Subgroups — C₃₁×#24991 #21481750

Order 7936.

706 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2538	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2101	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2073	C₂	1	2	yes	maximal
5	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1926	C₂	1	2	yes	maximal
8	C₃₁×#1920	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#106	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#396	—	2	2	no	—
22	C₃₁×#391	—	2	1	no	—
23	C₃₁×#390	C₂²	1	2	yes	—
24	C₃₁×#389	C₂²	1	2	yes	—
25	C₃₁×#387	—	2	2	no	—
26	C₃₁×#384	C₂²	1	1	yes	characteristic
27	C₃₁×#384	C₂²	1	1	yes	characteristic
28	C₃₁×#382	C₂²	1	2	yes	—
29	C₃₁×#380	—	2	1	no	—
30	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	1	yes	characteristic
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	—	2	1	no	—
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₃₁×C₂²⋊Q₈	—	4	4	no	—
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₄×D₄×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#106	—	2	4	no	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₄×C₈×C₃₁	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	1	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₂⁴×C₃₁	—	2	1	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	2	1	no	—
89	C₂×D₄×C₃₁	—	2	1	no	—
90	C₂×D₄×C₃₁	—	2	1	no	—
91	C₂×D₄×C₃₁	#41	1	2	yes	—
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	4	2	no	—
98	C₂²×C₄×C₃₁	—	4	2	no	—
99	C₂²×C₄×C₃₁	—	4	2	no	—
100	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	1	no	—
117	C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	1	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	4	4	no	—
126	C₄⋊C₄×C₃₁	—	4	2	no	—
127	C₄⋊C₄×C₃₁	—	4	2	no	—
128	C₄⋊C₄×C₃₁	—	8	1	no	—
129	C₄⋊C₄×C₃₁	—	8	1	no	—
130	C₄⋊C₄×C₃₁	#41	1	2	yes	—
131	C₂²⋊C₄×C₃₁	—	4	4	no	—
132	C₂²⋊C₄×C₃₁	—	4	2	no	—
133	C₂²⋊C₄×C₃₁	—	8	2	no	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	—	8	2	no	—
138	C₄²×C₃₁	#41	1	2	yes	—
139	C₄²×C₃₁	#41	1	2	yes	—
140	#24991	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	4	1	no	—
143	C₂³×C₃₁	—	4	1	no	—
144	C₂³×C₃₁	—	4	1	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	Q₈×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	2	2	no	—
151	D₄×C₃₁	—	2	2	no	—
152	D₄×C₃₁	—	4	1	no	—
153	D₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	8	4	no	—
178	C₂×C₄×C₃₁	#123	1	2	yes	—
179	C₂×C₄×C₃₁	#123	1	2	yes	—
180	C₂×C₄×C₃₁	#135	1	2	yes	—
181	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
182	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
187	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
188	C₈×C₃₁	—	4	2	no	—
189	C₈×C₃₁	—	4	1	no	—
190	C₈×C₃₁	—	4	1	no	—
191	#2538	C₂×C₃₁	1	1	yes	characteristic
192	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#2101	C₂×C₃₁	1	2	yes	complemented
194	#2073	C₂×C₃₁	1	2	yes	—
195	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1926	C₂×C₃₁	1	2	yes	—
198	#1920	C₂×C₃₁	1	2	yes	complemented
199	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1802	C₂×C₃₁	1	1	yes	characteristic
201	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
202	C₂²×C₃₁	—	4	2	no	—
203	C₂²×C₃₁	—	4	2	no	—
204	C₂²×C₃₁	—	4	2	no	—
205	C₂²×C₃₁	—	8	1	no	—
206	C₂²×C₃₁	—	8	1	no	—
207	C₂²×C₃₁	—	8	1	no	—
208	C₂²×C₃₁	—	8	1	no	—
209	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
210	C₂²×C₃₁	#548	1	1	yes	characteristic
211	C₂²×C₃₁	#548	1	1	yes	characteristic
212	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
213	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₂²×C₃₁	#577	1	1	yes	characteristic
215	C₂²×C₃₁	#577	1	1	yes	characteristic
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	4	2	no	—
218	C₄×C₃₁	—	4	2	no	—
219	C₄×C₃₁	—	4	2	no	—
220	C₄×C₃₁	—	8	1	no	—
221	C₄×C₃₁	—	8	1	no	—
222	C₄×C₃₁	—	8	2	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	#482	1	4	yes	—
225	C₄×C₃₁	#529	1	1	yes	characteristic
226	C₄×C₃₁	#541	1	1	yes	characteristic
227	C₄×C₃₁	#552	1	1	yes	characteristic
228	C₄×C₃₁	#556	1	1	yes	characteristic
229	C₂×C₂²⋊Q₈	—	2	1	no	—
230	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#106	C₂²×C₃₁	1	2	yes	—
233	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#102	C₂²×C₃₁	1	2	yes	—
235	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
238	#396	—	2	2	no	—
239	#391	—	2	1	no	—
240	#390	C₂²×C₃₁	1	2	yes	—
241	#389	C₂²×C₃₁	1	2	yes	—
242	#387	—	2	2	no	—
243	#384	C₂²×C₃₁	1	1	yes	characteristic
244	#384	C₂²×C₃₁	1	1	yes	characteristic
245	#382	C₂²×C₃₁	1	2	yes	—
246	#380	—	2	1	no	—
247	C₄×C₄⋊C₄	—	2	1	no	—
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
250	#340	C₂²×C₃₁	1	2	yes	—
251	#340	C₂²×C₃₁	1	2	yes	—
252	#340	C₂²×C₃₁	1	2	yes	—
253	#340	C₂²×C₃₁	1	2	yes	—
254	#340	C₂²×C₃₁	1	1	yes	characteristic
255	#340	C₂²×C₃₁	1	1	yes	characteristic
256	#336	C₂²×C₃₁	1	2	yes	—
257	#336	C₂²×C₃₁	1	2	yes	—
258	C₂×C₃₁	—	8	2	no	—
259	C₂×C₃₁	#2421	1	1	yes	characteristic
260	C₂×C₃₁	C₂×#482	1	1	yes	characteristic
261	C₂×C₃₁	#3099	1	1	yes	characteristic
262	C₂×C₃₁	#3360	1	1	yes	characteristic
263	C₂×C₃₁	#3360	1	1	yes	characteristic
264	C₂×C₃₁	#3368	1	1	yes	characteristic
265	C₂×C₃₁	#3373	1	1	yes	characteristic
266	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
267	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
268	C₂³×C₄	—	2	1	no	—
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
272	C₂²⋊Q₈	—	4	4	no	—
273	C₄×Q₈	—	2	2	no	—
274	C₄×Q₈	—	2	2	no	—
275	C₄×D₄	—	2	2	no	—
276	C₄×D₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
295	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
296	#106	—	2	4	no	—
297	C₄⋊C₈	—	2	2	no	—
298	#102	—	2	4	no	—
299	#102	—	2	2	no	—
300	#102	—	2	2	no	—
301	C₄×C₈	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	—	2	1	no	—
304	#95	—	2	2	no	—
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	C₃₁	#24991	1	1	yes	characteristic, direct product factor, Sylow
308	C₂⁴	—	2	1	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	1	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	#41×C₃₁	1	2	yes	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	4	2	no	—
324	C₂²×C₄	—	4	2	no	—
325	C₂²×C₄	—	4	2	no	—
326	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
327	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
328	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	4	no	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	4	4	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	8	1	no	—
355	C₄⋊C₄	—	8	1	no	—
356	C₄⋊C₄	#41×C₃₁	1	2	yes	—
357	C₂²⋊C₄	—	4	4	no	—
358	C₂²⋊C₄	—	4	2	no	—
359	C₂²⋊C₄	—	8	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	4	2	no	—
363	C₄²	—	8	2	no	—
364	C₄²	#41×C₃₁	1	2	yes	—
365	C₄²	#41×C₃₁	1	2	yes	—
366	C₂³	—	2	2	no	—
367	C₂³	—	4	1	no	—
368	C₂³	—	4	1	no	—
369	C₂³	—	4	1	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	1	no	—
373	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	Q₈	—	4	4	no	—
375	D₄	—	2	2	no	—
376	D₄	—	2	2	no	—
377	D₄	—	4	1	no	—
378	D₄	—	4	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	8	4	no	—
403	C₂×C₄	C₃₁×#123	1	2	yes	—
404	C₂×C₄	C₃₁×#123	1	2	yes	—
405	C₂×C₄	C₃₁×#135	1	2	yes	—
406	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
407	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×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	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
412	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
413	C₈	—	4	2	no	—
414	C₈	—	4	1	no	—
415	C₈	—	4	1	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	8	1	no	—
420	C₂²	—	8	1	no	—
421	C₂²	—	8	1	no	—
422	C₂²	—	8	1	no	—
423	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
424	C₂²	C₃₁×#548	1	1	yes	characteristic
425	C₂²	C₃₁×#548	1	1	yes	characteristic
426	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
427	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
428	C₂²	C₃₁×#577	1	1	yes	characteristic
429	C₂²	C₃₁×#577	1	1	yes	characteristic
430	C₄	—	2	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	8	1	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	2	no	—
437	C₄	—	8	2	no	—
438	C₄	C₃₁×#482	1	4	yes	—
439	C₄	C₃₁×#529	1	1	yes	characteristic
440	C₄	C₃₁×#541	1	1	yes	characteristic
441	C₄	C₃₁×#552	1	1	yes	characteristic
442	C₄	C₃₁×#556	1	1	yes	characteristic
443	C₂	—	8	2	no	—
444	C₂	C₃₁×#2421	1	1	yes	characteristic
445	C₂	C₂×C₃₁×#482	1	1	yes	characteristic
446	C₂	C₃₁×#3099	1	1	yes	characteristic
447	C₂	C₃₁×#3360	1	1	yes	characteristic
448	C₂	C₃₁×#3360	1	1	yes	characteristic
449	C₂	C₃₁×#3368	1	1	yes	characteristic
450	C₂	C₃₁×#3373	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 ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2355	C₂	semidirect
3. C₃₁×#2101	C₂	semidirect
5. C₃₁×#1984	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
8. C₃₁×#1920	C₂	semidirect
9. C₃₁×#1834	C₂	semidirect
140. #24991	C₃₁	direct (N × H)
192. #2355	C₂×C₃₁	semidirect
193. #2101	C₂×C₃₁	semidirect
195. #1984	C₂×C₃₁	semidirect
196. #1948	C₂×C₃₁	semidirect
198. #1920	C₂×C₃₁	semidirect
199. #1834	C₂×C₃₁	semidirect
307. C₃₁	#24991	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×#2538	2
C₃₁×#2355	2
C₃₁×#2101	2
C₃₁×#2073	2
C₃₁×#1984	2
C₃₁×#1948	2
C₃₁×#1926	2
C₃₁×#1920	2
C₃₁×#1834	2
C₃₁×#1802	2
C₄×C₃₁×#102	2
#24991	31

Subgroups — C31×#24991 #21481750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24991 #21481750