Subgroups — C₃₁×#3325 #16843258

Order 3968.

460 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×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#482	C₂	1	2	yes	maximal, complemented
5	C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#469	C₂	1	2	yes	maximal, complemented
7	C₃₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#447	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#418	C₂	1	2	yes	maximal, complemented
12	C₂²×D₄×C₃₁	—	2	1	no	—
13	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₃₁	C₂²	1	2	yes	complemented
16	C₂×M₁₆×C₃₁	C₂²	1	2	yes	complemented
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#127	C₂²	1	1	yes	characteristic
19	C₃₁×#124	—	2	2	no	—
20	C₃₁×#121	—	2	1	no	—
21	C₃₁×#121	—	2	1	no	—
22	C₃₁×#121	C₂²	1	2	yes	—
23	C₃₁×#121	C₂²	1	2	yes	complemented
24	C₃₁×#120	—	2	2	no	—
25	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#117	—	2	1	no	—
29	C₃₁×#117	C₂²	1	2	yes	complemented
30	C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#106	C₂²	1	2	yes	complemented
32	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#102	C₂²	1	1	yes	characteristic
34	C₃₁×#102	C₂²	1	2	yes	—
35	C₃₁×#102	C₂²	1	2	yes	complemented
36	C₃₁×#102	C₂²	1	2	yes	complemented
37	C₃₁×#102	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#102	C₂²	1	2	yes	complemented
39	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₃₁	—	2	2	no	—
42	C₂×Q₈×C₃₁	—	2	1	no	—
43	C₂×D₄×C₃₁	—	2	4	no	—
44	C₂×D₄×C₃₁	—	2	2	no	—
45	C₂×D₄×C₃₁	—	2	2	no	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	—	2	1	no	—
49	C₂×D₄×C₃₁	—	4	2	no	—
50	C₂×D₄×C₃₁	—	4	4	no	—
51	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂×D₄×C₃₁	C₂³	1	2	yes	—
53	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	—	2	1	no	—
57	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
58	D₈×C₃₁	—	2	2	no	—
59	D₈×C₃₁	—	4	2	no	—
60	M₁₆×C₃₁	—	4	2	no	—
61	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₃₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
75	C₄²×C₃₁	—	2	2	no	—
76	C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂³×C₃₁	—	2	1	no	—
78	C₂³×C₃₁	—	2	1	no	—
79	C₂³×C₃₁	—	2	2	no	—
80	C₂³×C₃₁	—	4	4	no	—
81	C₂³×C₃₁	—	4	4	no	—
82	C₂³×C₃₁	—	8	2	no	—
83	C₂³×C₃₁	C₂×D₄	1	2	yes	—
84	Q₈×C₃₁	—	8	1	no	—
85	D₄×C₃₁	—	2	2	no	—
86	D₄×C₃₁	—	2	1	no	—
87	D₄×C₃₁	—	2	2	no	—
88	D₄×C₃₁	—	2	1	no	—
89	D₄×C₃₁	—	2	1	no	—
90	D₄×C₃₁	—	2	2	no	—
91	D₄×C₃₁	—	4	4	no	—
92	D₄×C₃₁	—	4	4	no	—
93	D₄×C₃₁	—	4	2	no	—
94	D₄×C₃₁	—	8	1	no	—
95	D₄×C₃₁	—	8	2	no	—
96	D₄×C₃₁	#41	1	2	yes	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	2	no	—
111	#3325	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₃₁	—	2	2	no	—
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	—	2	2	no	—
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	4	1	no	—
117	C₂²×C₃₁	—	4	1	no	—
118	C₂²×C₃₁	—	4	2	no	—
119	C₂²×C₃₁	—	4	1	no	—
120	C₂²×C₃₁	—	4	1	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	—	4	1	no	—
123	C₂²×C₃₁	—	4	2	no	—
124	C₂²×C₃₁	—	8	8	no	—
125	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	4	1	no	—
129	C₄×C₃₁	—	4	1	no	—
130	C₄×C₃₁	—	4	2	no	—
131	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
132	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
133	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
134	#555	C₂×C₃₁	1	1	yes	characteristic, complemented
135	D₄²	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#486	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#482	C₂×C₃₁	1	2	yes	complemented
138	#476	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#469	C₂×C₃₁	1	2	yes	complemented
140	#459	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#447	C₂×C₃₁	1	2	yes	complemented
142	C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#436	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#418	C₂×C₃₁	1	2	yes	complemented
145	C₂×C₃₁	—	4	2	no	—
146	C₂×C₃₁	—	4	2	no	—
147	C₂×C₃₁	—	8	1	no	—
148	C₂×C₃₁	—	8	2	no	—
149	C₂×C₃₁	#546	1	1	yes	characteristic
150	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₃₁	#576	1	1	yes	characteristic
152	C₂²×D₄	—	2	1	no	—
153	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
154	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×D₈	C₂²×C₃₁	1	2	yes	complemented
156	C₂×M₁₆	C₂²×C₃₁	1	2	yes	complemented
157	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
158	#127	C₂²×C₃₁	1	1	yes	characteristic
159	#124	—	2	2	no	—
160	#121	—	2	1	no	—
161	#121	—	2	1	no	—
162	#121	C₂²×C₃₁	1	2	yes	—
163	#121	C₂²×C₃₁	1	2	yes	complemented
164	#120	—	2	2	no	—
165	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
166	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
167	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
168	#117	—	2	1	no	—
169	#117	C₂²×C₃₁	1	2	yes	complemented
170	#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
171	#106	C₂²×C₃₁	1	2	yes	complemented
172	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
173	#102	C₂²×C₃₁	1	1	yes	characteristic
174	#102	C₂²×C₃₁	1	2	yes	—
175	#102	C₂²×C₃₁	1	2	yes	complemented
176	#102	C₂²×C₃₁	1	2	yes	complemented
177	#102	C₂²×C₃₁	1	1	yes	characteristic, complemented
178	#102	C₂²×C₃₁	1	2	yes	complemented
179	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
180	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
181	C₃₁	#3325	1	1	yes	characteristic, direct product factor, Sylow
182	C₂⁴	—	2	2	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
193	C₂×D₄	C₂³×C₃₁	1	2	yes	—
194	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
195	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
199	D₈	—	2	2	no	—
200	D₈	—	4	2	no	—
201	M₁₆	—	4	2	no	—
202	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
204	C₂×C₈	C₂³×C₃₁	1	2	yes	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
207	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
208	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
209	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	C₂³×C₃₁	1	1	yes	characteristic
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	8	2	no	—
224	C₂³	C₂×D₄×C₃₁	1	2	yes	—
225	Q₈	—	8	1	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	1	no	—
228	D₄	—	2	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	2	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	1	no	—
236	D₄	—	8	2	no	—
237	D₄	#41×C₃₁	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
246	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	8	8	no	—
265	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
272	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
273	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂	—	4	2	no	—
275	C₂	—	4	2	no	—
276	C₂	—	8	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₃₁×#546	1	1	yes	characteristic
279	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂	C₃₁×#576	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#555	C₂	semidirect
2. D₄²×C₃₁	C₂	semidirect
3. C₃₁×#486	C₂	semidirect
4. C₃₁×#482	C₂	semidirect
5. C₃₁×#476	C₂	semidirect
6. C₃₁×#469	C₂	semidirect
7. C₃₁×#459	C₂	semidirect
8. C₃₁×#447	C₂	semidirect
9. C₄×D₈×C₃₁	C₂	semidirect
10. C₃₁×#436	C₂	semidirect
11. C₃₁×#418	C₂	semidirect
15. C₂×D₈×C₃₁	C₂²	semidirect
16. C₂×M₁₆×C₃₁	C₂²	semidirect
17. C₃₁×C₄⋊Q₈	C₂²	semidirect
23. C₃₁×#121	C₂²	semidirect
27. C₄×D₄×C₃₁	C₂²	semidirect
29. C₃₁×#117	C₂²	semidirect
30. C₃₁×#107	C₂²	semidirect
31. C₃₁×#106	C₂²	semidirect
32. C₃₁×C₄⋊C₈	C₂²	semidirect
35. C₃₁×#102	C₂²	semidirect
36. C₃₁×#102	C₂²	semidirect
37. C₃₁×#102	C₂²	semidirect
38. C₃₁×#102	C₂²	semidirect
39. C₃₁×C₂²⋊C₈	C₂²	semidirect
40. C₄×C₈×C₃₁	C₂²	semidirect
111. #3325	C₃₁	direct (N × H)
134. #555	C₂×C₃₁	semidirect
135. D₄²	C₂×C₃₁	semidirect
136. #486	C₂×C₃₁	semidirect
137. #482	C₂×C₃₁	semidirect
138. #476	C₂×C₃₁	semidirect
139. #469	C₂×C₃₁	semidirect
140. #459	C₂×C₃₁	semidirect
141. #447	C₂×C₃₁	semidirect
142. C₄×D₈	C₂×C₃₁	semidirect
143. #436	C₂×C₃₁	semidirect
144. #418	C₂×C₃₁	semidirect
155. C₂×D₈	C₂²×C₃₁	semidirect
156. C₂×M₁₆	C₂²×C₃₁	semidirect
157. C₄⋊Q₈	C₂²×C₃₁	semidirect
163. #121	C₂²×C₃₁	semidirect
167. C₄×D₄	C₂²×C₃₁	semidirect
169. #117	C₂²×C₃₁	semidirect
170. #107	C₂²×C₃₁	semidirect
171. #106	C₂²×C₃₁	semidirect
172. C₄⋊C₈	C₂²×C₃₁	semidirect
175. #102	C₂²×C₃₁	semidirect
176. #102	C₂²×C₃₁	semidirect
177. #102	C₂²×C₃₁	semidirect
178. #102	C₂²×C₃₁	semidirect
179. C₂²⋊C₈	C₂²×C₃₁	semidirect
180. C₄×C₈	C₂²×C₃₁	semidirect
181. C₃₁	#3325	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×#555	2
D₄²×C₃₁	2
C₃₁×#486	2
C₃₁×#482	2
C₃₁×#476	2
C₃₁×#469	2
C₃₁×#459	2
C₃₁×#447	2
C₄×D₈×C₃₁	2
C₃₁×#436	2
C₃₁×#418	2
#3325	31

Subgroups — C31×#3325 #16843258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3325 #16843258