Subgroups — C₃₁×#3010 #16842943

Order 3968.

732 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#449	C₂	1	4	yes	maximal, complemented
6	C₃₁×#448	C₂	1	4	yes	maximal, complemented
7	C₃₁×#419	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	C₃₁×2₋¹⁺⁴	—	2	4	no	—
11	C₃₁×2₋¹⁺⁴	—	2	4	no	—
12	C₂×#41×C₃₁	—	2	2	no	—
13	C₂×#41×C₃₁	—	2	1	no	—
14	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
17	C₂²×Q₈×C₃₁	—	2	1	no	—
18	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₃₁	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
22	C₂×SD₁₆×C₃₁	—	2	4	no	—
23	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
24	C₂×SD₁₆×C₃₁	C₂²	1	4	yes	—
25	C₂×D₈×C₃₁	C₂²	1	2	yes	—
26	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#127	—	2	1	no	—
29	C₃₁×#124	—	2	1	no	—
30	C₃₁×#121	C₂²	1	4	yes	—
31	C₃₁×#120	—	2	2	no	—
32	C₃₁×#117	C₂²	1	1	yes	characteristic
33	C₃₁×#103	C₂²	1	2	yes	complemented
34	C₃₁×#103	C₂²	1	4	yes	complemented
35	C₃₁×#102	C₂²	1	2	yes	complemented
36	C₃₁×C₂²⋊C₈	C₂²	1	4	yes	complemented
37	C₂⁴×C₃₁	—	2	1	no	—
38	#41×C₃₁	—	2	16	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	complemented
46	C₂×Q₈×C₃₁	—	2	8	no	—
47	C₂×Q₈×C₃₁	—	2	4	no	—
48	C₂×Q₈×C₃₁	—	2	4	no	—
49	C₂×Q₈×C₃₁	—	2	1	no	—
50	C₂×Q₈×C₃₁	—	2	2	no	—
51	C₂×Q₈×C₃₁	—	2	2	no	—
52	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂×Q₈×C₃₁	D₄	1	4	yes	complemented
54	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
55	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₃₁	—	2	1	no	—
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	4	no	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	—	2	1	no	—
62	C₂×D₄×C₃₁	D₄	1	2	yes	—
63	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	1	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	SD₁₆×C₃₁	—	2	8	no	—
73	SD₁₆×C₃₁	—	2	4	no	—
74	D₈×C₃₁	—	2	4	no	—
75	M₁₆×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	C₂³	1	2	yes	—
78	C₂×C₈×C₃₁	C₂³	1	2	yes	—
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	1	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	4	2	no	—
86	C₂³×C₃₁	—	8	2	no	—
87	C₂³×C₃₁	C₂×D₄	1	2	yes	—
88	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₃₁	—	2	8	no	—
90	Q₈×C₃₁	—	2	2	no	—
91	Q₈×C₃₁	—	2	4	no	—
92	Q₈×C₃₁	—	2	1	no	—
93	Q₈×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	C₂×D₄	1	2	yes	—
95	Q₈×C₃₁	C₂×D₄	1	4	yes	—
96	D₄×C₃₁	—	2	8	no	—
97	D₄×C₃₁	—	2	4	no	—
98	D₄×C₃₁	—	2	4	no	—
99	D₄×C₃₁	—	2	2	no	—
100	D₄×C₃₁	—	2	1	no	—
101	D₄×C₃₁	—	2	2	no	—
102	D₄×C₃₁	—	2	2	no	—
103	D₄×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	8	4	no	—
105	D₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₃₁	—	2	8	no	—
107	C₂×C₄×C₃₁	—	2	8	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	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	#3010	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	2	no	—
126	C₂²×C₃₁	—	2	4	no	—
127	C₂²×C₃₁	—	4	2	no	—
128	C₂²×C₃₁	—	4	2	no	—
129	C₂²×C₃₁	—	4	4	no	—
130	C₂²×C₃₁	—	8	4	no	—
131	C₂²×C₃₁	#120	1	2	yes	—
132	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₃₁	—	2	4	no	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	#120	1	2	yes	—
138	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×2₋¹⁺⁴	C₂×C₃₁	1	1	yes	characteristic, complemented
141	C₂×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂²×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#535	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#449	C₂×C₃₁	1	4	yes	complemented
145	#448	C₂×C₃₁	1	4	yes	complemented
146	#419	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#416	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#408	C₂×C₃₁	1	1	yes	characteristic, complemented
149	C₂×C₃₁	—	2	2	no	—
150	C₂×C₃₁	—	4	2	no	—
151	C₂×C₃₁	—	4	2	no	—
152	C₂×C₃₁	—	8	2	no	—
153	C₂×C₃₁	C₂×#120	1	1	yes	characteristic
154	C₂×C₃₁	#577	1	2	yes	—
155	2₋¹⁺⁴	—	2	4	no	—
156	2₋¹⁺⁴	—	2	4	no	—
157	C₂×#41	—	2	2	no	—
158	C₂×#41	—	2	1	no	—
159	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
164	C₂²×Q₈	C₂²×C₃₁	1	2	yes	complemented
165	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
166	AGL(1,ℤ/8)	—	2	4	no	—
167	C₂×SD₁₆	—	2	4	no	—
168	C₂×SD₁₆	C₂²×C₃₁	1	4	yes	—
169	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
170	C₂×D₈	C₂²×C₃₁	1	2	yes	—
171	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
172	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
173	#127	—	2	1	no	—
174	#124	—	2	1	no	—
175	#121	C₂²×C₃₁	1	4	yes	—
176	#120	—	2	2	no	—
177	#117	C₂²×C₃₁	1	1	yes	characteristic
178	#103	C₂²×C₃₁	1	2	yes	complemented
179	#103	C₂²×C₃₁	1	4	yes	complemented
180	#102	C₂²×C₃₁	1	2	yes	complemented
181	C₂²⋊C₈	C₂²×C₃₁	1	4	yes	complemented
182	C₃₁	#3010	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	—	2	1	no	—
184	#41	—	2	16	no	—
185	#41	—	2	8	no	—
186	#41	—	2	4	no	—
187	#41	—	2	4	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	D₄×C₃₁	1	4	yes	complemented
192	C₂×Q₈	—	2	8	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
199	C₂×Q₈	D₄×C₃₁	1	4	yes	complemented
200	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
201	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	D₄×C₃₁	1	2	yes	—
209	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
210	C₂×D₄	C₂³×C₃₁	1	2	yes	—
211	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
218	SD₁₆	—	2	8	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	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	C₂³×C₃₁	1	2	yes	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	8	2	no	—
233	C₂³	C₂×D₄×C₃₁	1	2	yes	—
234	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
235	Q₈	—	2	8	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	2	4	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	2	2	no	—
240	Q₈	C₂×D₄×C₃₁	1	2	yes	—
241	Q₈	C₂×D₄×C₃₁	1	4	yes	—
242	D₄	—	2	8	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	4	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	4	no	—
251	D₄	C₂×D₄×C₃₁	1	2	yes	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
262	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
264	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
265	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	C₃₁×#120	1	2	yes	—
277	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	C₃₁×#120	1	2	yes	—
283	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
284	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	—	4	2	no	—
288	C₂	—	8	2	no	—
289	C₂	C₂×C₃₁×#120	1	1	yes	characteristic
290	C₂	C₃₁×#577	1	2	yes	—

	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 ◁ C₂×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×2₋¹⁺⁴	C₂	semidirect
2. C₂×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
3. C₂²×SD₁₆×C₃₁	C₂	semidirect
4. C₃₁×#535	C₂	semidirect
5. C₃₁×#449	C₂	semidirect
6. C₃₁×#448	C₂	semidirect
7. C₃₁×#419	C₂	semidirect
8. C₃₁×#416	C₂	semidirect
9. C₃₁×#408	C₂	semidirect
16. C₂×#41×C₃₁	C₂²	semidirect
19. C₂²×Q₈×C₃₁	C₂²	semidirect
33. C₃₁×#103	C₂²	semidirect
34. C₃₁×#103	C₂²	semidirect
35. C₃₁×#102	C₂²	semidirect
36. C₃₁×C₂²⋊C₈	C₂²	semidirect
45. #41×C₃₁	D₄	semidirect
53. C₂×Q₈×C₃₁	D₄	semidirect
122. #3010	C₃₁	direct (N × H)
140. C₂×2₋¹⁺⁴	C₂×C₃₁	semidirect
141. C₂×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
142. C₂²×SD₁₆	C₂×C₃₁	semidirect
143. #535	C₂×C₃₁	semidirect
144. #449	C₂×C₃₁	semidirect
145. #448	C₂×C₃₁	semidirect
146. #419	C₂×C₃₁	semidirect
147. #416	C₂×C₃₁	semidirect
148. #408	C₂×C₃₁	semidirect
161. C₂×#41	C₂²×C₃₁	semidirect
164. C₂²×Q₈	C₂²×C₃₁	semidirect
178. #103	C₂²×C₃₁	semidirect
179. #103	C₂²×C₃₁	semidirect
180. #102	C₂²×C₃₁	semidirect
181. C₂²⋊C₈	C₂²×C₃₁	semidirect
182. C₃₁	#3010	direct (N × H)
191. #41	D₄×C₃₁	semidirect
199. C₂×Q₈	D₄×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×2₋¹⁺⁴	2
C₂×C₃₁×AGL(1,ℤ/8)	2
C₂²×SD₁₆×C₃₁	2
C₃₁×#535	2
C₃₁×#449	2
C₃₁×#448	2
C₃₁×#419	2
C₃₁×#416	2
C₃₁×#408	2
#3010	31

Subgroups — C31×#3010 #16842943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3010 #16842943