Subgroups — C₃₁×#24809 #21481568

Order 7936.

830 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3068	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2955	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2069	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2048	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1903	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1813	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#575	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
14	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
15	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#568	C₂²	1	2	yes	—
17	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₃₁×#472	—	2	4	no	—
20	C₃₁×#472	—	2	2	no	—
21	C₃₁×#472	—	2	2	no	—
22	C₃₁×#472	—	2	2	no	—
23	C₃₁×#472	—	2	2	no	—
24	C₃₁×#471	—	2	4	no	—
25	C₃₁×#444	—	2	2	no	—
26	C₃₁×#443	—	2	2	no	—
27	C₃₁×#430	—	2	2	no	—
28	C₃₁×#430	C₂²	1	1	yes	characteristic
29	C₃₁×#430	C₂²	1	2	yes	—
30	C₃₁×#430	C₂²	1	2	yes	—
31	C₂×C₃₁×#108	—	2	2	no	—
32	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
35	C₃₁×#421	C₂²	1	1	yes	characteristic
36	C₃₁×#421	C₂²	1	1	yes	characteristic
37	C₃₁×#418	C₂²	1	2	yes	—
38	C₃₁×#413	C₂²	1	2	yes	—
39	C₂×C₃₁×#100	C₂²	1	2	yes	—
40	C₂×C₃₁×#100	C₂²	1	2	yes	—
41	C₂×C₃₁×#100	C₂²	1	2	yes	—
42	C₃₁×#405	C₂²	1	1	yes	characteristic
43	C₃₁×#344	C₂²	1	2	yes	—
44	C₃₁×#338	C₂²	1	2	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₃₁×#137	—	2	1	no	—
49	C₃₁×#137	D₄	1	2	yes	—
50	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
51	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
52	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
53	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
54	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
55	C₃₁×AGL(1,ℤ/8)	D₄	1	2	yes	—
56	C₃₁×#135	—	2	2	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	2	no	—
60	C₂×D₈×C₃₁	—	2	2	no	—
61	C₂×D₈×C₃₁	—	2	1	no	—
62	C₂×D₈×C₃₁	—	2	1	no	—
63	C₂×D₈×C₃₁	—	2	1	no	—
64	C₂×D₈×C₃₁	—	2	1	no	—
65	C₂×D₈×C₃₁	—	2	1	no	—
66	C₂×D₈×C₃₁	—	2	1	no	—
67	C₃₁×#131	—	2	1	no	—
68	C₃₁×#131	—	2	1	no	—
69	C₃₁×#131	—	2	4	no	—
70	C₃₁×#131	—	2	1	no	—
71	C₃₁×#131	—	2	1	no	—
72	C₃₁×#131	—	2	1	no	—
73	C₃₁×#131	—	2	1	no	—
74	C₃₁×#131	—	4	2	no	—
75	C₃₁×#131	D₄	1	2	yes	—
76	C₃₁×#131	D₄	1	2	yes	—
77	C₂×M₁₆×C₃₁	—	2	1	no	—
78	C₂×M₁₆×C₃₁	—	2	1	no	—
79	C₂×M₁₆×C₃₁	—	2	4	no	—
80	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
82	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
83	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂²×C₈×C₃₁	—	2	1	no	—
87	C₂²×C₈×C₃₁	—	2	1	no	—
88	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₃₁×#117	C₂³	1	1	yes	characteristic
90	C₃₁×#108	—	2	4	no	—
91	C₃₁×#108	—	2	2	no	—
92	C₃₁×#108	—	2	4	no	—
93	C₃₁×#108	—	2	1	no	—
94	C₃₁×#108	—	2	1	no	—
95	C₃₁×#108	—	2	1	no	—
96	C₃₁×#108	—	2	1	no	—
97	C₃₁×#108	D₄	1	2	yes	—
98	C₃₁×#108	D₄	1	2	yes	—
99	C₃₁×#104	—	2	2	no	—
100	C₃₁×#104	—	2	2	no	—
101	C₃₁×#102	—	4	2	no	—
102	C₃₁×#101	—	2	2	no	—
103	C₃₁×#100	—	2	2	no	—
104	C₃₁×#100	—	2	4	no	—
105	C₃₁×#100	—	2	4	no	—
106	C₃₁×#100	—	2	4	no	—
107	C₃₁×#97	—	2	1	no	—
108	C₄×C₈×C₃₁	—	2	1	no	—
109	C₂⁴×C₃₁	—	2	2	no	—
110	#41×C₃₁	—	2	1	no	—
111	#41×C₃₁	—	2	1	no	—
112	#41×C₃₁	—	2	1	no	—
113	#41×C₃₁	—	2	1	no	—
114	#41×C₃₁	—	2	1	no	—
115	#41×C₃₁	—	2	1	no	—
116	#41×C₃₁	C₂×D₄	1	2	yes	—
117	#41×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	Q₁₆×C₃₁	—	2	2	no	—
134	SD₁₆×C₃₁	—	2	2	no	—
135	SD₁₆×C₃₁	—	2	2	no	—
136	SD₁₆×C₃₁	—	4	4	no	—
137	SD₁₆×C₃₁	—	4	4	no	—
138	D₈×C₃₁	—	2	2	no	—
139	D₈×C₃₁	—	4	4	no	—
140	D₈×C₃₁	—	4	4	no	—
141	D₈×C₃₁	—	4	1	no	—
142	D₈×C₃₁	—	4	1	no	—
143	D₈×C₃₁	—	4	1	no	—
144	D₈×C₃₁	—	4	1	no	—
145	M₁₆×C₃₁	—	2	1	no	—
146	M₁₆×C₃₁	—	2	1	no	—
147	M₁₆×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	M₁₆×C₃₁	—	2	1	no	—
151	M₁₆×C₃₁	—	2	2	no	—
152	M₁₆×C₃₁	—	2	2	no	—
153	M₁₆×C₃₁	—	2	2	no	—
154	M₁₆×C₃₁	—	2	1	no	—
155	M₁₆×C₃₁	—	2	2	no	—
156	M₁₆×C₃₁	—	4	4	no	—
157	M₁₆×C₃₁	—	4	1	no	—
158	M₁₆×C₃₁	—	4	1	no	—
159	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
160	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
161	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	2	2	no	—
170	C₂×C₈×C₃₁	—	4	4	no	—
171	C₂×C₈×C₃₁	—	4	1	no	—
172	C₂×C₈×C₃₁	—	4	1	no	—
173	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
178	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₂²⋊C₄×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	—	2	1	no	—
182	#24809	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	1	no	—
184	C₂³×C₃₁	—	2	1	no	—
185	C₂³×C₃₁	—	4	2	no	—
186	C₂³×C₃₁	—	4	2	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	16	2	no	—
189	C₂³×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
190	Q₈×C₃₁	—	2	1	no	—
191	Q₈×C₃₁	—	2	1	no	—
192	Q₈×C₃₁	—	2	1	no	—
193	Q₈×C₃₁	#121	1	2	yes	—
194	D₄×C₃₁	—	2	2	no	—
195	D₄×C₃₁	—	2	2	no	—
196	D₄×C₃₁	—	2	1	no	—
197	D₄×C₃₁	—	2	1	no	—
198	D₄×C₃₁	—	2	1	no	—
199	D₄×C₃₁	—	4	1	no	—
200	D₄×C₃₁	—	4	1	no	—
201	D₄×C₃₁	—	4	2	no	—
202	D₄×C₃₁	—	4	2	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	2	no	—
205	D₄×C₃₁	#121	1	2	yes	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	2	4	no	—
222	C₈×C₃₁	—	4	1	no	—
223	C₈×C₃₁	—	4	2	no	—
224	C₈×C₃₁	—	4	2	no	—
225	C₈×C₃₁	#121	1	1	yes	characteristic
226	C₈×C₃₁	#121	1	1	yes	characteristic
227	C₈×C₃₁	#121	1	1	yes	characteristic
228	C₈×C₃₁	#121	1	1	yes	characteristic
229	#3068	C₂×C₃₁	1	2	yes	complemented
230	C₂×#472	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂×#472	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2955	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#2069	C₂×C₃₁	1	2	yes	complemented
235	#2048	C₂×C₃₁	1	2	yes	complemented
236	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1903	C₂×C₃₁	1	2	yes	complemented
239	#1813	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	4	1	no	—
242	C₂²×C₃₁	—	4	1	no	—
243	C₂²×C₃₁	—	8	1	no	—
244	C₂²×C₃₁	—	8	1	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	—	16	2	no	—
247	C₂²×C₃₁	—	16	2	no	—
248	C₂²×C₃₁	—	16	2	no	—
249	C₂²×C₃₁	#537	1	1	yes	characteristic
250	C₂²×C₃₁	#547	1	1	yes	characteristic
251	C₂²×C₃₁	#548	1	1	yes	characteristic
252	C₄×C₃₁	—	2	2	no	—
253	C₄×C₃₁	—	4	1	no	—
254	C₄×C₃₁	—	8	1	no	—
255	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
256	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
257	C₄×C₃₁	D₄²	1	1	yes	characteristic
258	C₄×C₃₁	#553	1	1	yes	characteristic
259	#575	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
261	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
262	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
263	#568	C₂²×C₃₁	1	2	yes	—
264	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
266	#472	—	2	4	no	—
267	#472	—	2	2	no	—
268	#472	—	2	2	no	—
269	#472	—	2	2	no	—
270	#472	—	2	2	no	—
271	#471	—	2	4	no	—
272	#444	—	2	2	no	—
273	#443	—	2	2	no	—
274	#430	—	2	2	no	—
275	#430	C₂²×C₃₁	1	2	yes	—
276	#430	C₂²×C₃₁	1	1	yes	characteristic
277	#430	C₂²×C₃₁	1	2	yes	—
278	C₂×#108	—	2	2	no	—
279	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
282	#421	C₂²×C₃₁	1	1	yes	characteristic
283	#421	C₂²×C₃₁	1	1	yes	characteristic
284	#418	C₂²×C₃₁	1	2	yes	—
285	#413	C₂²×C₃₁	1	2	yes	—
286	C₂×#100	C₂²×C₃₁	1	2	yes	—
287	C₂×#100	C₂²×C₃₁	1	2	yes	—
288	C₂×#100	C₂²×C₃₁	1	2	yes	—
289	#405	C₂²×C₃₁	1	1	yes	characteristic
290	#344	C₂²×C₃₁	1	2	yes	—
291	#338	C₂²×C₃₁	1	2	yes	—
292	C₂×C₃₁	—	2	1	no	—
293	C₂×C₃₁	—	2	1	no	—
294	C₂×C₃₁	—	2	1	no	—
295	C₂×C₃₁	—	4	2	no	—
296	C₂×C₃₁	—	8	1	no	—
297	C₂×C₃₁	—	16	2	no	—
298	C₂×C₃₁	#2416	1	1	yes	characteristic
299	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
301	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
302	#137	—	2	1	no	—
303	#137	D₄×C₃₁	1	2	yes	—
304	AGL(1,ℤ/8)	—	2	1	no	—
305	AGL(1,ℤ/8)	—	2	4	no	—
306	AGL(1,ℤ/8)	—	2	4	no	—
307	AGL(1,ℤ/8)	—	2	4	no	—
308	AGL(1,ℤ/8)	—	4	2	no	—
309	AGL(1,ℤ/8)	D₄×C₃₁	1	2	yes	—
310	#135	—	2	2	no	—
311	C₂×SD₁₆	—	2	2	no	—
312	C₂×SD₁₆	—	2	2	no	—
313	C₂×D₈	—	2	2	no	—
314	C₂×D₈	—	2	2	no	—
315	C₂×D₈	—	2	1	no	—
316	C₂×D₈	—	2	1	no	—
317	C₂×D₈	—	2	1	no	—
318	C₂×D₈	—	2	1	no	—
319	C₂×D₈	—	2	1	no	—
320	C₂×D₈	—	2	1	no	—
321	#131	—	2	1	no	—
322	#131	—	2	1	no	—
323	#131	—	2	4	no	—
324	#131	—	2	1	no	—
325	#131	—	2	1	no	—
326	#131	—	2	1	no	—
327	#131	—	2	1	no	—
328	#131	—	4	2	no	—
329	#131	D₄×C₃₁	1	2	yes	—
330	#131	D₄×C₃₁	1	2	yes	—
331	C₂×M₁₆	—	2	1	no	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	2	4	no	—
334	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
336	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
337	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
339	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
340	C₂²×C₈	—	2	1	no	—
341	C₂²×C₈	—	2	1	no	—
342	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
343	#117	C₂³×C₃₁	1	1	yes	characteristic
344	#108	—	2	4	no	—
345	#108	—	2	2	no	—
346	#108	—	2	4	no	—
347	#108	—	2	1	no	—
348	#108	—	2	1	no	—
349	#108	—	2	1	no	—
350	#108	—	2	1	no	—
351	#108	D₄×C₃₁	1	2	yes	—
352	#108	D₄×C₃₁	1	2	yes	—
353	#104	—	2	2	no	—
354	#104	—	2	2	no	—
355	#102	—	4	2	no	—
356	#101	—	2	2	no	—
357	#100	—	2	2	no	—
358	#100	—	2	4	no	—
359	#100	—	2	4	no	—
360	#100	—	2	4	no	—
361	#97	—	2	1	no	—
362	C₄×C₈	—	2	1	no	—
363	C₃₁	#24809	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	2	no	—
365	#41	—	2	1	no	—
366	#41	—	2	1	no	—
367	#41	—	2	1	no	—
368	#41	—	2	1	no	—
369	#41	—	2	1	no	—
370	#41	—	2	1	no	—
371	#41	C₂×D₄×C₃₁	1	2	yes	—
372	#41	C₂×D₄×C₃₁	1	2	yes	—
373	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
388	Q₁₆	—	2	2	no	—
389	SD₁₆	—	2	2	no	—
390	SD₁₆	—	2	2	no	—
391	SD₁₆	—	4	4	no	—
392	SD₁₆	—	4	4	no	—
393	D₈	—	2	2	no	—
394	D₈	—	4	4	no	—
395	D₈	—	4	4	no	—
396	D₈	—	4	1	no	—
397	D₈	—	4	1	no	—
398	D₈	—	4	1	no	—
399	D₈	—	4	1	no	—
400	M₁₆	—	2	1	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	2	1	no	—
406	M₁₆	—	2	2	no	—
407	M₁₆	—	2	2	no	—
408	M₁₆	—	2	2	no	—
409	M₁₆	—	2	1	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	4	4	no	—
412	M₁₆	—	4	1	no	—
413	M₁₆	—	4	1	no	—
414	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
415	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
416	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	4	4	no	—
426	C₂×C₈	—	4	1	no	—
427	C₂×C₈	—	4	1	no	—
428	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
433	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
434	C₄⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	1	no	—
436	C₄²	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	4	2	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	16	2	no	—
443	C₂³	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
444	Q₈	—	2	1	no	—
445	Q₈	—	2	1	no	—
446	Q₈	—	2	1	no	—
447	Q₈	C₃₁×#121	1	2	yes	—
448	D₄	—	2	2	no	—
449	D₄	—	2	2	no	—
450	D₄	—	2	1	no	—
451	D₄	—	2	1	no	—
452	D₄	—	2	1	no	—
453	D₄	—	4	1	no	—
454	D₄	—	4	1	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	C₃₁×#121	1	2	yes	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
473	C₈	—	2	2	no	—
474	C₈	—	2	2	no	—
475	C₈	—	2	4	no	—
476	C₈	—	4	1	no	—
477	C₈	—	4	2	no	—
478	C₈	—	4	2	no	—
479	C₈	C₃₁×#121	1	1	yes	characteristic
480	C₈	C₃₁×#121	1	1	yes	characteristic
481	C₈	C₃₁×#121	1	1	yes	characteristic
482	C₈	C₃₁×#121	1	1	yes	characteristic
483	C₂²	—	2	2	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	4	1	no	—
486	C₂²	—	8	1	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	16	2	no	—
490	C₂²	—	16	2	no	—
491	C₂²	—	16	2	no	—
492	C₂²	C₃₁×#537	1	1	yes	characteristic
493	C₂²	C₃₁×#547	1	1	yes	characteristic
494	C₂²	C₃₁×#548	1	1	yes	characteristic
495	C₄	—	2	2	no	—
496	C₄	—	4	1	no	—
497	C₄	—	8	1	no	—
498	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
499	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
500	C₄	D₄²×C₃₁	1	1	yes	characteristic
501	C₄	C₃₁×#553	1	1	yes	characteristic
502	C₂	—	2	1	no	—
503	C₂	—	2	1	no	—
504	C₂	—	2	1	no	—
505	C₂	—	4	2	no	—
506	C₂	—	8	1	no	—
507	C₂	—	16	2	no	—
508	C₂	C₃₁×#2416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2416
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3068	C₂	semidirect
2. C₂×C₃₁×#472	C₂	semidirect
3. C₂×C₃₁×#472	C₂	semidirect
4. C₃₁×#2955	C₂	semidirect
5. C₃₁×#2916	C₂	semidirect
6. C₃₁×#2069	C₂	semidirect
7. C₃₁×#2048	C₂	semidirect
8. C₃₁×#1978	C₂	semidirect
9. C₃₁×#1953	C₂	semidirect
10. C₃₁×#1903	C₂	semidirect
11. C₃₁×#1813	C₂	semidirect
182. #24809	C₃₁	direct (N × H)
229. #3068	C₂×C₃₁	semidirect
230. C₂×#472	C₂×C₃₁	semidirect
231. C₂×#472	C₂×C₃₁	semidirect
232. #2955	C₂×C₃₁	semidirect
233. #2916	C₂×C₃₁	semidirect
234. #2069	C₂×C₃₁	semidirect
235. #2048	C₂×C₃₁	semidirect
236. #1978	C₂×C₃₁	semidirect
237. #1953	C₂×C₃₁	semidirect
238. #1903	C₂×C₃₁	semidirect
239. #1813	C₂×C₃₁	semidirect
363. C₃₁	#24809	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3068	2
C₂×C₃₁×#472	2
C₃₁×#2955	2
C₃₁×#2916	2
C₃₁×#2069	2
C₃₁×#2048	2
C₃₁×#1978	2
C₃₁×#1953	2
C₃₁×#1903	2
C₃₁×#1813	2
#24809	31

Subgroups — C31×#24809 #21481568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24809 #21481568