Subgroups — C₃₁×#25048 #21481807

Order 7936.

888 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×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2420	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2073	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2033	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1809	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#121	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#469	—	2	4	no	—
17	C₃₁×#442	—	2	4	no	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#397	—	2	2	no	—
29	C₃₁×#394	—	2	2	no	—
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#386	—	2	2	no	—
33	C₃₁×#382	C₂²	1	2	yes	—
34	C₃₁×#381	—	2	2	no	—
35	C₃₁×#380	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
43	C₂³×C₄×C₃₁	—	2	2	no	—
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	D₄	1	4	yes	—
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#121	—	2	4	no	—
56	C₃₁×#121	—	4	2	no	—
57	C₄×D₄×C₃₁	—	2	4	no	—
58	C₄×D₄×C₃₁	—	4	2	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×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#107	—	2	2	no	—
72	C₃₁×#107	—	4	2	no	—
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×#102	—	2	4	no	—
76	C₃₁×#102	—	2	4	no	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₃₁×#97	—	2	2	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	D₄	1	2	yes	—
86	C₂⁴×C₃₁	—	2	2	no	—
87	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×D₄×C₃₁	—	2	2	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	—	4	4	no	—
94	C₂×D₄×C₃₁	—	4	4	no	—
95	C₂×D₄×C₃₁	—	8	2	no	—
96	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×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	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	4	4	no	—
107	C₂²×C₄×C₃₁	—	4	4	no	—
108	C₂²×C₄×C₃₁	—	8	2	no	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
113	D₈×C₃₁	—	2	8	no	—
114	M₁₆×C₃₁	—	4	4	no	—
115	C₂×C₈×C₃₁	—	2	1	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	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₃₁	#41	1	2	yes	—
125	C₂×C₈×C₃₁	#41	1	2	yes	—
126	C₂×C₈×C₃₁	#41	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	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	8	2	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	4	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	1	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	#25048	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	2	4	no	—
147	C₂³×C₃₁	—	2	1	no	—
148	C₂³×C₃₁	—	2	1	no	—
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	—	2	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	—	4	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
158	D₄×C₃₁	—	2	4	no	—
159	D₄×C₃₁	—	2	4	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	8	8	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	1	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	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	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	—	8	8	no	—
186	C₂×C₄×C₃₁	C₂²×D₄	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	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
192	C₈×C₃₁	—	2	2	no	—
193	C₈×C₃₁	—	4	1	no	—
194	C₈×C₃₁	—	4	1	no	—
195	C₈×C₃₁	#123	1	4	yes	—
196	C₂×#469	C₂×C₃₁	1	1	yes	characteristic
197	C₂×#442	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2420	C₂×C₃₁	1	2	yes	complemented
199	#2073	C₂×C₃₁	1	2	yes	complemented
200	#2033	C₂×C₃₁	1	2	yes	complemented
201	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1943	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1862	C₂×C₃₁	1	2	yes	complemented
204	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1809	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1753	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	—	4	1	no	—
210	C₂²×C₃₁	—	4	1	no	—
211	C₂²×C₃₁	—	4	4	no	—
212	C₂²×C₃₁	—	4	4	no	—
213	C₂²×C₃₁	—	4	2	no	—
214	C₂²×C₃₁	—	4	4	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	#532	1	1	yes	characteristic
220	C₂²×C₃₁	D₄²	1	1	yes	characteristic
221	C₂²×C₃₁	#547	1	1	yes	characteristic
222	C₂²×C₃₁	#575	1	2	yes	—
223	C₂²×C₃₁	#576	1	2	yes	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	—	4	1	no	—
226	C₄×C₃₁	—	4	2	no	—
227	C₄×C₃₁	—	8	2	no	—
228	C₄×C₃₁	—	8	2	no	—
229	C₄×C₃₁	—	8	2	no	—
230	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
231	C₄×C₃₁	#542	1	1	yes	characteristic
232	C₄×C₃₁	#547	1	1	yes	characteristic
233	C₄×C₃₁	#553	1	1	yes	characteristic
234	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
235	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#121	C₂²×C₃₁	1	2	yes	—
237	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
238	#469	—	2	4	no	—
239	#442	—	2	4	no	—
240	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#107	C₂²×C₃₁	1	2	yes	—
242	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#102	C₂²×C₃₁	1	2	yes	—
245	C₂×#102	C₂²×C₃₁	1	2	yes	—
246	C₂×#102	C₂²×C₃₁	1	2	yes	—
247	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
250	#397	—	2	2	no	—
251	#394	—	2	2	no	—
252	#390	C₂²×C₃₁	1	2	yes	—
253	#389	C₂²×C₃₁	1	2	yes	—
254	#386	—	2	2	no	—
255	#382	C₂²×C₃₁	1	2	yes	—
256	#381	—	2	2	no	—
257	#380	C₂²×C₃₁	1	2	yes	—
258	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
259	#340	C₂²×C₃₁	1	2	yes	—
260	#340	C₂²×C₃₁	1	2	yes	—
261	#336	C₂²×C₃₁	1	1	yes	characteristic
262	#336	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₃₁	—	8	1	no	—
264	C₂×C₃₁	—	8	4	no	—
265	C₂×C₃₁	#2422	1	1	yes	characteristic
266	C₂×C₃₁	#3176	1	1	yes	characteristic
267	C₂×C₃₁	#3183	1	1	yes	characteristic
268	C₂×C₃₁	#3273	1	2	yes	—
269	C₂×C₃₁	#3347	1	2	yes	—
270	C₂²×D₄	—	2	2	no	—
271	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
272	C₂³×C₄	—	2	2	no	—
273	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×D₈	—	2	2	no	—
275	C₂×D₈	—	2	2	no	—
276	C₂×D₈	D₄×C₃₁	1	4	yes	—
277	C₂×M₁₆	—	2	2	no	—
278	C₂×M₁₆	—	2	2	no	—
279	C₂×M₁₆	—	2	2	no	—
280	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
284	#121	—	2	4	no	—
285	#121	—	4	2	no	—
286	C₄×D₄	—	2	4	no	—
287	C₄×D₄	—	4	2	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	2	no	—
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
298	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
300	#107	—	2	2	no	—
301	#107	—	4	2	no	—
302	#106	—	2	2	no	—
303	#106	—	2	2	no	—
304	#102	—	2	4	no	—
305	#102	—	2	4	no	—
306	#102	—	2	4	no	—
307	C₂²⋊C₈	—	2	2	no	—
308	#97	—	2	2	no	—
309	C₄×C₈	—	2	2	no	—
310	#95	—	2	2	no	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	2	no	—
314	#95	D₄×C₃₁	1	2	yes	—
315	C₃₁	#25048	1	1	yes	characteristic, direct product factor, Sylow
316	C₂⁴	—	2	2	no	—
317	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	4	4	no	—
324	C₂×D₄	—	4	4	no	—
325	C₂×D₄	—	8	2	no	—
326	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	4	4	no	—
337	C₂²×C₄	—	4	4	no	—
338	C₂²×C₄	—	8	2	no	—
339	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
342	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
343	D₈	—	2	8	no	—
344	M₁₆	—	4	4	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
354	C₂×C₈	#41×C₃₁	1	2	yes	—
355	C₂×C₈	#41×C₃₁	1	2	yes	—
356	C₂×C₈	#41×C₃₁	1	2	yes	—
357	C₄⋊C₄	—	2	2	no	—
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	—	2	2	no	—
363	C₄⋊C₄	—	8	2	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₂²⋊C₄	—	4	4	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	1	no	—
371	C₄²	—	2	1	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	4	no	—
376	C₂³	—	2	1	no	—
377	C₂³	—	2	1	no	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	4	2	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	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	D₄	—	2	4	no	—
388	D₄	—	2	4	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	2	no	—
391	D₄	—	8	8	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	8	8	no	—
415	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
420	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
421	C₈	—	2	2	no	—
422	C₈	—	4	1	no	—
423	C₈	—	4	1	no	—
424	C₈	C₃₁×#123	1	4	yes	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	1	no	—
429	C₂²	—	4	4	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	4	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	C₃₁×#532	1	1	yes	characteristic
438	C₂²	D₄²×C₃₁	1	1	yes	characteristic
439	C₂²	C₃₁×#547	1	1	yes	characteristic
440	C₂²	C₃₁×#575	1	2	yes	—
441	C₂²	C₃₁×#576	1	2	yes	—
442	C₄	—	4	2	no	—
443	C₄	—	4	1	no	—
444	C₄	—	4	2	no	—
445	C₄	—	8	2	no	—
446	C₄	—	8	2	no	—
447	C₄	—	8	2	no	—
448	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
449	C₄	C₃₁×#542	1	1	yes	characteristic
450	C₄	C₃₁×#547	1	1	yes	characteristic
451	C₄	C₃₁×#553	1	1	yes	characteristic
452	C₂	—	8	1	no	—
453	C₂	—	8	4	no	—
454	C₂	C₃₁×#2422	1	1	yes	characteristic
455	C₂	C₃₁×#3176	1	1	yes	characteristic
456	C₂	C₃₁×#3183	1	1	yes	characteristic
457	C₂	C₃₁×#3273	1	2	yes	—
458	C₂	C₃₁×#3347	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#442	C₂	semidirect
3. C₃₁×#2420	C₂	semidirect
4. C₃₁×#2073	C₂	semidirect
5. C₃₁×#2033	C₂	semidirect
6. C₃₁×#1970	C₂	semidirect
7. C₃₁×#1943	C₂	semidirect
8. C₃₁×#1862	C₂	semidirect
9. C₃₁×#1846	C₂	semidirect
10. C₃₁×#1809	C₂	semidirect
11. C₃₁×#1753	C₂	semidirect
144. #25048	C₃₁	direct (N × H)
197. C₂×#442	C₂×C₃₁	semidirect
198. #2420	C₂×C₃₁	semidirect
199. #2073	C₂×C₃₁	semidirect
200. #2033	C₂×C₃₁	semidirect
201. #1970	C₂×C₃₁	semidirect
202. #1943	C₂×C₃₁	semidirect
203. #1862	C₂×C₃₁	semidirect
204. #1846	C₂×C₃₁	semidirect
205. #1809	C₂×C₃₁	semidirect
206. #1753	C₂×C₃₁	semidirect
315. C₃₁	#25048	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×#469	2
C₂×C₃₁×#442	2
C₃₁×#2420	2
C₃₁×#2073	2
C₃₁×#2033	2
C₃₁×#1970	2
C₃₁×#1943	2
C₃₁×#1862	2
C₃₁×#1846	2
C₃₁×#1809	2
C₃₁×#1753	2
#25048	31

Subgroups — C31×#25048 #21481807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25048 #21481807