Subgroups — C₃₁×#20521 #21477280

Order 7936.

866 conjugacy classes of subgroups.

The subgroup lattice has 352 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×#1771	C₂	1	2	yes	maximal
5	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1770	C₂	1	2	yes	maximal
7	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1768	C₂	1	2	yes	maximal
9	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1726	C₂	1	2	yes	maximal
11	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#446	C₄	1	8	yes	complemented
14	C₈×Q₈×C₃₁	C₄	1	8	yes	complemented
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₂×C₃₁×#97	C₂²	1	2	yes	—
24	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#391	C₄	1	2	yes	—
31	C₃₁×#389	C₄	1	2	yes	—
32	C₃₁×#387	C₄	1	4	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₃₁×#376	C₂²	1	2	yes	—
36	C₄³×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
61	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
62	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
64	C₃₁×#97	—	2	2	no	—
65	C₃₁×#97	—	2	2	no	—
66	C₃₁×#97	—	2	4	no	—
67	C₃₁×#97	—	2	2	no	—
68	C₃₁×#97	C₂×C₄	1	8	yes	—
69	C₃₁×#97	Q₈	1	4	yes	complemented
70	C₄×C₈×C₃₁	—	2	4	no	—
71	C₄×C₈×C₃₁	—	2	4	no	—
72	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
73	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
74	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
75	C₃₁×#95	C₂×C₄	1	4	yes	—
76	C₃₁×#95	C₂×C₄	1	4	yes	—
77	C₃₁×#95	C₂×C₄	1	2	yes	—
78	C₃₁×#95	C₂×C₄	1	4	yes	—
79	C₂×Q₈×C₃₁	—	2	4	no	—
80	C₂×Q₈×C₃₁	C₄²	1	4	yes	—
81	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	1	no	—
94	C₂×C₈×C₃₁	—	2	1	no	—
95	C₂×C₈×C₃₁	—	2	1	no	—
96	C₂×C₈×C₃₁	—	2	1	no	—
97	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
104	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
105	C₂×C₈×C₃₁	#41	1	2	yes	—
106	C₂×C₈×C₃₁	#41	1	2	yes	—
107	C₄⋊C₄×C₃₁	—	2	4	no	—
108	C₄⋊C₄×C₃₁	—	4	4	no	—
109	C₄⋊C₄×C₃₁	C₄²	1	8	yes	—
110	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
111	C₄²×C₃₁	—	2	4	no	—
112	C₄²×C₃₁	—	2	2	no	—
113	C₄²×C₃₁	—	2	2	no	—
114	C₄²×C₃₁	—	2	1	no	—
115	C₄²×C₃₁	—	2	1	no	—
116	C₄²×C₃₁	—	4	2	no	—
117	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
118	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
119	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
120	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
121	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
122	C₄²×C₃₁	#41	1	2	yes	—
123	#20521	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
125	Q₈×C₃₁	—	2	8	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	4	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	4	no	—
131	C₂×C₄×C₃₁	—	2	4	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	4	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	4	no	—
137	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
139	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
140	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
142	C₂×C₄×C₃₁	#131	1	4	yes	—
143	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
149	C₈×C₃₁	—	2	4	no	—
150	C₈×C₃₁	—	2	4	no	—
151	C₈×C₃₁	—	2	2	no	—
152	C₈×C₃₁	—	2	2	no	—
153	C₈×C₃₁	C₄×Q₈	1	4	yes	—
154	C₈×C₃₁	C₄×Q₈	1	4	yes	—
155	C₂×#446	C₂×C₃₁	1	1	yes	characteristic
156	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
157	#2295	C₂×C₃₁	1	1	yes	characteristic
158	#1771	C₂×C₃₁	1	2	yes	—
159	#1770	C₂×C₃₁	1	2	yes	—
160	#1770	C₂×C₃₁	1	1	yes	characteristic
161	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
162	#1768	C₂×C₃₁	1	2	yes	—
163	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
164	#1726	C₂×C₃₁	1	2	yes	—
165	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
166	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
167	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
169	C₂²×C₃₁	C₂×#131	1	2	yes	—
170	C₂²×C₃₁	#568	1	2	yes	—
171	C₄×C₃₁	—	2	4	no	—
172	C₄×C₃₁	—	2	2	no	—
173	C₄×C₃₁	—	2	4	no	—
174	C₄×C₃₁	—	2	4	no	—
175	C₄×C₃₁	—	4	2	no	—
176	C₄×C₃₁	—	4	4	no	—
177	C₄×C₃₁	#405	1	4	yes	—
178	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
181	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
182	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
183	#446	C₄×C₃₁	1	8	yes	complemented
184	C₈×Q₈	C₄×C₃₁	1	8	yes	complemented
185	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
187	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
189	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×#97	C₂²×C₃₁	1	2	yes	—
193	C₂×#97	C₂²×C₃₁	1	2	yes	—
194	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
196	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
197	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
199	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
200	#391	C₄×C₃₁	1	2	yes	—
201	#389	C₄×C₃₁	1	2	yes	—
202	#387	C₄×C₃₁	1	4	yes	—
203	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
204	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
205	#376	C₂²×C₃₁	1	2	yes	—
206	C₄³	C₂²×C₃₁	1	1	yes	characteristic
207	#336	C₂²×C₃₁	1	2	yes	—
208	#336	C₂²×C₃₁	1	2	yes	—
209	#336	C₂²×C₃₁	1	2	yes	—
210	C₂×C₃₁	C₄²×Q₈	1	1	yes	characteristic
211	C₂×C₃₁	C₂×#405	1	1	yes	characteristic
212	C₂×C₃₁	#2875	1	1	yes	characteristic
213	C₂×C₃₁	#2962	1	2	yes	—
214	C₂×C₃₁	#2969	1	2	yes	—
215	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
217	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
218	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
222	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
223	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
224	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
225	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
226	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
229	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
230	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
232	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
234	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
235	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
236	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
237	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
238	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
239	#97	—	2	2	no	—
240	#97	—	2	2	no	—
241	#97	—	2	4	no	—
242	#97	—	2	2	no	—
243	#97	C₂×C₄×C₃₁	1	8	yes	—
244	#97	Q₈×C₃₁	1	4	yes	complemented
245	C₄×C₈	—	2	4	no	—
246	C₄×C₈	—	2	4	no	—
247	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
248	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
249	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
250	#95	C₂×C₄×C₃₁	1	4	yes	—
251	#95	C₂×C₄×C₃₁	1	4	yes	—
252	#95	C₂×C₄×C₃₁	1	2	yes	—
253	#95	C₂×C₄×C₃₁	1	4	yes	—
254	C₃₁	#20521	1	1	yes	characteristic, direct product factor, Sylow
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	C₄²×C₃₁	1	4	yes	—
257	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
258	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
259	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
260	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
262	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
264	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
274	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
275	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
277	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
278	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
279	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
280	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
281	C₂×C₈	#41×C₃₁	1	2	yes	—
282	C₂×C₈	#41×C₃₁	1	2	yes	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	C₄²×C₃₁	1	8	yes	—
286	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	4	2	no	—
293	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
294	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
295	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
296	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
297	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
298	C₄²	#41×C₃₁	1	2	yes	—
299	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
313	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
314	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
315	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
316	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
317	C₂×C₄	C₃₁×#131	1	4	yes	—
318	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
319	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
320	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
321	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
322	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	C₄×Q₈×C₃₁	1	4	yes	—
329	C₈	C₄×Q₈×C₃₁	1	4	yes	—
330	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
331	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
332	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
333	C₂²	C₂×C₃₁×#131	1	2	yes	—
334	C₂²	C₃₁×#568	1	2	yes	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	4	no	—
341	C₄	C₃₁×#405	1	4	yes	—
342	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
343	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
344	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
345	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
346	C₂	C₄²×Q₈×C₃₁	1	1	yes	characteristic
347	C₂	C₂×C₃₁×#405	1	1	yes	characteristic
348	C₂	C₃₁×#2875	1	1	yes	characteristic
349	C₂	C₃₁×#2962	1	2	yes	—
350	C₂	C₃₁×#2969	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle, prime radical	C₂³×C₃₁	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₃₁×#446	C₄	semidirect
14. C₈×Q₈×C₃₁	C₄	semidirect
69. C₃₁×#97	Q₈	semidirect
123. #20521	C₃₁	direct (N × H)
183. #446	C₄×C₃₁	semidirect
184. C₈×Q₈	C₄×C₃₁	semidirect
244. #97	Q₈×C₃₁	semidirect
254. C₃₁	#20521	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×#1771	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1768	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1726	2
C₄×C₃₁×#97	2
#20521	31

Subgroups — C31×#20521 #21477280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20521 #21477280