Subgroups — C₃₁×#33858 #21490617

Order 7936.

668 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3415	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3292	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2221	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2213	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2192	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#576	—	2	2	no	—
10	C₃₁×#576	C₂²	1	4	yes	complemented
11	C₃₁×#509	—	2	4	no	—
12	C₃₁×#509	C₂²	1	4	yes	complemented
13	C₃₁×#508	—	2	2	no	—
14	C₃₁×#508	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×D₁₆	—	2	2	no	—
16	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
17	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
18	C₃₁×#504	C₂²	1	2	yes	complemented
19	C₃₁×#493	C₂²	1	2	yes	complemented
20	C₃₁×#472	—	2	2	no	—
21	C₃₁×#472	C₂²	1	4	yes	complemented
22	C₃₁×#453	—	2	4	no	—
23	C₃₁×#443	C₂²	1	1	yes	characteristic
24	C₃₁×#443	C₂²	1	2	yes	complemented
25	C₃₁×#368	C₂²	1	1	yes	characteristic, complemented
26	C₃₁×#364	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#361	C₂²	1	4	yes	complemented
28	C₃₁×#359	C₂²	1	2	yes	complemented
29	C₃₁×#350	C₂²	1	2	yes	complemented
30	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×H(5,𝔽₂)	—	2	4	no	—
32	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
33	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
34	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
35	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
36	C₃₁×#135	—	2	4	no	—
37	C₃₁×#135	—	4	2	no	—
38	C₃₁×#135	D₄	1	1	yes	characteristic
39	C₃₁×#135	C₂³	1	2	yes	—
40	C₂×D₈×C₃₁	—	2	4	no	—
41	C₂×D₈×C₃₁	—	2	2	no	—
42	C₂×D₈×C₃₁	—	4	2	no	—
43	C₂×D₈×C₃₁	C₂³	1	4	yes	—
44	C₃₁×#131	—	2	2	no	—
45	C₃₁×#131	C₂³	1	2	yes	—
46	C₃₁×#127	—	2	2	no	—
47	C₃₁×Q₃₂	D₄	1	1	yes	characteristic, complemented
48	C₃₁×SD₃₂	—	2	4	no	—
49	C₃₁×SD₃₂	D₄	1	2	yes	complemented
50	C₃₁×D₁₆	—	2	4	no	—
51	C₃₁×D₁₆	—	2	2	no	—
52	C₃₁×D₁₆	—	2	2	no	—
53	C₃₁×D₁₆	—	4	2	no	—
54	C₃₁×D₁₆	D₄	1	1	yes	characteristic, complemented
55	C₃₁×M₃₂	—	2	2	no	—
56	C₃₁×M₃₂	C₂³	1	2	yes	—
57	C₂×C₁₆×C₃₁	—	2	2	no	—
58	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#108	D₄	1	1	yes	characteristic
61	C₃₁×#108	C₂³	1	2	yes	—
62	C₃₁×#104	—	2	2	no	—
63	C₃₁×#104	D₄	1	2	yes	—
64	C₃₁×#100	—	2	4	no	—
65	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	#41×C₃₁	—	2	2	no	—
67	#41×C₃₁	—	4	4	no	—
68	#41×C₃₁	—	4	4	no	—
69	#41×C₃₁	C₂×D₄	1	2	yes	—
70	C₂×D₄×C₃₁	—	2	4	no	—
71	C₂×D₄×C₃₁	—	4	4	no	—
72	C₂×D₄×C₃₁	—	4	4	no	—
73	C₂×D₄×C₃₁	—	8	2	no	—
74	C₂×D₄×C₃₁	—	8	4	no	—
75	Q₁₆×C₃₁	D₈	1	1	yes	characteristic
76	Q₁₆×C₃₁	C₂×D₄	1	2	yes	—
77	SD₁₆×C₃₁	—	2	2	no	—
78	SD₁₆×C₃₁	—	4	4	no	—
79	SD₁₆×C₃₁	—	4	4	no	—
80	SD₁₆×C₃₁	D₈	1	2	yes	—
81	D₈×C₃₁	—	2	4	no	—
82	D₈×C₃₁	—	2	4	no	—
83	D₈×C₃₁	—	4	4	no	—
84	D₈×C₃₁	—	4	4	no	—
85	D₈×C₃₁	—	4	2	no	—
86	D₈×C₃₁	—	4	2	no	—
87	D₈×C₃₁	D₈	1	1	yes	characteristic
88	D₈×C₃₁	C₂×D₄	1	2	yes	—
89	M₁₆×C₃₁	—	2	2	no	—
90	M₁₆×C₃₁	—	2	2	no	—
91	M₁₆×C₃₁	—	4	2	no	—
92	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	4	2	no	—
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₁₆×C₃₁	—	2	2	no	—
99	C₁₆×C₃₁	—	2	1	no	—
100	C₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
102	#33858	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	4	4	no	—
104	C₂³×C₃₁	—	8	4	no	—
105	Q₈×C₃₁	—	2	2	no	—
106	Q₈×C₃₁	C₂×D₈	1	2	yes	—
107	D₄×C₃₁	—	2	2	no	—
108	D₄×C₃₁	—	2	2	no	—
109	D₄×C₃₁	—	4	2	no	—
110	D₄×C₃₁	—	4	4	no	—
111	D₄×C₃₁	—	4	4	no	—
112	D₄×C₃₁	—	8	4	no	—
113	D₄×C₃₁	—	8	4	no	—
114	D₄×C₃₁	—	16	4	no	—
115	D₄×C₃₁	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	8	4	no	—
120	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₃₁	—	2	2	no	—
122	C₈×C₃₁	—	4	2	no	—
123	C₈×C₃₁	C₂×D₈	1	1	yes	characteristic
124	C₈×C₃₁	C₂×D₈	1	1	yes	characteristic
125	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
127	#3415	C₂×C₃₁	1	2	yes	complemented
128	#3292	C₂×C₃₁	1	2	yes	complemented
129	#2246	C₂×C₃₁	1	1	yes	characteristic, complemented
130	#2221	C₂×C₃₁	1	2	yes	complemented
131	#2213	C₂×C₃₁	1	2	yes	complemented
132	#2192	C₂×C₃₁	1	4	yes	complemented
133	#2178	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#2170	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂²×C₃₁	—	2	2	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	8	4	no	—
138	C₂²×C₃₁	—	16	4	no	—
139	C₂²×C₃₁	—	32	4	no	—
140	C₂²×C₃₁	#576	1	1	yes	characteristic
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₄×C₃₁	D₄²	1	1	yes	characteristic
145	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
146	#576	—	2	2	no	—
147	#576	C₂²×C₃₁	1	4	yes	complemented
148	#509	—	2	4	no	—
149	#509	C₂²×C₃₁	1	4	yes	complemented
150	#508	—	2	2	no	—
151	#508	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×D₁₆	—	2	2	no	—
153	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
154	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
155	#504	C₂²×C₃₁	1	2	yes	complemented
156	#493	C₂²×C₃₁	1	2	yes	complemented
157	#472	—	2	2	no	—
158	#472	C₂²×C₃₁	1	4	yes	complemented
159	#453	—	2	4	no	—
160	#443	C₂²×C₃₁	1	1	yes	characteristic
161	#443	C₂²×C₃₁	1	2	yes	complemented
162	#368	C₂²×C₃₁	1	1	yes	characteristic, complemented
163	#364	C₂²×C₃₁	1	1	yes	characteristic, complemented
164	#361	C₂²×C₃₁	1	4	yes	complemented
165	#359	C₂²×C₃₁	1	2	yes	complemented
166	#350	C₂²×C₃₁	1	2	yes	complemented
167	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
168	C₂×C₃₁	—	2	1	no	—
169	C₂×C₃₁	—	4	2	no	—
170	C₂×C₃₁	—	8	2	no	—
171	C₂×C₃₁	—	16	4	no	—
172	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
173	H(5,𝔽₂)	—	2	4	no	—
174	AGL(1,ℤ/8)	—	2	4	no	—
175	AGL(1,ℤ/8)	—	2	4	no	—
176	AGL(1,ℤ/8)	—	4	4	no	—
177	AGL(1,ℤ/8)	—	4	4	no	—
178	#135	—	2	4	no	—
179	#135	—	4	2	no	—
180	#135	D₄×C₃₁	1	1	yes	characteristic
181	#135	C₂³×C₃₁	1	2	yes	—
182	C₂×D₈	—	2	4	no	—
183	C₂×D₈	—	2	2	no	—
184	C₂×D₈	—	4	2	no	—
185	C₂×D₈	C₂³×C₃₁	1	4	yes	—
186	#131	—	2	2	no	—
187	#131	C₂³×C₃₁	1	2	yes	—
188	#127	—	2	2	no	—
189	Q₃₂	D₄×C₃₁	1	1	yes	characteristic, complemented
190	SD₃₂	—	2	4	no	—
191	SD₃₂	D₄×C₃₁	1	2	yes	complemented
192	D₁₆	—	2	4	no	—
193	D₁₆	—	2	2	no	—
194	D₁₆	—	2	2	no	—
195	D₁₆	—	4	2	no	—
196	D₁₆	D₄×C₃₁	1	1	yes	characteristic, complemented
197	M₃₂	—	2	2	no	—
198	M₃₂	C₂³×C₃₁	1	2	yes	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
201	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
202	#108	D₄×C₃₁	1	1	yes	characteristic
203	#108	C₂³×C₃₁	1	2	yes	—
204	#104	—	2	2	no	—
205	#104	D₄×C₃₁	1	2	yes	—
206	#100	—	2	4	no	—
207	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₃₁	#33858	1	1	yes	characteristic, direct product factor, Sylow
209	#41	—	2	2	no	—
210	#41	—	4	4	no	—
211	#41	—	4	4	no	—
212	#41	C₂×D₄×C₃₁	1	2	yes	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	—	8	2	no	—
217	C₂×D₄	—	8	4	no	—
218	Q₁₆	D₈×C₃₁	1	1	yes	characteristic
219	Q₁₆	C₂×D₄×C₃₁	1	2	yes	—
220	SD₁₆	—	2	2	no	—
221	SD₁₆	—	4	4	no	—
222	SD₁₆	—	4	4	no	—
223	SD₁₆	D₈×C₃₁	1	2	yes	—
224	D₈	—	2	4	no	—
225	D₈	—	2	4	no	—
226	D₈	—	4	4	no	—
227	D₈	—	4	4	no	—
228	D₈	—	4	2	no	—
229	D₈	—	4	2	no	—
230	D₈	D₈×C₃₁	1	1	yes	characteristic
231	D₈	C₂×D₄×C₃₁	1	2	yes	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	1	no	—
243	C₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂³	—	4	4	no	—
246	C₂³	—	8	4	no	—
247	Q₈	—	2	2	no	—
248	Q₈	C₂×D₈×C₃₁	1	2	yes	—
249	D₄	—	2	2	no	—
250	D₄	—	2	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	8	4	no	—
255	D₄	—	8	4	no	—
256	D₄	—	16	4	no	—
257	D₄	C₂×D₈×C₃₁	1	2	yes	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	C₂×D₈×C₃₁	1	1	yes	characteristic
266	C₈	C₂×D₈×C₃₁	1	1	yes	characteristic
267	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
268	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	16	4	no	—
273	C₂²	—	32	4	no	—
274	C₂²	C₃₁×#576	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	D₄²×C₃₁	1	1	yes	characteristic
279	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	4	2	no	—
282	C₂	—	8	2	no	—
283	C₂	—	16	4	no	—
284	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3415	C₂	semidirect
2. C₃₁×#3292	C₂	semidirect
3. C₃₁×#2246	C₂	semidirect
4. C₃₁×#2221	C₂	semidirect
5. C₃₁×#2213	C₂	semidirect
6. C₃₁×#2192	C₂	semidirect
7. C₃₁×#2178	C₂	semidirect
8. C₃₁×#2170	C₂	semidirect
10. C₃₁×#576	C₂²	semidirect
12. C₃₁×#509	C₂²	semidirect
16. C₂×C₃₁×D₁₆	C₂²	semidirect
17. C₂×C₃₁×D₁₆	C₂²	semidirect
18. C₃₁×#504	C₂²	semidirect
19. C₃₁×#493	C₂²	semidirect
21. C₃₁×#472	C₂²	semidirect
24. C₃₁×#443	C₂²	semidirect
25. C₃₁×#368	C₂²	semidirect
26. C₃₁×#364	C₂²	semidirect
27. C₃₁×#361	C₂²	semidirect
28. C₃₁×#359	C₂²	semidirect
29. C₃₁×#350	C₂²	semidirect
30. C₄×C₁₆×C₃₁	C₂²	semidirect
47. C₃₁×Q₃₂	D₄	semidirect
49. C₃₁×SD₃₂	D₄	semidirect
54. C₃₁×D₁₆	D₄	semidirect
102. #33858	C₃₁	direct (N × H)
127. #3415	C₂×C₃₁	semidirect
128. #3292	C₂×C₃₁	semidirect
129. #2246	C₂×C₃₁	semidirect
130. #2221	C₂×C₃₁	semidirect
131. #2213	C₂×C₃₁	semidirect
132. #2192	C₂×C₃₁	semidirect
133. #2178	C₂×C₃₁	semidirect
134. #2170	C₂×C₃₁	semidirect
147. #576	C₂²×C₃₁	semidirect
149. #509	C₂²×C₃₁	semidirect
153. C₂×D₁₆	C₂²×C₃₁	semidirect
154. C₂×D₁₆	C₂²×C₃₁	semidirect
155. #504	C₂²×C₃₁	semidirect
156. #493	C₂²×C₃₁	semidirect
158. #472	C₂²×C₃₁	semidirect
161. #443	C₂²×C₃₁	semidirect
162. #368	C₂²×C₃₁	semidirect
163. #364	C₂²×C₃₁	semidirect
164. #361	C₂²×C₃₁	semidirect
165. #359	C₂²×C₃₁	semidirect
166. #350	C₂²×C₃₁	semidirect
167. C₄×C₁₆	C₂²×C₃₁	semidirect
189. Q₃₂	D₄×C₃₁	semidirect
191. SD₃₂	D₄×C₃₁	semidirect
196. D₁₆	D₄×C₃₁	semidirect
208. C₃₁	#33858	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×#3415	2
C₃₁×#3292	2
C₃₁×#2246	2
C₃₁×#2221	2
C₃₁×#2213	2
C₃₁×#2192	2
C₃₁×#2178	2
C₃₁×#2170	2
#33858	31

Subgroups — C31×#33858 #21490617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33858 #21490617