Subgroups — C₃₁×#35132 #21491891

Order 7936.

1666 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2864	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2666	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2612	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2570	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2550	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2442	C₂	1	2	yes	maximal
10	C₃₁×#2440	C₂	1	2	yes	maximal
11	C₃₁×#2438	C₂	1	4	yes	maximal
12	C₃₁×#2329	C₂	1	4	yes	maximal
13	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic
14	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic
15	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#125	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₂×C₃₁×#117	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#400	C₂²	1	4	yes	—
23	C₃₁×#400	C₂²	1	4	yes	—
24	C₃₁×#398	C₂²	1	4	yes	—
25	C₃₁×#398	C₂²	1	4	yes	—
26	C₃₁×#398	C₂²	1	4	yes	—
27	C₃₁×#398	C₂²	1	4	yes	—
28	C₃₁×#395	C₂²	1	4	yes	—
29	C₃₁×#395	C₂²	1	4	yes	—
30	C₃₁×#391	C₂²	1	4	yes	—
31	C₃₁×#391	C₂²	1	4	yes	—
32	C₃₁×#391	C₂²	1	4	yes	—
33	C₃₁×#389	C₂²	1	2	yes	—
34	C₃₁×#389	C₂²	1	2	yes	—
35	C₃₁×#389	C₂²	1	4	yes	—
36	C₃₁×#389	C₂²	1	4	yes	—
37	C₃₁×#389	C₂²	1	4	yes	—
38	C₃₁×#389	C₂²	1	2	yes	—
39	C₃₁×#389	C₂²	1	2	yes	—
40	C₃₁×#387	C₂²	1	4	yes	—
41	C₃₁×#387	C₂²	1	4	yes	—
42	C₃₁×#387	C₂²	1	4	yes	—
43	C₃₁×#387	C₂²	1	4	yes	—
44	C₃₁×#387	C₂²	1	4	yes	—
45	C₃₁×#387	C₂²	1	4	yes	—
46	C₃₁×#385	C₂²	1	4	yes	—
47	C₃₁×#385	C₂²	1	4	yes	—
48	C₃₁×#384	C₂²	1	2	yes	—
49	C₃₁×#383	C₂²	1	2	yes	—
50	C₃₁×#382	C₂²	1	2	yes	—
51	C₃₁×#382	C₂²	1	2	yes	—
52	C₃₁×#382	C₂²	1	1	yes	characteristic
53	C₃₁×#382	C₂²	1	1	yes	characteristic
54	C₃₁×#382	C₂²	1	1	yes	characteristic
55	C₃₁×#382	C₂²	1	1	yes	characteristic
56	C₃₁×#382	C₂²	1	2	yes	—
57	C₃₁×#382	C₂²	1	2	yes	—
58	C₃₁×#381	C₂²	1	2	yes	—
59	C₃₁×#380	C₂²	1	4	yes	—
60	C₃₁×#380	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
68	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
69	C₃₁×#376	C₂²	1	2	yes	—
70	C₃₁×#376	C₂²	1	2	yes	—
71	C₂×C₃₁×#95	C₂²	1	2	yes	—
72	C₂×C₃₁×#95	C₂²	1	2	yes	—
73	C₄³×C₃₁	C₂²	1	1	yes	characteristic
74	C₄³×C₃₁	C₂²	1	1	yes	characteristic
75	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	—	4	4	no	—
81	C₃₁×#125	—	4	4	no	—
82	C₃₁×C₂²⋊Q₈	—	2	16	no	—
83	C₃₁×#117	—	2	8	no	—
84	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄²×C₃₁	Q₈	1	8	yes	complemented
105	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
106	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
113	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₃₁×#95	C₂³	1	2	yes	—
119	C₃₁×#95	C₂³	1	4	yes	—
120	C₃₁×#95	C₂³	1	4	yes	—
121	C₃₁×#95	C₂³	1	4	yes	—
122	C₃₁×#95	C₂³	1	4	yes	—
123	C₃₁×#95	C₂³	1	4	yes	—
124	C₃₁×#95	C₂³	1	2	yes	—
125	C₃₁×#95	C₂³	1	4	yes	—
126	C₃₁×#95	C₂³	1	4	yes	—
127	C₃₁×#95	C₂³	1	4	yes	—
128	C₃₁×#95	C₂³	1	4	yes	—
129	C₃₁×#95	C₂³	1	2	yes	—
130	C₃₁×#95	C₂³	1	2	yes	—
131	C₃₁×#95	C₂³	1	4	yes	—
132	C₃₁×#95	C₂³	1	2	yes	—
133	C₃₁×#95	C₂³	1	2	yes	—
134	C₃₁×#95	C₂³	1	2	yes	—
135	C₃₁×#95	C₂³	1	2	yes	—
136	C₃₁×#95	C₂³	1	2	yes	—
137	C₃₁×#95	C₂³	1	2	yes	—
138	C₃₁×#95	C₂³	1	2	yes	—
139	C₃₁×#95	C₂³	1	2	yes	—
140	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
141	C₂×Q₈×C₃₁	—	2	8	no	—
142	C₂×Q₈×C₃₁	—	4	4	no	—
143	C₂×Q₈×C₃₁	—	4	4	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	8	no	—
148	C₂²×C₄×C₃₁	—	4	2	no	—
149	C₂²×C₄×C₃₁	—	4	4	no	—
150	C₂²×C₄×C₃₁	—	4	4	no	—
151	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	2	8	no	—
172	C₄⋊C₄×C₃₁	—	2	8	no	—
173	C₄⋊C₄×C₃₁	—	2	4	no	—
174	C₄⋊C₄×C₃₁	—	2	4	no	—
175	C₄⋊C₄×C₃₁	—	2	4	no	—
176	C₄⋊C₄×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	4	4	no	—
178	C₄⋊C₄×C₃₁	—	4	4	no	—
179	C₄⋊C₄×C₃₁	—	4	4	no	—
180	C₄⋊C₄×C₃₁	—	4	4	no	—
181	C₄⋊C₄×C₃₁	—	4	4	no	—
182	C₄⋊C₄×C₃₁	—	4	4	no	—
183	C₄⋊C₄×C₃₁	—	4	4	no	—
184	C₂²⋊C₄×C₃₁	—	2	4	no	—
185	C₂²⋊C₄×C₃₁	—	2	8	no	—
186	C₂²⋊C₄×C₃₁	#41	1	8	yes	—
187	C₄²×C₃₁	—	2	4	no	—
188	C₄²×C₃₁	—	2	4	no	—
189	C₄²×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	—	2	2	no	—
191	C₄²×C₃₁	—	2	2	no	—
192	C₄²×C₃₁	—	2	2	no	—
193	C₄²×C₃₁	—	4	2	no	—
194	C₄²×C₃₁	—	4	2	no	—
195	C₄²×C₃₁	—	4	4	no	—
196	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
197	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
198	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
199	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
200	#35132	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂³×C₃₁	—	2	1	no	—
202	C₂³×C₃₁	—	2	1	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
205	C₂³×C₃₁	C₂×#41	1	4	yes	—
206	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
207	Q₈×C₃₁	—	8	8	no	—
208	C₂×C₄×C₃₁	—	2	8	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	8	no	—
220	C₂×C₄×C₃₁	—	2	2	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	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	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	2	no	—
234	C₂×C₄×C₃₁	—	2	4	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	4	4	no	—
239	C₂×C₄×C₃₁	—	8	8	no	—
240	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
241	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
242	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
243	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
244	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
245	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
246	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
247	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
248	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
249	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
250	#2864	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#2864	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#2862	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#2666	C₂×C₃₁	1	4	yes	complemented
255	#2612	C₂×C₃₁	1	4	yes	complemented
256	#2570	C₂×C₃₁	1	2	yes	complemented
257	#2550	C₂×C₃₁	1	2	yes	complemented
258	#2442	C₂×C₃₁	1	2	yes	—
259	#2440	C₂×C₃₁	1	2	yes	—
260	#2438	C₂×C₃₁	1	4	yes	—
261	#2329	C₂×C₃₁	1	4	yes	—
262	#2283	C₂×C₃₁	1	1	yes	characteristic
263	C₂×#382	C₂×C₃₁	1	1	yes	characteristic
264	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
265	C₂²×C₃₁	—	2	2	no	—
266	C₂²×C₃₁	—	2	2	no	—
267	C₂²×C₃₁	—	2	4	no	—
268	C₂²×C₃₁	—	2	4	no	—
269	C₂²×C₃₁	#533	1	4	yes	—
270	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
271	C₂²×C₃₁	C₂²×#41	1	2	yes	—
272	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
273	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
274	C₂²×C₃₁	#585	1	2	yes	—
275	C₄×C₃₁	—	2	2	no	—
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	2	2	no	—
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	4	no	—
281	C₄×C₃₁	—	2	4	no	—
282	C₄×C₃₁	—	4	2	no	—
283	C₄×C₃₁	—	4	4	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	—	4	4	no	—
286	C₄×C₃₁	—	4	2	no	—
287	C₄×C₃₁	—	4	4	no	—
288	C₄×C₃₁	#544	1	4	yes	—
289	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
290	C₂×#125	C₂²×C₃₁	1	2	yes	—
291	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
292	C₂×#117	C₂²×C₃₁	1	2	yes	—
293	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
294	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
295	#400	C₂²×C₃₁	1	4	yes	—
296	#400	C₂²×C₃₁	1	4	yes	—
297	#398	C₂²×C₃₁	1	4	yes	—
298	#398	C₂²×C₃₁	1	4	yes	—
299	#398	C₂²×C₃₁	1	4	yes	—
300	#398	C₂²×C₃₁	1	4	yes	—
301	#395	C₂²×C₃₁	1	4	yes	—
302	#395	C₂²×C₃₁	1	4	yes	—
303	#391	C₂²×C₃₁	1	4	yes	—
304	#391	C₂²×C₃₁	1	4	yes	—
305	#391	C₂²×C₃₁	1	4	yes	—
306	#389	C₂²×C₃₁	1	2	yes	—
307	#389	C₂²×C₃₁	1	2	yes	—
308	#389	C₂²×C₃₁	1	4	yes	—
309	#389	C₂²×C₃₁	1	4	yes	—
310	#389	C₂²×C₃₁	1	4	yes	—
311	#389	C₂²×C₃₁	1	2	yes	—
312	#389	C₂²×C₃₁	1	2	yes	—
313	#387	C₂²×C₃₁	1	4	yes	—
314	#387	C₂²×C₃₁	1	4	yes	—
315	#387	C₂²×C₃₁	1	4	yes	—
316	#387	C₂²×C₃₁	1	4	yes	—
317	#387	C₂²×C₃₁	1	4	yes	—
318	#387	C₂²×C₃₁	1	4	yes	—
319	#385	C₂²×C₃₁	1	4	yes	—
320	#385	C₂²×C₃₁	1	4	yes	—
321	#384	C₂²×C₃₁	1	2	yes	—
322	#383	C₂²×C₃₁	1	2	yes	—
323	#382	C₂²×C₃₁	1	2	yes	—
324	#382	C₂²×C₃₁	1	2	yes	—
325	#382	C₂²×C₃₁	1	1	yes	characteristic
326	#382	C₂²×C₃₁	1	1	yes	characteristic
327	#382	C₂²×C₃₁	1	1	yes	characteristic
328	#382	C₂²×C₃₁	1	1	yes	characteristic
329	#382	C₂²×C₃₁	1	2	yes	—
330	#382	C₂²×C₃₁	1	2	yes	—
331	#381	C₂²×C₃₁	1	2	yes	—
332	#380	C₂²×C₃₁	1	4	yes	—
333	#380	C₂²×C₃₁	1	2	yes	—
334	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
335	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
336	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
337	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
338	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
339	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
340	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
341	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
342	#376	C₂²×C₃₁	1	2	yes	—
343	#376	C₂²×C₃₁	1	2	yes	—
344	C₂×#95	C₂²×C₃₁	1	2	yes	—
345	C₂×#95	C₂²×C₃₁	1	2	yes	—
346	C₄³	C₂²×C₃₁	1	1	yes	characteristic
347	C₄³	C₂²×C₃₁	1	1	yes	characteristic
348	C₂×C₃₁	—	2	4	no	—
349	C₂×C₃₁	C₂×#533	1	1	yes	characteristic
350	C₂×C₃₁	C₂×#544	1	1	yes	characteristic
351	C₂×C₃₁	#3458	1	1	yes	characteristic
352	C₂×C₃₁	#3514	1	2	yes	—
353	C₂×C₃₁	#3515	1	2	yes	—
354	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
358	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
359	C₄⋊Q₈	—	4	4	no	—
360	#125	—	4	4	no	—
361	C₂²⋊Q₈	—	2	16	no	—
362	#117	—	2	8	no	—
363	C₂×C₄⋊C₄	—	4	4	no	—
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
377	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
379	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
383	C₂×C₄²	Q₈×C₃₁	1	8	yes	complemented
384	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
385	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
391	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
392	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
393	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
397	#95	C₂³×C₃₁	1	2	yes	—
398	#95	C₂³×C₃₁	1	2	yes	—
399	#95	C₂³×C₃₁	1	4	yes	—
400	#95	C₂³×C₃₁	1	2	yes	—
401	#95	C₂³×C₃₁	1	2	yes	—
402	#95	C₂³×C₃₁	1	4	yes	—
403	#95	C₂³×C₃₁	1	2	yes	—
404	#95	C₂³×C₃₁	1	2	yes	—
405	#95	C₂³×C₃₁	1	2	yes	—
406	#95	C₂³×C₃₁	1	2	yes	—
407	#95	C₂³×C₃₁	1	4	yes	—
408	#95	C₂³×C₃₁	1	2	yes	—
409	#95	C₂³×C₃₁	1	2	yes	—
410	#95	C₂³×C₃₁	1	2	yes	—
411	#95	C₂³×C₃₁	1	2	yes	—
412	#95	C₂³×C₃₁	1	4	yes	—
413	#95	C₂³×C₃₁	1	4	yes	—
414	#95	C₂³×C₃₁	1	4	yes	—
415	#95	C₂³×C₃₁	1	4	yes	—
416	#95	C₂³×C₃₁	1	4	yes	—
417	#95	C₂³×C₃₁	1	4	yes	—
418	#95	C₂³×C₃₁	1	4	yes	—
419	C₃₁	#35132	1	1	yes	characteristic, direct product factor, Sylow
420	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
421	C₂×Q₈	—	2	8	no	—
422	C₂×Q₈	—	4	4	no	—
423	C₂×Q₈	—	4	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	4	2	no	—
429	C₂²×C₄	—	4	4	no	—
430	C₂²×C₄	—	4	4	no	—
431	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
432	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
433	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
436	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
437	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	8	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	4	4	no	—
458	C₄⋊C₄	—	4	4	no	—
459	C₄⋊C₄	—	4	4	no	—
460	C₄⋊C₄	—	4	4	no	—
461	C₄⋊C₄	—	4	4	no	—
462	C₄⋊C₄	—	4	4	no	—
463	C₄⋊C₄	—	4	4	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	#41×C₃₁	1	8	yes	—
467	C₄²	—	2	4	no	—
468	C₄²	—	2	4	no	—
469	C₄²	—	2	2	no	—
470	C₄²	—	2	2	no	—
471	C₄²	—	2	2	no	—
472	C₄²	—	2	2	no	—
473	C₄²	—	4	2	no	—
474	C₄²	—	4	2	no	—
475	C₄²	—	4	4	no	—
476	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
477	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
478	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
479	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
480	C₂³	—	2	1	no	—
481	C₂³	—	2	1	no	—
482	C₂³	—	2	2	no	—
483	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
484	C₂³	C₂×#41×C₃₁	1	4	yes	—
485	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
486	Q₈	—	8	8	no	—
487	C₂×C₄	—	2	8	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	8	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	—	8	8	no	—
519	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
520	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
521	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
522	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
523	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
524	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
525	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
526	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
527	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
528	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	2	4	no	—
533	C₂²	C₃₁×#533	1	4	yes	—
534	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
535	C₂²	C₂²×#41×C₃₁	1	2	yes	—
536	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
537	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
538	C₂²	C₃₁×#585	1	2	yes	—
539	C₄	—	2	2	no	—
540	C₄	—	2	2	no	—
541	C₄	—	2	2	no	—
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	2	4	no	—
545	C₄	—	2	4	no	—
546	C₄	—	4	4	no	—
547	C₄	—	4	4	no	—
548	C₄	—	4	4	no	—
549	C₄	—	4	2	no	—
550	C₄	—	4	2	no	—
551	C₄	—	4	4	no	—
552	C₄	C₃₁×#544	1	4	yes	—
553	C₂	—	2	4	no	—
554	C₂	C₂×C₃₁×#533	1	1	yes	characteristic
555	C₂	C₂×C₃₁×#544	1	1	yes	characteristic
556	C₂	C₃₁×#3458	1	1	yes	characteristic
557	C₂	C₃₁×#3514	1	2	yes	—
558	C₂	C₃₁×#3515	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁴×C₃₁	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2864	C₂	semidirect
2. C₃₁×#2864	C₂	semidirect
3. C₃₁×#2862	C₂	semidirect
4. C₃₁×#2859	C₂	semidirect
5. C₃₁×#2666	C₂	semidirect
6. C₃₁×#2612	C₂	semidirect
7. C₃₁×#2570	C₂	semidirect
8. C₃₁×#2550	C₂	semidirect
104. C₂×C₄²×C₃₁	Q₈	semidirect
200. #35132	C₃₁	direct (N × H)
250. #2864	C₂×C₃₁	semidirect
251. #2864	C₂×C₃₁	semidirect
252. #2862	C₂×C₃₁	semidirect
253. #2859	C₂×C₃₁	semidirect
254. #2666	C₂×C₃₁	semidirect
255. #2612	C₂×C₃₁	semidirect
256. #2570	C₂×C₃₁	semidirect
257. #2550	C₂×C₃₁	semidirect
383. C₂×C₄²	Q₈×C₃₁	semidirect
419. C₃₁	#35132	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×#2864	2
C₃₁×#2862	2
C₃₁×#2859	2
C₃₁×#2666	2
C₃₁×#2612	2
C₃₁×#2570	2
C₃₁×#2550	2
C₃₁×#2442	2
C₃₁×#2440	2
C₃₁×#2438	2
C₃₁×#2329	2
C₃₁×#2283	2
C₂×C₃₁×#382	2
C₄×C₃₁×#117	2
#35132	31

Subgroups — C31×#35132 #21491891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35132 #21491891