Subgroups — C₃₁×#25448 #21482207

Order 7936.

860 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3156	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3069	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3067	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2049	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2009	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1915	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#575	C₂²	1	2	yes	—
13	C₂×C₃₁×#137	C₂²	1	2	yes	—
14	C₂×C₃₁×#137	C₂²	1	2	yes	—
15	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#568	C₂²	1	2	yes	—
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₃₁×#536	C₂²	1	2	yes	—
20	C₃₁×#497	—	2	2	no	—
21	C₃₁×#495	—	2	1	no	—
22	C₃₁×#494	—	2	1	no	—
23	C₃₁×#473	—	2	1	no	—
24	C₃₁×#473	—	2	2	no	—
25	C₃₁×#473	—	2	2	no	—
26	C₃₁×#473	—	2	4	no	—
27	C₃₁×#472	—	2	1	no	—
28	C₃₁×#471	—	2	2	no	—
29	C₃₁×#471	—	2	4	no	—
30	C₃₁×#444	—	2	2	no	—
31	C₃₁×#443	—	2	2	no	—
32	C₃₁×#430	C₂²	1	2	yes	—
33	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
35	C₃₁×#421	C₂²	1	2	yes	—
36	C₃₁×#419	C₂²	1	2	yes	—
37	C₃₁×#413	C₂²	1	2	yes	—
38	C₃₁×#413	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#101	C₂²	1	2	yes	—
40	C₂×C₃₁×#101	C₂²	1	2	yes	—
41	C₃₁×#410	C₂²	1	1	yes	characteristic
42	C₃₁×#405	C₂²	1	1	yes	characteristic
43	C₃₁×#405	C₂²	1	1	yes	characteristic
44	C₃₁×#344	C₂²	1	1	yes	characteristic
45	C₃₁×#344	C₂²	1	2	yes	—
46	C₃₁×#344	C₂²	1	1	yes	characteristic
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₃₁	C₂³	1	2	yes	—
49	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#137	—	2	2	no	—
51	C₃₁×#137	—	2	2	no	—
52	C₃₁×#137	—	2	4	no	—
53	C₃₁×#137	—	2	4	no	—
54	C₃₁×#137	—	4	2	no	—
55	C₃₁×#137	—	4	2	no	—
56	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
57	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
58	C₃₁×#135	—	2	1	no	—
59	C₃₁×#135	—	2	1	no	—
60	C₃₁×#135	—	2	1	no	—
61	C₃₁×#135	—	2	1	no	—
62	C₃₁×#135	—	4	2	no	—
63	C₂×Q₁₆×C₃₁	—	2	2	no	—
64	C₂×Q₁₆×C₃₁	—	2	2	no	—
65	C₂×Q₁₆×C₃₁	—	2	1	no	—
66	C₂×Q₁₆×C₃₁	—	2	1	no	—
67	C₂×Q₁₆×C₃₁	—	2	1	no	—
68	C₂×Q₁₆×C₃₁	—	2	1	no	—
69	C₂×Q₁₆×C₃₁	—	2	1	no	—
70	C₂×Q₁₆×C₃₁	—	2	1	no	—
71	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₃₁	—	2	2	no	—
73	C₂×SD₁₆×C₃₁	—	2	2	no	—
74	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
75	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
76	C₃₁×#131	—	2	4	no	—
77	C₃₁×#131	—	2	4	no	—
78	C₃₁×#131	—	2	1	no	—
79	C₃₁×#131	—	2	1	no	—
80	C₃₁×#131	—	2	1	no	—
81	C₃₁×#131	—	2	1	no	—
82	C₂×M₁₆×C₃₁	—	2	1	no	—
83	C₂×M₁₆×C₃₁	—	4	2	no	—
84	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
89	C₂²×C₈×C₃₁	—	2	1	no	—
90	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊Q₈	—	2	2	no	—
92	C₃₁×#124	—	2	2	no	—
93	C₃₁×#123	—	2	4	no	—
94	C₃₁×C₂²⋊Q₈	—	4	2	no	—
95	C₃₁×#117	C₂³	1	1	yes	characteristic
96	C₃₁×#117	C₂³	1	1	yes	characteristic
97	C₃₁×#108	—	2	4	no	—
98	C₃₁×#108	—	2	1	no	—
99	C₃₁×#108	—	2	1	no	—
100	C₃₁×#108	D₄	1	2	yes	—
101	C₃₁×#108	D₄	1	2	yes	—
102	C₃₁×#104	—	2	4	no	—
103	C₃₁×#103	—	4	2	no	—
104	C₃₁×#101	—	2	2	no	—
105	C₃₁×#101	—	2	4	no	—
106	C₃₁×#101	—	2	4	no	—
107	C₃₁×#101	D₄	1	2	yes	—
108	C₃₁×#100	—	2	2	no	—
109	C₃₁×#100	D₄	1	2	yes	—
110	C₃₁×#99	—	2	2	no	—
111	C₃₁×#97	—	2	1	no	—
112	C₃₁×#97	—	2	1	no	—
113	C₄×C₈×C₃₁	—	2	1	no	—
114	C₄×C₈×C₃₁	—	2	1	no	—
115	#41×C₃₁	—	2	2	no	—
116	#41×C₃₁	—	2	1	no	—
117	#41×C₃₁	—	2	2	no	—
118	#41×C₃₁	—	2	1	no	—
119	#41×C₃₁	—	2	2	no	—
120	#41×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	2	1	no	—
123	C₂×Q₈×C₃₁	—	2	2	no	—
124	C₂×Q₈×C₃₁	—	2	2	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	2	2	no	—
127	C₂×Q₈×C₃₁	—	2	2	no	—
128	C₂×Q₈×C₃₁	—	2	2	no	—
129	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
130	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₃₁	—	2	1	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₃₁	—	2	1	no	—
140	Q₁₆×C₃₁	—	2	1	no	—
141	Q₁₆×C₃₁	—	4	4	no	—
142	Q₁₆×C₃₁	—	4	4	no	—
143	Q₁₆×C₃₁	—	4	2	no	—
144	Q₁₆×C₃₁	—	4	1	no	—
145	Q₁₆×C₃₁	—	4	1	no	—
146	Q₁₆×C₃₁	—	4	1	no	—
147	Q₁₆×C₃₁	—	4	1	no	—
148	SD₁₆×C₃₁	—	2	2	no	—
149	SD₁₆×C₃₁	—	2	2	no	—
150	SD₁₆×C₃₁	—	4	4	no	—
151	SD₁₆×C₃₁	—	4	4	no	—
152	SD₁₆×C₃₁	—	4	2	no	—
153	SD₁₆×C₃₁	—	4	2	no	—
154	D₈×C₃₁	—	2	1	no	—
155	D₈×C₃₁	—	2	1	no	—
156	D₈×C₃₁	—	4	2	no	—
157	M₁₆×C₃₁	—	2	1	no	—
158	M₁₆×C₃₁	—	2	2	no	—
159	M₁₆×C₃₁	—	2	1	no	—
160	M₁₆×C₃₁	—	2	1	no	—
161	M₁₆×C₃₁	—	2	2	no	—
162	M₁₆×C₃₁	—	2	1	no	—
163	M₁₆×C₃₁	—	2	2	no	—
164	M₁₆×C₃₁	—	4	4	no	—
165	M₁₆×C₃₁	—	4	1	no	—
166	M₁₆×C₃₁	—	4	1	no	—
167	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
168	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
169	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	1	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	2	no	—
175	C₂×C₈×C₃₁	—	4	4	no	—
176	C₂×C₈×C₃₁	—	4	1	no	—
177	C₂×C₈×C₃₁	—	4	1	no	—
178	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
183	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	4	4	no	—
187	C₂²⋊C₄×C₃₁	—	2	1	no	—
188	C₂²⋊C₄×C₃₁	—	4	4	no	—
189	C₂²⋊C₄×C₃₁	#41	1	2	yes	—
190	C₄²×C₃₁	—	2	1	no	—
191	C₄²×C₃₁	—	2	1	no	—
192	#25448	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₃₁	—	2	1	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
196	Q₈×C₃₁	—	2	2	no	—
197	Q₈×C₃₁	—	2	1	no	—
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	2	1	no	—
200	Q₈×C₃₁	—	4	2	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	2	2	no	—
205	D₄×C₃₁	—	2	1	no	—
206	D₄×C₃₁	—	2	2	no	—
207	D₄×C₃₁	—	2	1	no	—
208	D₄×C₃₁	—	4	2	no	—
209	D₄×C₃₁	—	4	1	no	—
210	D₄×C₃₁	—	4	1	no	—
211	D₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	2	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
227	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
229	C₈×C₃₁	—	2	2	no	—
230	C₈×C₃₁	—	2	4	no	—
231	C₈×C₃₁	—	2	2	no	—
232	C₈×C₃₁	—	4	1	no	—
233	C₈×C₃₁	—	4	2	no	—
234	C₈×C₃₁	#121	1	1	yes	characteristic
235	C₈×C₃₁	#121	1	1	yes	characteristic
236	C₈×C₃₁	#121	1	1	yes	characteristic
237	C₈×C₃₁	#121	1	1	yes	characteristic
238	#3156	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#3069	C₂×C₃₁	1	2	yes	complemented
240	#3067	C₂×C₃₁	1	1	yes	characteristic
241	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#2957	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#2049	C₂×C₃₁	1	2	yes	complemented
244	#2009	C₂×C₃₁	1	2	yes	complemented
245	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1915	C₂×C₃₁	1	2	yes	complemented
248	#1757	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₃₁	—	4	1	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	—	8	2	no	—
253	C₂²×C₃₁	—	8	1	no	—
254	C₂²×C₃₁	#532	1	1	yes	characteristic
255	C₂²×C₃₁	#546	1	1	yes	characteristic
256	C₂²×C₃₁	#550	1	1	yes	characteristic
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	1	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	8	1	no	—
261	C₄×C₃₁	—	8	2	no	—
262	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
263	C₄×C₃₁	#540	1	1	yes	characteristic
264	C₄×C₃₁	D₄²	1	1	yes	characteristic
265	C₄×C₃₁	D₄²	1	1	yes	characteristic
266	#575	C₂²×C₃₁	1	2	yes	—
267	C₂×#137	C₂²×C₃₁	1	2	yes	—
268	C₂×#137	C₂²×C₃₁	1	2	yes	—
269	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
270	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
271	#568	C₂²×C₃₁	1	2	yes	—
272	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
273	#536	C₂²×C₃₁	1	2	yes	—
274	#497	—	2	2	no	—
275	#495	—	2	1	no	—
276	#494	—	2	1	no	—
277	#473	—	2	1	no	—
278	#473	—	2	2	no	—
279	#473	—	2	2	no	—
280	#473	—	2	4	no	—
281	#472	—	2	1	no	—
282	#471	—	2	2	no	—
283	#471	—	2	4	no	—
284	#444	—	2	2	no	—
285	#443	—	2	2	no	—
286	#430	C₂²×C₃₁	1	2	yes	—
287	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
289	#421	C₂²×C₃₁	1	2	yes	—
290	#419	C₂²×C₃₁	1	2	yes	—
291	#413	C₂²×C₃₁	1	2	yes	—
292	#413	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#101	C₂²×C₃₁	1	2	yes	—
294	C₂×#101	C₂²×C₃₁	1	2	yes	—
295	#410	C₂²×C₃₁	1	1	yes	characteristic
296	#405	C₂²×C₃₁	1	1	yes	characteristic
297	#405	C₂²×C₃₁	1	1	yes	characteristic
298	#344	C₂²×C₃₁	1	1	yes	characteristic
299	#344	C₂²×C₃₁	1	1	yes	characteristic
300	#344	C₂²×C₃₁	1	2	yes	—
301	C₂×C₃₁	—	2	1	no	—
302	C₂×C₃₁	—	2	1	no	—
303	C₂×C₃₁	—	2	1	no	—
304	C₂×C₃₁	—	8	1	no	—
305	C₂×C₃₁	—	8	2	no	—
306	C₂×C₃₁	#2428	1	1	yes	characteristic
307	C₂×#41	C₂³×C₃₁	1	2	yes	—
308	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
310	#137	—	2	2	no	—
311	#137	—	2	2	no	—
312	#137	—	2	4	no	—
313	#137	—	2	4	no	—
314	#137	—	4	2	no	—
315	#137	—	4	2	no	—
316	AGL(1,ℤ/8)	—	2	2	no	—
317	AGL(1,ℤ/8)	—	2	2	no	—
318	#135	—	2	1	no	—
319	#135	—	2	1	no	—
320	#135	—	2	1	no	—
321	#135	—	2	1	no	—
322	#135	—	4	2	no	—
323	C₂×Q₁₆	—	2	2	no	—
324	C₂×Q₁₆	—	2	2	no	—
325	C₂×Q₁₆	—	2	1	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×Q₁₆	—	2	1	no	—
328	C₂×Q₁₆	—	2	1	no	—
329	C₂×Q₁₆	—	2	1	no	—
330	C₂×Q₁₆	—	2	1	no	—
331	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
332	C₂×SD₁₆	—	2	2	no	—
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
335	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
336	#131	—	2	4	no	—
337	#131	—	2	4	no	—
338	#131	—	2	1	no	—
339	#131	—	2	1	no	—
340	#131	—	2	1	no	—
341	#131	—	2	1	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₂×M₁₆	—	4	2	no	—
344	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
346	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
347	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
348	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
349	C₂²×C₈	—	2	1	no	—
350	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₄⋊Q₈	—	2	2	no	—
352	#124	—	2	2	no	—
353	#123	—	2	4	no	—
354	C₂²⋊Q₈	—	4	2	no	—
355	#117	C₂³×C₃₁	1	1	yes	characteristic
356	#117	C₂³×C₃₁	1	1	yes	characteristic
357	#108	—	2	4	no	—
358	#108	—	2	1	no	—
359	#108	—	2	1	no	—
360	#108	D₄×C₃₁	1	2	yes	—
361	#108	D₄×C₃₁	1	2	yes	—
362	#104	—	2	4	no	—
363	#103	—	4	2	no	—
364	#101	—	2	2	no	—
365	#101	—	2	4	no	—
366	#101	—	2	4	no	—
367	#101	D₄×C₃₁	1	2	yes	—
368	#100	—	2	2	no	—
369	#100	D₄×C₃₁	1	2	yes	—
370	#99	—	2	2	no	—
371	#97	—	2	1	no	—
372	#97	—	2	1	no	—
373	C₄×C₈	—	2	1	no	—
374	C₄×C₈	—	2	1	no	—
375	C₃₁	#25448	1	1	yes	characteristic, direct product factor, Sylow
376	#41	—	2	2	no	—
377	#41	—	2	1	no	—
378	#41	—	2	2	no	—
379	#41	—	2	1	no	—
380	#41	—	2	2	no	—
381	#41	—	2	1	no	—
382	#41	—	2	2	no	—
383	#41	—	2	1	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
391	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
395	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
400	Q₁₆	—	2	1	no	—
401	Q₁₆	—	2	1	no	—
402	Q₁₆	—	4	4	no	—
403	Q₁₆	—	4	4	no	—
404	Q₁₆	—	4	2	no	—
405	Q₁₆	—	4	1	no	—
406	Q₁₆	—	4	1	no	—
407	Q₁₆	—	4	1	no	—
408	Q₁₆	—	4	1	no	—
409	SD₁₆	—	2	2	no	—
410	SD₁₆	—	2	2	no	—
411	SD₁₆	—	4	4	no	—
412	SD₁₆	—	4	4	no	—
413	SD₁₆	—	4	2	no	—
414	SD₁₆	—	4	2	no	—
415	D₈	—	2	1	no	—
416	D₈	—	2	1	no	—
417	D₈	—	4	2	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	2	1	no	—
422	M₁₆	—	2	2	no	—
423	M₁₆	—	2	1	no	—
424	M₁₆	—	2	2	no	—
425	M₁₆	—	4	4	no	—
426	M₁₆	—	4	1	no	—
427	M₁₆	—	4	1	no	—
428	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
429	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
430	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	4	4	no	—
437	C₂×C₈	—	4	1	no	—
438	C₂×C₈	—	4	1	no	—
439	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
444	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	4	4	no	—
448	C₂²⋊C₄	—	2	1	no	—
449	C₂²⋊C₄	—	4	4	no	—
450	C₂²⋊C₄	#41×C₃₁	1	2	yes	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₂³	—	2	1	no	—
454	C₂³	—	2	2	no	—
455	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
456	Q₈	—	2	2	no	—
457	Q₈	—	2	1	no	—
458	Q₈	—	2	2	no	—
459	Q₈	—	2	1	no	—
460	Q₈	—	4	2	no	—
461	Q₈	—	4	2	no	—
462	Q₈	—	4	2	no	—
463	Q₈	—	4	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	2	1	no	—
466	D₄	—	2	2	no	—
467	D₄	—	2	1	no	—
468	D₄	—	4	2	no	—
469	D₄	—	4	1	no	—
470	D₄	—	4	1	no	—
471	D₄	—	4	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
488	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	4	no	—
491	C₈	—	2	2	no	—
492	C₈	—	4	1	no	—
493	C₈	—	4	2	no	—
494	C₈	C₃₁×#121	1	1	yes	characteristic
495	C₈	C₃₁×#121	1	1	yes	characteristic
496	C₈	C₃₁×#121	1	1	yes	characteristic
497	C₈	C₃₁×#121	1	1	yes	characteristic
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	8	2	no	—
502	C₂²	—	8	1	no	—
503	C₂²	C₃₁×#532	1	1	yes	characteristic
504	C₂²	C₃₁×#546	1	1	yes	characteristic
505	C₂²	C₃₁×#550	1	1	yes	characteristic
506	C₄	—	4	2	no	—
507	C₄	—	4	1	no	—
508	C₄	—	4	2	no	—
509	C₄	—	8	1	no	—
510	C₄	—	8	2	no	—
511	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
512	C₄	C₃₁×#540	1	1	yes	characteristic
513	C₄	D₄²×C₃₁	1	1	yes	characteristic
514	C₄	D₄²×C₃₁	1	1	yes	characteristic
515	C₂	—	2	1	no	—
516	C₂	—	2	1	no	—
517	C₂	—	2	1	no	—
518	C₂	—	8	1	no	—
519	C₂	—	8	2	no	—
520	C₂	C₃₁×#2428	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2428
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3067	C₂
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×#3156	C₂	semidirect
2. C₃₁×#3069	C₂	semidirect
4. C₂×C₃₁×#473	C₂	semidirect
5. C₃₁×#2957	C₂	semidirect
6. C₃₁×#2049	C₂	semidirect
7. C₃₁×#2009	C₂	semidirect
8. C₃₁×#1979	C₂	semidirect
9. C₃₁×#1953	C₂	semidirect
10. C₃₁×#1915	C₂	semidirect
11. C₃₁×#1757	C₂	semidirect
192. #25448	C₃₁	direct (N × H)
238. #3156	C₂×C₃₁	semidirect
239. #3069	C₂×C₃₁	semidirect
241. C₂×#473	C₂×C₃₁	semidirect
242. #2957	C₂×C₃₁	semidirect
243. #2049	C₂×C₃₁	semidirect
244. #2009	C₂×C₃₁	semidirect
245. #1979	C₂×C₃₁	semidirect
246. #1953	C₂×C₃₁	semidirect
247. #1915	C₂×C₃₁	semidirect
248. #1757	C₂×C₃₁	semidirect
375. C₃₁	#25448	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×#3156	2
C₃₁×#3069	2
C₃₁×#3067	2
C₂×C₃₁×#473	2
C₃₁×#2957	2
C₃₁×#2049	2
C₃₁×#2009	2
C₃₁×#1979	2
C₃₁×#1953	2
C₃₁×#1915	2
C₃₁×#1757	2
#25448	31

Subgroups — C31×#25448 #21482207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25448 #21482207