Subgroups — C₃₁×#24647 #21481406

Order 7936.

932 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	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2054	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2012	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1983	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#459	—	2	4	no	—
19	C₃₁×#440	—	2	4	no	—
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₃₁×#394	—	2	1	no	—
31	C₃₁×#394	—	2	1	no	—
32	C₃₁×#390	—	2	2	no	—
33	C₃₁×#390	C₂²	1	2	yes	—
34	C₃₁×#390	C₂²	1	2	yes	—
35	C₃₁×#388	—	2	2	no	—
36	C₃₁×#382	C₂²	1	1	yes	characteristic
37	C₃₁×#381	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	4	no	—
53	C₂×D₈×C₃₁	—	4	2	no	—
54	C₂×D₈×C₃₁	—	4	2	no	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#127	—	2	2	no	—
59	C₃₁×#127	—	2	2	no	—
60	C₃₁×#124	—	2	4	no	—
61	C₃₁×#124	—	4	2	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	D₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	—	2	1	no	—
79	C₂×C₄²×C₃₁	—	2	2	no	—
80	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#107	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×C₄⋊C₈	—	4	2	no	—
85	C₃₁×#103	—	2	2	no	—
86	C₃₁×#103	—	4	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	—	2	4	no	—
89	C₃₁×#102	—	2	4	no	—
90	C₃₁×#102	—	4	2	no	—
91	C₃₁×#97	—	2	2	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	D₄	1	1	yes	characteristic
96	C₃₁×#95	D₄	1	1	yes	characteristic
97	C₂⁴×C₃₁	—	2	1	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	1	no	—
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	#41	1	2	yes	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	4	4	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	8	2	no	—
120	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	8	2	no	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	SD₁₆×C₃₁	—	2	8	no	—
133	D₈×C₃₁	—	4	8	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	#41	1	2	yes	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	4	2	no	—
149	C₄⋊C₄×C₃₁	—	4	2	no	—
150	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₃₁	—	2	4	no	—
152	C₂²⋊C₄×C₃₁	—	4	4	no	—
153	C₂²⋊C₄×C₃₁	—	4	4	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₂²⋊C₄×C₃₁	—	8	2	no	—
156	C₂²⋊C₄×C₃₁	—	8	2	no	—
157	C₂²⋊C₄×C₃₁	—	8	2	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	4	2	no	—
161	C₄²×C₃₁	—	8	2	no	—
162	C₄²×C₃₁	C₂×D₄	1	2	yes	—
163	C₄²×C₃₁	#41	1	2	yes	—
164	#24647	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₃₁	—	2	2	no	—
179	Q₈×C₃₁	—	2	2	no	—
180	Q₈×C₃₁	—	4	1	no	—
181	Q₈×C₃₁	—	4	1	no	—
182	D₄×C₃₁	—	2	2	no	—
183	D₄×C₃₁	—	2	2	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	4	1	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	8	4	no	—
191	D₄×C₃₁	—	8	4	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	8	4	no	—
211	C₂×C₄×C₃₁	#121	1	2	yes	—
212	C₂×C₄×C₃₁	#121	1	2	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	4	1	no	—
221	C₈×C₃₁	—	4	2	no	—
222	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
223	C₂×#440	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2548	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#2357	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#2054	C₂×C₃₁	1	2	yes	complemented
227	#2012	C₂×C₃₁	1	2	yes	complemented
228	#1983	C₂×C₃₁	1	2	yes	complemented
229	#1961	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1925	C₂×C₃₁	1	2	yes	complemented
232	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	8	2	no	—
237	C₂²×C₃₁	—	8	2	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	—	8	2	no	—
240	C₂²×C₃₁	—	8	2	no	—
241	C₂²×C₃₁	—	8	2	no	—
242	C₂²×C₃₁	—	8	2	no	—
243	C₂²×C₃₁	—	8	2	no	—
244	C₂²×C₃₁	—	8	2	no	—
245	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
246	C₂²×C₃₁	#546	1	1	yes	characteristic
247	C₂²×C₃₁	#550	1	1	yes	characteristic
248	C₂²×C₃₁	#575	1	2	yes	—
249	C₂²×C₃₁	#576	1	2	yes	—
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	2	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	8	2	no	—
257	C₄×C₃₁	—	8	1	no	—
258	C₄×C₃₁	#525	1	1	yes	characteristic
259	C₄×C₃₁	#535	1	1	yes	characteristic
260	C₄×C₃₁	#553	1	1	yes	characteristic
261	C₄×C₃₁	#555	1	1	yes	characteristic
262	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
263	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
264	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#124	C₂²×C₃₁	1	2	yes	—
266	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
268	#459	—	2	4	no	—
269	#440	—	2	4	no	—
270	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
273	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#103	C₂²×C₃₁	1	2	yes	—
275	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×#102	C₂²×C₃₁	1	2	yes	—
277	C₂×#102	C₂²×C₃₁	1	2	yes	—
278	C₂×#102	C₂²×C₃₁	1	2	yes	—
279	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
280	#394	—	2	1	no	—
281	#394	—	2	1	no	—
282	#390	—	2	2	no	—
283	#390	C₂²×C₃₁	1	2	yes	—
284	#390	C₂²×C₃₁	1	2	yes	—
285	#388	—	2	2	no	—
286	#382	C₂²×C₃₁	1	1	yes	characteristic
287	#381	—	2	1	no	—
288	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
289	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₄×C₂²⋊C₄	—	2	1	no	—
291	#340	C₂²×C₃₁	1	2	yes	—
292	#336	C₂²×C₃₁	1	2	yes	—
293	C₂×C₃₁	—	8	2	no	—
294	C₂×C₃₁	—	16	2	no	—
295	C₂×C₃₁	#2414	1	1	yes	characteristic
296	C₂×C₃₁	#3040	1	1	yes	characteristic
297	C₂×C₃₁	#3041	1	1	yes	characteristic
298	C₂×C₃₁	#3309	1	2	yes	—
299	C₂×C₃₁	#3403	1	2	yes	—
300	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
304	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂³×C₄	—	2	1	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	2	no	—
308	C₂×SD₁₆	—	2	2	no	—
309	C₂×D₈	—	2	4	no	—
310	C₂×D₈	—	4	2	no	—
311	C₂×D₈	—	4	2	no	—
312	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
313	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	#127	—	2	2	no	—
316	#127	—	2	2	no	—
317	#124	—	2	4	no	—
318	#124	—	4	2	no	—
319	C₄×Q₈	—	2	2	no	—
320	C₄×Q₈	—	2	2	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	D₄×C₃₁	1	4	yes	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	2	1	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₄²	—	2	1	no	—
336	C₂×C₄²	—	2	2	no	—
337	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
338	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
339	#107	—	2	2	no	—
340	C₄⋊C₈	—	2	2	no	—
341	C₄⋊C₈	—	4	2	no	—
342	#103	—	2	2	no	—
343	#103	—	4	2	no	—
344	#102	—	2	2	no	—
345	#102	—	2	4	no	—
346	#102	—	2	4	no	—
347	#102	—	4	2	no	—
348	#97	—	2	2	no	—
349	#95	—	2	1	no	—
350	#95	—	2	2	no	—
351	#95	—	2	1	no	—
352	#95	D₄×C₃₁	1	1	yes	characteristic
353	#95	D₄×C₃₁	1	1	yes	characteristic
354	C₃₁	#24647	1	1	yes	characteristic, direct product factor, Sylow
355	C₂⁴	—	2	1	no	—
356	C₂⁴	—	2	2	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	#41×C₃₁	1	2	yes	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	—	4	2	no	—
376	C₂×D₄	—	4	2	no	—
377	C₂×D₄	—	8	2	no	—
378	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	8	2	no	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
390	SD₁₆	—	2	8	no	—
391	D₈	—	4	8	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	#41×C₃₁	1	2	yes	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	4	2	no	—
407	C₄⋊C₄	—	4	2	no	—
408	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
409	C₂²⋊C₄	—	2	4	no	—
410	C₂²⋊C₄	—	4	4	no	—
411	C₂²⋊C₄	—	4	4	no	—
412	C₂²⋊C₄	—	4	4	no	—
413	C₂²⋊C₄	—	8	2	no	—
414	C₂²⋊C₄	—	8	2	no	—
415	C₂²⋊C₄	—	8	2	no	—
416	C₄²	—	2	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	4	2	no	—
419	C₄²	—	8	2	no	—
420	C₄²	C₂×D₄×C₃₁	1	2	yes	—
421	C₄²	#41×C₃₁	1	2	yes	—
422	C₂³	—	2	2	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	2	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
435	Q₈	—	2	2	no	—
436	Q₈	—	2	2	no	—
437	Q₈	—	4	1	no	—
438	Q₈	—	4	1	no	—
439	D₄	—	2	2	no	—
440	D₄	—	2	2	no	—
441	D₄	—	4	2	no	—
442	D₄	—	4	2	no	—
443	D₄	—	4	1	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	1	no	—
446	D₄	—	4	2	no	—
447	D₄	—	8	4	no	—
448	D₄	—	8	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	8	4	no	—
468	C₂×C₄	C₃₁×#121	1	2	yes	—
469	C₂×C₄	C₃₁×#121	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
474	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
475	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
476	C₈	—	2	2	no	—
477	C₈	—	4	1	no	—
478	C₈	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	8	2	no	—
483	C₂²	—	8	2	no	—
484	C₂²	—	8	2	no	—
485	C₂²	—	8	2	no	—
486	C₂²	—	8	2	no	—
487	C₂²	—	8	2	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	2	no	—
490	C₂²	—	8	2	no	—
491	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
492	C₂²	C₃₁×#546	1	1	yes	characteristic
493	C₂²	C₃₁×#550	1	1	yes	characteristic
494	C₂²	C₃₁×#575	1	2	yes	—
495	C₂²	C₃₁×#576	1	2	yes	—
496	C₄	—	2	2	no	—
497	C₄	—	2	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	8	2	no	—
503	C₄	—	8	1	no	—
504	C₄	C₃₁×#525	1	1	yes	characteristic
505	C₄	C₃₁×#535	1	1	yes	characteristic
506	C₄	C₃₁×#553	1	1	yes	characteristic
507	C₄	C₃₁×#555	1	1	yes	characteristic
508	C₂	—	8	2	no	—
509	C₂	—	16	2	no	—
510	C₂	C₃₁×#2414	1	1	yes	characteristic
511	C₂	C₃₁×#3040	1	1	yes	characteristic
512	C₂	C₃₁×#3041	1	1	yes	characteristic
513	C₂	C₃₁×#3309	1	2	yes	—
514	C₂	C₃₁×#3403	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#459	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#440	C₂	semidirect
3. C₃₁×#2548	C₂	semidirect
4. C₃₁×#2357	C₂	semidirect
5. C₃₁×#2054	C₂	semidirect
6. C₃₁×#2012	C₂	semidirect
7. C₃₁×#1983	C₂	semidirect
8. C₃₁×#1961	C₂	semidirect
9. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1925	C₂	semidirect
11. C₃₁×#1805	C₂	semidirect
164. #24647	C₃₁	direct (N × H)
223. C₂×#440	C₂×C₃₁	semidirect
224. #2548	C₂×C₃₁	semidirect
225. #2357	C₂×C₃₁	semidirect
226. #2054	C₂×C₃₁	semidirect
227. #2012	C₂×C₃₁	semidirect
228. #1983	C₂×C₃₁	semidirect
229. #1961	C₂×C₃₁	semidirect
230. #1940	C₂×C₃₁	semidirect
231. #1925	C₂×C₃₁	semidirect
232. #1805	C₂×C₃₁	semidirect
354. C₃₁	#24647	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×#459	2
C₂×C₃₁×#440	2
C₃₁×#2548	2
C₃₁×#2357	2
C₃₁×#2054	2
C₃₁×#2012	2
C₃₁×#1983	2
C₃₁×#1961	2
C₃₁×#1940	2
C₃₁×#1925	2
C₃₁×#1805	2
#24647	31

Subgroups — C31×#24647 #21481406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24647 #21481406