Subgroups — C₃₁×#3212 #16843145

Order 3968.

400 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#495	C₂	1	2	yes	maximal, complemented
4	C₃₁×#488	C₂	1	2	yes	maximal, complemented
5	C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#452	C₂	1	2	yes	maximal, complemented
10	C₃₁×#449	C₂	1	2	yes	maximal, complemented
11	C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂×D₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×#127	C₂²	1	1	yes	characteristic
19	C₃₁×#126	—	2	1	no	—
20	C₃₁×#125	C₂²	1	1	yes	characteristic
21	C₃₁×#124	C₂²	1	2	yes	—
22	C₃₁×#124	C₂²	1	1	yes	characteristic
23	C₃₁×#123	—	2	1	no	—
24	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×#121	—	2	1	no	—
26	C₃₁×#121	—	2	1	no	—
27	C₃₁×#121	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₃₁	—	2	1	no	—
30	C₄×D₄×C₃₁	—	2	1	no	—
31	C₄×D₄×C₃₁	—	2	1	no	—
32	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#117	—	2	1	no	—
34	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#103	C₂²	1	2	yes	—
38	C₃₁×#103	C₂²	1	2	yes	—
39	C₃₁×#102	C₂²	1	2	yes	—
40	C₃₁×#102	C₂²	1	2	yes	—
41	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₄×C₈×C₃₁	C₂²	1	2	yes	—
43	#41×C₃₁	—	4	2	no	—
44	#41×C₃₁	—	4	2	no	—
45	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	—	2	1	no	—
50	C₂×D₄×C₃₁	—	2	1	no	—
51	C₂×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	Q₁₆×C₃₁	—	4	2	no	—
59	SD₁₆×C₃₁	—	4	2	no	—
60	SD₁₆×C₃₁	—	4	2	no	—
61	D₈×C₃₁	—	4	2	no	—
62	C₂×C₈×C₃₁	C₂³	1	2	yes	—
63	C₂×C₈×C₃₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂²⋊C₄×C₃₁	—	2	1	no	—
79	C₄²×C₃₁	—	2	1	no	—
80	C₄²×C₃₁	D₄	1	1	yes	characteristic
81	C₄²×C₃₁	D₄	1	1	yes	characteristic
82	C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂³×C₃₁	—	2	1	no	—
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₃₁	—	2	1	no	—
88	Q₈×C₃₁	—	2	2	no	—
89	Q₈×C₃₁	—	2	1	no	—
90	Q₈×C₃₁	—	2	2	no	—
91	D₄×C₃₁	—	2	1	no	—
92	D₄×C₃₁	—	2	2	no	—
93	D₄×C₃₁	—	2	1	no	—
94	D₄×C₃₁	—	2	2	no	—
95	D₄×C₃₁	—	4	2	no	—
96	D₄×C₃₁	—	4	2	no	—
97	D₄×C₃₁	—	4	4	no	—
98	D₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	4	1	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₈×C₃₁	—	2	2	no	—
116	C₈×C₃₁	—	2	2	no	—
117	#3212	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₃₁	—	2	1	no	—
119	C₂²×C₃₁	—	2	1	no	—
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	—	4	1	no	—
123	C₂²×C₃₁	—	4	1	no	—
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	—	4	2	no	—
126	C₂²×C₃₁	—	4	1	no	—
127	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	2	1	no	—
132	C₄×C₃₁	—	4	2	no	—
133	C₄×C₃₁	—	4	1	no	—
134	C₄×C₃₁	—	4	1	no	—
135	C₄×C₃₁	—	4	1	no	—
136	C₄×C₃₁	#135	1	2	yes	—
137	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
139	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#495	C₂×C₃₁	1	2	yes	complemented
142	#488	C₂×C₃₁	1	2	yes	complemented
143	#462	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#461	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#460	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#459	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#452	C₂×C₃₁	1	2	yes	complemented
148	#449	C₂×C₃₁	1	2	yes	complemented
149	#431	C₂×C₃₁	1	1	yes	characteristic, complemented
150	C₂×C₃₁	—	4	1	no	—
151	C₂×C₃₁	—	8	2	no	—
152	C₂×C₃₁	—	8	1	no	—
153	C₂×C₃₁	#535	1	1	yes	characteristic
154	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
155	C₂×C₃₁	#575	1	1	yes	characteristic
156	C₂×#41	C₂²×C₃₁	1	2	yes	—
157	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
158	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
159	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
160	C₂×D₈	C₂²×C₃₁	1	2	yes	—
161	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
162	#127	C₂²×C₃₁	1	1	yes	characteristic
163	#126	—	2	1	no	—
164	#125	C₂²×C₃₁	1	1	yes	characteristic
165	#124	C₂²×C₃₁	1	1	yes	characteristic
166	#124	C₂²×C₃₁	1	2	yes	—
167	#123	—	2	1	no	—
168	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
169	#121	—	2	1	no	—
170	#121	—	2	1	no	—
171	#121	C₂²×C₃₁	1	1	yes	characteristic
172	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
177	#117	—	2	1	no	—
178	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
179	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
180	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
181	#103	C₂²×C₃₁	1	2	yes	—
182	#103	C₂²×C₃₁	1	2	yes	—
183	#102	C₂²×C₃₁	1	2	yes	—
184	#102	C₂²×C₃₁	1	2	yes	—
185	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
186	C₄×C₈	C₂²×C₃₁	1	2	yes	—
187	C₃₁	#3212	1	1	yes	characteristic, direct product factor, Sylow
188	#41	—	4	2	no	—
189	#41	—	4	2	no	—
190	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
191	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂³×C₃₁	1	2	yes	—
197	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
203	Q₁₆	—	4	2	no	—
204	SD₁₆	—	4	2	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	4	2	no	—
207	C₂×C₈	C₂³×C₃₁	1	2	yes	—
208	C₂×C₈	C₂³×C₃₁	1	2	yes	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
216	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	D₄×C₃₁	1	1	yes	characteristic
226	C₄²	D₄×C₃₁	1	1	yes	characteristic
227	C₄²	C₂³×C₃₁	1	1	yes	characteristic
228	C₄²	C₂³×C₃₁	1	1	yes	characteristic
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	2	1	no	—
235	Q₈	—	2	2	no	—
236	D₄	—	2	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	2	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	4	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	C₃₁×#135	1	2	yes	—
281	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
282	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	8	2	no	—
285	C₂	—	8	1	no	—
286	C₂	C₃₁×#535	1	1	yes	characteristic
287	C₂	C₂×C₃₁×#135	1	1	yes	characteristic
288	C₂	C₃₁×#575	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 ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#532	C₂	semidirect
2. C₃₁×#529	C₂	semidirect
3. C₃₁×#495	C₂	semidirect
4. C₃₁×#488	C₂	semidirect
5. C₃₁×#462	C₂	semidirect
6. C₃₁×#461	C₂	semidirect
7. C₃₁×#460	C₂	semidirect
8. C₃₁×#459	C₂	semidirect
9. C₃₁×#452	C₂	semidirect
10. C₃₁×#449	C₂	semidirect
11. C₃₁×#431	C₂	semidirect
117. #3212	C₃₁	direct (N × H)
139. #532	C₂×C₃₁	semidirect
140. #529	C₂×C₃₁	semidirect
141. #495	C₂×C₃₁	semidirect
142. #488	C₂×C₃₁	semidirect
143. #462	C₂×C₃₁	semidirect
144. #461	C₂×C₃₁	semidirect
145. #460	C₂×C₃₁	semidirect
146. #459	C₂×C₃₁	semidirect
147. #452	C₂×C₃₁	semidirect
148. #449	C₂×C₃₁	semidirect
149. #431	C₂×C₃₁	semidirect
187. C₃₁	#3212	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×#532	2
C₃₁×#529	2
C₃₁×#495	2
C₃₁×#488	2
C₃₁×#462	2
C₃₁×#461	2
C₃₁×#460	2
C₃₁×#459	2
C₃₁×#452	2
C₃₁×#449	2
C₃₁×#431	2
#3212	31

Subgroups — C31×#3212 #16843145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3212 #16843145