Subgroups — C₃₁×#22181 #21478940

Order 7936.

1062 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#121	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1893	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#121	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	—	2	2	no	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#466	C₄	1	8	yes	complemented
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
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	C₃₁×#390	C₂²	1	2	yes	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#388	—	2	2	no	—
27	C₃₁×#386	—	2	2	no	—
28	C₃₁×#380	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₃₁	—	4	1	no	—
43	C₂×D₈×C₃₁	—	4	1	no	—
44	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	—	4	1	no	—
47	C₂²×C₈×C₃₁	—	4	1	no	—
48	C₂²×C₈×C₃₁	—	4	1	no	—
49	C₂²×C₈×C₃₁	—	4	1	no	—
50	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#121	—	2	4	no	—
55	C₃₁×#121	C₂×C₄	1	8	yes	complemented
56	C₄×D₄×C₃₁	—	4	2	no	—
57	C₄×D₄×C₃₁	—	4	2	no	—
58	C₄×D₄×C₃₁	—	4	8	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	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	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	4	1	no	—
70	C₂×C₄²×C₃₁	—	4	1	no	—
71	C₂×C₄²×C₃₁	D₄	1	2	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
73	C₃₁×#107	—	4	1	no	—
74	C₃₁×#107	—	4	1	no	—
75	C₃₁×#107	C₂×C₄	1	4	yes	complemented
76	C₃₁×#102	—	4	2	no	—
77	C₃₁×#102	—	4	2	no	—
78	C₃₁×#102	C₂×C₄	1	8	yes	complemented
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₂⁴×C₃₁	—	2	2	no	—
83	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×D₄×C₃₁	—	2	8	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	4	4	no	—
91	C₂×D₄×C₃₁	—	4	4	no	—
92	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	4	1	no	—
102	C₂²×C₄×C₃₁	—	4	1	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	—	4	1	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	—	4	4	no	—
107	C₂²×C₄×C₃₁	—	4	4	no	—
108	C₂²×C₄×C₃₁	D₈	1	4	yes	complemented
109	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	#41	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
115	D₈×C₃₁	—	4	4	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	4	4	no	—
124	C₂×C₈×C₃₁	—	4	4	no	—
125	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
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	4	no	—
133	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
134	C₂²⋊C₄×C₃₁	—	2	8	no	—
135	C₂²⋊C₄×C₃₁	—	4	4	no	—
136	C₂²⋊C₄×C₃₁	—	4	4	no	—
137	C₂²⋊C₄×C₃₁	—	4	4	no	—
138	C₂²⋊C₄×C₃₁	—	4	4	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	4	2	no	—
142	C₄²×C₃₁	—	4	4	no	—
143	#22181	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	1	no	—
148	C₂³×C₃₁	—	2	4	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	—	4	2	no	—
153	C₂³×C₃₁	—	4	2	no	—
154	C₂³×C₃₁	—	4	2	no	—
155	C₂³×C₃₁	C₄×D₄	1	2	yes	—
156	C₂³×C₃₁	C₂×D₈	1	2	yes	—
157	C₂³×C₃₁	#135	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	16	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	2	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	2	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	4	4	no	—
177	C₂×C₄×C₃₁	—	4	4	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	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	—	8	8	no	—
187	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
189	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	2	2	no	—
197	C₈×C₃₁	—	4	2	no	—
198	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
199	C₄×#121	C₂×C₃₁	1	2	yes	complemented
200	#1925	C₂×C₃₁	1	2	yes	complemented
201	#1918	C₂×C₃₁	1	2	yes	complemented
202	#1893	C₂×C₃₁	1	2	yes	complemented
203	#1879	C₂×C₃₁	1	2	yes	complemented
204	#1862	C₂×C₃₁	1	2	yes	complemented
205	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	2	2	no	—
209	C₂²×C₃₁	—	2	2	no	—
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	4	4	no	—
214	C₂²×C₃₁	—	4	4	no	—
215	C₂²×C₃₁	—	4	4	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	—	8	2	no	—
219	C₂²×C₃₁	—	8	2	no	—
220	C₂²×C₃₁	C₄×D₈	1	4	yes	—
221	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₃₁	#534	1	1	yes	characteristic
223	C₂²×C₃₁	#537	1	1	yes	characteristic
224	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
225	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
226	C₂²×C₃₁	#575	1	1	yes	characteristic
227	C₂²×C₃₁	#575	1	1	yes	characteristic
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	2	4	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	4	no	—
232	C₄×C₃₁	—	8	2	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	#518	1	1	yes	characteristic
235	C₄×C₃₁	#520	1	1	yes	characteristic
236	C₄×C₃₁	#538	1	1	yes	characteristic
237	C₄×C₃₁	#539	1	1	yes	characteristic
238	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
240	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#121	C₂²×C₃₁	1	2	yes	—
242	C₂×C₄×D₄	—	2	2	no	—
243	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
244	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
245	#466	C₄×C₃₁	1	8	yes	complemented
246	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#102	C₂²×C₃₁	1	2	yes	—
250	C₂×#102	C₂²×C₃₁	1	2	yes	—
251	C₂×#102	C₂²×C₃₁	1	2	yes	—
252	#390	C₂²×C₃₁	1	2	yes	—
253	#389	C₂²×C₃₁	1	2	yes	—
254	#388	—	2	2	no	—
255	#386	—	2	2	no	—
256	#380	C₂²×C₃₁	1	2	yes	—
257	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
258	C₄×C₂²⋊C₄	—	2	2	no	—
259	#340	C₂²×C₃₁	1	2	yes	—
260	#340	C₂²×C₃₁	1	2	yes	—
261	#340	C₂²×C₃₁	1	2	yes	—
262	#336	C₂²×C₃₁	1	2	yes	—
263	#336	C₂²×C₃₁	1	2	yes	—
264	C₂×C₃₁	—	2	4	no	—
265	C₂×C₃₁	—	8	4	no	—
266	C₂×C₃₁	#2325	1	1	yes	characteristic
267	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
268	C₂×C₃₁	#2943	1	1	yes	characteristic
269	C₂×C₃₁	#3186	1	1	yes	characteristic
270	C₂×C₃₁	#3191	1	1	yes	characteristic
271	C₂×C₃₁	#3245	1	1	yes	characteristic
272	C₂×C₃₁	#3250	1	1	yes	characteristic
273	C₂²×D₄	—	2	2	no	—
274	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
275	C₂³×C₄	—	2	2	no	—
276	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
277	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
279	C₂×D₈	—	4	1	no	—
280	C₂×D₈	—	4	1	no	—
281	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
282	C₂²×C₈	—	2	2	no	—
283	C₂²×C₈	—	4	1	no	—
284	C₂²×C₈	—	4	1	no	—
285	C₂²×C₈	—	4	1	no	—
286	C₂²×C₈	—	4	1	no	—
287	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
288	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	#121	—	2	4	no	—
292	#121	C₂×C₄×C₃₁	1	8	yes	complemented
293	C₄×D₄	—	4	2	no	—
294	C₄×D₄	—	4	2	no	—
295	C₄×D₄	—	4	8	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₄²	—	2	2	no	—
306	C₂×C₄²	—	4	1	no	—
307	C₂×C₄²	—	4	1	no	—
308	C₂×C₄²	D₄×C₃₁	1	2	yes	—
309	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
310	#107	—	4	1	no	—
311	#107	—	4	1	no	—
312	#107	C₂×C₄×C₃₁	1	4	yes	complemented
313	#102	—	4	2	no	—
314	#102	—	4	2	no	—
315	#102	C₂×C₄×C₃₁	1	8	yes	complemented
316	#95	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	2	no	—
319	C₃₁	#22181	1	1	yes	characteristic, direct product factor, Sylow
320	C₂⁴	—	2	2	no	—
321	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂×D₄	—	2	8	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	4	4	no	—
329	C₂×D₄	—	4	4	no	—
330	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	4	1	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	—	4	1	no	—
342	C₂²×C₄	—	4	1	no	—
343	C₂²×C₄	—	4	4	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	—	4	4	no	—
346	C₂²×C₄	D₈×C₃₁	1	4	yes	complemented
347	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
351	C₂²×C₄	#41×C₃₁	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
353	D₈	—	4	4	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	4	4	no	—
362	C₂×C₈	—	4	4	no	—
363	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
364	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
372	C₂²⋊C₄	—	2	8	no	—
373	C₂²⋊C₄	—	4	4	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	4	2	no	—
380	C₄²	—	4	4	no	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	4	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	C₄×D₄×C₃₁	1	2	yes	—
393	C₂³	C₂×D₈×C₃₁	1	2	yes	—
394	C₂³	C₃₁×#135	1	2	yes	—
395	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	D₄	—	2	4	no	—
397	D₄	—	2	4	no	—
398	D₄	—	4	16	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	4	4	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	8	8	no	—
424	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
425	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
426	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
429	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
430	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
431	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
432	C₈	—	2	2	no	—
433	C₈	—	2	2	no	—
434	C₈	—	4	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	2	2	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	C₄×D₈×C₃₁	1	4	yes	—
449	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
450	C₂²	C₃₁×#534	1	1	yes	characteristic
451	C₂²	C₃₁×#537	1	1	yes	characteristic
452	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
453	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
454	C₂²	C₃₁×#575	1	1	yes	characteristic
455	C₂²	C₃₁×#575	1	1	yes	characteristic
456	C₄	—	2	2	no	—
457	C₄	—	2	4	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	4	no	—
460	C₄	—	8	2	no	—
461	C₄	—	8	2	no	—
462	C₄	C₃₁×#518	1	1	yes	characteristic
463	C₄	C₃₁×#520	1	1	yes	characteristic
464	C₄	C₃₁×#538	1	1	yes	characteristic
465	C₄	C₃₁×#539	1	1	yes	characteristic
466	C₂	—	2	4	no	—
467	C₂	—	8	4	no	—
468	C₂	C₃₁×#2325	1	1	yes	characteristic
469	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
470	C₂	C₃₁×#2943	1	1	yes	characteristic
471	C₂	C₃₁×#3186	1	1	yes	characteristic
472	C₂	C₃₁×#3191	1	1	yes	characteristic
473	C₂	C₃₁×#3245	1	1	yes	characteristic
474	C₂	C₃₁×#3250	1	1	yes	characteristic

	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×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×#121	C₂	semidirect
3. C₃₁×#1925	C₂	semidirect
4. C₃₁×#1918	C₂	semidirect
5. C₃₁×#1893	C₂	semidirect
6. C₃₁×#1879	C₂	semidirect
7. C₃₁×#1862	C₂	semidirect
8. C₃₁×#1808	C₂	semidirect
9. C₂×C₃₁×#336	C₂	semidirect
17. C₃₁×#466	C₄	semidirect
44. C₂×D₈×C₃₁	C₂×C₄	semidirect
50. C₂²×C₈×C₃₁	C₂×C₄	semidirect
55. C₃₁×#121	C₂×C₄	semidirect
75. C₃₁×#107	C₂×C₄	semidirect
78. C₃₁×#102	C₂×C₄	semidirect
108. C₂²×C₄×C₃₁	D₈	semidirect
143. #22181	C₃₁	direct (N × H)
199. C₄×#121	C₂×C₃₁	semidirect
200. #1925	C₂×C₃₁	semidirect
201. #1918	C₂×C₃₁	semidirect
202. #1893	C₂×C₃₁	semidirect
203. #1879	C₂×C₃₁	semidirect
204. #1862	C₂×C₃₁	semidirect
205. #1808	C₂×C₃₁	semidirect
206. C₂×#336	C₂×C₃₁	semidirect
245. #466	C₄×C₃₁	semidirect
281. C₂×D₈	C₂×C₄×C₃₁	semidirect
287. C₂²×C₈	C₂×C₄×C₃₁	semidirect
292. #121	C₂×C₄×C₃₁	semidirect
312. #107	C₂×C₄×C₃₁	semidirect
315. #102	C₂×C₄×C₃₁	semidirect
319. C₃₁	#22181	direct (N × H)
346. C₂²×C₄	D₈×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#466	2
C₄×C₃₁×#121	2
C₃₁×#1925	2
C₃₁×#1918	2
C₃₁×#1893	2
C₃₁×#1879	2
C₃₁×#1862	2
C₃₁×#1808	2
C₂×C₃₁×#336	2
#22181	31

Subgroups — C31×#22181 #21478940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22181 #21478940