Subgroups — C₃₁×#3220 #16843153

Order 3968.

420 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#463	C₂	1	2	yes	maximal, complemented
8	C₃₁×#459	C₂	1	2	yes	maximal, complemented
9	C₃₁×#449	C₂	1	2	yes	maximal, complemented
10	C₃₁×#447	C₂	1	2	yes	maximal, complemented
11	C₃₁×#433	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×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
15	C₂×D₈×C₃₁	C₂²	1	2	yes	—
16	C₂×D₈×C₃₁	C₂²	1	2	yes	—
17	C₂×D₈×C₃₁	C₂²	1	2	yes	complemented
18	C₃₁×#127	C₂²	1	1	yes	characteristic
19	C₃₁×#127	C₂²	1	1	yes	characteristic
20	C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#124	C₂²	1	1	yes	characteristic
23	C₃₁×#124	C₂²	1	1	yes	characteristic
24	C₃₁×#123	—	2	2	no	—
25	C₃₁×#121	—	2	2	no	—
26	C₃₁×#121	—	2	2	no	—
27	C₃₁×#121	C₂²	1	2	yes	—
28	C₃₁×#120	—	2	2	no	—
29	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
30	C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
32	C₃₁×#103	C₂²	1	2	yes	complemented
33	C₃₁×#102	C₂²	1	2	yes	—
34	C₃₁×#102	C₂²	1	2	yes	—
35	C₃₁×#102	C₂²	1	2	yes	complemented
36	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₃₁	—	2	1	no	—
40	#41×C₃₁	—	4	2	no	—
41	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×D₄×C₃₁	—	2	1	no	—
43	C₂×D₄×C₃₁	—	2	2	no	—
44	C₂×D₄×C₃₁	—	2	1	no	—
45	C₂×D₄×C₃₁	—	2	2	no	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	2	2	no	—
48	C₂×D₄×C₃₁	—	4	2	no	—
49	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	SD₁₆×C₃₁	—	4	2	no	—
57	D₈×C₃₁	—	4	2	no	—
58	D₈×C₃₁	—	4	2	no	—
59	D₈×C₃₁	—	4	2	no	—
60	C₂×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂×C₈×C₃₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
64	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
70	C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	2	1	no	—
74	C₂³×C₃₁	—	2	1	no	—
75	C₂³×C₃₁	—	2	1	no	—
76	C₂³×C₃₁	—	4	2	no	—
77	C₂³×C₃₁	—	8	2	no	—
78	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
79	Q₈×C₃₁	—	2	2	no	—
80	D₄×C₃₁	—	2	2	no	—
81	D₄×C₃₁	—	2	2	no	—
82	D₄×C₃₁	—	2	2	no	—
83	D₄×C₃₁	—	2	2	no	—
84	D₄×C₃₁	—	2	2	no	—
85	D₄×C₃₁	—	4	2	no	—
86	D₄×C₃₁	—	4	2	no	—
87	D₄×C₃₁	—	8	2	no	—
88	D₄×C₃₁	—	8	2	no	—
89	D₄×C₃₁	—	8	2	no	—
90	D₄×C₃₁	—	8	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	8	2	no	—
96	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₈×C₃₁	—	2	2	no	—
101	C₈×C₃₁	—	2	2	no	—
102	#3220	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₃₁	—	2	1	no	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	4	1	no	—
106	C₂²×C₃₁	—	4	1	no	—
107	C₂²×C₃₁	—	4	1	no	—
108	C₂²×C₃₁	—	4	2	no	—
109	C₂²×C₃₁	—	4	2	no	—
110	C₂²×C₃₁	—	4	2	no	—
111	C₂²×C₃₁	—	4	2	no	—
112	C₂²×C₃₁	—	4	2	no	—
113	C₂²×C₃₁	—	4	2	no	—
114	C₂²×C₃₁	—	8	4	no	—
115	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	2	2	no	—
118	C₄×C₃₁	—	2	2	no	—
119	C₄×C₃₁	—	4	1	no	—
120	C₄×C₃₁	—	4	2	no	—
121	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
122	#540	C₂×C₃₁	1	1	yes	characteristic, complemented
123	#535	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#496	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#493	C₂×C₃₁	1	1	yes	characteristic, complemented
126	#490	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#488	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#463	C₂×C₃₁	1	2	yes	complemented
129	#459	C₂×C₃₁	1	2	yes	complemented
130	#449	C₂×C₃₁	1	2	yes	complemented
131	#447	C₂×C₃₁	1	2	yes	complemented
132	#433	C₂×C₃₁	1	1	yes	characteristic, complemented
133	C₂×C₃₁	—	4	1	no	—
134	C₂×C₃₁	—	8	1	no	—
135	C₂×C₃₁	—	8	2	no	—
136	C₂×C₃₁	—	8	1	no	—
137	C₂×C₃₁	—	8	1	no	—
138	C₂×C₃₁	#535	1	1	yes	characteristic
139	C₂×C₃₁	#576	1	2	yes	—
140	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
141	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
143	C₂×D₈	C₂²×C₃₁	1	2	yes	—
144	C₂×D₈	C₂²×C₃₁	1	2	yes	—
145	C₂×D₈	C₂²×C₃₁	1	2	yes	complemented
146	#127	C₂²×C₃₁	1	1	yes	characteristic
147	#127	C₂²×C₃₁	1	1	yes	characteristic
148	#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
149	#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
150	#124	C₂²×C₃₁	1	1	yes	characteristic
151	#124	C₂²×C₃₁	1	1	yes	characteristic
152	#123	—	2	2	no	—
153	#121	—	2	2	no	—
154	#121	—	2	2	no	—
155	#121	C₂²×C₃₁	1	2	yes	—
156	#120	—	2	2	no	—
157	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
158	#117	C₂²×C₃₁	1	1	yes	characteristic
159	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
160	#103	C₂²×C₃₁	1	2	yes	complemented
161	#102	C₂²×C₃₁	1	2	yes	—
162	#102	C₂²×C₃₁	1	2	yes	—
163	#102	C₂²×C₃₁	1	2	yes	complemented
164	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
165	#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
166	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
167	C₃₁	#3220	1	1	yes	characteristic, direct product factor, Sylow
168	C₂⁴	—	2	1	no	—
169	#41	—	4	2	no	—
170	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
180	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
181	C₂×D₄	C₂³×C₃₁	1	2	yes	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
185	SD₁₆	—	4	2	no	—
186	D₈	—	4	2	no	—
187	D₈	—	4	2	no	—
188	D₈	—	4	2	no	—
189	C₂×C₈	C₂³×C₃₁	1	2	yes	—
190	C₂×C₈	C₂³×C₃₁	1	2	yes	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
193	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
199	C₄²	C₂³×C₃₁	1	1	yes	characteristic
200	C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	C₂³	—	2	2	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	8	2	no	—
207	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
208	Q₈	—	2	2	no	—
209	D₄	—	2	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	2	no	—
212	D₄	—	2	2	no	—
213	D₄	—	2	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	D₄	—	8	2	no	—
219	D₄	—	8	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
227	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	8	4	no	—
243	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	2	no	—
249	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
250	C₂	—	4	1	no	—
251	C₂	—	8	1	no	—
252	C₂	—	8	1	no	—
253	C₂	—	8	1	no	—
254	C₂	—	8	2	no	—
255	C₂	C₃₁×#535	1	1	yes	characteristic
256	C₂	C₃₁×#576	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 ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#540	C₂	semidirect
2. C₃₁×#535	C₂	semidirect
3. C₃₁×#496	C₂	semidirect
4. C₃₁×#493	C₂	semidirect
5. C₃₁×#490	C₂	semidirect
6. C₃₁×#488	C₂	semidirect
7. C₃₁×#463	C₂	semidirect
8. C₃₁×#459	C₂	semidirect
9. C₃₁×#449	C₂	semidirect
10. C₃₁×#447	C₂	semidirect
11. C₃₁×#433	C₂	semidirect
14. C₂×SD₁₆×C₃₁	C₂²	semidirect
17. C₂×D₈×C₃₁	C₂²	semidirect
20. C₃₁×#127	C₂²	semidirect
21. C₃₁×#125	C₂²	semidirect
29. C₄×D₄×C₃₁	C₂²	semidirect
31. C₃₁×C₄⋊C₈	C₂²	semidirect
32. C₃₁×#103	C₂²	semidirect
35. C₃₁×#102	C₂²	semidirect
37. C₃₁×#97	C₂²	semidirect
38. C₄×C₈×C₃₁	C₂²	semidirect
102. #3220	C₃₁	direct (N × H)
122. #540	C₂×C₃₁	semidirect
123. #535	C₂×C₃₁	semidirect
124. #496	C₂×C₃₁	semidirect
125. #493	C₂×C₃₁	semidirect
126. #490	C₂×C₃₁	semidirect
127. #488	C₂×C₃₁	semidirect
128. #463	C₂×C₃₁	semidirect
129. #459	C₂×C₃₁	semidirect
130. #449	C₂×C₃₁	semidirect
131. #447	C₂×C₃₁	semidirect
132. #433	C₂×C₃₁	semidirect
142. C₂×SD₁₆	C₂²×C₃₁	semidirect
145. C₂×D₈	C₂²×C₃₁	semidirect
148. #127	C₂²×C₃₁	semidirect
149. #125	C₂²×C₃₁	semidirect
157. C₄×D₄	C₂²×C₃₁	semidirect
159. C₄⋊C₈	C₂²×C₃₁	semidirect
160. #103	C₂²×C₃₁	semidirect
163. #102	C₂²×C₃₁	semidirect
165. #97	C₂²×C₃₁	semidirect
166. C₄×C₈	C₂²×C₃₁	semidirect
167. C₃₁	#3220	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×#540	2
C₃₁×#535	2
C₃₁×#496	2
C₃₁×#493	2
C₃₁×#490	2
C₃₁×#488	2
C₃₁×#463	2
C₃₁×#459	2
C₃₁×#449	2
C₃₁×#447	2
C₃₁×#433	2
#3220	31

Subgroups — C31×#3220 #16843153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3220 #16843153