Subgroups — C₃₁×#25432 #21482191

Order 7936.

912 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3058	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3043	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2046	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2004	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1888	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
17	C₃₁×#535	C₂²	1	2	yes	—
18	C₂×C₃₁×#124	C₂²	1	2	yes	—
19	C₃₁×#519	C₂²	1	2	yes	—
20	C₃₁×#495	—	2	4	no	—
21	C₃₁×#473	—	2	4	no	—
22	C₃₁×#469	—	2	2	no	—
23	C₃₁×#466	—	2	2	no	—
24	C₃₁×#464	—	2	4	no	—
25	C₃₁×#460	—	2	4	no	—
26	C₃₁×#441	—	2	2	no	—
27	C₄×Q₁₆×C₃₁	—	2	2	no	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₃₁×#427	C₂²	1	1	yes	characteristic
30	C₃₁×#424	C₂²	1	2	yes	—
31	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
32	C₃₁×#419	C₂²	1	2	yes	—
33	C₃₁×#418	C₂²	1	2	yes	—
34	C₃₁×#418	C₂²	1	2	yes	—
35	C₃₁×#416	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#101	C₂²	1	2	yes	—
37	C₂×C₃₁×#100	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#341	C₂²	1	1	yes	characteristic
41	C₃₁×#339	C₂²	1	1	yes	characteristic
42	C₃₁×#339	C₂²	1	2	yes	—
43	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
46	C₃₁×#137	—	2	4	no	—
47	C₃₁×#137	—	4	2	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	1	no	—
51	C₂×Q₁₆×C₃₁	—	2	2	no	—
52	C₂×Q₁₆×C₃₁	D₄	1	4	yes	complemented
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	4	no	—
56	C₂×SD₁₆×C₃₁	—	4	2	no	—
57	C₂×D₈×C₃₁	—	2	1	no	—
58	C₂×D₈×C₃₁	—	2	1	no	—
59	C₂×D₈×C₃₁	—	2	2	no	—
60	C₂×D₈×C₃₁	—	2	2	no	—
61	C₃₁×#131	—	2	2	no	—
62	C₃₁×#131	—	2	2	no	—
63	C₂×M₁₆×C₃₁	—	2	1	no	—
64	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
66	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	—	2	1	no	—
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#127	—	2	2	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#124	—	2	4	no	—
73	C₃₁×#124	—	2	4	no	—
74	C₃₁×#121	—	2	4	no	—
75	C₃₁×#120	—	4	2	no	—
76	C₄×Q₈×C₃₁	—	2	4	no	—
77	C₄×Q₈×C₃₁	—	2	4	no	—
78	C₃₁×#117	—	4	2	no	—
79	C₃₁×#117	C₂³	1	1	yes	characteristic
80	C₃₁×#117	C₂³	1	2	yes	—
81	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#108	—	2	2	no	—
84	C₃₁×#107	D₄	1	2	yes	—
85	C₃₁×#106	D₄	1	2	yes	—
86	C₃₁×C₄⋊C₈	—	2	4	no	—
87	C₃₁×#104	—	2	2	no	—
88	C₃₁×#103	—	2	4	no	—
89	C₃₁×#103	D₄	1	2	yes	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	D₄	1	2	yes	—
93	C₃₁×#101	—	2	4	no	—
94	C₃₁×#100	—	4	2	no	—
95	C₃₁×#97	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	2	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	#41×C₃₁	—	2	1	no	—
100	#41×C₃₁	—	2	1	no	—
101	#41×C₃₁	—	2	1	no	—
102	#41×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
109	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	4	4	no	—
118	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	Q₁₆×C₃₁	—	2	4	no	—
124	Q₁₆×C₃₁	—	2	4	no	—
125	Q₁₆×C₃₁	—	4	4	no	—
126	SD₁₆×C₃₁	—	4	8	no	—
127	SD₁₆×C₃₁	—	4	4	no	—
128	D₈×C₃₁	—	4	2	no	—
129	D₈×C₃₁	—	4	2	no	—
130	M₁₆×C₃₁	—	2	1	no	—
131	M₁₆×C₃₁	—	2	2	no	—
132	M₁₆×C₃₁	—	2	1	no	—
133	M₁₆×C₃₁	—	2	1	no	—
134	M₁₆×C₃₁	—	2	1	no	—
135	M₁₆×C₃₁	—	2	2	no	—
136	M₁₆×C₃₁	—	4	2	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	4	2	no	—
143	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₃₁	#41	1	2	yes	—
148	C₄⋊C₄×C₃₁	—	4	4	no	—
149	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
151	C₂²⋊C₄×C₃₁	—	2	1	no	—
152	C₂²⋊C₄×C₃₁	—	2	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	8	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	4	4	no	—
159	C₄²×C₃₁	#41	1	2	yes	—
160	#25432	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	8	4	no	—
166	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
167	Q₈×C₃₁	—	2	4	no	—
168	Q₈×C₃₁	—	2	1	no	—
169	Q₈×C₃₁	—	2	4	no	—
170	Q₈×C₃₁	—	2	1	no	—
171	Q₈×C₃₁	—	4	2	no	—
172	Q₈×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	2	1	no	—
174	D₄×C₃₁	—	2	1	no	—
175	D₄×C₃₁	—	4	1	no	—
176	D₄×C₃₁	—	4	1	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	16	4	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	4	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
198	C₈×C₃₁	—	2	2	no	—
199	C₈×C₃₁	—	4	1	no	—
200	C₈×C₃₁	—	4	2	no	—
201	C₈×C₃₁	—	4	1	no	—
202	C₈×C₃₁	#121	1	2	yes	—
203	C₈×C₃₁	#121	1	2	yes	—
204	C₂×#495	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#3058	C₂×C₃₁	1	1	yes	characteristic
207	#3043	C₂×C₃₁	1	2	yes	complemented
208	#2950	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#2046	C₂×C₃₁	1	2	yes	complemented
210	#2004	C₂×C₃₁	1	2	yes	complemented
211	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1888	C₂×C₃₁	1	2	yes	complemented
214	#1764	C₂×C₃₁	1	1	yes	characteristic, complemented
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	8	4	no	—
219	C₂²×C₃₁	—	8	2	no	—
220	C₂²×C₃₁	—	8	4	no	—
221	C₂²×C₃₁	—	16	2	no	—
222	C₂²×C₃₁	#540	1	1	yes	characteristic
223	C₂²×C₃₁	D₄²	1	1	yes	characteristic
224	C₂²×C₃₁	#546	1	1	yes	characteristic
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	4	1	no	—
227	C₄×C₃₁	—	4	4	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	4	4	no	—
230	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
231	C₄×C₃₁	#532	1	1	yes	characteristic
232	C₄×C₃₁	D₄²	1	1	yes	characteristic
233	C₄×C₃₁	#550	1	1	yes	characteristic
234	C₂×#137	C₂²×C₃₁	1	2	yes	—
235	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
236	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
237	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
239	#535	C₂²×C₃₁	1	2	yes	—
240	C₂×#124	C₂²×C₃₁	1	2	yes	—
241	#519	C₂²×C₃₁	1	2	yes	—
242	#495	—	2	4	no	—
243	#473	—	2	4	no	—
244	#469	—	2	2	no	—
245	#466	—	2	2	no	—
246	#464	—	2	4	no	—
247	#460	—	2	4	no	—
248	#441	—	2	2	no	—
249	C₄×Q₁₆	—	2	2	no	—
250	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
251	#427	C₂²×C₃₁	1	1	yes	characteristic
252	#424	C₂²×C₃₁	1	2	yes	—
253	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
254	#419	C₂²×C₃₁	1	2	yes	—
255	#418	C₂²×C₃₁	1	2	yes	—
256	#418	C₂²×C₃₁	1	2	yes	—
257	#416	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#101	C₂²×C₃₁	1	2	yes	—
259	C₂×#100	C₂²×C₃₁	1	2	yes	—
260	#405	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
262	#341	C₂²×C₃₁	1	1	yes	characteristic
263	#339	C₂²×C₃₁	1	1	yes	characteristic
264	#339	C₂²×C₃₁	1	2	yes	—
265	C₂×C₃₁	—	2	1	no	—
266	C₂×C₃₁	—	2	1	no	—
267	C₂×C₃₁	—	8	1	no	—
268	C₂×C₃₁	—	16	2	no	—
269	C₂×C₃₁	#2428	1	1	yes	characteristic
270	C₂×C₃₁	#3290	1	2	yes	—
271	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
272	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
273	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
274	#137	—	2	4	no	—
275	#137	—	4	2	no	—
276	C₂×Q₁₆	—	2	2	no	—
277	C₂×Q₁₆	—	2	1	no	—
278	C₂×Q₁₆	—	2	1	no	—
279	C₂×Q₁₆	—	2	2	no	—
280	C₂×Q₁₆	D₄×C₃₁	1	4	yes	complemented
281	C₂×SD₁₆	—	2	4	no	—
282	C₂×SD₁₆	—	2	2	no	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×SD₁₆	—	4	2	no	—
285	C₂×D₈	—	2	1	no	—
286	C₂×D₈	—	2	1	no	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	—	2	2	no	—
289	#131	—	2	2	no	—
290	#131	—	2	2	no	—
291	C₂×M₁₆	—	2	1	no	—
292	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
294	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×C₈	—	2	1	no	—
296	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	#127	—	2	2	no	—
299	#124	—	2	4	no	—
300	#124	—	2	2	no	—
301	#124	—	2	4	no	—
302	#121	—	2	4	no	—
303	#120	—	4	2	no	—
304	C₄×Q₈	—	2	4	no	—
305	C₄×Q₈	—	2	4	no	—
306	#117	—	4	2	no	—
307	#117	C₂³×C₃₁	1	1	yes	characteristic
308	#117	C₂³×C₃₁	1	2	yes	—
309	C₂×C₂²⋊C₄	—	4	2	no	—
310	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
311	#108	—	2	2	no	—
312	#107	D₄×C₃₁	1	2	yes	—
313	#106	D₄×C₃₁	1	2	yes	—
314	C₄⋊C₈	—	2	4	no	—
315	#104	—	2	2	no	—
316	#103	—	2	4	no	—
317	#103	D₄×C₃₁	1	2	yes	—
318	#102	—	2	4	no	—
319	#102	—	2	4	no	—
320	#102	D₄×C₃₁	1	2	yes	—
321	#101	—	2	4	no	—
322	#100	—	4	2	no	—
323	#97	—	2	1	no	—
324	C₄×C₈	—	2	2	no	—
325	C₄×C₈	—	2	1	no	—
326	C₃₁	#25432	1	1	yes	characteristic, direct product factor, Sylow
327	C₂⁴	—	2	2	no	—
328	#41	—	2	1	no	—
329	#41	—	2	1	no	—
330	#41	—	2	1	no	—
331	#41	—	2	1	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
338	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	4	4	no	—
347	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
352	Q₁₆	—	2	4	no	—
353	Q₁₆	—	2	4	no	—
354	Q₁₆	—	4	4	no	—
355	SD₁₆	—	4	8	no	—
356	SD₁₆	—	4	4	no	—
357	D₈	—	4	2	no	—
358	D₈	—	4	2	no	—
359	M₁₆	—	2	1	no	—
360	M₁₆	—	2	2	no	—
361	M₁₆	—	2	1	no	—
362	M₁₆	—	2	1	no	—
363	M₁₆	—	2	1	no	—
364	M₁₆	—	2	2	no	—
365	M₁₆	—	4	2	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	4	2	no	—
372	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
373	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
375	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂×C₈	#41×C₃₁	1	2	yes	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
379	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
380	C₂²⋊C₄	—	2	1	no	—
381	C₂²⋊C₄	—	2	2	no	—
382	C₂²⋊C₄	—	4	8	no	—
383	C₂²⋊C₄	—	4	4	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	4	4	no	—
388	C₄²	#41×C₃₁	1	2	yes	—
389	C₂³	—	2	1	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	8	4	no	—
394	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
395	Q₈	—	2	4	no	—
396	Q₈	—	2	1	no	—
397	Q₈	—	2	4	no	—
398	Q₈	—	2	1	no	—
399	Q₈	—	4	2	no	—
400	Q₈	—	4	2	no	—
401	D₄	—	2	1	no	—
402	D₄	—	2	1	no	—
403	D₄	—	4	1	no	—
404	D₄	—	4	1	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	2	no	—
408	D₄	—	4	2	no	—
409	D₄	—	16	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
426	C₈	—	2	2	no	—
427	C₈	—	4	1	no	—
428	C₈	—	4	2	no	—
429	C₈	—	4	1	no	—
430	C₈	C₃₁×#121	1	2	yes	—
431	C₈	C₃₁×#121	1	2	yes	—
432	C₂²	—	2	2	no	—
433	C₂²	—	4	1	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	8	4	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	4	no	—
438	C₂²	—	16	2	no	—
439	C₂²	C₃₁×#540	1	1	yes	characteristic
440	C₂²	D₄²×C₃₁	1	1	yes	characteristic
441	C₂²	C₃₁×#546	1	1	yes	characteristic
442	C₄	—	4	2	no	—
443	C₄	—	4	1	no	—
444	C₄	—	4	4	no	—
445	C₄	—	4	4	no	—
446	C₄	—	4	2	no	—
447	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
448	C₄	C₃₁×#532	1	1	yes	characteristic
449	C₄	D₄²×C₃₁	1	1	yes	characteristic
450	C₄	C₃₁×#550	1	1	yes	characteristic
451	C₂	—	2	1	no	—
452	C₂	—	2	1	no	—
453	C₂	—	8	1	no	—
454	C₂	—	16	2	no	—
455	C₂	C₃₁×#2428	1	1	yes	characteristic
456	C₂	C₃₁×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#495	C₂	semidirect
2. C₂×C₃₁×#473	C₂	semidirect
4. C₃₁×#3043	C₂	semidirect
5. C₃₁×#2950	C₂	semidirect
6. C₃₁×#2046	C₂	semidirect
7. C₃₁×#2004	C₂	semidirect
8. C₃₁×#1978	C₂	semidirect
9. C₃₁×#1945	C₂	semidirect
10. C₃₁×#1888	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
52. C₂×Q₁₆×C₃₁	D₄	semidirect
160. #25432	C₃₁	direct (N × H)
204. C₂×#495	C₂×C₃₁	semidirect
205. C₂×#473	C₂×C₃₁	semidirect
207. #3043	C₂×C₃₁	semidirect
208. #2950	C₂×C₃₁	semidirect
209. #2046	C₂×C₃₁	semidirect
210. #2004	C₂×C₃₁	semidirect
211. #1978	C₂×C₃₁	semidirect
212. #1945	C₂×C₃₁	semidirect
213. #1888	C₂×C₃₁	semidirect
214. #1764	C₂×C₃₁	semidirect
280. C₂×Q₁₆	D₄×C₃₁	semidirect
326. C₃₁	#25432	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×#495	2
C₂×C₃₁×#473	2
C₃₁×#3058	2
C₃₁×#3043	2
C₃₁×#2950	2
C₃₁×#2046	2
C₃₁×#2004	2
C₃₁×#1978	2
C₃₁×#1945	2
C₃₁×#1888	2
C₃₁×#1764	2
#25432	31

Subgroups — C31×#25432 #21482191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25432 #21482191