263. #535	C₄×C₃₁	semidirect
265. #532	C₄×C₃₁	semidirect
267. C₂×#124	C₄×C₃₁	semidirect
269. #519	C₂²×C₃₁	semidirect
276. C₂×#101	C₂²×C₃₁	semidirect
279. #410	C₂²×C₃₁	semidirect
280. #339	C₂²×C₃₁	semidirect
281. H(3,ℤ/4)	C₂²×C₃₁	semidirect
352. C₃₁	#21785	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×#3500	2
C₃₁×#2889	2
C₃₁×#2884	2
C₃₁×#1912	2
C₃₁×#1910	2
C₃₁×#1888	2
C₃₁×#1884	2
C₃₁×#1827	2
C₃₁×#1795	2
#21785	31

Subgroups — C31×#21785 #21478544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21785 #21478544