Subgroups — C₃₁×#3009 #16842942

Order 3968.

752 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#452	C₂	1	4	yes	maximal, complemented
6	C₃₁×#450	C₂	1	4	yes	maximal, complemented
7	C₃₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×H(5,𝔽₂)	—	2	4	no	—
11	C₃₁×H(5,𝔽₂)	—	2	4	no	—
12	C₂×#41×C₃₁	—	2	1	no	—
13	C₂×#41×C₃₁	C₂²	1	2	yes	—
14	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₄×C₃₁	—	2	2	no	—
18	C₂²×D₄×C₃₁	—	2	1	no	—
19	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#137	—	2	4	no	—
22	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
23	C₂×SD₁₆×C₃₁	—	2	4	no	—
24	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
25	C₂×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
26	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×C₄⋊Q₈	—	2	1	no	—
29	C₃₁×#124	—	2	1	no	—
30	C₃₁×#123	—	2	2	no	—
31	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
32	C₃₁×#117	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#103	C₂²	1	2	yes	complemented
34	C₃₁×#102	C₂²	1	2	yes	complemented
35	C₃₁×#102	C₂²	1	4	yes	complemented
36	C₃₁×C₂²⋊C₈	C₂²	1	4	yes	complemented
37	C₂⁴×C₃₁	—	2	2	no	—
38	C₂⁴×C₃₁	—	2	1	no	—
39	#41×C₃₁	—	2	8	no	—
40	#41×C₃₁	—	2	4	no	—
41	#41×C₃₁	—	2	4	no	—
42	#41×C₃₁	—	2	2	no	—
43	#41×C₃₁	—	2	2	no	—
44	#41×C₃₁	—	2	2	no	—
45	#41×C₃₁	D₄	1	4	yes	—
46	C₂×Q₈×C₃₁	—	2	1	no	—
47	C₂×Q₈×C₃₁	—	2	2	no	—
48	C₂×Q₈×C₃₁	—	2	2	no	—
49	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₃₁	—	2	16	no	—
53	C₂×D₄×C₃₁	—	2	8	no	—
54	C₂×D₄×C₃₁	—	2	2	no	—
55	C₂×D₄×C₃₁	—	2	1	no	—
56	C₂×D₄×C₃₁	—	2	4	no	—
57	C₂×D₄×C₃₁	—	2	4	no	—
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	2	1	no	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	D₄	1	2	yes	—
62	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂×D₄×C₃₁	D₄	1	4	yes	—
64	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	1	no	—
68	C₂²×C₄×C₃₁	—	2	1	no	—
69	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	Q₁₆×C₃₁	—	2	4	no	—
72	SD₁₆×C₃₁	—	2	8	no	—
73	SD₁₆×C₃₁	—	2	4	no	—
74	M₁₆×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	C₂³	1	2	yes	—
77	C₂×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂²⋊C₄×C₃₁	—	2	4	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	—	2	8	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	2	1	no	—
86	C₂³×C₃₁	—	2	4	no	—
87	C₂³×C₃₁	—	8	2	no	—
88	C₂³×C₃₁	—	8	4	no	—
89	C₂³×C₃₁	C₂×D₄	1	2	yes	—
90	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₃₁	—	2	2	no	—
92	Q₈×C₃₁	—	2	1	no	—
93	Q₈×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	—	2	2	no	—
95	Q₈×C₃₁	—	4	2	no	—
96	Q₈×C₃₁	C₂×D₄	1	2	yes	—
97	D₄×C₃₁	—	2	8	no	—
98	D₄×C₃₁	—	2	8	no	—
99	D₄×C₃₁	—	2	4	no	—
100	D₄×C₃₁	—	2	4	no	—
101	D₄×C₃₁	—	2	2	no	—
102	D₄×C₃₁	—	2	4	no	—
103	D₄×C₃₁	—	2	2	no	—
104	D₄×C₃₁	—	2	1	no	—
105	D₄×C₃₁	C₂×D₄	1	2	yes	—
106	D₄×C₃₁	C₂×D₄	1	4	yes	—
107	C₂×C₄×C₃₁	—	2	8	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	C₈×C₃₁	—	2	2	no	—
121	C₈×C₃₁	—	2	2	no	—
122	#3009	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₃₁	—	2	2	no	—
124	C₂²×C₃₁	—	2	2	no	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	4	no	—
128	C₂²×C₃₁	—	4	4	no	—
129	C₂²×C₃₁	—	4	2	no	—
130	C₂²×C₃₁	—	8	4	no	—
131	C₂²×C₃₁	—	8	8	no	—
132	C₂²×C₃₁	#120	1	2	yes	—
133	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	—	4	2	no	—
138	C₄×C₃₁	#120	1	2	yes	—
139	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×H(5,𝔽₂)	C₂×C₃₁	1	1	yes	characteristic
142	C₂×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂²×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#536	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#452	C₂×C₃₁	1	4	yes	complemented
146	#450	C₂×C₃₁	1	4	yes	complemented
147	#418	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#416	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#408	C₂×C₃₁	1	1	yes	characteristic, complemented
150	C₂×C₃₁	—	2	2	no	—
151	C₂×C₃₁	—	4	2	no	—
152	C₂×C₃₁	—	4	4	no	—
153	C₂×C₃₁	—	4	2	no	—
154	C₂×C₃₁	C₂×#120	1	1	yes	characteristic
155	C₂×C₃₁	#577	1	2	yes	—
156	H(5,𝔽₂)	—	2	4	no	—
157	H(5,𝔽₂)	—	2	4	no	—
158	C₂×#41	—	2	1	no	—
159	C₂×#41	C₂²×C₃₁	1	2	yes	—
160	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	—	2	1	no	—
165	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
166	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
167	#137	—	2	4	no	—
168	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
169	C₂×SD₁₆	—	2	4	no	—
170	C₂×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
171	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
172	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
173	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
174	C₄⋊Q₈	—	2	1	no	—
175	#124	—	2	1	no	—
176	#123	—	2	2	no	—
177	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
178	#117	C₂²×C₃₁	1	1	yes	characteristic, complemented
179	#103	C₂²×C₃₁	1	2	yes	complemented
180	#102	C₂²×C₃₁	1	2	yes	complemented
181	#102	C₂²×C₃₁	1	4	yes	complemented
182	C₂²⋊C₈	C₂²×C₃₁	1	4	yes	complemented
183	C₃₁	#3009	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	—	2	2	no	—
185	C₂⁴	—	2	1	no	—
186	#41	—	2	8	no	—
187	#41	—	2	4	no	—
188	#41	—	2	4	no	—
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	D₄×C₃₁	1	4	yes	—
193	C₂×Q₈	—	2	1	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
197	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
198	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
199	C₂×D₄	—	2	16	no	—
200	C₂×D₄	—	2	8	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	D₄×C₃₁	1	2	yes	—
209	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
210	C₂×D₄	D₄×C₃₁	1	4	yes	—
211	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
212	C₂×D₄	C₂³×C₃₁	1	2	yes	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	SD₁₆	—	2	8	no	—
220	SD₁₆	—	2	4	no	—
221	M₁₆	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	C₂³×C₃₁	1	2	yes	—
224	C₂×C₈	C₂³×C₃₁	1	2	yes	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₄²	—	2	1	no	—
230	C₂³	—	2	8	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	8	2	no	—
235	C₂³	—	8	4	no	—
236	C₂³	C₂×D₄×C₃₁	1	2	yes	—
237	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
238	Q₈	—	2	2	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	2	2	no	—
242	Q₈	—	4	2	no	—
243	Q₈	C₂×D₄×C₃₁	1	2	yes	—
244	D₄	—	2	8	no	—
245	D₄	—	2	8	no	—
246	D₄	—	2	4	no	—
247	D₄	—	2	4	no	—
248	D₄	—	2	2	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	2	no	—
251	D₄	—	2	1	no	—
252	D₄	C₂×D₄×C₃₁	1	2	yes	—
253	D₄	C₂×D₄×C₃₁	1	4	yes	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
263	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
265	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
266	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	8	4	no	—
277	C₂²	—	8	8	no	—
278	C₂²	C₃₁×#120	1	2	yes	—
279	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	C₃₁×#120	1	2	yes	—
285	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
286	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	4	4	no	—
290	C₂	—	4	2	no	—
291	C₂	C₂×C₃₁×#120	1	1	yes	characteristic
292	C₂	C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×H(5,𝔽₂)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#137	C₂	semidirect
3. C₂²×SD₁₆×C₃₁	C₂	semidirect
4. C₃₁×#536	C₂	semidirect
5. C₃₁×#452	C₂	semidirect
6. C₃₁×#450	C₂	semidirect
7. C₃₁×#418	C₂	semidirect
8. C₃₁×#416	C₂	semidirect
9. C₃₁×#408	C₂	semidirect
16. C₂²×Q₈×C₃₁	C₂²	semidirect
22. C₂×Q₁₆×C₃₁	C₂²	semidirect
24. C₂×SD₁₆×C₃₁	C₂²	semidirect
25. C₂×SD₁₆×C₃₁	C₂²	semidirect
26. C₂×M₁₆×C₃₁	C₂²	semidirect
27. C₂²×C₈×C₃₁	C₂²	semidirect
31. C₃₁×C₂²⋊Q₈	C₂²	semidirect
32. C₃₁×#117	C₂²	semidirect
33. C₃₁×#103	C₂²	semidirect
34. C₃₁×#102	C₂²	semidirect
35. C₃₁×#102	C₂²	semidirect
36. C₃₁×C₂²⋊C₈	C₂²	semidirect
122. #3009	C₃₁	direct (N × H)
142. C₂×#137	C₂×C₃₁	semidirect
143. C₂²×SD₁₆	C₂×C₃₁	semidirect
144. #536	C₂×C₃₁	semidirect
145. #452	C₂×C₃₁	semidirect
146. #450	C₂×C₃₁	semidirect
147. #418	C₂×C₃₁	semidirect
148. #416	C₂×C₃₁	semidirect
149. #408	C₂×C₃₁	semidirect
162. C₂²×Q₈	C₂²×C₃₁	semidirect
168. C₂×Q₁₆	C₂²×C₃₁	semidirect
170. C₂×SD₁₆	C₂²×C₃₁	semidirect
171. C₂×SD₁₆	C₂²×C₃₁	semidirect
172. C₂×M₁₆	C₂²×C₃₁	semidirect
173. C₂²×C₈	C₂²×C₃₁	semidirect
177. C₂²⋊Q₈	C₂²×C₃₁	semidirect
178. #117	C₂²×C₃₁	semidirect
179. #103	C₂²×C₃₁	semidirect
180. #102	C₂²×C₃₁	semidirect
181. #102	C₂²×C₃₁	semidirect
182. C₂²⋊C₈	C₂²×C₃₁	semidirect
183. C₃₁	#3009	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×H(5,𝔽₂)	2
C₂×C₃₁×#137	2
C₂²×SD₁₆×C₃₁	2
C₃₁×#536	2
C₃₁×#452	2
C₃₁×#450	2
C₃₁×#418	2
C₃₁×#416	2
C₃₁×#408	2
#3009	31

Subgroups — C31×#3009 #16842942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3009 #16842942