Subgroups — C₃₁×#17329 #21474088

Order 7936.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
2	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1454	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1453	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1452	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1451	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#446	C₄	1	2	yes	—
14	C₈×Q₈×C₃₁	C₄	1	2	yes	—
15	C₃₁×#436	C₄	1	2	yes	—
16	C₃₁×#435	C₄	1	4	yes	—
17	C₈×D₄×C₃₁	C₄	1	2	yes	—
18	C₈×D₄×C₃₁	C₄	1	4	yes	—
19	C₃₁×#433	C₄	1	4	yes	—
20	C₃₁×#431	C₂²	1	2	yes	—
21	C₃₁×#431	C₂²	1	2	yes	—
22	C₃₁×#431	C₂²	1	1	yes	characteristic
23	C₃₁×#431	C₂²	1	2	yes	—
24	C₃₁×#431	C₂²	1	1	yes	characteristic
25	C₃₁×#431	C₂²	1	2	yes	—
26	C₃₁×#424	C₄	1	2	yes	—
27	C₃₁×#408	C₄	1	2	yes	—
28	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#322	C₂²	1	1	yes	characteristic
35	C₃₁×#322	C₂²	1	1	yes	characteristic
36	C₃₁×#322	C₂²	1	1	yes	characteristic
37	C₃₁×#322	C₂²	1	1	yes	characteristic
38	C₃₁×#322	C₂²	1	2	yes	—
39	C₃₁×#322	C₂²	1	2	yes	—
40	C₃₁×#322	C₂²	1	2	yes	—
41	C₃₁×#322	C₂²	1	2	yes	—
42	C₃₁×#322	C₂²	1	2	yes	—
43	C₈²×C₃₁	C₂²	1	2	yes	—
44	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
58	C₃₁×#117	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
61	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
64	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
65	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
66	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
67	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
70	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
71	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
72	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
73	C₃₁×#97	C₂×C₄	1	2	yes	—
74	C₃₁×#97	C₂×C₄	1	4	yes	—
75	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₃₁	C₂³	1	2	yes	—
85	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₃₁	C₂³	1	2	yes	—
87	C₄×C₈×C₃₁	C₂³	1	2	yes	—
88	#41×C₃₁	—	2	4	no	—
89	C₂×Q₈×C₃₁	C₄²	1	1	yes	characteristic
90	C₂×D₄×C₃₁	C₄²	1	2	yes	—
91	C₂×D₄×C₃₁	C₄²	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	M₁₆×C₃₁	—	2	4	no	—
99	M₁₆×C₃₁	—	4	4	no	—
100	C₂×C₈×C₃₁	—	2	4	no	—
101	C₂×C₈×C₃₁	—	4	2	no	—
102	C₂×C₈×C₃₁	—	4	4	no	—
103	C₂×C₈×C₃₁	—	4	2	no	—
104	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
110	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₃₁	C₄²	1	2	yes	—
114	C₄⋊C₄×C₃₁	C₄²	1	2	yes	—
115	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
116	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
117	C₂²⋊C₄×C₃₁	C₄²	1	2	yes	—
118	C₂²⋊C₄×C₃₁	C₄²	1	2	yes	—
119	C₂²⋊C₄×C₃₁	C₄²	1	2	yes	—
120	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
122	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
125	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
126	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
127	#17329	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
129	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
130	Q₈×C₃₁	—	2	2	no	—
131	D₄×C₃₁	—	2	4	no	—
132	D₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	4	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	4	no	—
137	C₂×C₄×C₃₁	—	2	4	no	—
138	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
143	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₃₁	#131	1	4	yes	—
145	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
148	C₈×C₃₁	—	2	2	no	—
149	C₈×C₃₁	—	2	2	no	—
150	C₈×C₃₁	—	2	4	no	—
151	C₈×C₃₁	—	2	2	no	—
152	C₈×C₃₁	—	2	2	no	—
153	C₈×C₃₁	—	2	4	no	—
154	C₈×C₃₁	—	2	2	no	—
155	C₈×C₃₁	—	2	2	no	—
156	C₈×C₃₁	—	2	4	no	—
157	#2965	C₂×C₃₁	1	1	yes	characteristic
158	C₈×#41	C₂×C₃₁	1	1	yes	characteristic
159	#2922	C₂×C₃₁	1	1	yes	characteristic
160	#1454	C₂×C₃₁	1	2	yes	complemented
161	#1453	C₂×C₃₁	1	2	yes	complemented
162	#1452	C₂×C₃₁	1	2	yes	complemented
163	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#1451	C₂×C₃₁	1	2	yes	complemented
165	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
166	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
167	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
168	C₂²×C₃₁	—	2	1	no	—
169	C₂²×C₃₁	—	2	1	no	—
170	C₂²×C₃₁	—	2	2	no	—
171	C₂²×C₃₁	—	2	2	no	—
172	C₂²×C₃₁	—	2	2	no	—
173	C₂²×C₃₁	#405	1	2	yes	—
174	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
175	C₄×C₃₁	—	2	2	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	2	1	no	—
178	C₄×C₃₁	—	2	2	no	—
179	C₄×C₃₁	—	2	2	no	—
180	C₄×C₃₁	#405	1	2	yes	—
181	C₄×C₃₁	C₂×#131	1	2	yes	—
182	C₄×C₃₁	#568	1	2	yes	—
183	C₄×C₃₁	#568	1	2	yes	—
184	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
185	#446	C₄×C₃₁	1	2	yes	—
186	C₈×Q₈	C₄×C₃₁	1	2	yes	—
187	#436	C₄×C₃₁	1	2	yes	—
188	#435	C₄×C₃₁	1	4	yes	—
189	C₈×D₄	C₄×C₃₁	1	2	yes	—
190	C₈×D₄	C₄×C₃₁	1	4	yes	—
191	#433	C₄×C₃₁	1	4	yes	—
192	#431	C₂²×C₃₁	1	2	yes	—
193	#431	C₂²×C₃₁	1	2	yes	—
194	#431	C₂²×C₃₁	1	1	yes	characteristic
195	#431	C₂²×C₃₁	1	2	yes	—
196	#431	C₂²×C₃₁	1	1	yes	characteristic
197	#431	C₂²×C₃₁	1	2	yes	—
198	#424	C₄×C₃₁	1	2	yes	—
199	#408	C₄×C₃₁	1	2	yes	—
200	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
201	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
202	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
203	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
206	#322	C₂²×C₃₁	1	1	yes	characteristic
207	#322	C₂²×C₃₁	1	1	yes	characteristic
208	#322	C₂²×C₃₁	1	1	yes	characteristic
209	#322	C₂²×C₃₁	1	1	yes	characteristic
210	#322	C₂²×C₃₁	1	2	yes	—
211	#322	C₂²×C₃₁	1	2	yes	—
212	#322	C₂²×C₃₁	1	2	yes	—
213	#322	C₂²×C₃₁	1	2	yes	—
214	#322	C₂²×C₃₁	1	2	yes	—
215	C₈²	C₂²×C₃₁	1	2	yes	—
216	C₂×C₃₁	—	2	2	no	—
217	C₂×C₃₁	—	4	2	no	—
218	C₂×C₃₁	C₂×#405	1	1	yes	characteristic
219	C₂×C₃₁	#2875	1	1	yes	characteristic
220	C₂×C₃₁	#2875	1	1	yes	characteristic
221	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
222	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
223	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
224	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
225	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
226	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
227	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
228	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
229	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
230	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
231	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
232	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
233	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
234	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
235	#117	C₂×C₄×C₃₁	1	2	yes	—
236	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
238	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
239	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
240	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
241	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
242	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
243	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
244	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
245	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
246	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
247	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
248	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
249	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
250	#97	C₂×C₄×C₃₁	1	2	yes	—
251	#97	C₂×C₄×C₃₁	1	4	yes	—
252	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
253	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
254	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
255	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
256	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₃₁	1	2	yes	—
258	C₄×C₈	C₂³×C₃₁	1	2	yes	—
259	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₄×C₈	C₂³×C₃₁	1	2	yes	—
265	C₃₁	#17329	1	1	yes	characteristic, direct product factor, Sylow
266	#41	—	2	4	no	—
267	C₂×Q₈	C₄²×C₃₁	1	1	yes	characteristic
268	C₂×D₄	C₄²×C₃₁	1	2	yes	—
269	C₂×D₄	C₄²×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
271	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
272	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
273	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
275	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
276	M₁₆	—	2	4	no	—
277	M₁₆	—	4	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	4	2	no	—
280	C₂×C₈	—	4	4	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
283	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
284	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
285	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
286	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
287	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
288	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
289	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
290	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
291	C₄⋊C₄	C₄²×C₃₁	1	2	yes	—
292	C₄⋊C₄	C₄²×C₃₁	1	2	yes	—
293	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
294	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
295	C₂²⋊C₄	C₄²×C₃₁	1	2	yes	—
296	C₂²⋊C₄	C₄²×C₃₁	1	2	yes	—
297	C₂²⋊C₄	C₄²×C₃₁	1	2	yes	—
298	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
299	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
300	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
301	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
302	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
303	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
304	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
305	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
306	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	2	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
318	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
319	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
320	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
321	C₂×C₄	C₃₁×#131	1	4	yes	—
322	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	4	no	—
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	2	4	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	4	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	C₃₁×#405	1	2	yes	—
340	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	C₃₁×#405	1	2	yes	—
347	C₄	C₂×C₃₁×#131	1	2	yes	—
348	C₄	C₃₁×#568	1	2	yes	—
349	C₄	C₃₁×#568	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×C₃₁×#405	1	1	yes	characteristic
353	C₂	C₃₁×#2875	1	1	yes	characteristic
354	C₂	C₃₁×#2875	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄²
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1454	C₂	semidirect
5. C₃₁×#1453	C₂	semidirect
6. C₃₁×#1452	C₂	semidirect
7. C₃₁×#1452	C₂	semidirect
8. C₃₁×#1451	C₂	semidirect
9. C₃₁×#1451	C₂	semidirect
10. C₈×M₁₆×C₃₁	C₂	semidirect
11. C₂×C₃₁×#322	C₂	semidirect
127. #17329	C₃₁	direct (N × H)
160. #1454	C₂×C₃₁	semidirect
161. #1453	C₂×C₃₁	semidirect
162. #1452	C₂×C₃₁	semidirect
163. #1452	C₂×C₃₁	semidirect
164. #1451	C₂×C₃₁	semidirect
165. #1451	C₂×C₃₁	semidirect
166. C₈×M₁₆	C₂×C₃₁	semidirect
167. C₂×#322	C₂×C₃₁	semidirect
265. C₃₁	#17329	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×#2965	2
C₈×#41×C₃₁	2
C₃₁×#2922	2
C₃₁×#1454	2
C₃₁×#1453	2
C₃₁×#1452	2
C₃₁×#1451	2
C₈×M₁₆×C₃₁	2
C₂×C₃₁×#322	2
#17329	31

Subgroups — C31×#17329 #21474088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17329 #21474088