Subgroups — C₃₁×#12750 #21469509

Order 7936.

928 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#576	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2148	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2145	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2140	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2117	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1814	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×H(5,𝔽₂)	—	2	1	no	—
9	C₃₁×#576	—	2	2	no	—
10	C₃₁×#576	—	2	2	no	—
11	C₃₁×#576	—	2	2	no	—
12	C₃₁×#576	C₄	1	4	yes	complemented
13	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
14	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
15	C₂×C₃₁×#135	—	2	1	no	—
16	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₃₁	—	2	1	no	—
18	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#427	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#421	—	2	1	no	—
25	C₃₁×#362	—	2	1	no	—
26	C₃₁×#361	—	2	2	no	—
27	C₃₁×#361	—	2	1	no	—
28	C₃₁×#360	—	2	2	no	—
29	C₃₁×#357	—	2	2	no	—
30	C₃₁×#350	—	2	2	no	—
31	C₃₁×#344	—	2	1	no	—
32	C₃₁×H(5,𝔽₂)	—	2	4	no	—
33	C₃₁×H(5,𝔽₂)	—	2	4	no	—
34	C₃₁×H(5,𝔽₂)	—	4	2	no	—
35	C₃₁×H(5,𝔽₂)	—	4	2	no	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂×#41×C₃₁	—	2	1	no	—
38	C₂×#41×C₃₁	—	4	1	no	—
39	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	—	2	1	no	—
42	C₂²×D₄×C₃₁	—	4	2	no	—
43	C₂²×D₄×C₃₁	—	4	1	no	—
44	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
45	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
46	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
47	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
48	C₃₁×#135	—	2	4	no	—
49	C₃₁×#135	—	2	4	no	—
50	C₃₁×#135	—	2	1	no	—
51	C₃₁×#135	—	2	1	no	—
52	C₃₁×#135	C₂×C₄	1	2	yes	—
53	C₃₁×#135	D₄	1	2	yes	—
54	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	1	no	—
57	C₂×D₈×C₃₁	—	2	2	no	—
58	C₂×D₈×C₃₁	—	2	4	no	—
59	C₂×D₈×C₃₁	—	2	4	no	—
60	C₂×D₈×C₃₁	—	2	1	no	—
61	C₂×D₈×C₃₁	—	2	1	no	—
62	C₂×D₈×C₃₁	—	2	1	no	—
63	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
65	C₂×D₈×C₃₁	D₄	1	2	yes	—
66	C₂×D₈×C₃₁	D₄	1	2	yes	—
67	C₃₁×#131	—	2	1	no	—
68	C₃₁×#131	—	2	1	no	—
69	C₃₁×#131	C₂×C₄	1	2	yes	—
70	C₃₁×#131	D₄	1	2	yes	—
71	C₂×M₁₆×C₃₁	—	2	1	no	—
72	C₂×M₁₆×C₃₁	—	2	1	no	—
73	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#117	—	2	1	no	—
77	C₃₁×M₃₂	—	2	1	no	—
78	C₃₁×M₃₂	—	2	1	no	—
79	C₃₁×M₃₂	—	2	1	no	—
80	C₃₁×M₃₂	—	2	1	no	—
81	C₂×C₁₆×C₃₁	—	2	1	no	—
82	C₂×C₁₆×C₃₁	—	2	1	no	—
83	C₃₁×#108	—	2	1	no	—
84	C₃₁×#108	—	2	1	no	—
85	C₃₁×#107	—	2	1	no	—
86	C₃₁×#106	—	2	1	no	—
87	C₃₁×#104	—	4	2	no	—
88	C₂⁴×C₃₁	—	4	2	no	—
89	C₂⁴×C₃₁	—	4	1	no	—
90	#41×C₃₁	—	2	1	no	—
91	#41×C₃₁	—	2	1	no	—
92	#41×C₃₁	—	2	8	no	—
93	#41×C₃₁	—	2	2	no	—
94	#41×C₃₁	—	2	2	no	—
95	#41×C₃₁	—	4	4	no	—
96	#41×C₃₁	—	4	4	no	—
97	#41×C₃₁	—	4	4	no	—
98	#41×C₃₁	—	4	1	no	—
99	#41×C₃₁	—	4	1	no	—
100	#41×C₃₁	C₂²⋊C₄	1	2	yes	—
101	#41×C₃₁	D₈	1	2	yes	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	SD₁₆	1	1	yes	characteristic
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	—	2	8	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	4	8	no	—
113	C₂×D₄×C₃₁	—	4	8	no	—
114	C₂×D₄×C₃₁	—	4	4	no	—
115	C₂×D₄×C₃₁	—	4	2	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂×D₄×C₃₁	—	4	4	no	—
118	C₂×D₄×C₃₁	—	4	1	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	1	no	—
121	C₂×D₄×C₃₁	SD₁₆	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	4	2	no	—
124	C₂²×C₄×C₃₁	—	4	1	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	Q₁₆×C₃₁	—	2	1	no	—
127	Q₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
128	SD₁₆×C₃₁	—	2	8	no	—
129	SD₁₆×C₃₁	—	2	2	no	—
130	SD₁₆×C₃₁	—	2	2	no	—
131	D₈×C₃₁	—	2	8	no	—
132	D₈×C₃₁	—	2	2	no	—
133	D₈×C₃₁	—	2	1	no	—
134	D₈×C₃₁	—	2	2	no	—
135	D₈×C₃₁	—	2	2	no	—
136	D₈×C₃₁	C₂²⋊C₄	1	2	yes	—
137	D₈×C₃₁	C₂²⋊C₄	1	4	yes	—
138	M₁₆×C₃₁	—	2	1	no	—
139	M₁₆×C₃₁	—	2	2	no	—
140	M₁₆×C₃₁	—	2	2	no	—
141	M₁₆×C₃₁	—	4	1	no	—
142	M₁₆×C₃₁	—	4	1	no	—
143	M₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	4	1	no	—
148	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄⋊C₄×C₃₁	—	4	1	no	—
155	C₂²⋊C₄×C₃₁	—	4	1	no	—
156	C₄²×C₃₁	—	4	1	no	—
157	C₁₆×C₃₁	—	4	1	no	—
158	C₁₆×C₃₁	—	4	1	no	—
159	#12750	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	4	8	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	16	2	no	—
167	C₂³×C₃₁	—	16	4	no	—
168	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
169	Q₈×C₃₁	—	2	1	no	—
170	Q₈×C₃₁	—	2	2	no	—
171	Q₈×C₃₁	—	4	1	no	—
172	Q₈×C₃₁	#102	1	2	yes	—
173	D₄×C₃₁	—	2	2	no	—
174	D₄×C₃₁	—	2	2	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	2	4	no	—
177	D₄×C₃₁	—	2	1	no	—
178	D₄×C₃₁	—	4	8	no	—
179	D₄×C₃₁	—	4	8	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	4	1	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	#102	1	2	yes	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	4	8	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	8	1	no	—
195	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
197	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
200	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	—	8	1	no	—
203	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₈×C₃₁	#120	1	1	yes	characteristic
206	C₈×C₃₁	#120	1	1	yes	characteristic
207	C₂×#576	C₂×C₃₁	1	1	yes	characteristic
208	#2148	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#2145	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#2140	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#2117	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1814	C₂×C₃₁	1	1	yes	characteristic, complemented
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	4	1	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	8	4	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	16	2	no	—
222	C₂²×C₃₁	—	16	2	no	—
223	C₂²×C₃₁	—	16	8	no	—
224	C₂²×C₃₁	#418	1	1	yes	characteristic
225	C₂²×C₃₁	#447	1	1	yes	characteristic
226	C₂²×C₃₁	#450	1	1	yes	characteristic
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	16	1	no	—
230	C₄×C₃₁	#379	1	1	yes	characteristic
231	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
232	C₄×C₃₁	#447	1	1	yes	characteristic
233	C₄×C₃₁	#450	1	1	yes	characteristic
234	C₂×H(5,𝔽₂)	—	2	1	no	—
235	#576	—	2	2	no	—
236	#576	—	2	2	no	—
237	#576	—	2	2	no	—
238	#576	C₄×C₃₁	1	4	yes	complemented
239	C₂×AGL(1,ℤ/8)	—	2	2	no	—
240	C₂×AGL(1,ℤ/8)	—	2	1	no	—
241	C₂×#135	—	2	1	no	—
242	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
243	C₂²×D₈	—	2	1	no	—
244	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
247	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
248	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	#427	C₂²×C₃₁	1	1	yes	characteristic, complemented
250	#421	—	2	1	no	—
251	#362	—	2	1	no	—
252	#361	—	2	2	no	—
253	#361	—	2	1	no	—
254	#360	—	2	2	no	—
255	#357	—	2	2	no	—
256	#350	—	2	2	no	—
257	#344	—	2	1	no	—
258	C₂×C₃₁	—	2	1	no	—
259	C₂×C₃₁	—	2	1	no	—
260	C₂×C₃₁	—	2	1	no	—
261	C₂×C₃₁	—	4	2	no	—
262	C₂×C₃₁	—	8	4	no	—
263	C₂×C₃₁	—	8	2	no	—
264	C₂×C₃₁	#1786	1	1	yes	characteristic
265	H(5,𝔽₂)	—	2	4	no	—
266	H(5,𝔽₂)	—	2	4	no	—
267	H(5,𝔽₂)	—	4	2	no	—
268	H(5,𝔽₂)	—	4	2	no	—
269	C₂×#41	—	2	2	no	—
270	C₂×#41	—	2	1	no	—
271	C₂×#41	—	4	1	no	—
272	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
273	C₂²×D₄	—	2	2	no	—
274	C₂²×D₄	—	2	1	no	—
275	C₂²×D₄	—	4	2	no	—
276	C₂²×D₄	—	4	1	no	—
277	AGL(1,ℤ/8)	—	2	4	no	—
278	AGL(1,ℤ/8)	—	2	8	no	—
279	AGL(1,ℤ/8)	—	2	8	no	—
280	AGL(1,ℤ/8)	—	2	4	no	—
281	#135	—	2	4	no	—
282	#135	—	2	4	no	—
283	#135	—	2	1	no	—
284	#135	—	2	1	no	—
285	#135	C₂×C₄×C₃₁	1	2	yes	—
286	#135	D₄×C₃₁	1	2	yes	—
287	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
288	C₂×SD₁₆	—	2	2	no	—
289	C₂×SD₁₆	—	2	1	no	—
290	C₂×D₈	—	2	2	no	—
291	C₂×D₈	—	2	4	no	—
292	C₂×D₈	—	2	4	no	—
293	C₂×D₈	—	2	1	no	—
294	C₂×D₈	—	2	1	no	—
295	C₂×D₈	—	2	1	no	—
296	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
297	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
298	C₂×D₈	D₄×C₃₁	1	2	yes	—
299	C₂×D₈	D₄×C₃₁	1	2	yes	—
300	#131	—	2	1	no	—
301	#131	—	2	1	no	—
302	#131	C₂×C₄×C₃₁	1	2	yes	—
303	#131	D₄×C₃₁	1	2	yes	—
304	C₂×M₁₆	—	2	1	no	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	#117	—	2	1	no	—
310	M₃₂	—	2	1	no	—
311	M₃₂	—	2	1	no	—
312	M₃₂	—	2	1	no	—
313	M₃₂	—	2	1	no	—
314	C₂×C₁₆	—	2	1	no	—
315	C₂×C₁₆	—	2	1	no	—
316	#108	—	2	1	no	—
317	#108	—	2	1	no	—
318	#107	—	2	1	no	—
319	#106	—	2	1	no	—
320	#104	—	4	2	no	—
321	C₃₁	#12750	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	—	4	2	no	—
323	C₂⁴	—	4	1	no	—
324	#41	—	2	1	no	—
325	#41	—	2	1	no	—
326	#41	—	2	8	no	—
327	#41	—	2	2	no	—
328	#41	—	2	2	no	—
329	#41	—	4	4	no	—
330	#41	—	4	4	no	—
331	#41	—	4	4	no	—
332	#41	—	4	1	no	—
333	#41	—	4	1	no	—
334	#41	C₂²⋊C₄×C₃₁	1	2	yes	—
335	#41	D₈×C₃₁	1	2	yes	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	SD₁₆×C₃₁	1	1	yes	characteristic
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	8	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	4	2	no	—
345	C₂×D₄	—	4	4	no	—
346	C₂×D₄	—	4	8	no	—
347	C₂×D₄	—	4	8	no	—
348	C₂×D₄	—	4	4	no	—
349	C₂×D₄	—	4	2	no	—
350	C₂×D₄	—	4	1	no	—
351	C₂×D₄	—	4	4	no	—
352	C₂×D₄	—	4	1	no	—
353	C₂×D₄	—	4	1	no	—
354	C₂×D₄	—	4	1	no	—
355	C₂×D₄	SD₁₆×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	—	4	1	no	—
359	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	Q₁₆	—	2	1	no	—
361	Q₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
362	SD₁₆	—	2	8	no	—
363	SD₁₆	—	2	2	no	—
364	SD₁₆	—	2	2	no	—
365	D₈	—	2	8	no	—
366	D₈	—	2	2	no	—
367	D₈	—	2	1	no	—
368	D₈	—	2	2	no	—
369	D₈	—	2	2	no	—
370	D₈	C₂²⋊C₄×C₃₁	1	2	yes	—
371	D₈	C₂²⋊C₄×C₃₁	1	4	yes	—
372	M₁₆	—	2	1	no	—
373	M₁₆	—	2	2	no	—
374	M₁₆	—	2	2	no	—
375	M₁₆	—	4	1	no	—
376	M₁₆	—	4	1	no	—
377	M₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	4	1	no	—
382	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₄⋊C₄	—	4	1	no	—
389	C₂²⋊C₄	—	4	1	no	—
390	C₄²	—	4	1	no	—
391	C₁₆	—	4	1	no	—
392	C₁₆	—	4	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	4	8	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	16	2	no	—
400	C₂³	—	16	4	no	—
401	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
402	Q₈	—	2	1	no	—
403	Q₈	—	2	2	no	—
404	Q₈	—	4	1	no	—
405	Q₈	C₃₁×#102	1	2	yes	—
406	D₄	—	2	2	no	—
407	D₄	—	2	2	no	—
408	D₄	—	2	1	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	4	no	—
411	D₄	—	4	8	no	—
412	D₄	—	4	8	no	—
413	D₄	—	4	2	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	1	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	C₃₁×#102	1	2	yes	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	4	8	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	8	1	no	—
428	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
430	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
431	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
434	C₈	—	2	2	no	—
435	C₈	—	8	1	no	—
436	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
437	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
438	C₈	C₃₁×#120	1	1	yes	characteristic
439	C₈	C₃₁×#120	1	1	yes	characteristic
440	C₂²	—	2	2	no	—
441	C₂²	—	4	1	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	8	4	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	16	2	no	—
448	C₂²	—	16	2	no	—
449	C₂²	—	16	8	no	—
450	C₂²	C₃₁×#418	1	1	yes	characteristic
451	C₂²	C₃₁×#447	1	1	yes	characteristic
452	C₂²	C₃₁×#450	1	1	yes	characteristic
453	C₄	—	2	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	16	1	no	—
456	C₄	C₃₁×#379	1	1	yes	characteristic
457	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
458	C₄	C₃₁×#447	1	1	yes	characteristic
459	C₄	C₃₁×#450	1	1	yes	characteristic
460	C₂	—	2	1	no	—
461	C₂	—	2	1	no	—
462	C₂	—	2	1	no	—
463	C₂	—	4	2	no	—
464	C₂	—	8	4	no	—
465	C₂	—	8	2	no	—
466	C₂	C₃₁×#1786	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#1786
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2148	C₂	semidirect
3. C₂×C₃₁×#361	C₂	semidirect
4. C₃₁×#2145	C₂	semidirect
5. C₃₁×#2140	C₂	semidirect
6. C₃₁×#2117	C₂	semidirect
7. C₃₁×#1814	C₂	semidirect
12. C₃₁×#576	C₄	semidirect
20. C₂×C₃₁×M₃₂	C₂²	semidirect
21. C₂×C₃₁×M₃₂	C₂²	semidirect
22. C₂×C₃₁×#108	C₂²	semidirect
23. C₃₁×#427	C₂²	semidirect
159. #12750	C₃₁	direct (N × H)
208. #2148	C₂×C₃₁	semidirect
209. C₂×#361	C₂×C₃₁	semidirect
210. #2145	C₂×C₃₁	semidirect
211. #2140	C₂×C₃₁	semidirect
212. #2117	C₂×C₃₁	semidirect
213. #1814	C₂×C₃₁	semidirect
238. #576	C₄×C₃₁	semidirect
246. C₂×M₃₂	C₂²×C₃₁	semidirect
247. C₂×M₃₂	C₂²×C₃₁	semidirect
248. C₂×#108	C₂²×C₃₁	semidirect
249. #427	C₂²×C₃₁	semidirect
321. C₃₁	#12750	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×#576	2
C₃₁×#2148	2
C₂×C₃₁×#361	2
C₃₁×#2145	2
C₃₁×#2140	2
C₃₁×#2117	2
C₃₁×#1814	2
#12750	31

Subgroups — C31×#12750 #21469509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12750 #21469509