Subgroups — C₃₁×#21990 #21478749

Order 7936.

762 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2314	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1949	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1948	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1854	C₂	1	2	yes	maximal
7	C₃₁×#1853	C₂	1	2	yes	maximal
8	C₃₁×#1847	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#433	C₄	1	8	yes	complemented
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×#106	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#389	—	2	2	no	—
25	C₃₁×#388	—	2	2	no	—
26	C₃₁×#387	—	2	2	no	—
27	C₃₁×#385	—	2	2	no	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₃₁×#384	C₂²	1	1	yes	characteristic
31	C₃₁×#382	C₂²	1	1	yes	characteristic
32	C₃₁×#382	C₂²	1	1	yes	characteristic
33	C₃₁×#382	C₂²	1	1	yes	characteristic
34	C₃₁×#380	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₂³×C₄×C₃₁	—	2	1	no	—
41	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₃₁×#117	—	4	2	no	—
45	C₃₁×#117	C₂×C₄	1	4	yes	—
46	C₃₁×#117	Q₈	1	4	yes	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
52	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
62	C₂×C₄²×C₃₁	—	2	1	no	—
63	C₂×C₄²×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	4	2	no	—
67	C₃₁×#106	—	4	2	no	—
68	C₃₁×#106	—	4	2	no	—
69	C₃₁×#106	—	4	2	no	—
70	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
71	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
72	C₃₁×#97	C₂×C₄	1	4	yes	complemented
73	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
74	C₃₁×#95	—	2	2	no	—
75	C₃₁×#95	—	2	2	no	—
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	1	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	1	no	—
86	C₂²×C₄×C₃₁	—	2	1	no	—
87	C₂²×C₄×C₃₁	—	2	1	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	—	8	2	no	—
90	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	4	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	8	2	no	—
111	C₄⋊C₄×C₃₁	—	8	2	no	—
112	C₄⋊C₄×C₃₁	—	8	2	no	—
113	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₂²⋊C₄×C₃₁	—	8	2	no	—
116	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
117	C₄²×C₃₁	—	4	2	no	—
118	C₄²×C₃₁	—	8	2	no	—
119	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
120	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
121	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
122	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
123	#21990	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₃₁	—	2	1	no	—
125	C₂³×C₃₁	—	2	1	no	—
126	C₂³×C₃₁	—	2	2	no	—
127	C₂³×C₃₁	—	2	2	no	—
128	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	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	—	4	1	no	—
139	C₂×C₄×C₃₁	—	4	1	no	—
140	C₂×C₄×C₃₁	—	4	1	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	4	2	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	—	8	4	no	—
153	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
155	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
159	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
160	C₈×C₃₁	—	2	4	no	—
161	C₈×C₃₁	—	2	4	no	—
162	C₂×#433	C₂×C₃₁	1	1	yes	characteristic
163	#2317	C₂×C₃₁	1	1	yes	characteristic
164	#2314	C₂×C₃₁	1	1	yes	characteristic
165	#1949	C₂×C₃₁	1	2	yes	complemented
166	#1948	C₂×C₃₁	1	2	yes	complemented
167	#1854	C₂×C₃₁	1	2	yes	—
168	#1853	C₂×C₃₁	1	2	yes	—
169	#1847	C₂×C₃₁	1	1	yes	characteristic, complemented
170	#1844	C₂×C₃₁	1	1	yes	characteristic, complemented
171	#1836	C₂×C₃₁	1	1	yes	characteristic, complemented
172	#1835	C₂×C₃₁	1	1	yes	characteristic, complemented
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	—	2	4	no	—
175	C₂²×C₃₁	—	4	1	no	—
176	C₂²×C₃₁	—	4	1	no	—
177	C₂²×C₃₁	—	4	2	no	—
178	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₃₁	#535	1	1	yes	characteristic
180	C₂²×C₃₁	#536	1	1	yes	characteristic
181	C₂²×C₃₁	#577	1	2	yes	—
182	C₂²×C₃₁	#577	1	2	yes	—
183	C₄×C₃₁	—	2	2	no	—
184	C₄×C₃₁	—	2	4	no	—
185	C₄×C₃₁	—	2	4	no	—
186	C₄×C₃₁	—	8	1	no	—
187	C₄×C₃₁	—	8	2	no	—
188	C₄×C₃₁	—	8	1	no	—
189	C₄×C₃₁	—	8	1	no	—
190	C₄×C₃₁	—	8	1	no	—
191	C₄×C₃₁	—	8	2	no	—
192	C₄×C₃₁	#520	1	2	yes	—
193	C₄×C₃₁	#544	1	2	yes	—
194	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
196	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
197	#433	C₄×C₃₁	1	8	yes	complemented
198	C₂×#107	C₂²×C₃₁	1	2	yes	—
199	C₂×#106	C₂²×C₃₁	1	2	yes	—
200	C₂×#106	C₂²×C₃₁	1	2	yes	—
201	C₂×#106	C₂²×C₃₁	1	2	yes	—
202	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
203	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
204	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
206	#389	—	2	2	no	—
207	#388	—	2	2	no	—
208	#387	—	2	2	no	—
209	#385	—	2	2	no	—
210	#384	C₂²×C₃₁	1	1	yes	characteristic
211	#384	C₂²×C₃₁	1	1	yes	characteristic
212	#384	C₂²×C₃₁	1	1	yes	characteristic
213	#382	C₂²×C₃₁	1	1	yes	characteristic
214	#382	C₂²×C₃₁	1	1	yes	characteristic
215	#382	C₂²×C₃₁	1	1	yes	characteristic
216	#380	C₂²×C₃₁	1	2	yes	—
217	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
218	#340	C₂²×C₃₁	1	2	yes	—
219	#340	C₂²×C₃₁	1	2	yes	—
220	#340	C₂²×C₃₁	1	2	yes	—
221	#340	C₂²×C₃₁	1	2	yes	—
222	C₂×C₃₁	—	4	2	no	—
223	C₂×C₃₁	#2317	1	1	yes	characteristic
224	C₂×C₃₁	#2912	1	2	yes	—
225	C₂×C₃₁	#3221	1	1	yes	characteristic
226	C₂×C₃₁	#3222	1	1	yes	characteristic
227	C₂×C₃₁	#3244	1	2	yes	—
228	C₂³×C₄	—	2	1	no	—
229	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
230	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
231	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
232	#117	—	4	2	no	—
233	#117	C₂×C₄×C₃₁	1	4	yes	—
234	#117	Q₈×C₃₁	1	4	yes	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	4	2	no	—
240	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
241	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
243	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
244	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
246	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
247	C₂×C₂²⋊C₄	—	2	1	no	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
253	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
254	#107	—	4	2	no	—
255	#106	—	4	2	no	—
256	#106	—	4	2	no	—
257	#106	—	4	2	no	—
258	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
259	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
260	#97	C₂×C₄×C₃₁	1	4	yes	complemented
261	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
262	#95	—	2	2	no	—
263	#95	—	2	2	no	—
264	#95	—	2	2	no	—
265	#95	—	2	2	no	—
266	#95	—	2	2	no	—
267	C₃₁	#21990	1	1	yes	characteristic, direct product factor, Sylow
268	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	—	8	2	no	—
279	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
283	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
284	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
290	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	—	8	2	no	—
300	C₄⋊C₄	—	8	2	no	—
301	C₄⋊C₄	—	8	2	no	—
302	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	8	2	no	—
305	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
306	C₄²	—	4	2	no	—
307	C₄²	—	8	2	no	—
308	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
309	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
310	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
311	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
312	C₂³	—	2	1	no	—
313	C₂³	—	2	1	no	—
314	C₂³	—	2	2	no	—
315	C₂³	—	2	2	no	—
316	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
317	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	8	4	no	—
341	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
342	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
343	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
344	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
346	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
347	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
348	C₈	—	2	4	no	—
349	C₈	—	2	4	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	4	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	2	no	—
355	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
356	C₂²	C₃₁×#535	1	1	yes	characteristic
357	C₂²	C₃₁×#536	1	1	yes	characteristic
358	C₂²	C₃₁×#577	1	2	yes	—
359	C₂²	C₃₁×#577	1	2	yes	—
360	C₄	—	2	2	no	—
361	C₄	—	2	4	no	—
362	C₄	—	2	4	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	2	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	1	no	—
367	C₄	—	8	1	no	—
368	C₄	—	8	2	no	—
369	C₄	C₃₁×#520	1	2	yes	—
370	C₄	C₃₁×#544	1	2	yes	—
371	C₂	—	4	2	no	—
372	C₂	C₃₁×#2317	1	1	yes	characteristic
373	C₂	C₃₁×#2912	1	2	yes	—
374	C₂	C₃₁×#3221	1	1	yes	characteristic
375	C₂	C₃₁×#3222	1	1	yes	characteristic
376	C₂	C₃₁×#3244	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1949	C₂	semidirect
5. C₃₁×#1948	C₂	semidirect
8. C₃₁×#1847	C₂	semidirect
9. C₃₁×#1844	C₂	semidirect
10. C₃₁×#1836	C₂	semidirect
11. C₃₁×#1835	C₂	semidirect
15. C₃₁×#433	C₄	semidirect
70. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
72. C₃₁×#97	C₂×C₄	semidirect
73. C₄×C₈×C₃₁	C₂×C₄	semidirect
123. #21990	C₃₁	direct (N × H)
165. #1949	C₂×C₃₁	semidirect
166. #1948	C₂×C₃₁	semidirect
169. #1847	C₂×C₃₁	semidirect
170. #1844	C₂×C₃₁	semidirect
171. #1836	C₂×C₃₁	semidirect
172. #1835	C₂×C₃₁	semidirect
197. #433	C₄×C₃₁	semidirect
258. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
260. #97	C₂×C₄×C₃₁	semidirect
261. C₄×C₈	C₂×C₄×C₃₁	semidirect
267. C₃₁	#21990	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×#433	2
C₃₁×#2317	2
C₃₁×#2314	2
C₃₁×#1949	2
C₃₁×#1948	2
C₃₁×#1854	2
C₃₁×#1853	2
C₃₁×#1847	2
C₃₁×#1844	2
C₃₁×#1836	2
C₃₁×#1835	2
#21990	31

Subgroups — C31×#21990 #21478749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21990 #21478749