Subgroups — C₃₁×#33797 #21490556

Order 7936.

460 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2237	C₂	1	4	yes	maximal
8	C₃₁×#2229	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#533	—	2	1	no	—
11	C₃₁×#533	—	2	1	no	—
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×#499	—	2	1	no	—
16	C₃₁×#499	C₂²	1	2	yes	—
17	C₃₁×#498	—	2	1	no	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#487	—	2	1	no	—
20	C₃₁×#467	—	2	1	no	—
21	C₃₁×#467	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₃₁	—	2	1	no	—
23	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#427	—	2	1	no	—
25	C₃₁×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#367	C₂²	1	4	yes	—
28	C₃₁×#366	C₂²	1	4	yes	—
29	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₃₁×#358	C₂²	1	4	yes	—
31	C₃₁×#358	C₂²	1	4	yes	—
32	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₃₁	—	2	1	no	—
36	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×C₄⋊Q₈	—	2	1	no	—
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×C₄⋊Q₈	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×#125	—	2	2	no	—
44	C₃₁×#125	—	4	1	no	—
45	C₃₁×C₂²⋊Q₈	—	2	2	no	—
46	C₃₁×C₂²⋊Q₈	—	4	1	no	—
47	C₃₁×C₂²⋊Q₈	—	4	1	no	—
48	C₃₁×C₂²⋊Q₈	—	4	1	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₄×Q₈×C₃₁	—	4	1	no	—
51	C₄×Q₈×C₃₁	—	4	1	no	—
52	C₄×Q₈×C₃₁	—	4	1	no	—
53	C₃₁×#117	—	2	2	no	—
54	C₃₁×#117	—	4	1	no	—
55	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
58	C₃₁×#107	—	2	1	no	—
59	C₃₁×#107	—	2	1	no	—
60	C₃₁×#107	—	2	1	no	—
61	C₃₁×#107	C₂³	1	2	yes	—
62	C₃₁×#107	C₂³	1	2	yes	—
63	C₃₁×#107	C₂³	1	2	yes	—
64	C₃₁×#106	—	2	2	no	—
65	C₃₁×#106	—	2	2	no	—
66	C₃₁×#103	—	2	2	no	—
67	C₃₁×#103	—	2	2	no	—
68	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	4	1	no	—
74	C₂×Q₈×C₃₁	—	4	1	no	—
75	C₂×Q₈×C₃₁	—	4	1	no	—
76	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₃₁	—	2	4	no	—
80	Q₁₆×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	4	1	no	—
91	C₄⋊C₄×C₃₁	—	4	1	no	—
92	C₄⋊C₄×C₃₁	—	4	2	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	1	no	—
97	C₄⋊C₄×C₃₁	—	4	1	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₄²×C₃₁	—	4	2	no	—
102	C₄²×C₃₁	—	4	2	no	—
103	C₄²×C₃₁	D₈	1	1	yes	characteristic
104	C₄²×C₃₁	D₈	1	1	yes	characteristic
105	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₁₆×C₃₁	—	2	2	no	—
108	C₁₆×C₃₁	—	2	2	no	—
109	#33797	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
111	Q₈×C₃₁	—	4	2	no	—
112	Q₈×C₃₁	—	4	2	no	—
113	Q₈×C₃₁	—	8	2	no	—
114	Q₈×C₃₁	—	8	2	no	—
115	Q₈×C₃₁	—	8	2	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₃₁	—	2	1	no	—
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	#137	1	2	yes	—
133	C₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
134	#3170	C₂×C₃₁	1	1	yes	characteristic
135	#3160	C₂×C₃₁	1	1	yes	characteristic
136	#2254	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#2237	C₂×C₃₁	1	4	yes	—
141	#2229	C₂×C₃₁	1	4	yes	complemented
142	#2162	C₂×C₃₁	1	1	yes	characteristic
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	2	1	no	—
146	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₃₁	—	2	1	no	—
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	2	1	no	—
150	C₄×C₃₁	—	8	2	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	#508	1	2	yes	—
155	C₄×C₃₁	#510	1	2	yes	—
156	C₄×C₃₁	#536	1	1	yes	characteristic
157	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
158	#533	—	2	1	no	—
159	#533	—	2	1	no	—
160	#500	C₂²×C₃₁	1	1	yes	characteristic
161	#500	C₂²×C₃₁	1	1	yes	characteristic
162	#500	C₂²×C₃₁	1	1	yes	characteristic
163	#499	—	2	1	no	—
164	#499	C₂²×C₃₁	1	2	yes	—
165	#498	—	2	1	no	—
166	#494	C₂²×C₃₁	1	1	yes	characteristic
167	#487	—	2	1	no	—
168	#467	—	2	1	no	—
169	#467	C₂²×C₃₁	1	2	yes	—
170	C₄×Q₁₆	—	2	1	no	—
171	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
172	#427	—	2	1	no	—
173	#427	C₂²×C₃₁	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
175	#367	C₂²×C₃₁	1	4	yes	—
176	#366	C₂²×C₃₁	1	4	yes	—
177	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
178	#358	C₂²×C₃₁	1	4	yes	—
179	#358	C₂²×C₃₁	1	4	yes	—
180	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
181	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
182	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×C₃₁	—	4	1	no	—
184	C₂×C₃₁	#3226	1	1	yes	characteristic
185	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
186	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
189	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	#125	—	2	2	no	—
196	#125	—	4	1	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₂²⋊Q₈	—	4	1	no	—
199	C₂²⋊Q₈	—	4	1	no	—
200	C₂²⋊Q₈	—	4	1	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×Q₈	—	4	1	no	—
203	C₄×Q₈	—	4	1	no	—
204	C₄×Q₈	—	4	1	no	—
205	#117	—	2	2	no	—
206	#117	—	4	1	no	—
207	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
208	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
209	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
210	#107	—	2	1	no	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	C₂³×C₃₁	1	2	yes	—
214	#107	C₂³×C₃₁	1	2	yes	—
215	#107	C₂³×C₃₁	1	2	yes	—
216	#106	—	2	2	no	—
217	#106	—	2	2	no	—
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
221	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
222	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₃₁	#33797	1	1	yes	characteristic, direct product factor, Sylow
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	Q₁₆	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₄²	D₈×C₃₁	1	1	yes	characteristic
257	C₄²	D₈×C₃₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₁₆	—	2	2	no	—
261	C₁₆	—	2	2	no	—
262	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	8	2	no	—
267	Q₈	—	8	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	C₃₁×#137	1	2	yes	—
285	C₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	C₃₁×#508	1	2	yes	—
298	C₄	C₃₁×#510	1	2	yes	—
299	C₄	C₃₁×#536	1	1	yes	characteristic
300	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
301	C₂	—	4	1	no	—
302	C₂	C₃₁×#3226	1	1	yes	characteristic
303	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
304	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2254	C₂	semidirect
4. C₃₁×#2252	C₂	semidirect
5. C₃₁×#2241	C₂	semidirect
6. C₃₁×#2241	C₂	semidirect
8. C₃₁×#2229	C₂	semidirect
109. #33797	C₃₁	direct (N × H)
136. #2254	C₂×C₃₁	semidirect
137. #2252	C₂×C₃₁	semidirect
138. #2241	C₂×C₃₁	semidirect
139. #2241	C₂×C₃₁	semidirect
141. #2229	C₂×C₃₁	semidirect
224. C₃₁	#33797	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×#3170	2
C₃₁×#3160	2
C₃₁×#2254	2
C₃₁×#2252	2
C₃₁×#2241	2
C₃₁×#2237	2
C₃₁×#2229	2
C₃₁×#2162	2
#33797	31

Subgroups — C31×#33797 #21490556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33797 #21490556