Subgroups — C₃₁×#25447 #21482206

Order 7936.

892 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3155	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3068	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3067	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2048	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2008	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×#1914	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×AGL(1,ℤ/8)	C₂²	1	2	yes	—
14	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
15	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
16	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#568	C₂²	1	2	yes	—
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₃₁×#535	C₂²	1	2	yes	—
20	C₃₁×#496	—	2	2	no	—
21	C₃₁×#495	—	2	1	no	—
22	C₃₁×#493	—	2	1	no	—
23	C₃₁×#473	—	2	1	no	—
24	C₃₁×#472	—	2	1	no	—
25	C₃₁×#472	—	2	2	no	—
26	C₃₁×#472	—	2	2	no	—
27	C₃₁×#472	—	2	4	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×#418	C₂²	1	2	yes	—
37	C₃₁×#413	C₂²	1	2	yes	—
38	C₃₁×#413	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#100	C₂²	1	2	yes	—
40	C₂×C₃₁×#100	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×D₄×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#137	—	2	2	no	—
51	C₃₁×#137	—	2	2	no	—
52	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
53	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
54	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
55	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
56	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
57	C₃₁×AGL(1,ℤ/8)	—	4	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	D₄	1	1	yes	characteristic
64	C₂×SD₁₆×C₃₁	—	2	2	no	—
65	C₂×SD₁₆×C₃₁	—	2	2	no	—
66	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
67	C₂×D₈×C₃₁	—	2	2	no	—
68	C₂×D₈×C₃₁	—	2	2	no	—
69	C₂×D₈×C₃₁	—	2	1	no	—
70	C₂×D₈×C₃₁	—	2	1	no	—
71	C₂×D₈×C₃₁	—	2	1	no	—
72	C₂×D₈×C₃₁	—	2	1	no	—
73	C₂×D₈×C₃₁	—	2	1	no	—
74	C₂×D₈×C₃₁	—	2	1	no	—
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×#127	—	2	2	no	—
92	C₃₁×#124	—	2	2	no	—
93	C₃₁×#121	—	4	2	no	—
94	C₃₁×#120	—	2	4	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	complemented
101	C₃₁×#108	D₄	1	2	yes	complemented
102	C₃₁×#104	—	2	4	no	—
103	C₃₁×#102	—	4	2	no	—
104	C₃₁×#101	—	2	2	no	—
105	C₃₁×#101	D₄	1	2	yes	complemented
106	C₃₁×#100	—	2	2	no	—
107	C₃₁×#100	—	2	4	no	—
108	C₃₁×#100	—	2	4	no	—
109	C₃₁×#100	D₄	1	2	yes	complemented
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	C₂⁴×C₃₁	—	2	2	no	—
116	#41×C₃₁	—	2	2	no	—
117	#41×C₃₁	—	2	1	no	—
118	#41×C₃₁	—	2	2	no	—
119	#41×C₃₁	—	2	1	no	—
120	#41×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	2	no	—
123	#41×C₃₁	—	2	1	no	—
124	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
125	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×D₄×C₃₁	—	2	1	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	2	2	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	4	4	no	—
135	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₃₁	—	2	1	no	—
141	Q₁₆×C₃₁	—	2	1	no	—
142	Q₁₆×C₃₁	—	4	2	no	—
143	SD₁₆×C₃₁	—	2	2	no	—
144	SD₁₆×C₃₁	—	2	2	no	—
145	SD₁₆×C₃₁	—	4	4	no	—
146	SD₁₆×C₃₁	—	4	4	no	—
147	SD₁₆×C₃₁	—	4	2	no	—
148	SD₁₆×C₃₁	—	4	2	no	—
149	D₈×C₃₁	—	2	1	no	—
150	D₈×C₃₁	—	2	1	no	—
151	D₈×C₃₁	—	4	4	no	—
152	D₈×C₃₁	—	4	4	no	—
153	D₈×C₃₁	—	4	2	no	—
154	D₈×C₃₁	—	4	1	no	—
155	D₈×C₃₁	—	4	1	no	—
156	D₈×C₃₁	—	4	1	no	—
157	D₈×C₃₁	—	4	1	no	—
158	M₁₆×C₃₁	—	2	1	no	—
159	M₁₆×C₃₁	—	2	2	no	—
160	M₁₆×C₃₁	—	2	1	no	—
161	M₁₆×C₃₁	—	2	1	no	—
162	M₁₆×C₃₁	—	2	1	no	—
163	M₁₆×C₃₁	—	2	2	no	—
164	M₁₆×C₃₁	—	2	2	no	—
165	M₁₆×C₃₁	—	4	4	no	—
166	M₁₆×C₃₁	—	4	1	no	—
167	M₁₆×C₃₁	—	4	1	no	—
168	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
169	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
170	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₃₁	—	2	1	no	—
172	C₂×C₈×C₃₁	—	2	1	no	—
173	C₂×C₈×C₃₁	—	2	2	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	2	no	—
176	C₂×C₈×C₃₁	—	4	4	no	—
177	C₂×C₈×C₃₁	—	4	1	no	—
178	C₂×C₈×C₃₁	—	4	1	no	—
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	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
184	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	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	#25447	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	—	4	2	no	—
196	C₂³×C₃₁	—	4	2	no	—
197	C₂³×C₃₁	—	4	2	no	—
198	C₂³×C₃₁	—	16	2	no	—
199	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	2	1	no	—
202	Q₈×C₃₁	—	2	2	no	—
203	Q₈×C₃₁	—	2	1	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	D₄×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	16	4	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
230	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
231	C₈×C₃₁	—	2	2	no	—
232	C₈×C₃₁	—	2	4	no	—
233	C₈×C₃₁	—	2	2	no	—
234	C₈×C₃₁	—	4	1	no	—
235	C₈×C₃₁	—	4	2	no	—
236	C₈×C₃₁	#121	1	1	yes	characteristic
237	C₈×C₃₁	#121	1	1	yes	characteristic
238	C₈×C₃₁	#121	1	1	yes	characteristic
239	C₈×C₃₁	#121	1	1	yes	characteristic
240	#3155	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#3068	C₂×C₃₁	1	2	yes	complemented
242	#3067	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂×#472	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#2957	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#2048	C₂×C₃₁	1	2	yes	complemented
246	#2008	C₂×C₃₁	1	2	yes	complemented
247	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1914	C₂×C₃₁	1	2	yes	complemented
250	#1757	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	4	1	no	—
252	C₂²×C₃₁	—	4	1	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	8	2	no	—
255	C₂²×C₃₁	—	8	1	no	—
256	C₂²×C₃₁	—	8	4	no	—
257	C₂²×C₃₁	—	8	2	no	—
258	C₂²×C₃₁	—	16	2	no	—
259	C₂²×C₃₁	—	16	2	no	—
260	C₂²×C₃₁	#532	1	1	yes	characteristic
261	C₂²×C₃₁	#546	1	1	yes	characteristic
262	C₂²×C₃₁	#550	1	1	yes	characteristic
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	—	4	1	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	8	1	no	—
267	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
268	C₄×C₃₁	#540	1	1	yes	characteristic
269	C₄×C₃₁	D₄²	1	1	yes	characteristic
270	C₄×C₃₁	D₄²	1	1	yes	characteristic
271	#575	C₂²×C₃₁	1	2	yes	—
272	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
273	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
274	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
275	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
276	#568	C₂²×C₃₁	1	2	yes	—
277	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
278	#535	C₂²×C₃₁	1	2	yes	—
279	#496	—	2	2	no	—
280	#495	—	2	1	no	—
281	#493	—	2	1	no	—
282	#473	—	2	1	no	—
283	#472	—	2	1	no	—
284	#472	—	2	2	no	—
285	#472	—	2	2	no	—
286	#472	—	2	4	no	—
287	#471	—	2	2	no	—
288	#471	—	2	4	no	—
289	#444	—	2	2	no	—
290	#443	—	2	2	no	—
291	#430	C₂²×C₃₁	1	2	yes	—
292	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
294	#421	C₂²×C₃₁	1	2	yes	—
295	#418	C₂²×C₃₁	1	2	yes	—
296	#413	C₂²×C₃₁	1	2	yes	—
297	#413	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×#100	C₂²×C₃₁	1	2	yes	—
299	C₂×#100	C₂²×C₃₁	1	2	yes	—
300	#410	C₂²×C₃₁	1	1	yes	characteristic
301	#405	C₂²×C₃₁	1	1	yes	characteristic
302	#405	C₂²×C₃₁	1	1	yes	characteristic
303	#344	C₂²×C₃₁	1	1	yes	characteristic
304	#344	C₂²×C₃₁	1	1	yes	characteristic
305	#344	C₂²×C₃₁	1	2	yes	—
306	C₂×C₃₁	—	2	1	no	—
307	C₂×C₃₁	—	2	1	no	—
308	C₂×C₃₁	—	2	1	no	—
309	C₂×C₃₁	—	8	2	no	—
310	C₂×C₃₁	—	8	1	no	—
311	C₂×C₃₁	—	16	2	no	—
312	C₂×C₃₁	#2428	1	1	yes	characteristic
313	C₂×#41	C₂³×C₃₁	1	2	yes	—
314	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
316	#137	—	2	2	no	—
317	#137	—	2	2	no	—
318	AGL(1,ℤ/8)	—	2	2	no	—
319	AGL(1,ℤ/8)	—	2	2	no	—
320	AGL(1,ℤ/8)	—	2	4	no	—
321	AGL(1,ℤ/8)	—	2	4	no	—
322	AGL(1,ℤ/8)	—	4	2	no	—
323	AGL(1,ℤ/8)	—	4	2	no	—
324	#135	—	2	1	no	—
325	#135	—	2	1	no	—
326	#135	—	2	1	no	—
327	#135	—	2	1	no	—
328	#135	—	4	2	no	—
329	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
330	C₂×SD₁₆	—	2	2	no	—
331	C₂×SD₁₆	—	2	2	no	—
332	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
333	C₂×D₈	—	2	2	no	—
334	C₂×D₈	—	2	2	no	—
335	C₂×D₈	—	2	1	no	—
336	C₂×D₈	—	2	1	no	—
337	C₂×D₈	—	2	1	no	—
338	C₂×D₈	—	2	1	no	—
339	C₂×D₈	—	2	1	no	—
340	C₂×D₈	—	2	1	no	—
341	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
342	#131	—	2	4	no	—
343	#131	—	2	4	no	—
344	#131	—	2	1	no	—
345	#131	—	2	1	no	—
346	#131	—	2	1	no	—
347	#131	—	2	1	no	—
348	C₂×M₁₆	—	2	1	no	—
349	C₂×M₁₆	—	4	2	no	—
350	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
351	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
352	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
353	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
354	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
355	C₂²×C₈	—	2	1	no	—
356	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
357	#127	—	2	2	no	—
358	#124	—	2	2	no	—
359	#121	—	4	2	no	—
360	#120	—	2	4	no	—
361	#117	C₂³×C₃₁	1	1	yes	characteristic
362	#117	C₂³×C₃₁	1	1	yes	characteristic
363	#108	—	2	4	no	—
364	#108	—	2	1	no	—
365	#108	—	2	1	no	—
366	#108	D₄×C₃₁	1	2	yes	complemented
367	#108	D₄×C₃₁	1	2	yes	complemented
368	#104	—	2	4	no	—
369	#102	—	4	2	no	—
370	#101	—	2	2	no	—
371	#101	D₄×C₃₁	1	2	yes	complemented
372	#100	—	2	2	no	—
373	#100	—	2	4	no	—
374	#100	—	2	4	no	—
375	#100	D₄×C₃₁	1	2	yes	complemented
376	#99	—	2	2	no	—
377	#97	—	2	1	no	—
378	#97	—	2	1	no	—
379	C₄×C₈	—	2	1	no	—
380	C₄×C₈	—	2	1	no	—
381	C₃₁	#25447	1	1	yes	characteristic, direct product factor, Sylow
382	C₂⁴	—	2	2	no	—
383	#41	—	2	2	no	—
384	#41	—	2	1	no	—
385	#41	—	2	2	no	—
386	#41	—	2	1	no	—
387	#41	—	2	2	no	—
388	#41	—	2	1	no	—
389	#41	—	2	2	no	—
390	#41	—	2	1	no	—
391	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
392	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	4	4	no	—
402	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
403	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
407	Q₁₆	—	2	1	no	—
408	Q₁₆	—	2	1	no	—
409	Q₁₆	—	4	2	no	—
410	SD₁₆	—	2	2	no	—
411	SD₁₆	—	2	2	no	—
412	SD₁₆	—	4	4	no	—
413	SD₁₆	—	4	4	no	—
414	SD₁₆	—	4	2	no	—
415	SD₁₆	—	4	2	no	—
416	D₈	—	2	1	no	—
417	D₈	—	2	1	no	—
418	D₈	—	4	4	no	—
419	D₈	—	4	4	no	—
420	D₈	—	4	2	no	—
421	D₈	—	4	1	no	—
422	D₈	—	4	1	no	—
423	D₈	—	4	1	no	—
424	D₈	—	4	1	no	—
425	M₁₆	—	2	1	no	—
426	M₁₆	—	2	2	no	—
427	M₁₆	—	2	1	no	—
428	M₁₆	—	2	1	no	—
429	M₁₆	—	2	1	no	—
430	M₁₆	—	2	2	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	4	4	no	—
433	M₁₆	—	4	1	no	—
434	M₁₆	—	4	1	no	—
435	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
436	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
437	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	4	4	no	—
444	C₂×C₈	—	4	1	no	—
445	C₂×C₈	—	4	1	no	—
446	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
449	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
450	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
451	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₂²⋊C₄	—	2	1	no	—
455	C₂²⋊C₄	—	4	4	no	—
456	C₂²⋊C₄	#41×C₃₁	1	2	yes	—
457	C₄²	—	2	1	no	—
458	C₄²	—	2	1	no	—
459	C₂³	—	2	1	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	4	2	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	16	2	no	—
465	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
466	Q₈	—	2	2	no	—
467	Q₈	—	2	1	no	—
468	Q₈	—	2	2	no	—
469	Q₈	—	2	1	no	—
470	D₄	—	2	2	no	—
471	D₄	—	2	1	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	1	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	1	no	—
476	D₄	—	4	1	no	—
477	D₄	—	4	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	16	4	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
496	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	2	4	no	—
499	C₈	—	2	2	no	—
500	C₈	—	4	1	no	—
501	C₈	—	4	2	no	—
502	C₈	C₃₁×#121	1	1	yes	characteristic
503	C₈	C₃₁×#121	1	1	yes	characteristic
504	C₈	C₃₁×#121	1	1	yes	characteristic
505	C₈	C₃₁×#121	1	1	yes	characteristic
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	8	2	no	—
510	C₂²	—	8	1	no	—
511	C₂²	—	8	4	no	—
512	C₂²	—	8	2	no	—
513	C₂²	—	16	2	no	—
514	C₂²	—	16	2	no	—
515	C₂²	C₃₁×#532	1	1	yes	characteristic
516	C₂²	C₃₁×#546	1	1	yes	characteristic
517	C₂²	C₃₁×#550	1	1	yes	characteristic
518	C₄	—	4	2	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	2	no	—
521	C₄	—	8	1	no	—
522	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
523	C₄	C₃₁×#540	1	1	yes	characteristic
524	C₄	D₄²×C₃₁	1	1	yes	characteristic
525	C₄	D₄²×C₃₁	1	1	yes	characteristic
526	C₂	—	2	1	no	—
527	C₂	—	2	1	no	—
528	C₂	—	2	1	no	—
529	C₂	—	8	2	no	—
530	C₂	—	8	1	no	—
531	C₂	—	16	2	no	—
532	C₂	C₃₁×#2428	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2428
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×#3155	C₂	semidirect
2. C₃₁×#3068	C₂	semidirect
3. C₃₁×#3067	C₂	semidirect
4. C₂×C₃₁×#472	C₂	semidirect
5. C₃₁×#2957	C₂	semidirect
6. C₃₁×#2048	C₂	semidirect
7. C₃₁×#2008	C₂	semidirect
8. C₃₁×#1978	C₂	semidirect
9. C₃₁×#1953	C₂	semidirect
10. C₃₁×#1914	C₂	semidirect
11. C₃₁×#1757	C₂	semidirect
100. C₃₁×#108	D₄	semidirect
101. C₃₁×#108	D₄	semidirect
105. C₃₁×#101	D₄	semidirect
109. C₃₁×#100	D₄	semidirect
192. #25447	C₃₁	direct (N × H)
240. #3155	C₂×C₃₁	semidirect
241. #3068	C₂×C₃₁	semidirect
242. #3067	C₂×C₃₁	semidirect
243. C₂×#472	C₂×C₃₁	semidirect
244. #2957	C₂×C₃₁	semidirect
245. #2048	C₂×C₃₁	semidirect
246. #2008	C₂×C₃₁	semidirect
247. #1978	C₂×C₃₁	semidirect
248. #1953	C₂×C₃₁	semidirect
249. #1914	C₂×C₃₁	semidirect
250. #1757	C₂×C₃₁	semidirect
366. #108	D₄×C₃₁	semidirect
367. #108	D₄×C₃₁	semidirect
371. #101	D₄×C₃₁	semidirect
375. #100	D₄×C₃₁	semidirect
381. C₃₁	#25447	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×#3155	2
C₃₁×#3068	2
C₃₁×#3067	2
C₂×C₃₁×#472	2
C₃₁×#2957	2
C₃₁×#2048	2
C₃₁×#2008	2
C₃₁×#1978	2
C₃₁×#1953	2
C₃₁×#1914	2
C₃₁×#1757	2
#25447	31

Subgroups — C31×#25447 #21482206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25447 #21482206