Subgroups — C₃₁×#22721 #21479480

Order 7936.

760 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1940	C₂	1	2	yes	maximal
5	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1939	C₂	1	2	yes	maximal
7	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1843	C₂	1	2	yes	maximal
9	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1768	C₂	1	2	yes	maximal
11	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#446	—	2	4	no	—
14	C₈×Q₈×C₃₁	—	2	4	no	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#391	C₄	1	1	yes	characteristic
30	C₃₁×#391	C₄	1	1	yes	characteristic
31	C₃₁×#389	C₄	1	1	yes	characteristic
32	C₃₁×#389	C₄	1	1	yes	characteristic
33	C₃₁×#387	C₄	1	2	yes	—
34	C₃₁×#387	C₄	1	2	yes	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₃₁×#376	C₂²	1	2	yes	—
38	C₄³×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₃₁	D₄	1	4	yes	—
51	C₄×Q₈×C₃₁	Q₈	1	4	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	4	no	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×C₄⋊C₈	—	2	4	no	—
71	C₃₁×C₄⋊C₈	—	2	4	no	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×C₄⋊C₈	—	4	2	no	—
74	C₃₁×C₄⋊C₈	—	4	2	no	—
75	C₃₁×C₄⋊C₈	—	4	2	no	—
76	C₃₁×C₄⋊C₈	Q₈	1	4	yes	complemented
77	C₃₁×#97	—	2	4	no	—
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₄×C₈×C₃₁	—	2	4	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₃₁×#95	C₂×C₄	1	2	yes	—
82	C₃₁×#95	C₂×C₄	1	2	yes	—
83	C₃₁×#95	C₂×C₄	1	2	yes	—
84	C₃₁×#95	C₂×C₄	1	2	yes	—
85	C₃₁×#95	C₂×C₄	1	2	yes	—
86	C₃₁×#95	C₂×C₄	1	2	yes	—
87	C₃₁×#95	C₂×C₄	1	2	yes	—
88	C₂×Q₈×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
91	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	1	no	—
104	C₂×C₈×C₃₁	—	2	1	no	—
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
120	C₄⋊C₄×C₃₁	—	2	4	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
125	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	1	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	2	1	no	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	4	1	no	—
134	C₄²×C₃₁	C₂×D₄	1	2	yes	—
135	C₄²×C₃₁	C₂×D₄	1	4	yes	—
136	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
137	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
138	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
139	C₄²×C₃₁	#41	1	2	yes	—
140	#22721	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
142	Q₈×C₃₁	—	2	8	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
166	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
167	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
168	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
169	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
170	C₂×C₄×C₃₁	#131	1	2	yes	—
171	C₂×C₄×C₃₁	#131	1	2	yes	—
172	C₂×C₄×C₃₁	#131	1	2	yes	—
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
179	C₈×C₃₁	—	2	2	no	—
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	—	4	2	no	—
183	C₈×C₃₁	—	4	1	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₂×#446	C₂×C₃₁	1	1	yes	characteristic
186	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
187	#2295	C₂×C₃₁	1	1	yes	characteristic
188	#1940	C₂×C₃₁	1	2	yes	—
189	#1939	C₂×C₃₁	1	2	yes	—
190	#1939	C₂×C₃₁	1	1	yes	characteristic
191	#1939	C₂×C₃₁	1	1	yes	characteristic
192	#1843	C₂×C₃₁	1	2	yes	—
193	#1842	C₂×C₃₁	1	1	yes	characteristic
194	#1768	C₂×C₃₁	1	2	yes	—
195	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
196	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
198	C₂²×C₃₁	#520	1	1	yes	characteristic
199	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
200	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
201	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
202	C₂²×C₃₁	#568	1	1	yes	characteristic
203	C₄×C₃₁	—	2	4	no	—
204	C₄×C₃₁	—	2	2	no	—
205	C₄×C₃₁	—	2	4	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	2	no	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	4	1	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	#424	1	4	yes	—
213	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
214	C₄×C₃₁	#550	1	1	yes	characteristic
215	C₄×C₃₁	#557	1	1	yes	characteristic
216	C₄×C₃₁	Q₈²	1	1	yes	characteristic
217	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
218	#446	—	2	4	no	—
219	C₈×Q₈	—	2	4	no	—
220	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
221	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
225	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
226	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
229	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#97	C₂²×C₃₁	1	2	yes	—
231	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
233	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
234	#391	C₄×C₃₁	1	1	yes	characteristic
235	#391	C₄×C₃₁	1	1	yes	characteristic
236	#389	C₄×C₃₁	1	1	yes	characteristic
237	#389	C₄×C₃₁	1	1	yes	characteristic
238	#387	C₄×C₃₁	1	2	yes	—
239	#387	C₄×C₃₁	1	2	yes	—
240	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
241	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
242	#376	C₂²×C₃₁	1	2	yes	—
243	C₄³	C₂²×C₃₁	1	1	yes	characteristic
244	#336	C₂²×C₃₁	1	1	yes	characteristic
245	#336	C₂²×C₃₁	1	2	yes	—
246	#336	C₂²×C₃₁	1	2	yes	—
247	#336	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₃₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
249	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
250	C₂×C₃₁	#2906	1	1	yes	characteristic
251	C₂×C₃₁	#2960	1	1	yes	characteristic
252	C₂×C₃₁	#2962	1	1	yes	characteristic
253	C₂×C₃₁	#2993	1	1	yes	characteristic
254	C₂×C₃₁	#2994	1	1	yes	characteristic
255	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
258	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
260	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₄×Q₈	D₄×C₃₁	1	4	yes	—
263	C₄×Q₈	Q₈×C₃₁	1	4	yes	—
264	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
268	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
269	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
273	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
276	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
278	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
279	C₄⋊C₈	—	2	2	no	—
280	C₄⋊C₈	—	2	4	no	—
281	C₄⋊C₈	—	2	2	no	—
282	C₄⋊C₈	—	2	4	no	—
283	C₄⋊C₈	—	2	4	no	—
284	C₄⋊C₈	—	2	2	no	—
285	C₄⋊C₈	—	4	2	no	—
286	C₄⋊C₈	—	4	2	no	—
287	C₄⋊C₈	—	4	2	no	—
288	C₄⋊C₈	Q₈×C₃₁	1	4	yes	complemented
289	#97	—	2	4	no	—
290	C₄×C₈	—	2	2	no	—
291	C₄×C₈	—	2	4	no	—
292	C₄×C₈	—	2	2	no	—
293	#95	C₂×C₄×C₃₁	1	2	yes	—
294	#95	C₂×C₄×C₃₁	1	2	yes	—
295	#95	C₂×C₄×C₃₁	1	2	yes	—
296	#95	C₂×C₄×C₃₁	1	2	yes	—
297	#95	C₂×C₄×C₃₁	1	2	yes	—
298	#95	C₂×C₄×C₃₁	1	2	yes	—
299	#95	C₂×C₄×C₃₁	1	2	yes	—
300	C₃₁	#22721	1	1	yes	characteristic, direct product factor, Sylow
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
304	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
310	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
332	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
338	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	4	1	no	—
347	C₄²	C₂×D₄×C₃₁	1	2	yes	—
348	C₄²	C₂×D₄×C₃₁	1	4	yes	—
349	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
350	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
351	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
352	C₄²	#41×C₃₁	1	2	yes	—
353	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
354	Q₈	—	2	8	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
377	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
378	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
379	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
380	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
381	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
382	C₂×C₄	C₃₁×#131	1	2	yes	—
383	C₂×C₄	C₃₁×#131	1	2	yes	—
384	C₂×C₄	C₃₁×#131	1	2	yes	—
385	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
388	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
390	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
391	C₈	—	2	2	no	—
392	C₈	—	2	2	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	2	no	—
397	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
398	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
399	C₂²	C₃₁×#520	1	1	yes	characteristic
400	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
401	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
402	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
403	C₂²	C₃₁×#568	1	1	yes	characteristic
404	C₄	—	2	4	no	—
405	C₄	—	2	2	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	2	no	—
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	C₃₁×#424	1	4	yes	—
414	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
415	C₄	C₃₁×#550	1	1	yes	characteristic
416	C₄	C₃₁×#557	1	1	yes	characteristic
417	C₄	Q₈²×C₃₁	1	1	yes	characteristic
418	C₂	Q₈×C₄⋊C₄×C₃₁	1	1	yes	characteristic
419	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
420	C₂	C₃₁×#2906	1	1	yes	characteristic
421	C₂	C₃₁×#2960	1	1	yes	characteristic
422	C₂	C₃₁×#2962	1	1	yes	characteristic
423	C₂	C₃₁×#2993	1	1	yes	characteristic
424	C₂	C₃₁×#2994	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
76. C₃₁×C₄⋊C₈	Q₈	semidirect
140. #22721	C₃₁	direct (N × H)
288. C₄⋊C₈	Q₈×C₃₁	semidirect
300. C₃₁	#22721	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×#446	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#2295	2
C₃₁×#1940	2
C₃₁×#1939	2
C₃₁×#1843	2
C₃₁×#1842	2
C₃₁×#1768	2
C₄×C₃₁×C₄⋊C₈	2
#22721	31

Subgroups — C31×#22721 #21479480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22721 #21479480