Subgroups — C₃₁×#24646 #21481405

Order 7936.

864 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2546	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2357	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2055	C₂	1	2	yes	maximal
6	C₃₁×#2013	C₂	1	2	yes	maximal
7	C₃₁×#1981	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1960	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1928	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×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	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×#461	—	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×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₃₁×#398	—	2	1	no	—
31	C₃₁×#398	—	2	1	no	—
32	C₃₁×#391	C₂²	1	2	yes	—
33	C₃₁×#390	C₂²	1	2	yes	—
34	C₃₁×#389	—	2	2	no	—
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	2	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	4	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	4	2	no	—
52	C₂×SD₁₆×C₃₁	—	4	2	no	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	—	2	4	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	4	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₄×D₄×C₃₁	D₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄²×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#107	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	4	2	no	—
83	C₃₁×#103	—	2	4	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	—	2	4	no	—
86	C₃₁×#103	—	4	2	no	—
87	C₃₁×#103	—	4	2	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#97	—	2	2	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₃₁×#95	—	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	C₂×Q₈×C₃₁	—	2	1	no	—
99	C₂×Q₈×C₃₁	—	2	1	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	#41	1	2	yes	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	8	2	no	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
128	SD₁₆×C₃₁	—	2	8	no	—
129	SD₁₆×C₃₁	—	4	8	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	#41	1	2	yes	—
139	C₄⋊C₄×C₃₁	—	2	4	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	4	4	no	—
146	C₄⋊C₄×C₃₁	—	4	4	no	—
147	C₄⋊C₄×C₃₁	—	4	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	—	8	2	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	2	2	no	—
156	C₄²×C₃₁	—	4	2	no	—
157	C₄²×C₃₁	—	8	2	no	—
158	C₄²×C₃₁	C₂×D₄	1	2	yes	—
159	C₄²×C₃₁	#41	1	2	yes	—
160	#24646	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	4	1	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
167	Q₈×C₃₁	—	2	2	no	—
168	Q₈×C₃₁	—	2	2	no	—
169	Q₈×C₃₁	—	4	1	no	—
170	Q₈×C₃₁	—	4	1	no	—
171	Q₈×C₃₁	—	4	2	no	—
172	Q₈×C₃₁	—	4	2	no	—
173	Q₈×C₃₁	—	4	2	no	—
174	Q₈×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	2	2	no	—
177	D₄×C₃₁	—	4	1	no	—
178	D₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	8	4	no	—
200	C₂×C₄×C₃₁	#121	1	2	yes	—
201	C₂×C₄×C₃₁	#121	1	2	yes	—
202	C₂×C₄×C₃₁	#137	1	4	yes	—
203	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
209	C₈×C₃₁	—	2	2	no	—
210	C₈×C₃₁	—	4	1	no	—
211	C₈×C₃₁	—	4	2	no	—
212	C₂×#461	C₂×C₃₁	1	1	yes	characteristic
213	C₂×#440	C₂×C₃₁	1	1	yes	characteristic
214	#2546	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#2357	C₂×C₃₁	1	1	yes	characteristic
216	#2055	C₂×C₃₁	1	2	yes	—
217	#2013	C₂×C₃₁	1	2	yes	—
218	#1981	C₂×C₃₁	1	2	yes	complemented
219	#1960	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1928	C₂×C₃₁	1	2	yes	complemented
222	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	8	2	no	—
227	C₂²×C₃₁	—	8	2	no	—
228	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
229	C₂²×C₃₁	#546	1	1	yes	characteristic
230	C₂²×C₃₁	#550	1	1	yes	characteristic
231	C₂²×C₃₁	C₂×#137	1	2	yes	—
232	C₂²×C₃₁	#577	1	2	yes	—
233	C₄×C₃₁	—	2	2	no	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	2	4	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	8	2	no	—
240	C₄×C₃₁	—	8	2	no	—
241	C₄×C₃₁	—	8	1	no	—
242	C₄×C₃₁	#525	1	1	yes	characteristic
243	C₄×C₃₁	#535	1	1	yes	characteristic
244	C₄×C₃₁	#553	1	1	yes	characteristic
245	C₄×C₃₁	#555	1	1	yes	characteristic
246	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
247	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
248	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
250	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
252	#461	—	2	4	no	—
253	#440	—	2	4	no	—
254	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
257	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#103	C₂²×C₃₁	1	2	yes	—
259	C₂×#103	C₂²×C₃₁	1	2	yes	—
260	C₂×#103	C₂²×C₃₁	1	2	yes	—
261	C₂×#103	C₂²×C₃₁	1	2	yes	—
262	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
264	#398	—	2	1	no	—
265	#398	—	2	1	no	—
266	#391	C₂²×C₃₁	1	2	yes	—
267	#390	C₂²×C₃₁	1	2	yes	—
268	#389	—	2	2	no	—
269	#388	—	2	2	no	—
270	#382	C₂²×C₃₁	1	1	yes	characteristic
271	#381	—	2	1	no	—
272	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
273	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
274	C₄×C₂²⋊C₄	—	2	1	no	—
275	#340	C₂²×C₃₁	1	2	yes	—
276	#336	C₂²×C₃₁	1	2	yes	—
277	C₂×C₃₁	—	8	2	no	—
278	C₂×C₃₁	#2414	1	1	yes	characteristic
279	C₂×C₃₁	#3039	1	1	yes	characteristic
280	C₂×C₃₁	#3042	1	1	yes	characteristic
281	C₂×C₃₁	#3320	1	2	yes	—
282	C₂×C₃₁	#3400	1	2	yes	—
283	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
284	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂³×C₄	—	2	1	no	—
287	C₂×SD₁₆	—	2	2	no	—
288	C₂×SD₁₆	—	2	2	no	—
289	C₂×SD₁₆	—	2	4	no	—
290	C₂×SD₁₆	—	2	2	no	—
291	C₂×SD₁₆	—	4	2	no	—
292	C₂×SD₁₆	—	4	2	no	—
293	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
294	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₄⋊Q₈	—	2	4	no	—
297	C₄⋊Q₈	—	2	2	no	—
298	C₄⋊Q₈	—	2	2	no	—
299	C₄⋊Q₈	—	4	2	no	—
300	C₄×Q₈	—	2	2	no	—
301	C₄×Q₈	—	2	2	no	—
302	C₄×D₄	—	2	2	no	—
303	C₄×D₄	D₄×C₃₁	1	4	yes	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	—	2	2	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	#107	—	2	2	no	—
321	C₄⋊C₈	—	2	2	no	—
322	C₄⋊C₈	—	4	2	no	—
323	#103	—	2	4	no	—
324	#103	—	2	2	no	—
325	#103	—	2	4	no	—
326	#103	—	4	2	no	—
327	#103	—	4	2	no	—
328	#102	—	2	2	no	—
329	#97	—	2	2	no	—
330	#95	—	2	1	no	—
331	#95	—	2	2	no	—
332	#95	—	2	1	no	—
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	#95	D₄×C₃₁	1	1	yes	characteristic
336	#95	D₄×C₃₁	1	1	yes	characteristic
337	C₃₁	#24646	1	1	yes	characteristic, direct product factor, Sylow
338	C₂⁴	—	2	1	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	1	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	#41×C₃₁	1	2	yes	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	8	2	no	—
364	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
365	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
369	SD₁₆	—	2	8	no	—
370	SD₁₆	—	4	8	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	#41×C₃₁	1	2	yes	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	4	4	no	—
387	C₄⋊C₄	—	4	4	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	4	2	no	—
391	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
392	C₂²⋊C₄	—	2	4	no	—
393	C₂²⋊C₄	—	4	4	no	—
394	C₂²⋊C₄	—	8	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	4	2	no	—
398	C₄²	—	8	2	no	—
399	C₄²	C₂×D₄×C₃₁	1	2	yes	—
400	C₄²	#41×C₃₁	1	2	yes	—
401	C₂³	—	2	2	no	—
402	C₂³	—	4	1	no	—
403	C₂³	—	4	1	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
407	Q₈	—	2	2	no	—
408	Q₈	—	2	2	no	—
409	Q₈	—	4	1	no	—
410	Q₈	—	4	1	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	4	2	no	—
414	Q₈	—	4	2	no	—
415	D₄	—	2	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	4	1	no	—
418	D₄	—	4	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	8	4	no	—
440	C₂×C₄	C₃₁×#121	1	2	yes	—
441	C₂×C₄	C₃₁×#121	1	2	yes	—
442	C₂×C₄	C₃₁×#137	1	4	yes	—
443	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
445	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
448	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
449	C₈	—	2	2	no	—
450	C₈	—	4	1	no	—
451	C₈	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	8	2	no	—
456	C₂²	—	8	2	no	—
457	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
458	C₂²	C₃₁×#546	1	1	yes	characteristic
459	C₂²	C₃₁×#550	1	1	yes	characteristic
460	C₂²	C₂×C₃₁×#137	1	2	yes	—
461	C₂²	C₃₁×#577	1	2	yes	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	4	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	8	2	no	—
469	C₄	—	8	2	no	—
470	C₄	—	8	1	no	—
471	C₄	C₃₁×#525	1	1	yes	characteristic
472	C₄	C₃₁×#535	1	1	yes	characteristic
473	C₄	C₃₁×#553	1	1	yes	characteristic
474	C₄	C₃₁×#555	1	1	yes	characteristic
475	C₂	—	8	2	no	—
476	C₂	C₃₁×#2414	1	1	yes	characteristic
477	C₂	C₃₁×#3039	1	1	yes	characteristic
478	C₂	C₃₁×#3042	1	1	yes	characteristic
479	C₂	C₃₁×#3320	1	2	yes	—
480	C₂	C₃₁×#3400	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
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
3. C₃₁×#2546	C₂	semidirect
7. C₃₁×#1981	C₂	semidirect
8. C₃₁×#1960	C₂	semidirect
9. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1928	C₂	semidirect
11. C₃₁×#1805	C₂	semidirect
160. #24646	C₃₁	direct (N × H)
214. #2546	C₂×C₃₁	semidirect
218. #1981	C₂×C₃₁	semidirect
219. #1960	C₂×C₃₁	semidirect
220. #1940	C₂×C₃₁	semidirect
221. #1928	C₂×C₃₁	semidirect
222. #1805	C₂×C₃₁	semidirect
337. C₃₁	#24646	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×#461	2
C₂×C₃₁×#440	2
C₃₁×#2546	2
C₃₁×#2357	2
C₃₁×#2055	2
C₃₁×#2013	2
C₃₁×#1981	2
C₃₁×#1960	2
C₃₁×#1940	2
C₃₁×#1928	2
C₃₁×#1805	2
#24646	31

Subgroups — C31×#24646 #21481405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24646 #21481405