Subgroups — C₃₁×#3114 #16843047

Order 3968.

450 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#483	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#480	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#478	C₂	1	2	yes	maximal, complemented
6	C₃₁×#463	C₂	1	2	yes	maximal, complemented
7	C₃₁×#450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#447	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#442	C₂	1	2	yes	maximal, complemented
10	C₃₁×#440	C₂	1	2	yes	maximal, complemented
11	C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₁	—	2	1	no	—
14	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×D₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#126	—	2	1	no	—
17	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#123	—	2	1	no	—
20	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃₁×#121	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₄×D₄×C₃₁	—	2	1	no	—
24	C₄×D₄×C₃₁	—	2	1	no	—
25	C₄×D₄×C₃₁	—	2	1	no	—
26	C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#117	—	2	1	no	—
30	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
32	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#107	C₂²	1	2	yes	complemented
34	C₃₁×#106	C₂²	1	2	yes	complemented
35	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₃₁×#103	C₂²	1	2	yes	complemented
37	C₃₁×#102	C₂²	1	2	yes	—
38	C₃₁×#102	C₂²	1	2	yes	—
39	C₃₁×#102	C₂²	1	2	yes	complemented
40	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×#97	C₂²	1	2	yes	complemented
43	C₂⁴×C₃₁	—	2	1	no	—
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₁	—	2	4	no	—
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	4	4	no	—
55	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	SD₁₆×C₃₁	—	2	4	no	—
59	D₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂×C₈×C₃₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₄²×C₃₁	—	2	1	no	—
78	C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₄²×C₃₁	D₄	1	1	yes	characteristic
80	C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂³×C₃₁	—	2	4	no	—
83	C₂³×C₃₁	—	2	2	no	—
84	C₂³×C₃₁	—	2	1	no	—
85	C₂³×C₃₁	—	4	2	no	—
86	C₂³×C₃₁	—	4	2	no	—
87	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₃₁	—	2	2	no	—
89	D₄×C₃₁	—	2	4	no	—
90	D₄×C₃₁	—	2	2	no	—
91	D₄×C₃₁	—	2	2	no	—
92	D₄×C₃₁	—	4	2	no	—
93	D₄×C₃₁	#41	1	4	yes	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	1	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	C₈×C₃₁	—	2	2	no	—
112	C₈×C₃₁	—	2	2	no	—
113	#3114	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₃₁	—	2	1	no	—
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	4	no	—
117	C₂²×C₃₁	—	4	8	no	—
118	C₂²×C₃₁	—	4	2	no	—
119	C₂²×C₃₁	—	4	2	no	—
120	C₂²×C₃₁	—	4	1	no	—
121	C₂²×C₃₁	—	4	1	no	—
122	C₂²×C₃₁	—	4	1	no	—
123	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
124	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	4	1	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	—	4	1	no	—
133	C₄×C₃₁	—	4	1	no	—
134	C₄×C₃₁	C₂×#41	1	2	yes	—
135	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
136	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#483	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#480	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#478	C₂×C₃₁	1	2	yes	complemented
140	#463	C₂×C₃₁	1	2	yes	complemented
141	#450	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#447	C₂×C₃₁	1	1	yes	characteristic
143	#442	C₂×C₃₁	1	2	yes	complemented
144	#440	C₂×C₃₁	1	2	yes	complemented
145	#432	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₂×C₃₁	—	2	2	no	—
147	C₂×C₃₁	—	4	4	no	—
148	C₂×C₃₁	—	8	1	no	—
149	C₂×C₃₁	#525	1	1	yes	characteristic
150	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₃₁	#575	1	1	yes	characteristic
152	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
153	C₂³×C₄	—	2	1	no	—
154	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
155	C₂×D₈	C₂²×C₃₁	1	2	yes	—
156	#126	—	2	1	no	—
157	#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
158	#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
159	#123	—	2	1	no	—
160	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
161	#121	C₂²×C₃₁	1	1	yes	characteristic
162	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	C₂²×C₃₁	1	2	yes	—
167	C₄×D₄	C₂²×C₃₁	1	2	yes	—
168	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
169	#117	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	—	2	1	no	—
172	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
173	#107	C₂²×C₃₁	1	2	yes	complemented
174	#106	C₂²×C₃₁	1	2	yes	complemented
175	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
176	#103	C₂²×C₃₁	1	2	yes	complemented
177	#102	C₂²×C₃₁	1	2	yes	—
178	#102	C₂²×C₃₁	1	2	yes	—
179	#102	C₂²×C₃₁	1	2	yes	complemented
180	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
181	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
182	#97	C₂²×C₃₁	1	2	yes	complemented
183	C₃₁	#3114	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	—	2	1	no	—
185	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂×D₄	C₂³×C₃₁	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
199	SD₁₆	—	2	4	no	—
200	D₈	—	2	4	no	—
201	C₂×C₈	C₂³×C₃₁	1	2	yes	—
202	C₂×C₈	C₂³×C₃₁	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
209	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	D₄×C₃₁	1	1	yes	characteristic
220	C₄²	D₄×C₃₁	1	1	yes	characteristic
221	C₄²	C₂³×C₃₁	1	1	yes	characteristic
222	C₄²	C₂³×C₃₁	1	1	yes	characteristic
223	C₂³	—	2	4	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	4	2	no	—
234	D₄	#41×C₃₁	1	4	yes	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
264	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	C₂×#41×C₃₁	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	—	4	4	no	—
277	C₂	—	8	1	no	—
278	C₂	C₃₁×#525	1	1	yes	characteristic
279	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#529	C₂	semidirect
2. C₂×C₄×D₄×C₃₁	C₂	semidirect
3. C₃₁×#483	C₂	semidirect
4. C₃₁×#480	C₂	semidirect
5. C₃₁×#478	C₂	semidirect
6. C₃₁×#463	C₂	semidirect
7. C₃₁×#450	C₂	semidirect
9. C₃₁×#442	C₂	semidirect
10. C₃₁×#440	C₂	semidirect
11. C₃₁×#432	C₂	semidirect
17. C₃₁×#125	C₂²	semidirect
18. C₃₁×#124	C₂²	semidirect
22. C₄×Q₈×C₃₁	C₂²	semidirect
28. C₄×D₄×C₃₁	C₂²	semidirect
33. C₃₁×#107	C₂²	semidirect
34. C₃₁×#106	C₂²	semidirect
35. C₃₁×C₄⋊C₈	C₂²	semidirect
36. C₃₁×#103	C₂²	semidirect
39. C₃₁×#102	C₂²	semidirect
42. C₃₁×#97	C₂²	semidirect
113. #3114	C₃₁	direct (N × H)
135. #529	C₂×C₃₁	semidirect
136. C₂×C₄×D₄	C₂×C₃₁	semidirect
137. #483	C₂×C₃₁	semidirect
138. #480	C₂×C₃₁	semidirect
139. #478	C₂×C₃₁	semidirect
140. #463	C₂×C₃₁	semidirect
141. #450	C₂×C₃₁	semidirect
143. #442	C₂×C₃₁	semidirect
144. #440	C₂×C₃₁	semidirect
145. #432	C₂×C₃₁	semidirect
157. #125	C₂²×C₃₁	semidirect
158. #124	C₂²×C₃₁	semidirect
162. C₄×Q₈	C₂²×C₃₁	semidirect
168. C₄×D₄	C₂²×C₃₁	semidirect
173. #107	C₂²×C₃₁	semidirect
174. #106	C₂²×C₃₁	semidirect
175. C₄⋊C₈	C₂²×C₃₁	semidirect
176. #103	C₂²×C₃₁	semidirect
179. #102	C₂²×C₃₁	semidirect
182. #97	C₂²×C₃₁	semidirect
183. C₃₁	#3114	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×#529	2
C₂×C₄×D₄×C₃₁	2
C₃₁×#483	2
C₃₁×#480	2
C₃₁×#478	2
C₃₁×#463	2
C₃₁×#450	2
C₃₁×#447	2
C₃₁×#442	2
C₃₁×#440	2
C₃₁×#432	2
#3114	31

Subgroups — C31×#3114 #16843047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3114 #16843047