Subgroups — C₃₁×#23560 #21480319

Order 7936.

1082 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3458	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2911	C₂	1	2	yes	maximal
3	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2061	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2060	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1742	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#339	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#544	—	2	4	no	—
10	C₃₁×#544	C₂²	1	4	yes	—
11	C₃₁×#543	—	2	2	no	—
12	C₃₁×#533	—	2	4	no	—
13	C₃₁×#533	—	2	2	no	—
14	C₂×C₃₁×#117	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#428	—	2	4	no	—
19	C₃₁×#428	C₂²	1	8	yes	—
20	C₃₁×#427	—	2	4	no	—
21	C₃₁×#384	C₂²	1	2	yes	—
22	C₃₁×#382	C₂²	1	2	yes	—
23	C₃₁×#380	—	2	2	no	—
24	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₃₁×#343	C₂²	1	4	yes	—
26	C₃₁×#339	C₂²	1	4	yes	—
27	C₃₁×#339	C₂²	1	4	yes	—
28	C₂³×C₄×C₃₁	—	2	1	no	—
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₃₁	—	2	2	no	—
31	C₂×M₁₆×C₃₁	—	2	4	no	—
32	C₂×M₁₆×C₃₁	Q₈	1	8	yes	complemented
33	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₃₁×C₄⋊Q₈	—	2	4	no	—
36	C₃₁×C₄⋊Q₈	—	2	8	no	—
37	C₃₁×#125	—	2	4	no	—
38	C₃₁×#125	—	2	8	no	—
39	C₃₁×C₂²⋊Q₈	—	2	16	no	—
40	C₃₁×C₂²⋊Q₈	—	2	8	no	—
41	C₄×Q₈×C₃₁	—	2	8	no	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₃₁×#117	—	2	2	no	—
45	C₃₁×#117	—	2	2	no	—
46	C₃₁×#117	—	2	2	no	—
47	C₃₁×#117	—	2	2	no	—
48	C₃₁×#117	D₄	1	4	yes	—
49	C₃₁×#117	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
57	C₂×C₄²×C₃₁	—	2	1	no	—
58	C₂×C₄²×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	—	2	2	no	—
60	C₂×C₄²×C₃₁	—	2	1	no	—
61	C₂×C₄²×C₃₁	D₄	1	2	yes	—
62	C₂×C₄²×C₃₁	Q₈	1	4	yes	complemented
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#107	—	2	8	no	—
66	C₃₁×#106	—	2	8	no	—
67	C₃₁×#95	—	2	4	no	—
68	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×Q₈×C₃₁	—	2	8	no	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	—	2	4	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	4	4	no	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₃₁	#41	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₃₁	—	2	8	no	—
85	M₁₆×C₃₁	—	2	4	no	—
86	M₁₆×C₃₁	C₂×Q₈	1	8	yes	—
87	C₂×C₈×C₃₁	—	2	8	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	2	16	no	—
90	C₄⋊C₄×C₃₁	—	2	8	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	4	8	no	—
95	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
96	C₂²⋊C₄×C₃₁	—	2	4	no	—
97	C₂²⋊C₄×C₃₁	—	2	4	no	—
98	C₂²⋊C₄×C₃₁	—	2	4	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	2	2	no	—
101	C₄²×C₃₁	—	2	4	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	4	no	—
104	C₄²×C₃₁	—	2	4	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	2	1	no	—
107	C₄²×C₃₁	—	4	1	no	—
108	C₄²×C₃₁	—	4	1	no	—
109	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
110	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
111	#23560	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₃₁	—	4	2	no	—
113	C₂³×C₃₁	#120	1	2	yes	—
114	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
115	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
116	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
117	Q₈×C₃₁	—	8	8	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	2	8	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	4	8	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	8	4	no	—
129	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
130	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
132	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
133	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
135	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
136	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
139	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₃₁	—	2	8	no	—
142	#3458	C₂×C₃₁	1	1	yes	characteristic
143	#2911	C₂×C₃₁	1	2	yes	—
144	#2285	C₂×C₃₁	1	1	yes	characteristic
145	#2061	C₂×C₃₁	1	4	yes	complemented
146	#2060	C₂×C₃₁	1	4	yes	complemented
147	#1742	C₂×C₃₁	1	1	yes	characteristic
148	C₂×#339	C₂×C₃₁	1	2	yes	—
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	2	4	no	—
151	C₂²×C₃₁	—	4	2	no	—
152	C₂²×C₃₁	—	4	2	no	—
153	C₂²×C₃₁	#395	1	4	yes	—
154	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
155	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₄×C₃₁	—	2	4	no	—
158	C₄×C₃₁	—	4	2	no	—
159	C₄×C₃₁	—	4	4	no	—
160	C₄×C₃₁	—	4	8	no	—
161	C₄×C₃₁	—	8	2	no	—
162	C₄×C₃₁	#395	1	2	yes	—
163	C₄×C₃₁	#395	1	2	yes	—
164	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
165	C₄×C₃₁	C₂×C₄⋊Q₈	1	2	yes	—
166	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
167	#544	—	2	4	no	—
168	#544	C₂²×C₃₁	1	4	yes	—
169	#543	—	2	2	no	—
170	#533	—	2	4	no	—
171	#533	—	2	2	no	—
172	C₂×#117	C₂²×C₃₁	1	2	yes	—
173	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
175	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
176	#428	—	2	4	no	—
177	#428	C₂²×C₃₁	1	8	yes	—
178	#427	—	2	4	no	—
179	#384	C₂²×C₃₁	1	2	yes	—
180	#382	C₂²×C₃₁	1	2	yes	—
181	#380	—	2	2	no	—
182	C₄×C₄⋊C₄	—	2	2	no	—
183	#343	C₂²×C₃₁	1	4	yes	—
184	#339	C₂²×C₃₁	1	4	yes	—
185	#339	C₂²×C₃₁	1	4	yes	—
186	C₂×C₃₁	—	2	1	no	—
187	C₂×C₃₁	—	2	1	no	—
188	C₂×C₃₁	—	2	4	no	—
189	C₂×C₃₁	C₂×#395	1	1	yes	characteristic
190	C₂×C₃₁	#3018	1	2	yes	—
191	C₂³×C₄	—	2	1	no	—
192	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	4	no	—
195	C₂×M₁₆	Q₈×C₃₁	1	8	yes	complemented
196	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	4	no	—
199	C₄⋊Q₈	—	2	8	no	—
200	#125	—	2	4	no	—
201	#125	—	2	8	no	—
202	C₂²⋊Q₈	—	2	16	no	—
203	C₂²⋊Q₈	—	2	8	no	—
204	C₄×Q₈	—	2	8	no	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	—	2	2	no	—
210	#117	—	2	2	no	—
211	#117	D₄×C₃₁	1	4	yes	—
212	#117	C₂³×C₃₁	1	4	yes	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
215	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₂²⋊C₄	—	2	1	no	—
219	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	D₄×C₃₁	1	2	yes	—
225	C₂×C₄²	Q₈×C₃₁	1	4	yes	complemented
226	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
228	#107	—	2	8	no	—
229	#106	—	2	8	no	—
230	#95	—	2	4	no	—
231	C₃₁	#23560	1	1	yes	characteristic, direct product factor, Sylow
232	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂×Q₈	—	2	8	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
244	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
245	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
246	C₂²×C₄	#41×C₃₁	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
248	M₁₆	—	2	8	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	C₂×Q₈×C₃₁	1	8	yes	—
251	C₂×C₈	—	2	8	no	—
252	C₂×C₈	—	2	4	no	—
253	C₄⋊C₄	—	2	16	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	4	8	no	—
259	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
274	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
275	C₂³	—	4	2	no	—
276	C₂³	C₃₁×#120	1	2	yes	—
277	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
278	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
279	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
280	Q₈	—	8	8	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	8	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
293	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
294	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
295	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
296	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
297	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
298	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
299	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
300	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
301	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
302	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
304	C₈	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₃₁×#395	1	4	yes	—
310	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
311	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
312	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	8	no	—
317	C₄	—	8	2	no	—
318	C₄	C₃₁×#395	1	2	yes	—
319	C₄	C₃₁×#395	1	2	yes	—
320	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
321	C₄	C₂×C₃₁×C₄⋊Q₈	1	2	yes	—
322	C₂	—	2	1	no	—
323	C₂	—	2	4	no	—
324	C₂	—	2	1	no	—
325	C₂	C₂×C₃₁×#395	1	1	yes	characteristic
326	C₂	C₃₁×#3018	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#120
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#2061	C₂	semidirect
5. C₃₁×#2060	C₂	semidirect
32. C₂×M₁₆×C₃₁	Q₈	semidirect
62. C₂×C₄²×C₃₁	Q₈	semidirect
111. #23560	C₃₁	direct (N × H)
145. #2061	C₂×C₃₁	semidirect
146. #2060	C₂×C₃₁	semidirect
195. C₂×M₁₆	Q₈×C₃₁	semidirect
225. C₂×C₄²	Q₈×C₃₁	semidirect
231. C₃₁	#23560	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×#3458	2
C₃₁×#2911	2
C₃₁×#2285	2
C₃₁×#2061	2
C₃₁×#2060	2
C₃₁×#1742	2
C₂×C₃₁×#339	2
#23560	31

Subgroups — C31×#23560 #21480319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23560 #21480319