Subgroups — C₃₁×#34281 #21491040

Order 7936.

2070 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#541	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2373	C₂	1	4	yes	maximal
3	C₃₁×#2373	C₂	1	4	yes	maximal
4	C₃₁×#2371	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2370	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2327	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2323	C₂	1	2	yes	maximal
9	C₃₁×#2314	C₂	1	1	yes	maximal, characteristic
10	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal
13	C₃₁×#2295	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2276	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
16	C₃₁×#541	C₄	1	16	yes	complemented
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#126	C₂²	1	4	yes	—
19	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#125	C₂²	1	4	yes	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
25	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#391	C₂²	1	2	yes	—
29	C₃₁×#391	C₂²	1	2	yes	—
30	C₃₁×#389	C₂²	1	4	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	4	yes	—
33	C₃₁×#389	C₂²	1	4	yes	—
34	C₃₁×#389	C₂²	1	2	yes	—
35	C₃₁×#388	C₂²	1	4	yes	—
36	C₃₁×#388	C₂²	1	4	yes	—
37	C₃₁×#388	C₂²	1	4	yes	—
38	C₃₁×#387	C₂²	1	4	yes	—
39	C₃₁×#387	C₂²	1	4	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×#387	C₂²	1	4	yes	—
47	C₃₁×#385	C₂²	1	4	yes	—
48	C₃₁×#384	C₂²	1	1	yes	characteristic
49	C₃₁×#382	C₂²	1	1	yes	characteristic
50	C₃₁×#380	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
59	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
60	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
66	C₃₁×#376	C₂²	1	1	yes	characteristic
67	C₃₁×#376	C₂²	1	4	yes	—
68	C₃₁×#376	C₂²	1	4	yes	—
69	C₃₁×#376	C₂²	1	1	yes	characteristic
70	C₄³×C₃₁	C₂²	1	1	yes	characteristic
71	C₄³×C₃₁	C₂²	1	1	yes	characteristic
72	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
76	C₃₁×#126	C₂×C₄	1	32	yes	—
77	C₃₁×#125	C₂×C₄	1	8	yes	complemented
78	C₃₁×#125	C₂×C₄	1	32	yes	complemented
79	C₃₁×C₂²⋊Q₈	C₂×C₄	1	16	yes	—
80	C₄×Q₈×C₃₁	—	2	8	no	—
81	C₄×Q₈×C₃₁	—	2	8	no	—
82	C₄×Q₈×C₃₁	C₂×C₄	1	16	yes	complemented
83	C₃₁×#117	—	2	4	no	—
84	C₃₁×#117	C₂×C₄	1	8	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	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×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
105	C₂×C₄²×C₃₁	—	2	4	no	—
106	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
111	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
113	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
116	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₃₁×#95	C₂³	1	2	yes	—
124	C₃₁×#95	C₂³	1	2	yes	—
125	C₃₁×#95	C₂³	1	2	yes	—
126	C₃₁×#95	C₂³	1	4	yes	—
127	C₃₁×#95	C₂³	1	2	yes	—
128	C₃₁×#95	C₂³	1	4	yes	—
129	C₃₁×#95	C₂³	1	4	yes	—
130	C₃₁×#95	C₂³	1	4	yes	—
131	C₃₁×#95	C₂³	1	4	yes	—
132	C₃₁×#95	C₂³	1	4	yes	—
133	C₃₁×#95	C₂³	1	4	yes	—
134	C₃₁×#95	C₂³	1	4	yes	—
135	C₃₁×#95	C₂³	1	4	yes	—
136	C₃₁×#95	C₂³	1	4	yes	—
137	C₃₁×#95	C₂³	1	4	yes	—
138	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
139	C₂×Q₈×C₃₁	—	2	4	no	—
140	C₂×Q₈×C₃₁	—	2	4	no	—
141	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	4	2	no	—
147	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	8	no	—
165	C₄⋊C₄×C₃₁	—	2	8	no	—
166	C₄⋊C₄×C₃₁	—	2	4	no	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	8	no	—
169	C₄⋊C₄×C₃₁	—	4	4	no	—
170	C₄⋊C₄×C₃₁	—	4	4	no	—
171	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
172	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
173	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
174	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
175	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
176	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₂²⋊C₄×C₃₁	—	2	2	no	—
179	C₂²⋊C₄×C₃₁	—	4	4	no	—
180	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
181	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
182	C₄²×C₃₁	—	2	4	no	—
183	C₄²×C₃₁	—	2	2	no	—
184	C₄²×C₃₁	—	2	2	no	—
185	C₄²×C₃₁	—	2	8	no	—
186	C₄²×C₃₁	—	2	8	no	—
187	C₄²×C₃₁	—	2	4	no	—
188	C₄²×C₃₁	—	2	4	no	—
189	C₄²×C₃₁	—	4	4	no	—
190	C₄²×C₃₁	—	4	2	no	—
191	C₄²×C₃₁	—	4	2	no	—
192	C₄²×C₃₁	—	4	4	no	—
193	C₄²×C₃₁	—	4	4	no	—
194	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
195	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
196	C₄²×C₃₁	C₂²×C₄	1	16	yes	—
197	C₄²×C₃₁	#41	1	4	yes	—
198	C₄²×C₃₁	#41	1	4	yes	—
199	#34281	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₃₁	—	2	1	no	—
201	C₂³×C₃₁	—	2	1	no	—
202	C₂³×C₃₁	—	2	2	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
205	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
206	Q₈×C₃₁	—	2	16	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	4	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	4	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	8	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	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	—	4	8	no	—
235	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
236	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
237	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
238	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
239	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
242	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
243	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
244	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
245	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
246	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
247	C₂×#541	C₂×C₃₁	1	1	yes	characteristic
248	#2373	C₂×C₃₁	1	4	yes	—
249	#2373	C₂×C₃₁	1	4	yes	—
250	#2371	C₂×C₃₁	1	4	yes	complemented
251	#2370	C₂×C₃₁	1	4	yes	complemented
252	#2330	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#2327	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#2323	C₂×C₃₁	1	2	yes	—
255	#2314	C₂×C₃₁	1	1	yes	characteristic
256	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₄×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
258	C₄×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	—
259	#2295	C₂×C₃₁	1	2	yes	complemented
260	#2276	C₂×C₃₁	1	2	yes	complemented
261	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	4	no	—
264	C₂²×C₃₁	—	4	2	no	—
265	C₂²×C₃₁	—	4	1	no	—
266	C₂²×C₃₁	—	4	1	no	—
267	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
268	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
269	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
270	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
271	C₂²×C₃₁	#585	1	2	yes	—
272	C₄×C₃₁	—	2	2	no	—
273	C₄×C₃₁	—	2	8	no	—
274	C₄×C₃₁	—	2	4	no	—
275	C₄×C₃₁	—	2	4	no	—
276	C₄×C₃₁	—	2	8	no	—
277	C₄×C₃₁	—	2	4	no	—
278	C₄×C₃₁	—	2	4	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	4	4	no	—
282	C₄×C₃₁	—	4	1	no	—
283	C₄×C₃₁	—	4	1	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	C₄×#41	1	4	yes	—
286	#541	C₄×C₃₁	1	16	yes	complemented
287	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#126	C₂²×C₃₁	1	4	yes	—
289	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#125	C₂²×C₃₁	1	4	yes	—
291	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
292	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
293	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
294	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
295	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
297	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
298	#391	C₂²×C₃₁	1	2	yes	—
299	#391	C₂²×C₃₁	1	2	yes	—
300	#389	C₂²×C₃₁	1	2	yes	—
301	#389	C₂²×C₃₁	1	4	yes	—
302	#389	C₂²×C₃₁	1	2	yes	—
303	#389	C₂²×C₃₁	1	4	yes	—
304	#389	C₂²×C₃₁	1	4	yes	—
305	#388	C₂²×C₃₁	1	4	yes	—
306	#388	C₂²×C₃₁	1	4	yes	—
307	#388	C₂²×C₃₁	1	4	yes	—
308	#387	C₂²×C₃₁	1	4	yes	—
309	#387	C₂²×C₃₁	1	4	yes	—
310	#387	C₂²×C₃₁	1	4	yes	—
311	#387	C₂²×C₃₁	1	4	yes	—
312	#387	C₂²×C₃₁	1	4	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	#385	C₂²×C₃₁	1	4	yes	—
318	#384	C₂²×C₃₁	1	1	yes	characteristic
319	#382	C₂²×C₃₁	1	1	yes	characteristic
320	#380	C₂²×C₃₁	1	2	yes	—
321	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
322	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
323	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
324	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
325	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
326	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
327	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
328	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
329	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
330	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
331	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
332	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
333	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
334	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
335	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
336	#376	C₂²×C₃₁	1	1	yes	characteristic
337	#376	C₂²×C₃₁	1	4	yes	—
338	#376	C₂²×C₃₁	1	4	yes	—
339	#376	C₂²×C₃₁	1	1	yes	characteristic
340	C₄³	C₂²×C₃₁	1	1	yes	characteristic
341	C₄³	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₃₁	—	4	2	no	—
343	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
344	C₂×C₃₁	C₄×2₋¹⁺⁴	1	2	yes	—
345	C₂×C₃₁	#3495	1	2	yes	—
346	C₂×C₃₁	#3509	1	1	yes	characteristic
347	C₂×C₃₁	#3512	1	1	yes	characteristic
348	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
349	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
350	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
351	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
352	#126	C₂×C₄×C₃₁	1	32	yes	—
353	#125	C₂×C₄×C₃₁	1	8	yes	complemented
354	#125	C₂×C₄×C₃₁	1	32	yes	complemented
355	C₂²⋊Q₈	C₂×C₄×C₃₁	1	16	yes	—
356	C₄×Q₈	—	2	8	no	—
357	C₄×Q₈	—	2	8	no	—
358	C₄×Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
359	#117	—	2	4	no	—
360	#117	C₂×C₄×C₃₁	1	8	yes	—
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	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	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	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×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
381	C₂×C₄²	—	2	4	no	—
382	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
384	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
385	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
387	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
388	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
389	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
390	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
391	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
392	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
393	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
394	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
397	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
399	#95	C₂³×C₃₁	1	2	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	4	yes	—
405	#95	C₂³×C₃₁	1	4	yes	—
406	#95	C₂³×C₃₁	1	4	yes	—
407	#95	C₂³×C₃₁	1	4	yes	—
408	#95	C₂³×C₃₁	1	4	yes	—
409	#95	C₂³×C₃₁	1	4	yes	—
410	#95	C₂³×C₃₁	1	4	yes	—
411	#95	C₂³×C₃₁	1	4	yes	—
412	#95	C₂³×C₃₁	1	4	yes	—
413	#95	C₂³×C₃₁	1	4	yes	—
414	C₃₁	#34281	1	1	yes	characteristic, direct product factor, Sylow
415	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	4	2	no	—
424	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
428	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
429	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
430	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
432	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	8	no	—
446	C₄⋊C₄	—	4	4	no	—
447	C₄⋊C₄	—	4	4	no	—
448	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
449	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
450	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
451	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
452	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
453	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
454	C₂²⋊C₄	—	2	2	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	4	4	no	—
457	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
458	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
459	C₄²	—	2	4	no	—
460	C₄²	—	2	2	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	8	no	—
463	C₄²	—	2	8	no	—
464	C₄²	—	2	4	no	—
465	C₄²	—	2	4	no	—
466	C₄²	—	4	4	no	—
467	C₄²	—	4	2	no	—
468	C₄²	—	4	2	no	—
469	C₄²	—	4	4	no	—
470	C₄²	—	4	4	no	—
471	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
472	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
473	C₄²	C₂²×C₄×C₃₁	1	16	yes	—
474	C₄²	#41×C₃₁	1	4	yes	—
475	C₄²	#41×C₃₁	1	4	yes	—
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
481	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
482	Q₈	—	2	16	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	4	8	no	—
511	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
512	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
513	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
514	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
515	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
516	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
517	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
518	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
519	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
520	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
521	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
522	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	1	no	—
527	C₂²	—	4	1	no	—
528	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
529	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
530	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
531	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
532	C₂²	C₃₁×#585	1	2	yes	—
533	C₄	—	2	2	no	—
534	C₄	—	2	8	no	—
535	C₄	—	2	4	no	—
536	C₄	—	2	4	no	—
537	C₄	—	2	8	no	—
538	C₄	—	2	4	no	—
539	C₄	—	2	4	no	—
540	C₄	—	2	2	no	—
541	C₄	—	2	2	no	—
542	C₄	—	4	4	no	—
543	C₄	—	4	1	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	4	no	—
546	C₄	C₄×#41×C₃₁	1	4	yes	—
547	C₂	—	4	2	no	—
548	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
549	C₂	C₄×C₃₁×2₋¹⁺⁴	1	2	yes	—
550	C₂	C₃₁×#3495	1	2	yes	—
551	C₂	C₃₁×#3509	1	1	yes	characteristic
552	C₂	C₃₁×#3512	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁴×C₃₁	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#2371	C₂	semidirect
5. C₃₁×#2370	C₂	semidirect
6. C₃₁×#2330	C₂	semidirect
7. C₃₁×#2327	C₂	semidirect
10. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
11. C₄×C₃₁×#125	C₂	semidirect
13. C₃₁×#2295	C₂	semidirect
14. C₃₁×#2276	C₂	semidirect
16. C₃₁×#541	C₄	semidirect
75. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
77. C₃₁×#125	C₂×C₄	semidirect
78. C₃₁×#125	C₂×C₄	semidirect
82. C₄×Q₈×C₃₁	C₂×C₄	semidirect
199. #34281	C₃₁	direct (N × H)
250. #2371	C₂×C₃₁	semidirect
251. #2370	C₂×C₃₁	semidirect
252. #2330	C₂×C₃₁	semidirect
253. #2327	C₂×C₃₁	semidirect
256. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
257. C₄×#125	C₂×C₃₁	semidirect
259. #2295	C₂×C₃₁	semidirect
260. #2276	C₂×C₃₁	semidirect
286. #541	C₄×C₃₁	semidirect
351. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
353. #125	C₂×C₄×C₃₁	semidirect
354. #125	C₂×C₄×C₃₁	semidirect
358. C₄×Q₈	C₂×C₄×C₃₁	semidirect
414. C₃₁	#34281	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#541	2
C₃₁×#2373	2
C₃₁×#2371	2
C₃₁×#2370	2
C₃₁×#2330	2
C₃₁×#2327	2
C₃₁×#2323	2
C₃₁×#2314	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×#125	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#2295	2
C₃₁×#2276	2
C₄×C₃₁×#117	2
#34281	31

Subgroups — C31×#34281 #21491040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34281 #21491040