Subgroups — C₃₁×#19690 #21476449

Order 7936.

656 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3231	C₂	1	2	yes	maximal
2	C₃₁×#3185	C₂	1	2	yes	maximal
3	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1723	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1723	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1718	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1716	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#533	C₂²	1	2	yes	—
11	C₃₁×#533	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	2	yes	—
16	C₃₁×#499	—	2	2	no	—
17	C₃₁×#498	C₂²	1	2	yes	—
18	C₃₁×#498	C₂²	1	2	yes	—
19	C₃₁×#498	C₂²	1	2	yes	—
20	C₃₁×#494	C₂²	1	1	yes	characteristic
21	C₃₁×#494	C₂²	1	2	yes	—
22	C₃₁×#487	C₂²	1	2	yes	—
23	C₃₁×#487	C₂²	1	2	yes	—
24	C₃₁×#487	C₂²	1	2	yes	—
25	C₃₁×#487	C₂²	1	2	yes	—
26	C₃₁×#487	C₂²	1	2	yes	—
27	C₃₁×#485	—	2	4	no	—
28	C₃₁×#479	—	2	4	no	—
29	C₃₁×#470	—	2	4	no	—
30	C₃₁×#467	—	2	2	no	—
31	C₃₁×#441	—	2	4	no	—
32	C₄×Q₁₆×C₃₁	—	2	2	no	—
33	C₃₁×#431	C₂²	1	1	yes	characteristic
34	C₃₁×#427	—	2	2	no	—
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#322	C₂²	1	2	yes	—
38	C₃₁×#322	C₂²	1	2	yes	—
39	C₂×Q₁₆×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	—	2	4	no	—
41	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
42	C₂²×C₈×C₃₁	D₄	1	2	yes	—
43	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
44	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×#125	—	2	2	no	—
48	C₃₁×#125	—	2	2	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	2	2	no	—
52	C₃₁×C₂²⋊Q₈	—	2	2	no	—
53	C₄×Q₈×C₃₁	—	2	2	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₃₁×#117	—	2	2	no	—
58	C₃₁×#117	—	2	2	no	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#107	—	2	2	no	—
61	C₃₁×#107	—	2	2	no	—
62	C₃₁×#107	—	2	2	no	—
63	C₃₁×#107	—	2	4	no	—
64	C₃₁×#106	—	2	4	no	—
65	C₃₁×#106	—	2	4	no	—
66	C₃₁×#106	—	2	4	no	—
67	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
68	C₃₁×#103	—	2	4	no	—
69	C₃₁×#103	—	2	4	no	—
70	C₃₁×#103	—	2	4	no	—
71	C₃₁×#103	—	2	4	no	—
72	C₃₁×#103	—	2	4	no	—
73	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
74	C₃₁×#97	D₄	1	2	yes	—
75	C₄×C₈×C₃₁	D₄	1	2	yes	—
76	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	2	yes	—
79	C₄×C₈×C₃₁	C₂³	1	2	yes	—
80	C₂×Q₈×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₃₁	—	4	8	no	—
88	Q₁₆×C₃₁	—	4	4	no	—
89	M₁₆×C₃₁	—	4	2	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	4	2	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₄⋊C₄×C₃₁	—	4	2	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₂²⋊C₄×C₃₁	—	4	2	no	—
108	C₂²⋊C₄×C₃₁	—	4	2	no	—
109	C₄²×C₃₁	—	4	2	no	—
110	C₄²×C₃₁	—	4	2	no	—
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
115	#19690	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₃₁	#127	1	1	yes	characteristic
117	Q₈×C₃₁	—	4	4	no	—
118	Q₈×C₃₁	—	4	4	no	—
119	Q₈×C₃₁	—	4	4	no	—
120	Q₈×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
130	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₃₁	—	2	2	no	—
135	C₈×C₃₁	—	2	4	no	—
136	C₈×C₃₁	—	2	4	no	—
137	C₈×C₃₁	#137	1	4	yes	—
138	#3231	C₂×C₃₁	1	2	yes	—
139	#3185	C₂×C₃₁	1	2	yes	—
140	#3170	C₂×C₃₁	1	1	yes	characteristic
141	#3160	C₂×C₃₁	1	1	yes	characteristic
142	#1723	C₂×C₃₁	1	2	yes	complemented
143	#1723	C₂×C₃₁	1	2	yes	complemented
144	#1718	C₂×C₃₁	1	2	yes	complemented
145	#1716	C₂×C₃₁	1	2	yes	complemented
146	#1451	C₂×C₃₁	1	1	yes	characteristic
147	C₂²×C₃₁	—	2	1	no	—
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
151	C₄×C₃₁	—	2	1	no	—
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	8	2	no	—
155	C₄×C₃₁	—	8	2	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	—	8	2	no	—
158	C₄×C₃₁	C₂×#137	1	2	yes	—
159	C₄×C₃₁	#575	1	2	yes	—
160	C₄×C₃₁	#578	1	2	yes	—
161	#533	C₂²×C₃₁	1	2	yes	—
162	#533	C₂²×C₃₁	1	2	yes	—
163	#500	C₂²×C₃₁	1	1	yes	characteristic
164	#500	C₂²×C₃₁	1	1	yes	characteristic
165	#500	C₂²×C₃₁	1	1	yes	characteristic
166	#500	C₂²×C₃₁	1	2	yes	—
167	#499	—	2	2	no	—
168	#498	C₂²×C₃₁	1	2	yes	—
169	#498	C₂²×C₃₁	1	2	yes	—
170	#498	C₂²×C₃₁	1	2	yes	—
171	#494	C₂²×C₃₁	1	1	yes	characteristic
172	#494	C₂²×C₃₁	1	2	yes	—
173	#487	C₂²×C₃₁	1	2	yes	—
174	#487	C₂²×C₃₁	1	2	yes	—
175	#487	C₂²×C₃₁	1	2	yes	—
176	#487	C₂²×C₃₁	1	2	yes	—
177	#487	C₂²×C₃₁	1	2	yes	—
178	#485	—	2	4	no	—
179	#479	—	2	4	no	—
180	#470	—	2	4	no	—
181	#467	—	2	2	no	—
182	#441	—	2	4	no	—
183	C₄×Q₁₆	—	2	2	no	—
184	#431	C₂²×C₃₁	1	1	yes	characteristic
185	#427	—	2	2	no	—
186	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
188	#322	C₂²×C₃₁	1	2	yes	—
189	#322	C₂²×C₃₁	1	2	yes	—
190	C₂×C₃₁	—	4	1	no	—
191	C₂×C₃₁	#3150	1	1	yes	characteristic
192	C₂×C₃₁	#3152	1	1	yes	characteristic
193	C₂×C₃₁	#3156	1	1	yes	characteristic
194	C₂×Q₁₆	—	2	2	no	—
195	C₂×Q₁₆	—	2	4	no	—
196	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
197	C₂²×C₈	D₄×C₃₁	1	2	yes	—
198	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
201	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
202	#125	—	2	2	no	—
203	#125	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	#117	—	2	2	no	—
213	#117	—	2	2	no	—
214	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
215	#107	—	2	2	no	—
216	#107	—	2	2	no	—
217	#107	—	2	2	no	—
218	#107	—	2	4	no	—
219	#106	—	2	4	no	—
220	#106	—	2	4	no	—
221	#106	—	2	4	no	—
222	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
223	#103	—	2	4	no	—
224	#103	—	2	4	no	—
225	#103	—	2	4	no	—
226	#103	—	2	4	no	—
227	#103	—	2	4	no	—
228	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
229	#97	D₄×C₃₁	1	2	yes	—
230	C₄×C₈	D₄×C₃₁	1	2	yes	—
231	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₃₁	1	2	yes	—
234	C₄×C₈	C₂³×C₃₁	1	2	yes	—
235	C₃₁	#19690	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	Q₁₆	—	4	8	no	—
244	Q₁₆	—	4	4	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
249	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
250	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
271	C₂³	C₃₁×#127	1	1	yes	characteristic
272	Q₈	—	4	4	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
284	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
285	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	4	no	—
291	C₈	—	2	4	no	—
292	C₈	C₃₁×#137	1	4	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₂×C₃₁×#137	1	2	yes	—
305	C₄	C₃₁×#575	1	2	yes	—
306	C₄	C₃₁×#578	1	2	yes	—
307	C₂	—	4	1	no	—
308	C₂	C₃₁×#3150	1	1	yes	characteristic
309	C₂	C₃₁×#3152	1	1	yes	characteristic
310	C₂	C₃₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	#127
LCS	G ▷ C₄² ▷ C₂² ▷ 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
5. C₃₁×#1723	C₂	semidirect
6. C₃₁×#1723	C₂	semidirect
7. C₃₁×#1718	C₂	semidirect
8. C₃₁×#1716	C₂	semidirect
115. #19690	C₃₁	direct (N × H)
142. #1723	C₂×C₃₁	semidirect
143. #1723	C₂×C₃₁	semidirect
144. #1718	C₂×C₃₁	semidirect
145. #1716	C₂×C₃₁	semidirect
235. C₃₁	#19690	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×#3231	2
C₃₁×#3185	2
C₃₁×#3170	2
C₃₁×#3160	2
C₃₁×#1723	2
C₃₁×#1718	2
C₃₁×#1716	2
C₃₁×#1451	2
#19690	31

Subgroups — C31×#19690 #21476449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19690 #21476449