Subgroups — C₃₁×#32950 #21489709

Order 7936.

1282 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#471	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3057	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3022	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3021	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1915	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1914	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1903	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1902	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1814	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1813	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#585	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#137	C₂²	1	2	yes	—
14	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#568	—	2	4	no	—
17	C₃₁×#568	—	2	2	no	—
18	C₃₁×#568	C₂²	1	2	yes	—
19	C₃₁×#568	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#131	—	2	4	no	—
21	C₂×C₃₁×#131	—	2	2	no	—
22	C₂×C₃₁×#131	C₂²	1	2	yes	—
23	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
24	C₃₁×#536	C₂²	1	1	yes	characteristic
25	C₃₁×#535	C₂²	1	1	yes	characteristic
26	C₃₁×#471	—	2	4	no	—
27	C₃₁×#471	—	2	4	no	—
28	C₃₁×#468	—	2	2	no	—
29	C₃₁×#465	—	2	2	no	—
30	C₃₁×#456	—	2	2	no	—
31	C₃₁×#455	—	2	2	no	—
32	C₃₁×#454	—	2	2	no	—
33	C₃₁×#453	—	2	2	no	—
34	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
35	C₃₁×#427	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#421	C₂²	1	2	yes	—
37	C₃₁×#421	C₂²	1	2	yes	—
38	C₃₁×#419	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#418	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#413	C₂²	1	1	yes	characteristic
41	C₃₁×#413	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#101	C₂²	1	2	yes	complemented
43	C₂×C₃₁×#100	C₂²	1	2	yes	complemented
44	C₃₁×#344	C₂²	1	2	yes	complemented
45	C₃₁×#344	C₂²	1	2	yes	complemented
46	C₃₁×#338	C₂²	1	2	yes	complemented
47	C₃₁×2₋¹⁺⁴	—	2	2	no	—
48	C₃₁×2₋¹⁺⁴	D₄	1	2	yes	—
49	C₃₁×H(5,𝔽₂)	—	2	2	no	—
50	C₃₁×H(5,𝔽₂)	—	2	2	no	—
51	C₃₁×H(5,𝔽₂)	D₄	1	2	yes	—
52	C₂×#41×C₃₁	—	2	4	no	—
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	C₂³	1	2	yes	—
55	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#137	—	2	4	no	—
59	C₃₁×#137	—	2	4	no	—
60	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
61	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
62	C₂×Q₁₆×C₃₁	—	2	2	no	—
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×SD₁₆×C₃₁	—	2	2	no	—
65	C₂×SD₁₆×C₃₁	—	2	1	no	—
66	C₂×SD₁₆×C₃₁	—	2	1	no	—
67	C₂×SD₁₆×C₃₁	—	2	2	no	—
68	C₂×SD₁₆×C₃₁	—	2	2	no	—
69	C₂×D₈×C₃₁	—	2	2	no	—
70	C₃₁×#131	—	2	4	no	—
71	C₃₁×#131	—	2	8	no	—
72	C₃₁×#131	—	2	4	no	—
73	C₃₁×#131	—	2	4	no	—
74	C₃₁×#131	—	2	4	no	—
75	C₃₁×#131	—	2	2	no	—
76	C₃₁×#131	—	2	2	no	—
77	C₃₁×#131	—	2	4	no	—
78	C₃₁×#131	—	2	2	no	—
79	C₃₁×#131	—	2	2	no	—
80	C₃₁×#131	D₄	1	4	yes	complemented
81	C₃₁×#131	D₄	1	4	yes	complemented
82	C₂×M₁₆×C₃₁	—	2	8	no	—
83	C₂×M₁₆×C₃₁	—	2	4	no	—
84	C₂×M₁₆×C₃₁	—	2	2	no	—
85	C₂×M₁₆×C₃₁	—	2	4	no	—
86	C₂×M₁₆×C₃₁	—	2	2	no	—
87	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
88	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
89	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
90	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
91	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₃₁	—	2	4	no	—
93	C₂²×C₈×C₃₁	—	2	2	no	—
94	C₂²×C₈×C₃₁	D₄	1	2	yes	—
95	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₃₁×C₄⋊Q₈	—	2	1	no	—
97	C₃₁×#127	—	2	1	no	—
98	C₃₁×#124	—	2	1	no	—
99	C₃₁×#124	—	2	1	no	—
100	C₃₁×#123	—	2	2	no	—
101	C₃₁×C₂²⋊Q₈	—	2	2	no	—
102	C₃₁×#121	—	2	2	no	—
103	C₃₁×#120	—	2	2	no	—
104	C₃₁×#117	C₂³	1	1	yes	characteristic
105	C₃₁×#117	C₂³	1	1	yes	characteristic
106	C₃₁×#108	—	2	2	no	—
107	C₃₁×#108	—	2	2	no	—
108	C₃₁×#107	—	2	1	no	—
109	C₃₁×#106	—	2	1	no	—
110	C₃₁×#104	—	2	4	no	—
111	C₃₁×#104	—	2	4	no	—
112	C₃₁×#103	—	2	2	no	—
113	C₃₁×#102	—	2	2	no	—
114	C₃₁×#101	—	2	1	no	—
115	C₃₁×#101	—	2	1	no	—
116	C₃₁×#101	—	2	4	no	—
117	C₃₁×#100	—	2	1	no	—
118	C₃₁×#100	—	2	1	no	—
119	C₃₁×#100	—	2	4	no	—
120	C₂⁴×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	4	no	—
122	#41×C₃₁	—	2	8	no	—
123	#41×C₃₁	—	2	4	no	—
124	#41×C₃₁	—	2	4	no	—
125	#41×C₃₁	—	2	4	no	—
126	#41×C₃₁	—	2	4	no	—
127	#41×C₃₁	—	2	2	no	—
128	#41×C₃₁	—	2	2	no	—
129	#41×C₃₁	—	2	2	no	—
130	#41×C₃₁	—	2	2	no	—
131	#41×C₃₁	C₂×D₄	1	4	yes	—
132	#41×C₃₁	C₂×D₄	1	4	yes	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	—	2	4	no	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	2	2	no	—
139	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×Q₈×C₃₁	#41	1	2	yes	—
141	C₂×D₄×C₃₁	—	2	8	no	—
142	C₂×D₄×C₃₁	—	2	2	no	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	—	2	2	no	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	4	2	no	—
151	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂×D₄×C₃₁	#41	1	2	yes	—
153	C₂×D₄×C₃₁	#41	1	2	yes	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	1	no	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	Q₁₆×C₃₁	—	4	4	no	—
160	SD₁₆×C₃₁	—	4	4	no	—
161	SD₁₆×C₃₁	—	4	4	no	—
162	SD₁₆×C₃₁	—	4	1	no	—
163	SD₁₆×C₃₁	—	4	1	no	—
164	SD₁₆×C₃₁	—	4	2	no	—
165	D₈×C₃₁	—	4	4	no	—
166	M₁₆×C₃₁	—	2	4	no	—
167	M₁₆×C₃₁	—	2	8	no	—
168	M₁₆×C₃₁	—	2	2	no	—
169	M₁₆×C₃₁	—	2	2	no	—
170	M₁₆×C₃₁	—	2	4	no	—
171	M₁₆×C₃₁	—	2	4	no	—
172	M₁₆×C₃₁	—	2	4	no	—
173	M₁₆×C₃₁	—	2	2	no	—
174	M₁₆×C₃₁	—	2	2	no	—
175	M₁₆×C₃₁	—	2	2	no	—
176	M₁₆×C₃₁	—	2	2	no	—
177	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
178	C₂×C₈×C₃₁	—	2	4	no	—
179	C₂×C₈×C₃₁	—	2	8	no	—
180	C₂×C₈×C₃₁	—	2	2	no	—
181	C₂×C₈×C₃₁	—	2	4	no	—
182	C₂×C₈×C₃₁	—	2	2	no	—
183	C₂×C₈×C₃₁	—	2	4	no	—
184	C₂×C₈×C₃₁	—	2	2	no	—
185	C₂×C₈×C₃₁	—	2	2	no	—
186	C₂×C₈×C₃₁	—	2	2	no	—
187	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
188	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
191	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	4	2	no	—
195	C₂²⋊C₄×C₃₁	—	2	1	no	—
196	C₂²⋊C₄×C₃₁	—	2	1	no	—
197	C₂²⋊C₄×C₃₁	—	4	2	no	—
198	C₂²⋊C₄×C₃₁	—	4	2	no	—
199	C₄²×C₃₁	—	2	1	no	—
200	C₄²×C₃₁	—	2	1	no	—
201	#32950	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	4	1	no	—
205	C₂³×C₃₁	—	4	2	no	—
206	C₂³×C₃₁	—	16	1	no	—
207	C₂³×C₃₁	#121	1	2	yes	—
208	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
209	Q₈×C₃₁	—	2	4	no	—
210	Q₈×C₃₁	—	2	2	no	—
211	Q₈×C₃₁	—	2	2	no	—
212	Q₈×C₃₁	—	4	1	no	—
213	Q₈×C₃₁	—	4	2	no	—
214	Q₈×C₃₁	—	4	1	no	—
215	Q₈×C₃₁	#121	1	4	yes	—
216	D₄×C₃₁	—	2	8	no	—
217	D₄×C₃₁	—	2	4	no	—
218	D₄×C₃₁	—	2	4	no	—
219	D₄×C₃₁	—	2	4	no	—
220	D₄×C₃₁	—	2	4	no	—
221	D₄×C₃₁	—	2	2	no	—
222	D₄×C₃₁	—	2	2	no	—
223	D₄×C₃₁	—	4	1	no	—
224	D₄×C₃₁	—	4	2	no	—
225	D₄×C₃₁	—	4	1	no	—
226	D₄×C₃₁	—	16	2	no	—
227	D₄×C₃₁	#121	1	4	yes	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	8	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	2	no	—
239	C₂×C₄×C₃₁	#121	1	2	yes	—
240	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
244	C₈×C₃₁	—	2	4	no	—
245	C₈×C₃₁	—	2	2	no	—
246	C₈×C₃₁	—	4	2	no	—
247	C₈×C₃₁	—	4	2	no	—
248	C₈×C₃₁	#120	1	1	yes	characteristic
249	C₈×C₃₁	#120	1	2	yes	—
250	C₈×C₃₁	#120	1	1	yes	characteristic
251	#3573	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂×#471	C₂×C₃₁	1	2	yes	complemented
253	#3057	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#3022	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#3021	C₂×C₃₁	1	1	yes	characteristic
256	#1915	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1914	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1903	C₂×C₃₁	1	2	yes	complemented
259	#1902	C₂×C₃₁	1	2	yes	complemented
260	#1814	C₂×C₃₁	1	2	yes	complemented
261	#1813	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	4	no	—
264	C₂²×C₃₁	—	4	1	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	8	2	no	—
267	C₂²×C₃₁	—	8	2	no	—
268	C₂²×C₃₁	—	8	4	no	—
269	C₂²×C₃₁	—	8	1	no	—
270	C₂²×C₃₁	—	16	2	no	—
271	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
272	C₂²×C₃₁	#525	1	2	yes	—
273	C₄×C₃₁	—	2	4	no	—
274	C₄×C₃₁	—	2	2	no	—
275	C₄×C₃₁	—	8	1	no	—
276	C₄×C₃₁	—	8	1	no	—
277	C₄×C₃₁	—	8	1	no	—
278	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
279	C₄×C₃₁	C₂×#121	1	2	yes	—
280	C₄×C₃₁	#525	1	1	yes	characteristic
281	#585	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#137	C₂²×C₃₁	1	2	yes	—
283	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
284	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
285	#568	—	2	4	no	—
286	#568	—	2	2	no	—
287	#568	C₂²×C₃₁	1	2	yes	—
288	#568	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#131	—	2	4	no	—
290	C₂×#131	—	2	2	no	—
291	C₂×#131	C₂²×C₃₁	1	2	yes	—
292	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
293	#536	C₂²×C₃₁	1	1	yes	characteristic
294	#535	C₂²×C₃₁	1	1	yes	characteristic
295	#471	—	2	4	no	—
296	#471	—	2	4	no	—
297	#468	—	2	2	no	—
298	#465	—	2	2	no	—
299	#456	—	2	2	no	—
300	#455	—	2	2	no	—
301	#454	—	2	2	no	—
302	#453	—	2	2	no	—
303	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
304	#427	C₂²×C₃₁	1	1	yes	characteristic, complemented
305	#421	C₂²×C₃₁	1	2	yes	—
306	#421	C₂²×C₃₁	1	2	yes	—
307	#419	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	#418	C₂²×C₃₁	1	1	yes	characteristic, complemented
309	#413	C₂²×C₃₁	1	1	yes	characteristic
310	#413	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×#101	C₂²×C₃₁	1	2	yes	complemented
312	C₂×#100	C₂²×C₃₁	1	2	yes	complemented
313	#344	C₂²×C₃₁	1	2	yes	complemented
314	#344	C₂²×C₃₁	1	2	yes	complemented
315	#338	C₂²×C₃₁	1	2	yes	complemented
316	C₂×C₃₁	—	2	1	no	—
317	C₂×C₃₁	—	2	2	no	—
318	C₂×C₃₁	—	4	2	no	—
319	C₂×C₃₁	—	4	4	no	—
320	C₂×C₃₁	—	16	1	no	—
321	C₂×C₃₁	#2847	1	1	yes	characteristic
322	2₋¹⁺⁴	—	2	2	no	—
323	2₋¹⁺⁴	D₄×C₃₁	1	2	yes	—
324	H(5,𝔽₂)	—	2	2	no	—
325	H(5,𝔽₂)	—	2	2	no	—
326	H(5,𝔽₂)	D₄×C₃₁	1	2	yes	—
327	C₂×#41	—	2	4	no	—
328	C₂×#41	—	2	2	no	—
329	C₂×#41	C₂³×C₃₁	1	2	yes	—
330	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
331	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
333	#137	—	2	4	no	—
334	#137	—	2	4	no	—
335	AGL(1,ℤ/8)	—	2	4	no	—
336	AGL(1,ℤ/8)	—	2	4	no	—
337	C₂×Q₁₆	—	2	2	no	—
338	C₂×SD₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	1	no	—
341	C₂×SD₁₆	—	2	1	no	—
342	C₂×SD₁₆	—	2	2	no	—
343	C₂×SD₁₆	—	2	2	no	—
344	C₂×D₈	—	2	2	no	—
345	#131	—	2	4	no	—
346	#131	—	2	8	no	—
347	#131	—	2	4	no	—
348	#131	—	2	4	no	—
349	#131	—	2	4	no	—
350	#131	—	2	2	no	—
351	#131	—	2	2	no	—
352	#131	—	2	4	no	—
353	#131	—	2	2	no	—
354	#131	—	2	2	no	—
355	#131	D₄×C₃₁	1	4	yes	complemented
356	#131	D₄×C₃₁	1	4	yes	complemented
357	C₂×M₁₆	—	2	8	no	—
358	C₂×M₁₆	—	2	4	no	—
359	C₂×M₁₆	—	2	2	no	—
360	C₂×M₁₆	—	2	4	no	—
361	C₂×M₁₆	—	2	2	no	—
362	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
363	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
364	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
365	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
366	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
367	C₂²×C₈	—	2	4	no	—
368	C₂²×C₈	—	2	2	no	—
369	C₂²×C₈	D₄×C₃₁	1	2	yes	—
370	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₄⋊Q₈	—	2	1	no	—
372	#127	—	2	1	no	—
373	#124	—	2	1	no	—
374	#124	—	2	1	no	—
375	#123	—	2	2	no	—
376	C₂²⋊Q₈	—	2	2	no	—
377	#121	—	2	2	no	—
378	#120	—	2	2	no	—
379	#117	C₂³×C₃₁	1	1	yes	characteristic
380	#117	C₂³×C₃₁	1	1	yes	characteristic
381	#108	—	2	2	no	—
382	#108	—	2	2	no	—
383	#107	—	2	1	no	—
384	#106	—	2	1	no	—
385	#104	—	2	4	no	—
386	#104	—	2	4	no	—
387	#103	—	2	2	no	—
388	#102	—	2	2	no	—
389	#101	—	2	1	no	—
390	#101	—	2	1	no	—
391	#101	—	2	4	no	—
392	#100	—	2	1	no	—
393	#100	—	2	1	no	—
394	#100	—	2	4	no	—
395	C₃₁	#32950	1	1	yes	characteristic, direct product factor, Sylow
396	C₂⁴	—	2	1	no	—
397	#41	—	2	4	no	—
398	#41	—	2	8	no	—
399	#41	—	2	4	no	—
400	#41	—	2	4	no	—
401	#41	—	2	4	no	—
402	#41	—	2	4	no	—
403	#41	—	2	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	2	no	—
407	#41	C₂×D₄×C₃₁	1	4	yes	—
408	#41	C₂×D₄×C₃₁	1	4	yes	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	4	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
416	C₂×Q₈	#41×C₃₁	1	2	yes	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂×D₄	#41×C₃₁	1	2	yes	—
429	C₂×D₄	#41×C₃₁	1	2	yes	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
435	Q₁₆	—	4	4	no	—
436	SD₁₆	—	4	4	no	—
437	SD₁₆	—	4	4	no	—
438	SD₁₆	—	4	1	no	—
439	SD₁₆	—	4	1	no	—
440	SD₁₆	—	4	2	no	—
441	D₈	—	4	4	no	—
442	M₁₆	—	2	4	no	—
443	M₁₆	—	2	8	no	—
444	M₁₆	—	2	2	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	4	no	—
447	M₁₆	—	2	4	no	—
448	M₁₆	—	2	4	no	—
449	M₁₆	—	2	2	no	—
450	M₁₆	—	2	2	no	—
451	M₁₆	—	2	2	no	—
452	M₁₆	—	2	2	no	—
453	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
454	C₂×C₈	—	2	4	no	—
455	C₂×C₈	—	2	8	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	4	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	4	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
464	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
467	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	4	2	no	—
471	C₂²⋊C₄	—	2	1	no	—
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	4	2	no	—
474	C₂²⋊C₄	—	4	2	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	4	1	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	16	1	no	—
482	C₂³	C₃₁×#121	1	2	yes	—
483	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
484	Q₈	—	2	4	no	—
485	Q₈	—	2	2	no	—
486	Q₈	—	2	2	no	—
487	Q₈	—	4	1	no	—
488	Q₈	—	4	2	no	—
489	Q₈	—	4	1	no	—
490	Q₈	C₃₁×#121	1	4	yes	—
491	D₄	—	2	8	no	—
492	D₄	—	2	4	no	—
493	D₄	—	2	4	no	—
494	D₄	—	2	4	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	2	no	—
497	D₄	—	2	2	no	—
498	D₄	—	4	1	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	1	no	—
501	D₄	—	16	2	no	—
502	D₄	C₃₁×#121	1	4	yes	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	C₃₁×#121	1	2	yes	—
515	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
516	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
517	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
518	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
519	C₈	—	2	4	no	—
520	C₈	—	2	2	no	—
521	C₈	—	4	2	no	—
522	C₈	—	4	2	no	—
523	C₈	C₃₁×#120	1	1	yes	characteristic
524	C₈	C₃₁×#120	1	2	yes	—
525	C₈	C₃₁×#120	1	1	yes	characteristic
526	C₂²	—	2	2	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	4	1	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	8	2	no	—
531	C₂²	—	8	2	no	—
532	C₂²	—	8	4	no	—
533	C₂²	—	8	1	no	—
534	C₂²	—	16	2	no	—
535	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
536	C₂²	C₃₁×#525	1	2	yes	—
537	C₄	—	2	4	no	—
538	C₄	—	2	2	no	—
539	C₄	—	8	1	no	—
540	C₄	—	8	1	no	—
541	C₄	—	8	1	no	—
542	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
543	C₄	C₂×C₃₁×#121	1	2	yes	—
544	C₄	C₃₁×#525	1	1	yes	characteristic
545	C₂	—	2	1	no	—
546	C₂	—	2	2	no	—
547	C₂	—	4	2	no	—
548	C₂	—	4	4	no	—
549	C₂	—	16	1	no	—
550	C₂	C₃₁×#2847	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3021	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×#3573	C₂	semidirect
2. C₂×C₃₁×#471	C₂	semidirect
3. C₃₁×#3057	C₂	semidirect
4. C₃₁×#3022	C₂	semidirect
6. C₃₁×#1915	C₂	semidirect
7. C₃₁×#1914	C₂	semidirect
8. C₃₁×#1903	C₂	semidirect
9. C₃₁×#1902	C₂	semidirect
10. C₃₁×#1814	C₂	semidirect
11. C₃₁×#1813	C₂	semidirect
15. C₂²×SD₁₆×C₃₁	C₂²	semidirect
34. C₂×C₃₁×#108	C₂²	semidirect
35. C₃₁×#427	C₂²	semidirect
38. C₃₁×#419	C₂²	semidirect
39. C₃₁×#418	C₂²	semidirect
42. C₂×C₃₁×#101	C₂²	semidirect
43. C₂×C₃₁×#100	C₂²	semidirect
44. C₃₁×#344	C₂²	semidirect
45. C₃₁×#344	C₂²	semidirect
46. C₃₁×#338	C₂²	semidirect
80. C₃₁×#131	D₄	semidirect
81. C₃₁×#131	D₄	semidirect
201. #32950	C₃₁	direct (N × H)
251. #3573	C₂×C₃₁	semidirect
252. C₂×#471	C₂×C₃₁	semidirect
253. #3057	C₂×C₃₁	semidirect
254. #3022	C₂×C₃₁	semidirect
256. #1915	C₂×C₃₁	semidirect
257. #1914	C₂×C₃₁	semidirect
258. #1903	C₂×C₃₁	semidirect
259. #1902	C₂×C₃₁	semidirect
260. #1814	C₂×C₃₁	semidirect
261. #1813	C₂×C₃₁	semidirect
284. C₂²×SD₁₆	C₂²×C₃₁	semidirect
303. C₂×#108	C₂²×C₃₁	semidirect
304. #427	C₂²×C₃₁	semidirect
307. #419	C₂²×C₃₁	semidirect
308. #418	C₂²×C₃₁	semidirect
311. C₂×#101	C₂²×C₃₁	semidirect
312. C₂×#100	C₂²×C₃₁	semidirect
313. #344	C₂²×C₃₁	semidirect
314. #344	C₂²×C₃₁	semidirect
315. #338	C₂²×C₃₁	semidirect
355. #131	D₄×C₃₁	semidirect
356. #131	D₄×C₃₁	semidirect
395. C₃₁	#32950	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×#3573	2
C₂×C₃₁×#471	2
C₃₁×#3057	2
C₃₁×#3022	2
C₃₁×#3021	2
C₃₁×#1915	2
C₃₁×#1914	2
C₃₁×#1903	2
C₃₁×#1902	2
C₃₁×#1814	2
C₃₁×#1813	2
#32950	31

Subgroups — C31×#32950 #21489709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32950 #21489709