Subgroups — C₃₁×#22279 #21479038

Order 7936.

986 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#496	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2328	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2326	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1958	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1880	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1877	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#496	C₄	1	8	yes	complemented
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#391	C₂²	1	1	yes	characteristic
27	C₃₁×#390	C₂²	1	1	yes	characteristic
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#388	—	2	2	no	—
30	C₃₁×#386	—	2	2	no	—
31	C₃₁×#386	—	2	2	no	—
32	C₃₁×#385	—	2	2	no	—
33	C₃₁×#384	C₂²	1	1	yes	characteristic
34	C₃₁×#382	C₂²	1	1	yes	characteristic
35	C₃₁×#376	C₂²	1	1	yes	characteristic
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	4	2	no	—
47	C₂×SD₁₆×C₃₁	—	4	2	no	—
48	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
49	C₂×D₈×C₃₁	—	4	2	no	—
50	C₂×D₈×C₃₁	—	4	2	no	—
51	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₃₁×#127	—	2	2	no	—
55	C₃₁×#127	C₂×C₄	1	4	yes	complemented
56	C₃₁×#124	—	2	2	no	—
57	C₃₁×#124	C₂×C₄	1	4	yes	complemented
58	C₄×Q₈×C₃₁	—	4	2	no	—
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₄×D₄×C₃₁	—	4	2	no	—
61	C₄×D₄×C₃₁	—	4	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#103	—	4	2	no	—
78	C₃₁×#102	—	4	2	no	—
79	C₃₁×#102	—	4	2	no	—
80	C₃₁×#102	—	4	2	no	—
81	C₃₁×#97	C₂×C₄	1	4	yes	complemented
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	D₄	1	2	yes	—
86	C₃₁×#95	D₄	1	2	yes	—
87	C₂⁴×C₃₁	—	2	1	no	—
88	C₂⁴×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₃₁	—	2	8	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	8	2	no	—
103	C₂×D₄×C₃₁	—	8	2	no	—
104	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	4	4	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	—	8	2	no	—
113	C₂²×C₄×C₃₁	—	8	2	no	—
114	C₂²×C₄×C₃₁	—	8	2	no	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	SD₁₆×C₃₁	—	4	8	no	—
121	D₈×C₃₁	—	4	8	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	8	2	no	—
134	C₂²⋊C₄×C₃₁	—	2	8	no	—
135	C₂²⋊C₄×C₃₁	—	4	2	no	—
136	C₂²⋊C₄×C₃₁	—	4	4	no	—
137	C₂²⋊C₄×C₃₁	—	8	2	no	—
138	C₂²⋊C₄×C₃₁	—	8	2	no	—
139	C₂²⋊C₄×C₃₁	—	8	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	4	2	no	—
143	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
144	C₄²×C₃₁	#41	1	2	yes	—
145	#22279	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	2	4	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	1	no	—
152	C₂³×C₃₁	—	4	2	no	—
153	C₂³×C₃₁	—	4	2	no	—
154	C₂³×C₃₁	—	4	2	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	4	no	—
160	Q₈×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	2	4	no	—
162	D₄×C₃₁	—	2	4	no	—
163	D₄×C₃₁	—	2	4	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	8	8	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	8	4	no	—
188	C₂×C₄×C₃₁	—	8	8	no	—
189	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
190	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
193	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
194	C₈×C₃₁	—	2	4	no	—
195	C₈×C₃₁	—	4	2	no	—
196	C₂×#496	C₂×C₃₁	1	1	yes	characteristic
197	#2328	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2326	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1958	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1880	C₂×C₃₁	1	2	yes	complemented
202	#1879	C₂×C₃₁	1	2	yes	complemented
203	#1879	C₂×C₃₁	1	2	yes	complemented
204	#1877	C₂×C₃₁	1	2	yes	complemented
205	#1836	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	4	no	—
209	C₂²×C₃₁	—	4	4	no	—
210	C₂²×C₃₁	—	4	4	no	—
211	C₂²×C₃₁	—	4	4	no	—
212	C₂²×C₃₁	—	8	1	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	—	8	1	no	—
215	C₂²×C₃₁	—	8	2	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	—	8	2	no	—
219	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₃₁	#535	1	1	yes	characteristic
221	C₂²×C₃₁	#535	1	1	yes	characteristic
222	C₂²×C₃₁	#576	1	2	yes	—
223	C₂²×C₃₁	#577	1	2	yes	—
224	C₄×C₃₁	—	2	4	no	—
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	4	1	no	—
227	C₄×C₃₁	—	4	1	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	8	1	no	—
230	C₄×C₃₁	—	8	1	no	—
231	C₄×C₃₁	—	8	2	no	—
232	C₄×C₃₁	#518	1	2	yes	—
233	C₄×C₃₁	#540	1	2	yes	—
234	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
235	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
236	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
241	#496	C₄×C₃₁	1	8	yes	complemented
242	C₂×#103	C₂²×C₃₁	1	2	yes	—
243	C₂×#102	C₂²×C₃₁	1	2	yes	—
244	C₂×#102	C₂²×C₃₁	1	2	yes	—
245	C₂×#102	C₂²×C₃₁	1	2	yes	—
246	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
248	#391	C₂²×C₃₁	1	1	yes	characteristic
249	#390	C₂²×C₃₁	1	1	yes	characteristic
250	#390	C₂²×C₃₁	1	2	yes	—
251	#388	—	2	2	no	—
252	#386	—	2	2	no	—
253	#386	—	2	2	no	—
254	#385	—	2	2	no	—
255	#384	C₂²×C₃₁	1	1	yes	characteristic
256	#382	C₂²×C₃₁	1	1	yes	characteristic
257	#376	C₂²×C₃₁	1	1	yes	characteristic
258	#340	C₂²×C₃₁	1	2	yes	—
259	#340	C₂²×C₃₁	1	2	yes	—
260	#336	C₂²×C₃₁	1	2	yes	—
261	#336	C₂²×C₃₁	1	2	yes	—
262	C₂×C₃₁	—	8	4	no	—
263	C₂×C₃₁	—	8	2	no	—
264	C₂×C₃₁	#2328	1	1	yes	characteristic
265	C₂×C₃₁	#2946	1	2	yes	—
266	C₂×C₃₁	#3220	1	1	yes	characteristic
267	C₂×C₃₁	#3221	1	1	yes	characteristic
268	C₂×C₃₁	#3224	1	2	yes	—
269	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×D₄	—	2	2	no	—
271	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
272	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
273	C₂³×C₄	—	2	1	no	—
274	C₂³×C₄	—	2	2	no	—
275	C₂×SD₁₆	—	4	2	no	—
276	C₂×SD₁₆	—	4	2	no	—
277	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
278	C₂×D₈	—	4	2	no	—
279	C₂×D₈	—	4	2	no	—
280	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
281	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
282	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
283	#127	—	2	2	no	—
284	#127	C₂×C₄×C₃₁	1	4	yes	complemented
285	#124	—	2	2	no	—
286	#124	C₂×C₄×C₃₁	1	4	yes	complemented
287	C₄×Q₈	—	4	2	no	—
288	C₄×D₄	—	4	2	no	—
289	C₄×D₄	—	4	2	no	—
290	C₄×D₄	—	4	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	—	2	2	no	—
298	C₂×C₂²⋊C₄	—	2	1	no	—
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
306	#103	—	4	2	no	—
307	#102	—	4	2	no	—
308	#102	—	4	2	no	—
309	#102	—	4	2	no	—
310	#97	C₂×C₄×C₃₁	1	4	yes	complemented
311	C₄×C₈	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	2	no	—
314	#95	D₄×C₃₁	1	2	yes	—
315	#95	D₄×C₃₁	1	2	yes	—
316	C₃₁	#22279	1	1	yes	characteristic, direct product factor, Sylow
317	C₂⁴	—	2	1	no	—
318	C₂⁴	—	2	2	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
323	C₂×D₄	—	2	8	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	8	2	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	4	4	no	—
341	C₂²×C₄	—	4	2	no	—
342	C₂²×C₄	—	8	2	no	—
343	C₂²×C₄	—	8	2	no	—
344	C₂²×C₄	—	8	2	no	—
345	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
350	SD₁₆	—	4	8	no	—
351	D₈	—	4	8	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₄⋊C₄	—	8	2	no	—
364	C₂²⋊C₄	—	2	8	no	—
365	C₂²⋊C₄	—	4	2	no	—
366	C₂²⋊C₄	—	4	4	no	—
367	C₂²⋊C₄	—	8	2	no	—
368	C₂²⋊C₄	—	8	2	no	—
369	C₂²⋊C₄	—	8	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	4	2	no	—
373	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
374	C₄²	#41×C₃₁	1	2	yes	—
375	C₂³	—	2	2	no	—
376	C₂³	—	2	4	no	—
377	C₂³	—	4	1	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	1	no	—
381	C₂³	—	4	2	no	—
382	C₂³	—	4	2	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	2	no	—
387	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	Q₈	—	2	4	no	—
389	Q₈	—	4	2	no	—
390	D₄	—	2	4	no	—
391	D₄	—	2	4	no	—
392	D₄	—	2	4	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	8	8	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	8	4	no	—
417	C₂×C₄	—	8	8	no	—
418	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
419	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
422	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
423	C₈	—	2	4	no	—
424	C₈	—	4	2	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	4	no	—
427	C₂²	—	4	4	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	4	4	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	1	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
438	C₂²	C₃₁×#535	1	1	yes	characteristic
439	C₂²	C₃₁×#535	1	1	yes	characteristic
440	C₂²	C₃₁×#576	1	2	yes	—
441	C₂²	C₃₁×#577	1	2	yes	—
442	C₄	—	2	4	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	1	no	—
445	C₄	—	4	1	no	—
446	C₄	—	4	2	no	—
447	C₄	—	8	1	no	—
448	C₄	—	8	1	no	—
449	C₄	—	8	2	no	—
450	C₄	C₃₁×#518	1	2	yes	—
451	C₄	C₃₁×#540	1	2	yes	—
452	C₂	—	8	4	no	—
453	C₂	—	8	2	no	—
454	C₂	C₃₁×#2328	1	1	yes	characteristic
455	C₂	C₃₁×#2946	1	2	yes	—
456	C₂	C₃₁×#3220	1	1	yes	characteristic
457	C₂	C₃₁×#3221	1	1	yes	characteristic
458	C₂	C₃₁×#3224	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#496	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2328	C₂	semidirect
3. C₃₁×#2326	C₂	semidirect
4. C₃₁×#1958	C₂	semidirect
5. C₃₁×#1957	C₂	semidirect
6. C₃₁×#1880	C₂	semidirect
7. C₃₁×#1879	C₂	semidirect
8. C₃₁×#1879	C₂	semidirect
9. C₃₁×#1877	C₂	semidirect
10. C₃₁×#1836	C₂	semidirect
11. C₃₁×#1746	C₂	semidirect
19. C₃₁×#496	C₄	semidirect
48. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
51. C₂×D₈×C₃₁	C₂×C₄	semidirect
55. C₃₁×#127	C₂×C₄	semidirect
57. C₃₁×#124	C₂×C₄	semidirect
81. C₃₁×#97	C₂×C₄	semidirect
145. #22279	C₃₁	direct (N × H)
197. #2328	C₂×C₃₁	semidirect
198. #2326	C₂×C₃₁	semidirect
199. #1958	C₂×C₃₁	semidirect
200. #1957	C₂×C₃₁	semidirect
201. #1880	C₂×C₃₁	semidirect
202. #1879	C₂×C₃₁	semidirect
203. #1879	C₂×C₃₁	semidirect
204. #1877	C₂×C₃₁	semidirect
205. #1836	C₂×C₃₁	semidirect
206. #1746	C₂×C₃₁	semidirect
241. #496	C₄×C₃₁	semidirect
277. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
280. C₂×D₈	C₂×C₄×C₃₁	semidirect
284. #127	C₂×C₄×C₃₁	semidirect
286. #124	C₂×C₄×C₃₁	semidirect
310. #97	C₂×C₄×C₃₁	semidirect
316. C₃₁	#22279	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×#496	2
C₃₁×#2328	2
C₃₁×#2326	2
C₃₁×#1958	2
C₃₁×#1957	2
C₃₁×#1880	2
C₃₁×#1879	2
C₃₁×#1877	2
C₃₁×#1836	2
C₃₁×#1746	2
#22279	31

Subgroups — C31×#22279 #21479038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22279 #21479038