Subgroups — C₃₁×#10811 #21467570

Order 7936.

644 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	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1740	C₂	1	2	yes	maximal
3	C₃₁×#1733	C₂	1	2	yes	maximal
4	C₂×C₃₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#117	C₂²	1	2	yes	—
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#117	C₂²	1	2	yes	—
11	C₃₁×#405	—	2	2	no	—
12	C₃₁×#405	C₄	1	4	yes	complemented
13	C₄×M₁₆×C₃₁	—	2	2	no	—
14	C₃₁×#342	—	2	4	no	—
15	C₃₁×#339	C₄	1	8	yes	complemented
16	C₃₁×H(3,ℤ/4)	—	2	2	no	—
17	C₃₁×H(3,ℤ/4)	—	2	2	no	—
18	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	—	2	1	no	—
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
25	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
26	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
27	C₂²×C₈×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₃₁×#117	—	2	1	no	—
30	C₃₁×#117	—	2	1	no	—
31	C₃₁×#117	—	2	2	no	—
32	C₃₁×#117	—	2	2	no	—
33	C₃₁×#117	—	2	2	no	—
34	C₃₁×#117	—	2	2	no	—
35	C₃₁×#117	—	2	2	no	—
36	C₃₁×#117	—	2	2	no	—
37	C₃₁×#117	—	2	2	no	—
38	C₃₁×#117	C₂×C₄	1	4	yes	—
39	C₃₁×#117	C₂×C₄	1	2	yes	—
40	C₃₁×#117	Q₈	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
47	C₂×C₄²×C₃₁	—	2	1	no	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
50	C₃₁×#97	—	2	4	no	—
51	C₄×C₈×C₃₁	—	2	4	no	—
52	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	#41	1	2	yes	—
64	C₂²×C₄×C₃₁	#41	1	2	yes	—
65	C₂²×C₄×C₃₁	#41	1	2	yes	—
66	M₁₆×C₃₁	—	2	2	no	—
67	M₁₆×C₃₁	—	2	4	no	—
68	M₁₆×C₃₁	—	2	4	no	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	C₄²	1	4	yes	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	2	2	no	—
83	C₂²⋊C₄×C₃₁	—	2	4	no	—
84	C₂²⋊C₄×C₃₁	—	4	2	no	—
85	C₂²⋊C₄×C₃₁	—	4	2	no	—
86	C₂²⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
88	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	—	2	2	no	—
90	C₄²×C₃₁	—	2	4	no	—
91	C₄²×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	2	4	no	—
93	C₄²×C₃₁	—	4	2	no	—
94	#10811	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₃₁	—	8	1	no	—
96	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₃₁	C₄×D₄	1	2	yes	—
98	C₂³×C₃₁	#121	1	1	yes	characteristic
99	C₂³×C₃₁	#121	1	1	yes	characteristic
100	C₂³×C₃₁	#124	1	2	yes	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	8	2	no	—
105	C₂×C₄×C₃₁	—	8	4	no	—
106	C₂×C₄×C₃₁	—	8	4	no	—
107	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
110	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#117	1	2	yes	—
112	C₂×C₄×C₃₁	#117	1	2	yes	—
113	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#123	1	2	yes	—
121	C₂×C₄×C₃₁	#123	1	2	yes	—
122	C₂×C₄×C₃₁	#125	1	2	yes	—
123	C₂×C₄×C₃₁	#126	1	2	yes	—
124	C₂×C₄×C₃₁	#126	1	2	yes	—
125	C₈×C₃₁	—	2	4	no	—
126	C₈×C₃₁	—	2	4	no	—
127	#2873	C₂×C₃₁	1	1	yes	characteristic
128	#1740	C₂×C₃₁	1	2	yes	—
129	#1733	C₂×C₃₁	1	2	yes	—
130	C₂×H(3,ℤ/4)	C₂×C₃₁	1	1	yes	characteristic
131	C₂×H(3,ℤ/4)	C₂×C₃₁	1	1	yes	characteristic
132	C₂²×C₃₁	—	4	2	no	—
133	C₂²×C₃₁	—	4	1	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	4	1	no	—
136	C₂²×C₃₁	—	4	1	no	—
137	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₃₁	#380	1	1	yes	characteristic
139	C₂²×C₃₁	#382	1	1	yes	characteristic
140	C₂²×C₃₁	#388	1	2	yes	—
141	C₂²×C₃₁	#388	1	2	yes	—
142	C₄×C₃₁	—	4	4	no	—
143	C₄×C₃₁	—	8	4	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	#376	1	1	yes	characteristic
146	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₄×C₃₁	#381	1	1	yes	characteristic
148	C₄×C₃₁	#383	1	1	yes	characteristic
149	C₄×C₃₁	#387	1	2	yes	—
150	C₄×C₃₁	#387	1	2	yes	—
151	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
152	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×#117	C₂²×C₃₁	1	2	yes	—
154	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×#117	C₂²×C₃₁	1	2	yes	—
156	#405	—	2	2	no	—
157	#405	C₄×C₃₁	1	4	yes	complemented
158	C₄×M₁₆	—	2	2	no	—
159	#342	—	2	4	no	—
160	#339	C₄×C₃₁	1	8	yes	complemented
161	H(3,ℤ/4)	—	2	2	no	—
162	H(3,ℤ/4)	—	2	2	no	—
163	C₂×C₃₁	—	2	1	no	—
164	C₂×C₃₁	—	2	2	no	—
165	C₂×C₃₁	—	2	2	no	—
166	C₂×C₃₁	—	2	1	no	—
167	C₂×C₃₁	—	2	1	no	—
168	C₂×C₃₁	#1434	1	1	yes	characteristic
169	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
175	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
176	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
177	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
180	#117	—	2	1	no	—
181	#117	—	2	1	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	C₂×C₄×C₃₁	1	4	yes	—
190	#117	C₂×C₄×C₃₁	1	2	yes	—
191	#117	Q₈×C₃₁	1	2	yes	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
201	#97	—	2	4	no	—
202	C₄×C₈	—	2	4	no	—
203	C₃₁	#10811	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	#41×C₃₁	1	2	yes	—
216	C₂²×C₄	#41×C₃₁	1	2	yes	—
217	C₂²×C₄	#41×C₃₁	1	2	yes	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	C₄²×C₃₁	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	4	2	no	—
246	C₂³	—	8	1	no	—
247	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
248	C₂³	C₄×D₄×C₃₁	1	2	yes	—
249	C₂³	C₃₁×#121	1	1	yes	characteristic
250	C₂³	C₃₁×#121	1	1	yes	characteristic
251	C₂³	C₃₁×#124	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
261	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
262	C₂×C₄	C₃₁×#117	1	2	yes	—
263	C₂×C₄	C₃₁×#117	1	2	yes	—
264	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
265	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
266	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
268	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
269	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
270	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#123	1	2	yes	—
272	C₂×C₄	C₃₁×#123	1	2	yes	—
273	C₂×C₄	C₃₁×#125	1	2	yes	—
274	C₂×C₄	C₃₁×#126	1	2	yes	—
275	C₂×C₄	C₃₁×#126	1	2	yes	—
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
284	C₂²	C₃₁×#380	1	1	yes	characteristic
285	C₂²	C₃₁×#382	1	1	yes	characteristic
286	C₂²	C₃₁×#388	1	2	yes	—
287	C₂²	C₃₁×#388	1	2	yes	—
288	C₄	—	4	4	no	—
289	C₄	—	8	4	no	—
290	C₄	—	8	4	no	—
291	C₄	C₃₁×#376	1	1	yes	characteristic
292	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
293	C₄	C₃₁×#381	1	1	yes	characteristic
294	C₄	C₃₁×#383	1	1	yes	characteristic
295	C₄	C₃₁×#387	1	2	yes	—
296	C₄	C₃₁×#387	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	C₃₁×#1434	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂×C₃₁	#1434
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₁×#405	C₄	semidirect
15. C₃₁×#339	C₄	semidirect
94. #10811	C₃₁	direct (N × H)
157. #405	C₄×C₃₁	semidirect
160. #339	C₄×C₃₁	semidirect
203. C₃₁	#10811	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×#2873	2
C₃₁×#1740	2
C₃₁×#1733	2
C₂×C₃₁×H(3,ℤ/4)	2
#10811	31

Subgroups — C31×#10811 #21467570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10811 #21467570