Subgroups — C₃₁×#21478 #21478237

Order 7936.

1282 conjugacy classes of subgroups.

The subgroup lattice has 542 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	C₃₁×#3018	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2955	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2932	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1884	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1881	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1861	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1858	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1794	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1765	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#585	C₂²	1	1	yes	characteristic
13	C₃₁×#575	C₂²	1	2	yes	—
14	C₃₁×#568	—	2	4	no	—
15	C₃₁×#568	—	2	2	no	—
16	C₃₁×#568	C₂²	1	1	yes	characteristic
17	C₃₁×#568	C₂²	1	2	yes	—
18	C₂×C₃₁×#131	—	2	4	no	—
19	C₂×C₃₁×#131	—	2	2	no	—
20	C₂×C₃₁×#131	C₂²	1	2	yes	—
21	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₃₁×#456	C₄	1	2	yes	complemented
25	C₃₁×#455	C₄	1	2	yes	complemented
26	C₃₁×#454	C₄	1	2	yes	complemented
27	C₃₁×#453	C₄	1	2	yes	complemented
28	C₃₁×#444	—	2	4	no	—
29	C₃₁×#443	—	2	4	no	—
30	C₃₁×#435	—	2	2	no	—
31	C₃₁×#435	—	2	2	no	—
32	C₃₁×#430	C₂²	1	2	yes	complemented
33	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#421	C₂²	1	2	yes	—
35	C₃₁×#421	C₂²	1	2	yes	complemented
36	C₂×C₃₁×#104	C₂²	1	2	yes	—
37	C₃₁×#416	C₂²	1	2	yes	—
38	C₃₁×#413	C₂²	1	1	yes	characteristic
39	C₃₁×#410	C₂²	1	1	yes	characteristic
40	C₃₁×#408	C₂²	1	1	yes	characteristic
41	C₃₁×#408	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#405	C₂²	1	2	yes	complemented
43	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#344	C₂²	1	2	yes	complemented
45	C₃₁×#341	C₂²	1	2	yes	complemented
46	C₃₁×#339	C₂²	1	2	yes	complemented
47	C₃₁×2₋¹⁺⁴	—	2	2	no	—
48	C₃₁×2₋¹⁺⁴	C₂×C₄	1	2	yes	—
49	C₃₁×H(5,𝔽₂)	—	2	2	no	—
50	C₃₁×H(5,𝔽₂)	—	2	2	no	—
51	C₃₁×H(5,𝔽₂)	C₂×C₄	1	2	yes	—
52	C₂×#41×C₃₁	—	2	4	no	—
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×#41×C₃₁	C₂³	1	2	yes	—
56	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×#137	—	2	2	no	—
58	C₃₁×#137	C₂×C₄	1	4	yes	—
59	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
60	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
61	C₃₁×#135	—	2	4	no	—
62	C₃₁×#131	—	2	4	no	—
63	C₃₁×#131	—	2	8	no	—
64	C₃₁×#131	—	2	4	no	—
65	C₃₁×#131	—	2	4	no	—
66	C₃₁×#131	—	2	4	no	—
67	C₃₁×#131	—	2	2	no	—
68	C₃₁×#131	—	2	2	no	—
69	C₃₁×#131	—	2	4	no	—
70	C₃₁×#131	—	2	2	no	—
71	C₃₁×#131	—	2	2	no	—
72	C₃₁×#131	—	2	2	no	—
73	C₃₁×#131	—	2	2	no	—
74	C₃₁×#131	D₄	1	4	yes	complemented
75	C₃₁×#131	D₄	1	4	yes	complemented
76	C₂×M₁₆×C₃₁	—	2	8	no	—
77	C₂×M₁₆×C₃₁	—	2	4	no	—
78	C₂×M₁₆×C₃₁	—	2	2	no	—
79	C₂×M₁₆×C₃₁	—	2	4	no	—
80	C₂×M₁₆×C₃₁	—	2	2	no	—
81	C₂×M₁₆×C₃₁	—	2	1	no	—
82	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
83	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
85	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₃₁	—	2	4	no	—
88	C₂²×C₈×C₃₁	—	2	2	no	—
89	C₂²×C₈×C₃₁	—	2	1	no	—
90	C₂²×C₈×C₃₁	D₄	1	2	yes	—
91	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
94	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
95	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
96	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
97	C₃₁×#121	—	2	2	no	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₃₁×#117	C₂³	1	2	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₃₁×#108	—	2	4	no	—
102	C₃₁×C₄⋊C₈	—	2	2	no	—
103	C₃₁×#104	—	2	4	no	—
104	C₃₁×#104	—	2	4	no	—
105	C₃₁×#104	C₂×C₄	1	4	yes	—
106	C₃₁×#104	C₂×C₄	1	4	yes	—
107	C₃₁×#103	—	2	2	no	—
108	C₃₁×#102	—	2	2	no	—
109	C₃₁×#101	C₂×C₄	1	2	yes	—
110	C₃₁×#100	C₂×C₄	1	2	yes	—
111	C₃₁×#99	—	2	2	no	—
112	C₃₁×C₂²⋊C₈	—	2	2	no	—
113	C₃₁×C₂²⋊C₈	—	2	2	no	—
114	C₃₁×#97	—	2	1	no	—
115	C₃₁×#97	—	2	1	no	—
116	C₃₁×#97	—	2	2	no	—
117	C₄×C₈×C₃₁	—	2	2	no	—
118	#41×C₃₁	—	2	4	no	—
119	#41×C₃₁	—	2	8	no	—
120	#41×C₃₁	—	2	4	no	—
121	#41×C₃₁	—	2	4	no	—
122	#41×C₃₁	—	2	4	no	—
123	#41×C₃₁	—	2	4	no	—
124	#41×C₃₁	—	2	2	no	—
125	#41×C₃₁	—	2	2	no	—
126	#41×C₃₁	—	2	2	no	—
127	#41×C₃₁	—	2	2	no	—
128	#41×C₃₁	—	2	2	no	—
129	#41×C₃₁	—	2	2	no	—
130	#41×C₃₁	C₂²×C₄	1	4	yes	—
131	#41×C₃₁	C₂×D₄	1	4	yes	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	2	4	no	—
134	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂×Q₈×C₃₁	#41	1	2	yes	—
137	C₂×D₄×C₃₁	—	2	8	no	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	4	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	2	no	—
144	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
146	C₂×D₄×C₃₁	#41	1	2	yes	—
147	C₂×D₄×C₃₁	#41	1	2	yes	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	1	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₃₁	—	2	4	no	—
155	SD₁₆×C₃₁	—	2	4	no	—
156	SD₁₆×C₃₁	—	2	4	no	—
157	D₈×C₃₁	—	2	4	no	—
158	M₁₆×C₃₁	—	2	4	no	—
159	M₁₆×C₃₁	—	2	8	no	—
160	M₁₆×C₃₁	—	2	2	no	—
161	M₁₆×C₃₁	—	2	2	no	—
162	M₁₆×C₃₁	—	2	2	no	—
163	M₁₆×C₃₁	—	2	4	no	—
164	M₁₆×C₃₁	—	2	4	no	—
165	M₁₆×C₃₁	—	2	4	no	—
166	M₁₆×C₃₁	—	2	2	no	—
167	M₁₆×C₃₁	—	4	1	no	—
168	M₁₆×C₃₁	—	4	1	no	—
169	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
170	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
171	C₂×C₈×C₃₁	—	2	4	no	—
172	C₂×C₈×C₃₁	—	2	8	no	—
173	C₂×C₈×C₃₁	—	2	2	no	—
174	C₂×C₈×C₃₁	—	2	2	no	—
175	C₂×C₈×C₃₁	—	2	4	no	—
176	C₂×C₈×C₃₁	—	2	2	no	—
177	C₂×C₈×C₃₁	—	2	4	no	—
178	C₂×C₈×C₃₁	—	2	2	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	2	no	—
182	C₂×C₈×C₃₁	—	4	2	no	—
183	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
184	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
185	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
187	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₂²⋊C₄×C₃₁	—	2	2	no	—
191	C₂²⋊C₄×C₃₁	—	2	2	no	—
192	C₂²⋊C₄×C₃₁	—	2	2	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	2	no	—
195	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
197	#21478	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₃₁	—	2	4	no	—
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	1	no	—
201	C₂³×C₃₁	C₄×D₄	1	2	yes	—
202	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
203	Q₈×C₃₁	—	2	4	no	—
204	Q₈×C₃₁	—	2	2	no	—
205	Q₈×C₃₁	—	2	2	no	—
206	Q₈×C₃₁	—	2	2	no	—
207	Q₈×C₃₁	C₄×D₄	1	4	yes	—
208	D₄×C₃₁	—	2	8	no	—
209	D₄×C₃₁	—	2	4	no	—
210	D₄×C₃₁	—	2	4	no	—
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	—	2	4	no	—
213	D₄×C₃₁	—	2	2	no	—
214	D₄×C₃₁	—	2	2	no	—
215	D₄×C₃₁	—	2	2	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	8	2	no	—
218	D₄×C₃₁	C₄×D₄	1	4	yes	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	8	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	8	1	no	—
233	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
234	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
238	C₈×C₃₁	—	2	4	no	—
239	C₈×C₃₁	—	2	4	no	—
240	C₈×C₃₁	—	2	2	no	—
241	C₈×C₃₁	—	4	1	no	—
242	C₈×C₃₁	—	4	1	no	—
243	C₈×C₃₁	#120	1	1	yes	characteristic
244	C₈×C₃₁	#120	1	2	yes	—
245	C₈×C₃₁	#120	1	1	yes	characteristic
246	#3573	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#3018	C₂×C₃₁	1	1	yes	characteristic
248	#2955	C₂×C₃₁	1	2	yes	complemented
249	#2932	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1884	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1881	C₂×C₃₁	1	2	yes	complemented
252	#1861	C₂×C₃₁	1	2	yes	complemented
253	#1858	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1794	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1765	C₂×C₃₁	1	2	yes	complemented
256	#1757	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂²×C₃₁	—	2	2	no	—
258	C₂²×C₃₁	—	2	4	no	—
259	C₂²×C₃₁	—	4	1	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	—	4	2	no	—
262	C₂²×C₃₁	—	4	1	no	—
263	C₂²×C₃₁	—	8	2	no	—
264	C₂²×C₃₁	—	8	4	no	—
265	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
266	C₂²×C₃₁	#525	1	2	yes	—
267	C₄×C₃₁	—	2	4	no	—
268	C₄×C₃₁	—	2	2	no	—
269	C₄×C₃₁	—	4	1	no	—
270	C₄×C₃₁	—	4	1	no	—
271	C₄×C₃₁	—	4	1	no	—
272	C₄×C₃₁	—	8	2	no	—
273	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
274	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
275	C₄×C₃₁	#525	1	1	yes	characteristic
276	#585	C₂²×C₃₁	1	1	yes	characteristic
277	#575	C₂²×C₃₁	1	2	yes	—
278	#568	—	2	4	no	—
279	#568	—	2	2	no	—
280	#568	C₂²×C₃₁	1	1	yes	characteristic
281	#568	C₂²×C₃₁	1	2	yes	—
282	C₂×#131	—	2	4	no	—
283	C₂×#131	—	2	2	no	—
284	C₂×#131	C₂²×C₃₁	1	2	yes	—
285	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic, complemented
287	#532	C₂²×C₃₁	1	1	yes	characteristic
288	#456	C₄×C₃₁	1	2	yes	complemented
289	#455	C₄×C₃₁	1	2	yes	complemented
290	#454	C₄×C₃₁	1	2	yes	complemented
291	#453	C₄×C₃₁	1	2	yes	complemented
292	#444	—	2	4	no	—
293	#443	—	2	4	no	—
294	#435	—	2	2	no	—
295	#435	—	2	2	no	—
296	#430	C₂²×C₃₁	1	2	yes	complemented
297	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
298	#421	C₂²×C₃₁	1	2	yes	—
299	#421	C₂²×C₃₁	1	2	yes	complemented
300	C₂×#104	C₂²×C₃₁	1	2	yes	—
301	#416	C₂²×C₃₁	1	2	yes	—
302	#413	C₂²×C₃₁	1	1	yes	characteristic
303	#410	C₂²×C₃₁	1	1	yes	characteristic
304	#408	C₂²×C₃₁	1	1	yes	characteristic
305	#408	C₂²×C₃₁	1	1	yes	characteristic, complemented
306	#405	C₂²×C₃₁	1	2	yes	complemented
307	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	#344	C₂²×C₃₁	1	2	yes	complemented
309	#341	C₂²×C₃₁	1	2	yes	complemented
310	#339	C₂²×C₃₁	1	2	yes	complemented
311	C₂×C₃₁	—	2	1	no	—
312	C₂×C₃₁	—	2	2	no	—
313	C₂×C₃₁	—	4	2	no	—
314	C₂×C₃₁	—	4	4	no	—
315	C₂×C₃₁	—	8	1	no	—
316	C₂×C₃₁	C₄×#120	1	1	yes	characteristic
317	2₋¹⁺⁴	—	2	2	no	—
318	2₋¹⁺⁴	C₂×C₄×C₃₁	1	2	yes	—
319	H(5,𝔽₂)	—	2	2	no	—
320	H(5,𝔽₂)	—	2	2	no	—
321	H(5,𝔽₂)	C₂×C₄×C₃₁	1	2	yes	—
322	C₂×#41	—	2	4	no	—
323	C₂×#41	—	2	2	no	—
324	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
325	C₂×#41	C₂³×C₃₁	1	2	yes	—
326	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
327	#137	—	2	2	no	—
328	#137	C₂×C₄×C₃₁	1	4	yes	—
329	AGL(1,ℤ/8)	—	2	2	no	—
330	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	4	yes	—
331	#135	—	2	4	no	—
332	#131	—	2	4	no	—
333	#131	—	2	8	no	—
334	#131	—	2	4	no	—
335	#131	—	2	4	no	—
336	#131	—	2	4	no	—
337	#131	—	2	2	no	—
338	#131	—	2	2	no	—
339	#131	—	2	4	no	—
340	#131	—	2	2	no	—
341	#131	—	2	2	no	—
342	#131	—	2	2	no	—
343	#131	—	2	2	no	—
344	#131	D₄×C₃₁	1	4	yes	complemented
345	#131	D₄×C₃₁	1	4	yes	complemented
346	C₂×M₁₆	—	2	8	no	—
347	C₂×M₁₆	—	2	4	no	—
348	C₂×M₁₆	—	2	2	no	—
349	C₂×M₁₆	—	2	4	no	—
350	C₂×M₁₆	—	2	2	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
353	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
355	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
356	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
357	C₂²×C₈	—	2	4	no	—
358	C₂²×C₈	—	2	2	no	—
359	C₂²×C₈	—	2	1	no	—
360	C₂²×C₈	D₄×C₃₁	1	2	yes	—
361	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
364	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
365	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
366	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
367	#121	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	#117	C₂³×C₃₁	1	2	yes	—
370	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
371	#108	—	2	4	no	—
372	C₄⋊C₈	—	2	2	no	—
373	#104	—	2	4	no	—
374	#104	—	2	4	no	—
375	#104	C₂×C₄×C₃₁	1	4	yes	—
376	#104	C₂×C₄×C₃₁	1	4	yes	—
377	#103	—	2	2	no	—
378	#102	—	2	2	no	—
379	#101	C₂×C₄×C₃₁	1	2	yes	—
380	#100	C₂×C₄×C₃₁	1	2	yes	—
381	#99	—	2	2	no	—
382	C₂²⋊C₈	—	2	2	no	—
383	C₂²⋊C₈	—	2	2	no	—
384	#97	—	2	1	no	—
385	#97	—	2	1	no	—
386	#97	—	2	2	no	—
387	C₄×C₈	—	2	2	no	—
388	C₃₁	#21478	1	1	yes	characteristic, direct product factor, Sylow
389	#41	—	2	4	no	—
390	#41	—	2	8	no	—
391	#41	—	2	4	no	—
392	#41	—	2	4	no	—
393	#41	—	2	4	no	—
394	#41	—	2	4	no	—
395	#41	—	2	2	no	—
396	#41	—	2	2	no	—
397	#41	—	2	2	no	—
398	#41	—	2	2	no	—
399	#41	—	2	2	no	—
400	#41	—	2	2	no	—
401	#41	C₂²×C₄×C₃₁	1	4	yes	—
402	#41	C₂×D₄×C₃₁	1	4	yes	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
406	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
407	C₂×Q₈	#41×C₃₁	1	2	yes	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
416	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
417	C₂×D₄	#41×C₃₁	1	2	yes	—
418	C₂×D₄	#41×C₃₁	1	2	yes	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
424	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
425	Q₁₆	—	2	4	no	—
426	SD₁₆	—	2	4	no	—
427	SD₁₆	—	2	4	no	—
428	D₈	—	2	4	no	—
429	M₁₆	—	2	4	no	—
430	M₁₆	—	2	8	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	2	2	no	—
433	M₁₆	—	2	2	no	—
434	M₁₆	—	2	4	no	—
435	M₁₆	—	2	4	no	—
436	M₁₆	—	2	4	no	—
437	M₁₆	—	2	2	no	—
438	M₁₆	—	4	1	no	—
439	M₁₆	—	4	1	no	—
440	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
441	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
442	C₂×C₈	—	2	4	no	—
443	C₂×C₈	—	2	8	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	4	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	2	4	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	4	2	no	—
454	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
455	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
456	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
458	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	2	no	—
466	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
467	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
468	C₂³	—	2	4	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	1	no	—
471	C₂³	C₄×D₄×C₃₁	1	2	yes	—
472	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
473	Q₈	—	2	4	no	—
474	Q₈	—	2	2	no	—
475	Q₈	—	2	2	no	—
476	Q₈	—	2	2	no	—
477	Q₈	C₄×D₄×C₃₁	1	4	yes	—
478	D₄	—	2	8	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	4	no	—
483	D₄	—	2	2	no	—
484	D₄	—	2	2	no	—
485	D₄	—	2	2	no	—
486	D₄	—	4	2	no	—
487	D₄	—	8	2	no	—
488	D₄	C₄×D₄×C₃₁	1	4	yes	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	8	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	8	1	no	—
503	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
504	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
508	C₈	—	2	4	no	—
509	C₈	—	2	4	no	—
510	C₈	—	2	2	no	—
511	C₈	—	4	1	no	—
512	C₈	—	4	1	no	—
513	C₈	C₃₁×#120	1	1	yes	characteristic
514	C₈	C₃₁×#120	1	2	yes	—
515	C₈	C₃₁×#120	1	1	yes	characteristic
516	C₂²	—	2	2	no	—
517	C₂²	—	2	4	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	1	no	—
522	C₂²	—	8	2	no	—
523	C₂²	—	8	4	no	—
524	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
525	C₂²	C₃₁×#525	1	2	yes	—
526	C₄	—	2	4	no	—
527	C₄	—	2	2	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	1	no	—
530	C₄	—	4	1	no	—
531	C₄	—	8	2	no	—
532	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
533	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
534	C₄	C₃₁×#525	1	1	yes	characteristic
535	C₂	—	2	1	no	—
536	C₂	—	2	2	no	—
537	C₂	—	4	2	no	—
538	C₂	—	4	4	no	—
539	C₂	—	8	1	no	—
540	C₂	C₄×C₃₁×#120	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂×#120
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	C₄×#120
prime radical	C₃₁×#3018	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3573	C₂	semidirect
3. C₃₁×#2955	C₂	semidirect
4. C₃₁×#2932	C₂	semidirect
5. C₃₁×#1884	C₂	semidirect
6. C₃₁×#1881	C₂	semidirect
7. C₃₁×#1861	C₂	semidirect
8. C₃₁×#1858	C₂	semidirect
9. C₃₁×#1794	C₂	semidirect
10. C₃₁×#1765	C₂	semidirect
11. C₃₁×#1757	C₂	semidirect
22. C₂×C₃₁×#131	C₂²	semidirect
24. C₃₁×#456	C₄	semidirect
25. C₃₁×#455	C₄	semidirect
26. C₃₁×#454	C₄	semidirect
27. C₃₁×#453	C₄	semidirect
32. C₃₁×#430	C₂²	semidirect
33. C₃₁×#423	C₂²	semidirect
35. C₃₁×#421	C₂²	semidirect
41. C₃₁×#408	C₂²	semidirect
42. C₃₁×#405	C₂²	semidirect
43. C₂×C₃₁×#97	C₂²	semidirect
44. C₃₁×#344	C₂²	semidirect
45. C₃₁×#341	C₂²	semidirect
46. C₃₁×#339	C₂²	semidirect
74. C₃₁×#131	D₄	semidirect
75. C₃₁×#131	D₄	semidirect
197. #21478	C₃₁	direct (N × H)
246. #3573	C₂×C₃₁	semidirect
248. #2955	C₂×C₃₁	semidirect
249. #2932	C₂×C₃₁	semidirect
250. #1884	C₂×C₃₁	semidirect
251. #1881	C₂×C₃₁	semidirect
252. #1861	C₂×C₃₁	semidirect
253. #1858	C₂×C₃₁	semidirect
254. #1794	C₂×C₃₁	semidirect
255. #1765	C₂×C₃₁	semidirect
256. #1757	C₂×C₃₁	semidirect
286. C₂×#131	C₂²×C₃₁	semidirect
288. #456	C₄×C₃₁	semidirect
289. #455	C₄×C₃₁	semidirect
290. #454	C₄×C₃₁	semidirect
291. #453	C₄×C₃₁	semidirect
296. #430	C₂²×C₃₁	semidirect
297. #423	C₂²×C₃₁	semidirect
299. #421	C₂²×C₃₁	semidirect
305. #408	C₂²×C₃₁	semidirect
306. #405	C₂²×C₃₁	semidirect
307. C₂×#97	C₂²×C₃₁	semidirect
308. #344	C₂²×C₃₁	semidirect
309. #341	C₂²×C₃₁	semidirect
310. #339	C₂²×C₃₁	semidirect
344. #131	D₄×C₃₁	semidirect
345. #131	D₄×C₃₁	semidirect
388. C₃₁	#21478	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
C₃₁×#3018	2
C₃₁×#2955	2
C₃₁×#2932	2
C₃₁×#1884	2
C₃₁×#1881	2
C₃₁×#1861	2
C₃₁×#1858	2
C₃₁×#1794	2
C₃₁×#1765	2
C₃₁×#1757	2
#21478	31

Subgroups — C31×#21478 #21478237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21478 #21478237