Subgroups — C₃₁×#23632 #21480391

Order 7936.

1122 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2309	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2073	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2071	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1736	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#418	—	2	4	no	—
18	C₂×C₃₁×#102	—	2	4	no	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	4	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₃₁×#390	—	2	2	no	—
23	C₃₁×#390	C₂²	1	4	yes	—
24	C₃₁×#389	—	2	2	no	—
25	C₃₁×#385	—	2	2	no	—
26	C₃₁×#384	C₂²	1	2	yes	complemented
27	C₃₁×#382	C₂²	1	2	yes	complemented
28	C₃₁×#379	—	2	2	no	—
29	C₃₁×#340	C₂²	1	2	yes	complemented
30	C₃₁×#340	C₂²	1	4	yes	complemented
31	C₃₁×#340	C₂²	1	2	yes	complemented
32	C₃₁×#336	C₂²	1	4	yes	complemented
33	C₂⁵×C₃₁	—	2	1	no	—
34	C₂²×D₄×C₃₁	—	2	1	no	—
35	C₂²×D₄×C₃₁	—	2	2	no	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	—	2	2	no	—
38	C₂²×D₄×C₃₁	—	2	2	no	—
39	C₂²×D₄×C₃₁	—	2	2	no	—
40	C₂²×D₄×C₃₁	D₄	1	4	yes	—
41	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	—	2	1	no	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	—	2	2	no	—
47	C₂²×C₈×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₃₁×#117	—	2	2	no	—
53	C₃₁×#117	—	2	2	no	—
54	C₃₁×#117	—	2	2	no	—
55	C₃₁×#117	—	2	2	no	—
56	C₃₁×#117	—	2	2	no	—
57	C₃₁×#117	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
62	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
75	C₃₁×#102	—	2	8	no	—
76	C₃₁×#102	—	2	4	no	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#95	—	2	4	no	—
79	C₂⁴×C₃₁	—	2	2	no	—
80	C₂⁴×C₃₁	—	2	4	no	—
81	C₂⁴×C₃₁	—	4	2	no	—
82	C₂⁴×C₃₁	—	4	2	no	—
83	C₂⁴×C₃₁	—	8	2	no	—
84	C₂⁴×C₃₁	—	8	2	no	—
85	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×D₄×C₃₁	—	2	4	no	—
87	C₂×D₄×C₃₁	—	2	8	no	—
88	C₂×D₄×C₃₁	—	2	8	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	4	2	no	—
95	C₂×D₄×C₃₁	—	4	2	no	—
96	C₂×D₄×C₃₁	—	4	4	no	—
97	C₂×D₄×C₃₁	—	4	4	no	—
98	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	4	2	no	—
104	C₂²×C₄×C₃₁	—	8	2	no	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₃₁	#41	1	2	yes	—
109	C₂²×C₄×C₃₁	#41	1	2	yes	—
110	C₂²×C₄×C₃₁	#41	1	2	yes	—
111	C₂²×C₄×C₃₁	#41	1	2	yes	—
112	C₂²×C₄×C₃₁	#41	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	M₁₆×C₃₁	—	4	4	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	4	4	no	—
122	C₄⋊C₄×C₃₁	—	2	4	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	—	8	4	no	—
133	C₂²⋊C₄×C₃₁	—	2	4	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	2	no	—
136	C₂²⋊C₄×C₃₁	—	8	4	no	—
137	C₂²⋊C₄×C₃₁	—	8	4	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	4	2	no	—
141	C₄²×C₃₁	—	4	2	no	—
142	#23632	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	4	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	4	no	—
149	C₂³×C₃₁	—	4	4	no	—
150	C₂³×C₃₁	—	4	8	no	—
151	C₂³×C₃₁	—	8	4	no	—
152	C₂³×C₃₁	—	8	4	no	—
153	C₂³×C₃₁	—	8	8	no	—
154	C₂³×C₃₁	—	8	8	no	—
155	C₂³×C₃₁	#121	1	2	yes	—
156	C₂³×C₃₁	#123	1	2	yes	—
157	C₂³×C₃₁	#124	1	2	yes	—
158	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	D₄×C₃₁	—	2	4	no	—
160	D₄×C₃₁	—	2	4	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	8	no	—
164	C₂×C₄×C₃₁	—	2	4	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	4	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	8	4	no	—
177	C₂×C₄×C₃₁	#121	1	2	yes	—
178	C₂×C₄×C₃₁	#123	1	4	yes	—
179	C₂×C₄×C₃₁	#123	1	4	yes	—
180	C₂×C₄×C₃₁	#123	1	2	yes	—
181	C₂×C₄×C₃₁	#124	1	2	yes	—
182	C₂×C₄×C₃₁	#126	1	4	yes	—
183	C₂×C₄×C₃₁	#126	1	4	yes	—
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
190	C₈×C₃₁	—	4	2	no	—
191	C₈×C₃₁	—	4	2	no	—
192	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
193	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic
194	#2317	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#2309	C₂×C₃₁	1	1	yes	characteristic
196	#2073	C₂×C₃₁	1	4	yes	complemented
197	#2071	C₂×C₃₁	1	4	yes	complemented
198	#1736	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1735	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂²×C₃₁	—	2	4	no	—
202	C₂²×C₃₁	—	2	4	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	4	4	no	—
207	C₂²×C₃₁	—	4	4	no	—
208	C₂²×C₃₁	—	8	16	no	—
209	C₂²×C₃₁	—	8	4	no	—
210	C₂²×C₃₁	—	8	4	no	—
211	C₂²×C₃₁	#397	1	4	yes	—
212	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
213	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
214	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
215	C₂²×C₃₁	#576	1	2	yes	—
216	C₂²×C₃₁	#577	1	2	yes	—
217	C₄×C₃₁	—	4	4	no	—
218	C₄×C₃₁	—	8	2	no	—
219	C₄×C₃₁	—	8	1	no	—
220	C₄×C₃₁	—	8	2	no	—
221	C₄×C₃₁	—	8	1	no	—
222	C₄×C₃₁	#397	1	4	yes	—
223	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
224	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
225	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
226	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
227	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
228	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
229	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
233	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
234	#418	—	2	4	no	—
235	C₂×#102	—	2	4	no	—
236	C₂×#102	C₂²×C₃₁	1	2	yes	—
237	C₂×#102	C₂²×C₃₁	1	4	yes	—
238	C₂×#102	C₂²×C₃₁	1	2	yes	—
239	#390	—	2	2	no	—
240	#390	C₂²×C₃₁	1	4	yes	—
241	#389	—	2	2	no	—
242	#385	—	2	2	no	—
243	#384	C₂²×C₃₁	1	2	yes	complemented
244	#382	C₂²×C₃₁	1	2	yes	complemented
245	#379	—	2	2	no	—
246	#340	C₂²×C₃₁	1	2	yes	complemented
247	#340	C₂²×C₃₁	1	4	yes	complemented
248	#340	C₂²×C₃₁	1	2	yes	complemented
249	#336	C₂²×C₃₁	1	4	yes	complemented
250	C₂×C₃₁	—	2	4	no	—
251	C₂×C₃₁	—	8	4	no	—
252	C₂×C₃₁	C₂×#397	1	1	yes	characteristic
253	C₂×C₃₁	#3041	1	1	yes	characteristic
254	C₂×C₃₁	#3042	1	1	yes	characteristic
255	C₂×C₃₁	#3096	1	2	yes	—
256	C₂×C₃₁	#3136	1	2	yes	—
257	C₂⁵	—	2	1	no	—
258	C₂²×D₄	—	2	1	no	—
259	C₂²×D₄	—	2	2	no	—
260	C₂²×D₄	—	2	2	no	—
261	C₂²×D₄	—	2	2	no	—
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	—	2	2	no	—
264	C₂²×D₄	D₄×C₃₁	1	4	yes	—
265	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
266	C₂³×C₄	—	2	1	no	—
267	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
268	C₂×M₁₆	—	2	2	no	—
269	C₂×M₁₆	—	2	2	no	—
270	C₂×M₁₆	—	2	2	no	—
271	C₂²×C₈	—	2	2	no	—
272	C₂²×C₈	—	2	2	no	—
273	C₂²×C₈	—	2	2	no	—
274	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
276	#117	—	2	2	no	—
277	#117	—	2	2	no	—
278	#117	—	2	2	no	—
279	#117	—	2	2	no	—
280	#117	—	2	2	no	—
281	#117	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	4	no	—
286	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	4	no	—
294	C₂×C₂²⋊C₄	—	2	4	no	—
295	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	—	2	1	no	—
298	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
299	#102	—	2	8	no	—
300	#102	—	2	4	no	—
301	#102	—	2	4	no	—
302	#95	—	2	4	no	—
303	C₃₁	#23632	1	1	yes	characteristic, direct product factor, Sylow
304	C₂⁴	—	2	2	no	—
305	C₂⁴	—	2	4	no	—
306	C₂⁴	—	4	2	no	—
307	C₂⁴	—	4	2	no	—
308	C₂⁴	—	8	2	no	—
309	C₂⁴	—	8	2	no	—
310	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	2	8	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	4	2	no	—
321	C₂×D₄	—	4	4	no	—
322	C₂×D₄	—	4	4	no	—
323	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	4	2	no	—
329	C₂²×C₄	—	8	2	no	—
330	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
331	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
333	C₂²×C₄	#41×C₃₁	1	2	yes	—
334	C₂²×C₄	#41×C₃₁	1	2	yes	—
335	C₂²×C₄	#41×C₃₁	1	2	yes	—
336	C₂²×C₄	#41×C₃₁	1	2	yes	—
337	C₂²×C₄	#41×C₃₁	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
339	M₁₆	—	4	4	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	4	4	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	8	4	no	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₂²⋊C₄	—	4	2	no	—
360	C₂²⋊C₄	—	4	2	no	—
361	C₂²⋊C₄	—	8	4	no	—
362	C₂²⋊C₄	—	8	4	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	4	2	no	—
366	C₄²	—	4	2	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	4	no	—
370	C₂³	—	4	8	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	4	no	—
374	C₂³	—	4	4	no	—
375	C₂³	—	8	4	no	—
376	C₂³	—	8	4	no	—
377	C₂³	—	8	8	no	—
378	C₂³	—	8	8	no	—
379	C₂³	C₃₁×#121	1	2	yes	—
380	C₂³	C₃₁×#123	1	2	yes	—
381	C₂³	C₃₁×#124	1	2	yes	—
382	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
383	D₄	—	2	4	no	—
384	D₄	—	2	4	no	—
385	D₄	—	4	8	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	2	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	8	4	no	—
401	C₂×C₄	C₃₁×#121	1	2	yes	—
402	C₂×C₄	C₃₁×#123	1	2	yes	—
403	C₂×C₄	C₃₁×#123	1	4	yes	—
404	C₂×C₄	C₃₁×#123	1	4	yes	—
405	C₂×C₄	C₃₁×#124	1	2	yes	—
406	C₂×C₄	C₃₁×#126	1	4	yes	—
407	C₂×C₄	C₃₁×#126	1	4	yes	—
408	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
409	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
410	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
411	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
414	C₈	—	4	2	no	—
415	C₈	—	4	2	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	2	4	no	—
418	C₂²	—	2	2	no	—
419	C₂²	—	2	2	no	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	4	no	—
423	C₂²	—	8	16	no	—
424	C₂²	—	8	4	no	—
425	C₂²	—	8	4	no	—
426	C₂²	C₃₁×#397	1	4	yes	—
427	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
428	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
429	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
430	C₂²	C₃₁×#576	1	2	yes	—
431	C₂²	C₃₁×#577	1	2	yes	—
432	C₄	—	4	4	no	—
433	C₄	—	8	2	no	—
434	C₄	—	8	2	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	1	no	—
437	C₄	C₃₁×#397	1	4	yes	—
438	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
439	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
440	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
441	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
442	C₂	—	2	4	no	—
443	C₂	—	8	4	no	—
444	C₂	C₂×C₃₁×#397	1	1	yes	characteristic
445	C₂	C₃₁×#3041	1	1	yes	characteristic
446	C₂	C₃₁×#3042	1	1	yes	characteristic
447	C₂	C₃₁×#3096	1	2	yes	—
448	C₂	C₃₁×#3136	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2317	C₂	semidirect
5. C₃₁×#2073	C₂	semidirect
6. C₃₁×#2071	C₂	semidirect
7. C₃₁×#1736	C₂	semidirect
8. C₃₁×#1735	C₂	semidirect
9. C₃₁×#1729	C₂	semidirect
26. C₃₁×#384	C₂²	semidirect
27. C₃₁×#382	C₂²	semidirect
29. C₃₁×#340	C₂²	semidirect
30. C₃₁×#340	C₂²	semidirect
31. C₃₁×#340	C₂²	semidirect
32. C₃₁×#336	C₂²	semidirect
142. #23632	C₃₁	direct (N × H)
194. #2317	C₂×C₃₁	semidirect
196. #2073	C₂×C₃₁	semidirect
197. #2071	C₂×C₃₁	semidirect
198. #1736	C₂×C₃₁	semidirect
199. #1735	C₂×C₃₁	semidirect
200. #1729	C₂×C₃₁	semidirect
243. #384	C₂²×C₃₁	semidirect
244. #382	C₂²×C₃₁	semidirect
246. #340	C₂²×C₃₁	semidirect
247. #340	C₂²×C₃₁	semidirect
248. #340	C₂²×C₃₁	semidirect
249. #336	C₂²×C₃₁	semidirect
303. C₃₁	#23632	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×#418	2
C₂²×C₃₁×#102	2
C₃₁×#2317	2
C₃₁×#2309	2
C₃₁×#2073	2
C₃₁×#2071	2
C₃₁×#1736	2
C₃₁×#1735	2
C₃₁×#1729	2
#23632	31

Subgroups — C31×#23632 #21480391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23632 #21480391