Subgroups — C₃₁×#23046 #21479805

Order 7936.

896 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2946	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1931	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1883	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1856	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
15	C₃₁×#520	C₂²	1	2	yes	—
16	C₃₁×#518	C₂²	1	2	yes	—
17	C₃₁×#472	C₄	1	8	yes	complemented
18	C₃₁×#442	—	2	2	no	—
19	C₃₁×#442	—	2	2	no	—
20	C₃₁×#440	—	2	2	no	—
21	C₄×SD₁₆×C₃₁	—	2	2	no	—
22	C₄×D₈×C₃₁	—	2	1	no	—
23	C₄×D₈×C₃₁	—	2	1	no	—
24	C₄×D₈×C₃₁	—	2	2	no	—
25	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
26	C₃₁×#428	—	2	2	no	—
27	C₃₁×#428	C₂²	1	2	yes	—
28	C₃₁×#428	C₂²	1	2	yes	—
29	C₃₁×#427	—	2	1	no	—
30	C₃₁×#427	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#107	—	2	1	no	—
32	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
34	C₃₁×#418	C₂²	1	2	yes	—
35	C₃₁×#416	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#102	C₂²	1	2	yes	—
37	C₂×C₃₁×#100	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	2	yes	—
39	C₃₁×#405	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#390	C₂²	1	2	yes	—
42	C₃₁×#343	C₂²	1	2	yes	—
43	C₃₁×#341	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₃₁×#339	C₂²	1	2	yes	—
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
49	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	8	yes	complemented
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
56	C₃₁×#131	—	2	2	no	—
57	C₃₁×#131	C₂×C₄	1	4	yes	complemented
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	4	2	no	—
72	C₃₁×#117	—	2	2	no	—
73	C₃₁×#117	C₂³	1	1	yes	characteristic
74	C₃₁×#117	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#108	C₂×C₄	1	4	yes	complemented
81	C₃₁×#107	—	2	1	no	—
82	C₃₁×#107	—	2	1	no	—
83	C₃₁×#107	—	2	2	no	—
84	C₃₁×#107	—	2	2	no	—
85	C₃₁×#107	D₄	1	1	yes	characteristic
86	C₃₁×#107	D₄	1	1	yes	characteristic
87	C₃₁×#106	—	2	2	no	—
88	C₃₁×#106	—	2	2	no	—
89	C₃₁×#106	D₄	1	2	yes	—
90	C₃₁×#104	—	2	2	no	—
91	C₃₁×#103	D₄	1	1	yes	characteristic
92	C₃₁×#103	D₄	1	1	yes	characteristic
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×#102	—	2	2	no	—
97	C₃₁×#102	D₄	1	1	yes	characteristic
98	C₃₁×#102	D₄	1	1	yes	characteristic
99	C₃₁×#100	C₂×C₄	1	8	yes	complemented
100	C₃₁×#97	—	2	2	no	—
101	C₃₁×#97	—	2	1	no	—
102	C₄×C₈×C₃₁	—	2	1	no	—
103	C₄×C₈×C₃₁	—	2	2	no	—
104	C₄×C₈×C₃₁	—	2	2	no	—
105	C₂⁴×C₃₁	—	2	2	no	—
106	#41×C₃₁	—	2	1	no	—
107	#41×C₃₁	—	2	1	no	—
108	#41×C₃₁	C₂²×C₄	1	2	yes	—
109	#41×C₃₁	#41	1	2	yes	—
110	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×D₄×C₃₁	—	2	1	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	C₂²×C₄	1	4	yes	—
118	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	4	2	no	—
123	C₂²×C₄×C₃₁	—	4	2	no	—
124	C₂²×C₄×C₃₁	—	4	2	no	—
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	SD₁₆×C₃₁	—	2	8	no	—
127	D₈×C₃₁	—	2	8	no	—
128	D₈×C₃₁	—	2	4	no	—
129	D₈×C₃₁	—	2	4	no	—
130	M₁₆×C₃₁	—	2	1	no	—
131	M₁₆×C₃₁	—	2	2	no	—
132	M₁₆×C₃₁	—	2	4	no	—
133	M₁₆×C₃₁	—	2	1	no	—
134	M₁₆×C₃₁	—	2	1	no	—
135	M₁₆×C₃₁	C₂²×C₄	1	2	yes	—
136	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
137	C₂×C₈×C₃₁	—	2	2	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	4	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₃₁	#41	1	2	yes	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²⋊C₄×C₃₁	—	2	1	no	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	4	2	no	—
159	C₂²⋊C₄×C₃₁	—	4	2	no	—
160	C₂²⋊C₄×C₃₁	—	4	2	no	—
161	C₂²⋊C₄×C₃₁	—	8	4	no	—
162	C₄²×C₃₁	—	2	1	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	1	no	—
165	C₄²×C₃₁	—	2	1	no	—
166	C₄²×C₃₁	—	4	2	no	—
167	#23046	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	4	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	8	2	no	—
173	C₂³×C₃₁	—	8	2	no	—
174	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
175	Q₈×C₃₁	—	2	1	no	—
176	Q₈×C₃₁	C₄×D₄	1	2	yes	—
177	D₄×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	2	4	no	—
180	D₄×C₃₁	—	2	4	no	—
181	D₄×C₃₁	—	2	1	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	C₄×D₄	1	2	yes	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	8	2	no	—
197	C₂×C₄×C₃₁	—	16	2	no	—
198	C₂×C₄×C₃₁	—	16	2	no	—
199	C₂×C₄×C₃₁	—	16	2	no	—
200	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
206	C₈×C₃₁	—	2	2	no	—
207	C₈×C₃₁	—	2	4	no	—
208	C₈×C₃₁	—	2	1	no	—
209	C₈×C₃₁	—	2	1	no	—
210	C₈×C₃₁	—	4	1	no	—
211	C₈×C₃₁	C₄×D₄	1	2	yes	—
212	C₈×C₃₁	C₄×D₄	1	2	yes	—
213	C₂×#472	C₂×C₃₁	1	1	yes	characteristic
214	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#2946	C₂×C₃₁	1	2	yes	complemented
216	#2913	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1931	C₂×C₃₁	1	2	yes	complemented
220	#1883	C₂×C₃₁	1	2	yes	complemented
221	#1856	C₂×C₃₁	1	2	yes	complemented
222	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#1764	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂²×C₃₁	—	2	1	no	—
225	C₂²×C₃₁	—	2	1	no	—
226	C₂²×C₃₁	—	2	2	no	—
227	C₂²×C₃₁	—	4	1	no	—
228	C₂²×C₃₁	—	4	1	no	—
229	C₂²×C₃₁	—	4	4	no	—
230	C₂²×C₃₁	—	8	4	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	—	8	2	no	—
233	C₂²×C₃₁	—	16	2	no	—
234	C₂²×C₃₁	#520	1	1	yes	characteristic
235	C₂²×C₃₁	D₄²	1	1	yes	characteristic
236	C₂²×C₃₁	#553	1	1	yes	characteristic
237	C₄×C₃₁	—	2	2	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	4	1	no	—
241	C₄×C₃₁	—	4	1	no	—
242	C₄×C₃₁	—	4	2	no	—
243	C₄×C₃₁	—	8	2	no	—
244	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
245	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
246	C₄×C₃₁	#547	1	1	yes	characteristic
247	C₄×C₃₁	#548	1	1	yes	characteristic
248	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
249	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
251	#520	C₂²×C₃₁	1	2	yes	—
252	#518	C₂²×C₃₁	1	2	yes	—
253	#472	C₄×C₃₁	1	8	yes	complemented
254	#442	—	2	2	no	—
255	#442	—	2	2	no	—
256	#440	—	2	2	no	—
257	C₄×SD₁₆	—	2	2	no	—
258	C₄×D₈	—	2	1	no	—
259	C₄×D₈	—	2	1	no	—
260	C₄×D₈	—	2	2	no	—
261	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
262	#428	—	2	2	no	—
263	#428	C₂²×C₃₁	1	2	yes	—
264	#428	C₂²×C₃₁	1	2	yes	—
265	#427	—	2	1	no	—
266	#427	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#107	—	2	1	no	—
268	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
270	#418	C₂²×C₃₁	1	2	yes	—
271	#416	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#102	C₂²×C₃₁	1	2	yes	—
273	C₂×#100	C₂²×C₃₁	1	2	yes	—
274	#405	C₂²×C₃₁	1	2	yes	—
275	#405	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
277	#390	C₂²×C₃₁	1	2	yes	—
278	#343	C₂²×C₃₁	1	2	yes	—
279	#341	C₂²×C₃₁	1	1	yes	characteristic
280	#339	C₂²×C₃₁	1	1	yes	characteristic
281	#339	C₂²×C₃₁	1	2	yes	—
282	C₂×C₃₁	—	2	1	no	—
283	C₂×C₃₁	—	2	1	no	—
284	C₂×C₃₁	—	4	2	no	—
285	C₂×C₃₁	—	8	4	no	—
286	C₂×C₃₁	#2361	1	1	yes	characteristic
287	C₂×C₃₁	#3290	1	1	yes	characteristic
288	C₂×C₃₁	#3292	1	1	yes	characteristic
289	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
291	AGL(1,ℤ/8)	—	2	4	no	—
292	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	8	yes	complemented
293	C₂×SD₁₆	—	2	2	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	—	2	1	no	—
296	C₂×D₈	—	2	1	no	—
297	C₂×D₈	—	2	2	no	—
298	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
299	#131	—	2	2	no	—
300	#131	C₂×C₄×C₃₁	1	4	yes	complemented
301	C₂×M₁₆	—	2	1	no	—
302	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
303	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
305	C₂²×C₈	—	2	1	no	—
306	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₄×Q₈	—	2	2	no	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	—	2	2	no	—
311	C₄×D₄	—	2	2	no	—
312	C₄×D₄	—	2	2	no	—
313	C₄×D₄	—	2	2	no	—
314	C₄×D₄	—	4	2	no	—
315	#117	—	2	2	no	—
316	#117	C₂³×C₃₁	1	1	yes	characteristic
317	#117	C₂³×C₃₁	1	2	yes	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
320	C₂×C₂²⋊C₄	—	4	2	no	—
321	C₂×C₂²⋊C₄	—	4	2	no	—
322	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
323	#108	C₂×C₄×C₃₁	1	4	yes	complemented
324	#107	—	2	1	no	—
325	#107	—	2	1	no	—
326	#107	—	2	2	no	—
327	#107	—	2	2	no	—
328	#107	D₄×C₃₁	1	1	yes	characteristic
329	#107	D₄×C₃₁	1	1	yes	characteristic
330	#106	—	2	2	no	—
331	#106	—	2	2	no	—
332	#106	D₄×C₃₁	1	2	yes	—
333	#104	—	2	2	no	—
334	#103	D₄×C₃₁	1	1	yes	characteristic
335	#103	D₄×C₃₁	1	1	yes	characteristic
336	#102	—	2	2	no	—
337	#102	—	2	2	no	—
338	#102	—	2	2	no	—
339	#102	—	2	2	no	—
340	#102	D₄×C₃₁	1	1	yes	characteristic
341	#102	D₄×C₃₁	1	1	yes	characteristic
342	#100	C₂×C₄×C₃₁	1	8	yes	complemented
343	#97	—	2	2	no	—
344	#97	—	2	1	no	—
345	C₄×C₈	—	2	1	no	—
346	C₄×C₈	—	2	2	no	—
347	C₄×C₈	—	2	2	no	—
348	C₃₁	#23046	1	1	yes	characteristic, direct product factor, Sylow
349	C₂⁴	—	2	2	no	—
350	#41	—	2	1	no	—
351	#41	—	2	1	no	—
352	#41	C₂²×C₄×C₃₁	1	2	yes	—
353	#41	#41×C₃₁	1	2	yes	—
354	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
362	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
370	SD₁₆	—	2	8	no	—
371	D₈	—	2	8	no	—
372	D₈	—	2	4	no	—
373	D₈	—	2	4	no	—
374	M₁₆	—	2	1	no	—
375	M₁₆	—	2	2	no	—
376	M₁₆	—	2	4	no	—
377	M₁₆	—	2	1	no	—
378	M₁₆	—	2	1	no	—
379	M₁₆	C₂²×C₄×C₃₁	1	2	yes	—
380	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	4	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
388	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₈	#41×C₃₁	1	2	yes	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
397	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
398	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	2	2	no	—
402	C₂²⋊C₄	—	4	2	no	—
403	C₂²⋊C₄	—	4	2	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	8	4	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₄²	—	4	2	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	4	2	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	8	2	no	—
416	C₂³	—	8	2	no	—
417	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
418	Q₈	—	2	1	no	—
419	Q₈	C₄×D₄×C₃₁	1	2	yes	—
420	D₄	—	2	2	no	—
421	D₄	—	2	2	no	—
422	D₄	—	2	4	no	—
423	D₄	—	2	4	no	—
424	D₄	—	2	1	no	—
425	D₄	—	4	2	no	—
426	D₄	—	4	2	no	—
427	D₄	C₄×D₄×C₃₁	1	2	yes	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	8	2	no	—
440	C₂×C₄	—	16	2	no	—
441	C₂×C₄	—	16	2	no	—
442	C₂×C₄	—	16	2	no	—
443	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
445	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
448	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
449	C₈	—	2	2	no	—
450	C₈	—	2	4	no	—
451	C₈	—	2	1	no	—
452	C₈	—	2	1	no	—
453	C₈	—	4	1	no	—
454	C₈	C₄×D₄×C₃₁	1	2	yes	—
455	C₈	C₄×D₄×C₃₁	1	2	yes	—
456	C₂²	—	2	1	no	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	2	no	—
459	C₂²	—	4	1	no	—
460	C₂²	—	4	1	no	—
461	C₂²	—	4	4	no	—
462	C₂²	—	8	4	no	—
463	C₂²	—	8	2	no	—
464	C₂²	—	8	2	no	—
465	C₂²	—	16	2	no	—
466	C₂²	C₃₁×#520	1	1	yes	characteristic
467	C₂²	D₄²×C₃₁	1	1	yes	characteristic
468	C₂²	C₃₁×#553	1	1	yes	characteristic
469	C₄	—	2	2	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	1	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	1	no	—
474	C₄	—	4	2	no	—
475	C₄	—	8	2	no	—
476	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
477	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
478	C₄	C₃₁×#547	1	1	yes	characteristic
479	C₄	C₃₁×#548	1	1	yes	characteristic
480	C₂	—	2	1	no	—
481	C₂	—	2	1	no	—
482	C₂	—	4	2	no	—
483	C₂	—	8	4	no	—
484	C₂	C₃₁×#2361	1	1	yes	characteristic
485	C₂	C₃₁×#3290	1	1	yes	characteristic
486	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#472	C₂
LCS	G ▷ C₂×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
2. C₃₁×#2948	C₂	semidirect
3. C₃₁×#2946	C₂	semidirect
4. C₃₁×#2913	C₂	semidirect
5. C₃₁×#1970	C₂	semidirect
6. C₃₁×#1952	C₂	semidirect
7. C₃₁×#1931	C₂	semidirect
8. C₃₁×#1883	C₂	semidirect
9. C₃₁×#1856	C₂	semidirect
10. C₃₁×#1772	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
17. C₃₁×#472	C₄	semidirect
49. C₃₁×AGL(1,ℤ/8)	C₂×C₄	semidirect
55. C₂×D₈×C₃₁	C₂×C₄	semidirect
57. C₃₁×#131	C₂×C₄	semidirect
80. C₃₁×#108	C₂×C₄	semidirect
99. C₃₁×#100	C₂×C₄	semidirect
167. #23046	C₃₁	direct (N × H)
214. #2948	C₂×C₃₁	semidirect
215. #2946	C₂×C₃₁	semidirect
216. #2913	C₂×C₃₁	semidirect
217. #1970	C₂×C₃₁	semidirect
218. #1952	C₂×C₃₁	semidirect
219. #1931	C₂×C₃₁	semidirect
220. #1883	C₂×C₃₁	semidirect
221. #1856	C₂×C₃₁	semidirect
222. #1772	C₂×C₃₁	semidirect
223. #1764	C₂×C₃₁	semidirect
253. #472	C₄×C₃₁	semidirect
292. AGL(1,ℤ/8)	C₂×C₄×C₃₁	semidirect
298. C₂×D₈	C₂×C₄×C₃₁	semidirect
300. #131	C₂×C₄×C₃₁	semidirect
323. #108	C₂×C₄×C₃₁	semidirect
342. #100	C₂×C₄×C₃₁	semidirect
348. C₃₁	#23046	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×#472	2
C₃₁×#2948	2
C₃₁×#2946	2
C₃₁×#2913	2
C₃₁×#1970	2
C₃₁×#1952	2
C₃₁×#1931	2
C₃₁×#1883	2
C₃₁×#1856	2
C₃₁×#1772	2
C₃₁×#1764	2
#23046	31

Subgroups — C31×#23046 #21479805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23046 #21479805