Subgroups — C₃₁×#3136 #16843069

Order 3968.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#535	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#490	C₂	1	2	yes	maximal, complemented
4	C₃₁×#489	C₂	1	2	yes	maximal, complemented
5	C₃₁×#488	C₂	1	2	yes	maximal, complemented
6	C₃₁×#486	C₂	1	2	yes	maximal, complemented
7	C₃₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊Q₈	—	2	1	no	—
17	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
18	C₃₁×#127	C₂²	1	2	yes	—
19	C₃₁×#125	—	2	1	no	—
20	C₃₁×#125	C₂²	1	2	yes	complemented
21	C₃₁×#124	C₂²	1	2	yes	—
22	C₃₁×C₂²⋊Q₈	—	2	2	no	—
23	C₃₁×#121	—	2	1	no	—
24	C₃₁×#121	—	2	1	no	—
25	C₃₁×#120	—	2	2	no	—
26	C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₃₁×#117	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#103	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#103	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#103	C₂²	1	2	yes	complemented
32	C₃₁×#102	C₂²	1	2	yes	—
33	C₃₁×#102	C₂²	1	2	yes	—
34	C₃₁×#102	C₂²	1	2	yes	—
35	C₃₁×#102	C₂²	1	2	yes	—
36	C₃₁×#102	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#102	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#102	C₂²	1	2	yes	complemented
39	C₃₁×#97	C₂²	1	2	yes	complemented
40	C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
41	C₂⁴×C₃₁	—	2	1	no	—
42	#41×C₃₁	—	2	2	no	—
43	#41×C₃₁	—	2	2	no	—
44	C₂×Q₈×C₃₁	—	2	2	no	—
45	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	C₂³	1	2	yes	—
53	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	M₁₆×C₃₁	—	2	2	no	—
58	C₂×C₈×C₃₁	—	2	2	no	—
59	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
69	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
75	C₄²×C₃₁	—	2	1	no	—
76	C₄²×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₃₁	—	2	2	no	—
78	C₂³×C₃₁	—	2	1	no	—
79	C₂³×C₃₁	—	4	2	no	—
80	C₂³×C₃₁	—	8	1	no	—
81	C₂³×C₃₁	—	8	1	no	—
82	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₃₁	—	2	1	no	—
84	Q₈×C₃₁	—	2	1	no	—
85	Q₈×C₃₁	—	8	2	no	—
86	D₄×C₃₁	—	2	2	no	—
87	D₄×C₃₁	—	2	2	no	—
88	D₄×C₃₁	—	2	1	no	—
89	D₄×C₃₁	—	2	1	no	—
90	D₄×C₃₁	—	4	2	no	—
91	D₄×C₃₁	—	4	2	no	—
92	D₄×C₃₁	—	8	2	no	—
93	D₄×C₃₁	—	8	2	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	8	1	no	—
100	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#41	1	2	yes	—
104	C₂×C₄×C₃₁	#41	1	2	yes	—
105	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₈×C₃₁	—	2	1	no	—
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	2	2	no	—
109	#3136	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₃₁	—	2	1	no	—
111	C₂²×C₃₁	—	2	1	no	—
112	C₂²×C₃₁	—	4	1	no	—
113	C₂²×C₃₁	—	4	2	no	—
114	C₂²×C₃₁	—	4	1	no	—
115	C₂²×C₃₁	—	4	2	no	—
116	C₂²×C₃₁	—	4	1	no	—
117	C₂²×C₃₁	—	4	2	no	—
118	C₂²×C₃₁	—	8	4	no	—
119	C₂²×C₃₁	#124	1	2	yes	—
120	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₃₁	—	2	2	no	—
122	C₄×C₃₁	—	2	2	no	—
123	C₄×C₃₁	—	4	1	no	—
124	C₄×C₃₁	—	4	2	no	—
125	C₄×C₃₁	—	4	1	no	—
126	C₄×C₃₁	—	4	1	no	—
127	C₄×C₃₁	#124	1	2	yes	—
128	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
129	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
130	#544	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#535	C₂×C₃₁	1	1	yes	characteristic
132	#490	C₂×C₃₁	1	2	yes	complemented
133	#489	C₂×C₃₁	1	2	yes	complemented
134	#488	C₂×C₃₁	1	2	yes	complemented
135	#486	C₂×C₃₁	1	2	yes	complemented
136	#418	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#417	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#416	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#405	C₂×C₃₁	1	1	yes	characteristic, complemented
141	C₂×C₃₁	—	2	2	no	—
142	C₂×C₃₁	—	8	1	no	—
143	C₂×C₃₁	—	8	2	no	—
144	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
145	C₂×C₃₁	#576	1	1	yes	characteristic
146	C₂×C₃₁	#577	1	1	yes	characteristic
147	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
148	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
150	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
151	C₄⋊Q₈	—	2	1	no	—
152	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
153	#127	C₂²×C₃₁	1	2	yes	—
154	#125	—	2	1	no	—
155	#125	C₂²×C₃₁	1	2	yes	complemented
156	#124	C₂²×C₃₁	1	2	yes	—
157	C₂²⋊Q₈	—	2	2	no	—
158	#121	—	2	1	no	—
159	#121	—	2	1	no	—
160	#120	—	2	2	no	—
161	#117	C₂²×C₃₁	1	1	yes	characteristic
162	#117	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
164	#103	C₂²×C₃₁	1	1	yes	characteristic, complemented
165	#103	C₂²×C₃₁	1	1	yes	characteristic, complemented
166	#103	C₂²×C₃₁	1	2	yes	complemented
167	#102	C₂²×C₃₁	1	2	yes	—
168	#102	C₂²×C₃₁	1	2	yes	—
169	#102	C₂²×C₃₁	1	2	yes	—
170	#102	C₂²×C₃₁	1	2	yes	—
171	#102	C₂²×C₃₁	1	1	yes	characteristic, complemented
172	#102	C₂²×C₃₁	1	1	yes	characteristic, complemented
173	#102	C₂²×C₃₁	1	2	yes	complemented
174	#97	C₂²×C₃₁	1	2	yes	complemented
175	C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
176	C₃₁	#3136	1	1	yes	characteristic, direct product factor, Sylow
177	C₂⁴	—	2	1	no	—
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂³×C₃₁	1	2	yes	—
189	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	M₁₆	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
196	C₂×C₈	C₂³×C₃₁	1	2	yes	—
197	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
203	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
204	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
205	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
206	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
211	C₄²	—	2	1	no	—
212	C₄²	C₂³×C₃₁	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	4	2	no	—
216	C₂³	—	8	1	no	—
217	C₂³	—	8	1	no	—
218	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
219	Q₈	—	2	1	no	—
220	Q₈	—	2	1	no	—
221	Q₈	—	8	2	no	—
222	D₄	—	2	2	no	—
223	D₄	—	2	2	no	—
224	D₄	—	2	1	no	—
225	D₄	—	2	1	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	2	no	—
229	D₄	—	8	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
237	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
238	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	#41×C₃₁	1	2	yes	—
240	C₂×C₄	#41×C₃₁	1	2	yes	—
241	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	8	4	no	—
254	C₂²	C₃₁×#124	1	2	yes	—
255	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	C₃₁×#124	1	2	yes	—
263	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
264	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	8	2	no	—
267	C₂	—	8	1	no	—
268	C₂	C₂×C₃₁×#124	1	1	yes	characteristic
269	C₂	C₃₁×#576	1	1	yes	characteristic
270	C₂	C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#544	C₂	semidirect
3. C₃₁×#490	C₂	semidirect
4. C₃₁×#489	C₂	semidirect
5. C₃₁×#488	C₂	semidirect
6. C₃₁×#486	C₂	semidirect
7. C₃₁×#418	C₂	semidirect
8. C₃₁×#417	C₂	semidirect
9. C₃₁×#416	C₂	semidirect
10. C₂×C₃₁×#102	C₂	semidirect
11. C₃₁×#405	C₂	semidirect
17. C₃₁×C₄⋊Q₈	C₂²	semidirect
20. C₃₁×#125	C₂²	semidirect
29. C₃₁×#103	C₂²	semidirect
30. C₃₁×#103	C₂²	semidirect
31. C₃₁×#103	C₂²	semidirect
36. C₃₁×#102	C₂²	semidirect
37. C₃₁×#102	C₂²	semidirect
38. C₃₁×#102	C₂²	semidirect
39. C₃₁×#97	C₂²	semidirect
40. C₄×C₈×C₃₁	C₂²	semidirect
109. #3136	C₃₁	direct (N × H)
130. #544	C₂×C₃₁	semidirect
132. #490	C₂×C₃₁	semidirect
133. #489	C₂×C₃₁	semidirect
134. #488	C₂×C₃₁	semidirect
135. #486	C₂×C₃₁	semidirect
136. #418	C₂×C₃₁	semidirect
137. #417	C₂×C₃₁	semidirect
138. #416	C₂×C₃₁	semidirect
139. C₂×#102	C₂×C₃₁	semidirect
140. #405	C₂×C₃₁	semidirect
152. C₄⋊Q₈	C₂²×C₃₁	semidirect
155. #125	C₂²×C₃₁	semidirect
164. #103	C₂²×C₃₁	semidirect
165. #103	C₂²×C₃₁	semidirect
166. #103	C₂²×C₃₁	semidirect
171. #102	C₂²×C₃₁	semidirect
172. #102	C₂²×C₃₁	semidirect
173. #102	C₂²×C₃₁	semidirect
174. #97	C₂²×C₃₁	semidirect
175. C₄×C₈	C₂²×C₃₁	semidirect
176. C₃₁	#3136	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×#544	2
C₃₁×#535	2
C₃₁×#490	2
C₃₁×#489	2
C₃₁×#488	2
C₃₁×#486	2
C₃₁×#418	2
C₃₁×#417	2
C₃₁×#416	2
C₂×C₃₁×#102	2
C₃₁×#405	2
#3136	31

Subgroups — C31×#3136 #16843069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3136 #16843069