Subgroups — C₃₁×#33282 #21490041

Order 7936.

720 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2861	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1959	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1959	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1936	C₂	1	2	yes	maximal
8	C₃₁×#1935	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1934	C₂	1	2	yes	maximal
10	C₃₁×#1933	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#125	—	2	1	no	—
13	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#103	C₂²	1	2	yes	—
16	C₂×C₃₁×#103	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#391	C₂²	1	1	yes	characteristic
24	C₃₁×#391	C₂²	1	1	yes	characteristic
25	C₃₁×#390	C₂²	1	1	yes	characteristic
26	C₃₁×#390	C₂²	1	1	yes	characteristic
27	C₃₁×#389	—	2	1	no	—
28	C₃₁×#389	—	2	1	no	—
29	C₃₁×#389	C₂²	1	2	yes	—
30	C₃₁×#389	C₂²	1	2	yes	—
31	C₃₁×#388	—	2	1	no	—
32	C₃₁×#388	—	2	1	no	—
33	C₃₁×#388	—	2	1	no	—
34	C₃₁×#388	—	2	1	no	—
35	C₃₁×#384	C₂²	1	1	yes	characteristic
36	C₃₁×#384	C₂²	1	1	yes	characteristic
37	C₃₁×#382	—	2	1	no	—
38	C₄³×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₃₁×#125	—	2	2	no	—
48	C₃₁×#125	—	2	2	no	—
49	C₃₁×#125	—	4	4	no	—
50	C₃₁×#124	—	2	2	no	—
51	C₃₁×#124	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
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	—	2	4	no	—
78	C₃₁×#103	—	2	4	no	—
79	C₃₁×#102	—	2	4	no	—
80	C₃₁×#102	—	2	4	no	—
81	C₃₁×#97	—	2	4	no	—
82	C₃₁×#97	—	2	2	no	—
83	C₃₁×#97	—	2	2	no	—
84	C₄×C₈×C₃₁	—	2	4	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₂⁴×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	#41	1	4	yes	—
117	C₂×C₈×C₃₁	#41	1	4	yes	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	4	2	no	—
127	C₄⋊C₄×C₃₁	—	4	2	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	—	4	2	no	—
130	C₄⋊C₄×C₃₁	—	4	4	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	4	4	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	2	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₂²⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	4	1	no	—
145	C₄²×C₃₁	—	4	1	no	—
146	C₄²×C₃₁	#41	1	2	yes	—
147	C₄²×C₃₁	#41	1	2	yes	—
148	#33282	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	1	no	—
151	C₂³×C₃₁	—	4	1	no	—
152	C₂³×C₃₁	—	4	1	no	—
153	C₂³×C₃₁	—	4	1	no	—
154	C₂³×C₃₁	—	4	1	no	—
155	C₂³×C₃₁	—	4	1	no	—
156	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
157	Q₈×C₃₁	—	4	2	no	—
158	Q₈×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	#124	1	2	yes	—
191	C₂×C₄×C₃₁	#124	1	2	yes	—
192	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
193	C₂×C₄×C₃₁	#137	1	2	yes	—
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
197	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
198	C₈×C₃₁	—	2	4	no	—
199	C₈×C₃₁	—	2	4	no	—
200	#2861	C₂×C₃₁	1	1	yes	characteristic
201	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1992	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1992	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1959	C₂×C₃₁	1	1	yes	characteristic
205	#1959	C₂×C₃₁	1	1	yes	characteristic
206	#1936	C₂×C₃₁	1	2	yes	—
207	#1935	C₂×C₃₁	1	2	yes	complemented
208	#1934	C₂×C₃₁	1	2	yes	—
209	#1933	C₂×C₃₁	1	2	yes	complemented
210	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂²×C₃₁	—	8	1	no	—
212	C₂²×C₃₁	—	8	1	no	—
213	C₂²×C₃₁	—	8	1	no	—
214	C₂²×C₃₁	—	8	1	no	—
215	C₂²×C₃₁	—	8	1	no	—
216	C₂²×C₃₁	—	8	1	no	—
217	C₂²×C₃₁	—	8	1	no	—
218	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
219	C₂²×C₃₁	#532	1	1	yes	characteristic
220	C₂²×C₃₁	#532	1	1	yes	characteristic
221	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
223	C₂²×C₃₁	#575	1	1	yes	characteristic
224	C₂²×C₃₁	#575	1	1	yes	characteristic
225	C₄×C₃₁	—	2	2	no	—
226	C₄×C₃₁	—	2	2	no	—
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	4	1	no	—
230	C₄×C₃₁	—	4	1	no	—
231	C₄×C₃₁	—	8	1	no	—
232	C₄×C₃₁	—	8	1	no	—
233	C₄×C₃₁	—	8	1	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	#529	1	2	yes	—
237	C₄×C₃₁	#529	1	2	yes	—
238	C₂×#125	—	2	1	no	—
239	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#103	C₂²×C₃₁	1	2	yes	—
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×#97	C₂²×C₃₁	1	2	yes	—
246	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
249	#391	C₂²×C₃₁	1	1	yes	characteristic
250	#391	C₂²×C₃₁	1	1	yes	characteristic
251	#390	C₂²×C₃₁	1	1	yes	characteristic
252	#390	C₂²×C₃₁	1	1	yes	characteristic
253	#389	—	2	1	no	—
254	#389	—	2	1	no	—
255	#389	C₂²×C₃₁	1	2	yes	—
256	#389	C₂²×C₃₁	1	2	yes	—
257	#388	—	2	1	no	—
258	#388	—	2	1	no	—
259	#388	—	2	1	no	—
260	#388	—	2	1	no	—
261	#384	C₂²×C₃₁	1	1	yes	characteristic
262	#384	C₂²×C₃₁	1	1	yes	characteristic
263	#382	—	2	1	no	—
264	C₄³	C₂²×C₃₁	1	1	yes	characteristic
265	#340	C₂²×C₃₁	1	2	yes	—
266	#340	C₂²×C₃₁	1	2	yes	—
267	#340	C₂²×C₃₁	1	2	yes	—
268	#340	C₂²×C₃₁	1	2	yes	—
269	C₂×C₃₁	—	16	1	no	—
270	C₂×C₃₁	#2861	1	1	yes	characteristic
271	C₂×C₃₁	#3141	1	1	yes	characteristic
272	C₂×C₃₁	#3142	1	1	yes	characteristic
273	C₂×C₃₁	#3171	1	1	yes	characteristic
274	C₂×C₃₁	#3172	1	1	yes	characteristic
275	C₂×C₃₁	#3180	1	1	yes	characteristic
276	C₂×C₃₁	#3181	1	1	yes	characteristic
277	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
280	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
281	#125	—	2	2	no	—
282	#125	—	2	2	no	—
283	#125	—	4	4	no	—
284	#124	—	2	2	no	—
285	#124	—	2	2	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
298	C₂×C₂²⋊C₄	—	2	1	no	—
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
311	#103	—	2	4	no	—
312	#103	—	2	4	no	—
313	#102	—	2	4	no	—
314	#102	—	2	4	no	—
315	#97	—	2	4	no	—
316	#97	—	2	2	no	—
317	#97	—	2	2	no	—
318	C₄×C₈	—	2	4	no	—
319	#95	—	2	2	no	—
320	#95	—	2	2	no	—
321	#95	—	2	1	no	—
322	#95	—	2	1	no	—
323	#95	—	2	1	no	—
324	#95	—	2	1	no	—
325	C₃₁	#33282	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	#41×C₃₁	1	4	yes	—
352	C₂×C₈	#41×C₃₁	1	4	yes	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	4	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₄⋊C₄	—	4	2	no	—
364	C₄⋊C₄	—	4	2	no	—
365	C₄⋊C₄	—	4	4	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	4	no	—
369	C₂²⋊C₄	—	4	2	no	—
370	C₂²⋊C₄	—	4	2	no	—
371	C₂²⋊C₄	—	4	2	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₂²⋊C₄	—	4	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	4	1	no	—
380	C₄²	—	4	1	no	—
381	C₄²	#41×C₃₁	1	2	yes	—
382	C₄²	#41×C₃₁	1	2	yes	—
383	C₂³	—	4	1	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	1	no	—
388	C₂³	—	4	1	no	—
389	C₂³	—	4	1	no	—
390	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
391	Q₈	—	4	2	no	—
392	Q₈	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	C₃₁×#124	1	2	yes	—
425	C₂×C₄	C₃₁×#124	1	2	yes	—
426	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
427	C₂×C₄	C₃₁×#137	1	2	yes	—
428	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
431	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
432	C₈	—	2	4	no	—
433	C₈	—	2	4	no	—
434	C₂²	—	8	1	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	8	1	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	1	no	—
439	C₂²	—	8	1	no	—
440	C₂²	—	8	1	no	—
441	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
442	C₂²	C₃₁×#532	1	1	yes	characteristic
443	C₂²	C₃₁×#532	1	1	yes	characteristic
444	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
445	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
446	C₂²	C₃₁×#575	1	1	yes	characteristic
447	C₂²	C₃₁×#575	1	1	yes	characteristic
448	C₄	—	2	2	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	1	no	—
454	C₄	—	8	1	no	—
455	C₄	—	8	1	no	—
456	C₄	—	8	1	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	2	no	—
459	C₄	C₃₁×#529	1	2	yes	—
460	C₄	C₃₁×#529	1	2	yes	—
461	C₂	—	16	1	no	—
462	C₂	C₃₁×#2861	1	1	yes	characteristic
463	C₂	C₃₁×#3141	1	1	yes	characteristic
464	C₂	C₃₁×#3142	1	1	yes	characteristic
465	C₂	C₃₁×#3171	1	1	yes	characteristic
466	C₂	C₃₁×#3172	1	1	yes	characteristic
467	C₂	C₃₁×#3180	1	1	yes	characteristic
468	C₂	C₃₁×#3181	1	1	yes	characteristic

	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 ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2859	C₂	semidirect
3. C₃₁×#1992	C₂	semidirect
4. C₃₁×#1992	C₂	semidirect
8. C₃₁×#1935	C₂	semidirect
10. C₃₁×#1933	C₂	semidirect
11. C₄×C₃₁×#97	C₂	semidirect
148. #33282	C₃₁	direct (N × H)
201. #2859	C₂×C₃₁	semidirect
202. #1992	C₂×C₃₁	semidirect
203. #1992	C₂×C₃₁	semidirect
207. #1935	C₂×C₃₁	semidirect
209. #1933	C₂×C₃₁	semidirect
210. C₄×#97	C₂×C₃₁	semidirect
325. C₃₁	#33282	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×#2861	2
C₃₁×#2859	2
C₃₁×#1992	2
C₃₁×#1959	2
C₃₁×#1936	2
C₃₁×#1935	2
C₃₁×#1934	2
C₃₁×#1933	2
C₄×C₃₁×#97	2
#33282	31

Subgroups — C31×#33282 #21490041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33282 #21490041