Subgroups — C₃₁×#21172 #21477931

Order 7936.

1022 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#428	C₂	1	2	yes	maximal
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1949	C₂	1	8	yes	maximal, complemented
6	C₃₁×#1736	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#428	—	2	8	no	—
13	C₂×C₃₁×#107	C₂²	1	4	yes	—
14	C₂×C₃₁×#106	C₂²	1	4	yes	—
15	C₃₁×#423	C₄	1	8	yes	complemented
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
17	C₃₁×#382	—	2	1	no	—
18	C₃₁×#382	—	2	1	no	—
19	C₃₁×#382	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
21	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
22	C₃₁×#376	—	2	1	no	—
23	C₂×C₃₁×#95	—	2	2	no	—
24	C₄³×C₃₁	—	2	1	no	—
25	C₃₁×#340	C₂²	1	8	yes	—
26	C₂³×C₄×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	D₄	1	2	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₃₁×#117	—	2	2	no	—
34	C₃₁×#117	—	2	2	no	—
35	C₃₁×#117	D₄	1	4	yes	—
36	C₃₁×#117	Q₈	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	2	4	no	—
48	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
53	C₃₁×#107	—	2	8	no	—
54	C₃₁×#106	—	2	8	no	—
55	C₃₁×C₄⋊C₈	C₂×C₄	1	16	yes	complemented
56	C₃₁×#95	—	2	4	no	—
57	C₃₁×#95	—	2	4	no	—
58	C₃₁×#95	—	2	4	no	—
59	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	4	4	no	—
64	C₂²×C₄×C₃₁	—	4	8	no	—
65	C₂²×C₄×C₃₁	—	4	4	no	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₃₁	#41	1	2	yes	—
75	C₂²×C₄×C₃₁	#41	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₃₁	—	2	16	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	2	4	no	—
85	C₄⋊C₄×C₃₁	—	4	4	no	—
86	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
88	C₂²⋊C₄×C₃₁	—	2	4	no	—
89	C₂²⋊C₄×C₃₁	—	2	4	no	—
90	C₄²×C₃₁	—	2	4	no	—
91	C₄²×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	4	8	no	—
93	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
94	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
95	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
96	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
97	#21172	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	2	4	no	—
99	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂³×C₃₁	#121	1	2	yes	—
101	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
102	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	4	8	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	8	no	—
112	C₂×C₄×C₃₁	—	8	8	no	—
113	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₃₁	#121	1	2	yes	—
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₃₁	#125	1	4	yes	—
121	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
122	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
128	C₈×C₃₁	—	2	8	no	—
129	C₂×#428	C₂×C₃₁	1	2	yes	—
130	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
131	C₂×#382	C₂×C₃₁	1	1	yes	characteristic
132	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
133	#1949	C₂×C₃₁	1	8	yes	complemented
134	#1736	C₂×C₃₁	1	2	yes	—
135	C₂²×C₃₁	—	2	4	no	—
136	C₂²×C₃₁	—	2	4	no	—
137	C₂²×C₃₁	—	2	4	no	—
138	C₂²×C₃₁	#389	1	4	yes	—
139	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
141	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₃₁	#575	1	4	yes	—
143	C₄×C₃₁	—	2	4	no	—
144	C₄×C₃₁	—	2	4	no	—
145	C₄×C₃₁	—	4	4	no	—
146	C₄×C₃₁	—	4	4	no	—
147	C₄×C₃₁	—	8	4	no	—
148	C₄×C₃₁	#389	1	4	yes	—
149	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
152	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
154	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
156	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
157	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
158	#428	—	2	8	no	—
159	C₂×#107	C₂²×C₃₁	1	4	yes	—
160	C₂×#106	C₂²×C₃₁	1	4	yes	—
161	#423	C₄×C₃₁	1	8	yes	complemented
162	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
163	#382	—	2	1	no	—
164	#382	—	2	1	no	—
165	#382	C₂²×C₃₁	1	4	yes	—
166	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
167	C₄×C₂²⋊C₄	—	2	2	no	—
168	#376	—	2	1	no	—
169	C₂×#95	—	2	2	no	—
170	C₄³	—	2	1	no	—
171	#340	C₂²×C₃₁	1	8	yes	—
172	C₂×C₃₁	—	2	4	no	—
173	C₂×C₃₁	C₂×#389	1	1	yes	characteristic
174	C₂×C₃₁	#2911	1	2	yes	—
175	C₂×C₃₁	#3040	1	2	yes	—
176	C₂×C₃₁	#3080	1	2	yes	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	D₄×C₃₁	1	2	yes	—
179	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
183	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	D₄×C₃₁	1	4	yes	—
187	#117	Q₈×C₃₁	1	4	yes	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	—	2	1	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
200	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
203	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
204	#107	—	2	8	no	—
205	#106	—	2	8	no	—
206	C₄⋊C₈	C₂×C₄×C₃₁	1	16	yes	complemented
207	#95	—	2	4	no	—
208	#95	—	2	4	no	—
209	#95	—	2	4	no	—
210	C₃₁	#21172	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	8	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
219	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
226	C₂²×C₄	#41×C₃₁	1	2	yes	—
227	C₂²×C₄	#41×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
229	M₁₆	—	2	16	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
239	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	8	no	—
245	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
246	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
247	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
248	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
251	C₂³	C₃₁×#121	1	2	yes	—
252	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
253	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	8	8	no	—
264	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
266	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
268	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
269	C₂×C₄	C₃₁×#121	1	2	yes	—
270	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
271	C₂×C₄	C₃₁×#125	1	4	yes	—
272	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
273	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
279	C₈	—	2	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₃₁×#389	1	4	yes	—
284	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
285	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
286	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
287	C₂²	C₃₁×#575	1	4	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	4	no	—
293	C₄	C₃₁×#389	1	4	yes	—
294	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
295	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
296	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
297	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
298	C₂	—	2	4	no	—
299	C₂	C₂×C₃₁×#389	1	1	yes	characteristic
300	C₂	C₃₁×#2911	1	2	yes	—
301	C₂	C₃₁×#3040	1	2	yes	—
302	C₂	C₃₁×#3080	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 ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃₁×#1949	C₂	semidirect
15. C₃₁×#423	C₄	semidirect
55. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
97. #21172	C₃₁	direct (N × H)
133. #1949	C₂×C₃₁	semidirect
161. #423	C₄×C₃₁	semidirect
206. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
210. C₃₁	#21172	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×#428	2
C₂×C₃₁×#423	2
C₂×C₃₁×#382	2
C₄×C₃₁×#117	2
C₃₁×#1949	2
C₃₁×#1736	2
#21172	31

Subgroups — C31×#21172 #21477931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21172 #21477931