Subgroups — C₃₁×#33794 #21490553

Order 7936.

504 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2234	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2228	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#533	—	2	1	no	—
11	C₃₁×#532	—	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×#493	C₂²	1	1	yes	characteristic
19	C₃₁×#486	—	2	1	no	—
20	C₃₁×#466	—	2	1	no	—
21	C₃₁×#466	C₂²	1	2	yes	—
22	C₄×D₈×C₃₁	—	2	1	no	—
23	C₄×D₈×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×#357	C₂²	1	4	yes	—
31	C₃₁×#357	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×#41×C₃₁	—	4	1	no	—
36	C₂×D₈×C₃₁	—	2	1	no	—
37	C₂×D₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×C₄⋊Q₈	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×#127	—	2	1	no	—
44	C₃₁×#125	—	2	2	no	—
45	C₃₁×#124	—	4	1	no	—
46	C₃₁×C₂²⋊Q₈	—	4	1	no	—
47	C₃₁×C₂²⋊Q₈	—	4	1	no	—
48	C₃₁×#121	—	2	2	no	—
49	C₃₁×#121	—	4	1	no	—
50	C₄×Q₈×C₃₁	—	4	1	no	—
51	C₄×Q₈×C₃₁	—	4	1	no	—
52	C₄×D₄×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	4	1	no	—
54	C₃₁×#117	—	2	2	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×#102	—	2	2	no	—
67	C₃₁×#102	—	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	#41×C₃₁	—	8	4	no	—
73	C₂×Q₈×C₃₁	—	4	1	no	—
74	C₂×Q₈×C₃₁	—	4	1	no	—
75	C₂×Q₈×C₃₁	—	4	1	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	4	2	no	—
78	C₂×D₄×C₃₁	—	4	1	no	—
79	C₂×D₄×C₃₁	—	4	1	no	—
80	C₂²×C₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	D₈×C₃₁	—	2	4	no	—
85	D₈×C₃₁	—	4	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	4	1	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₄⋊C₄×C₃₁	—	4	1	no	—
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₃₁	—	4	2	no	—
103	C₄²×C₃₁	—	4	2	no	—
104	C₄²×C₃₁	D₈	1	1	yes	characteristic
105	C₄²×C₃₁	D₈	1	1	yes	characteristic
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₁₆×C₃₁	—	2	2	no	—
109	C₁₆×C₃₁	—	2	2	no	—
110	#33794	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₃₁	—	8	2	no	—
114	Q₈×C₃₁	—	8	2	no	—
115	Q₈×C₃₁	—	8	2	no	—
116	D₄×C₃₁	—	4	2	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	—	8	4	no	—
119	D₄×C₃₁	—	8	2	no	—
120	D₄×C₃₁	—	8	2	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	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	—	8	4	no	—
129	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	—	2	1	no	—
137	C₈×C₃₁	#137	1	2	yes	—
138	C₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
139	#3169	C₂×C₃₁	1	1	yes	characteristic
140	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#2254	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#2234	C₂×C₃₁	1	4	yes	complemented
146	#2228	C₂×C₃₁	1	4	yes	complemented
147	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	8	2	no	—
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	8	2	no	—
154	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	#508	1	2	yes	—
162	C₄×C₃₁	#509	1	2	yes	—
163	C₄×C₃₁	#536	1	1	yes	characteristic
164	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
165	#533	—	2	1	no	—
166	#532	—	2	1	no	—
167	#500	C₂²×C₃₁	1	1	yes	characteristic
168	#500	C₂²×C₃₁	1	1	yes	characteristic
169	#500	C₂²×C₃₁	1	1	yes	characteristic
170	#499	—	2	1	no	—
171	#499	C₂²×C₃₁	1	2	yes	—
172	#498	—	2	1	no	—
173	#493	C₂²×C₃₁	1	1	yes	characteristic
174	#486	—	2	1	no	—
175	#466	—	2	1	no	—
176	#466	C₂²×C₃₁	1	2	yes	—
177	C₄×D₈	—	2	1	no	—
178	C₄×D₈	C₂²×C₃₁	1	2	yes	—
179	#427	—	2	1	no	—
180	#427	C₂²×C₃₁	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
182	#367	C₂²×C₃₁	1	4	yes	—
183	#366	C₂²×C₃₁	1	4	yes	—
184	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
185	#357	C₂²×C₃₁	1	4	yes	—
186	#357	C₂²×C₃₁	1	4	yes	—
187	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
188	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
189	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₃₁	—	4	1	no	—
191	C₂×C₃₁	—	16	2	no	—
192	C₂×C₃₁	#3226	1	1	yes	characteristic
193	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
194	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
195	C₂×#41	—	4	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	C₂³×C₃₁	1	2	yes	—
198	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	#127	—	2	1	no	—
204	#125	—	2	2	no	—
205	#124	—	4	1	no	—
206	C₂²⋊Q₈	—	4	1	no	—
207	C₂²⋊Q₈	—	4	1	no	—
208	#121	—	2	2	no	—
209	#121	—	4	1	no	—
210	C₄×Q₈	—	4	1	no	—
211	C₄×Q₈	—	4	1	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	4	1	no	—
214	#117	—	2	2	no	—
215	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
216	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
217	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	C₂³×C₃₁	1	2	yes	—
222	#107	C₂³×C₃₁	1	2	yes	—
223	#107	C₂³×C₃₁	1	2	yes	—
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
229	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
232	C₃₁	#33794	1	1	yes	characteristic, direct product factor, Sylow
233	#41	—	8	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
245	D₈	—	2	4	no	—
246	D₈	—	4	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₄²	—	4	2	no	—
265	C₄²	D₈×C₃₁	1	1	yes	characteristic
266	C₄²	D₈×C₃₁	1	1	yes	characteristic
267	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
273	Q₈	—	8	2	no	—
274	Q₈	—	8	2	no	—
275	Q₈	—	8	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	2	no	—
280	D₄	—	8	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	C₃₁×#137	1	2	yes	—
298	C₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	C₃₁×#508	1	2	yes	—
313	C₄	C₃₁×#509	1	2	yes	—
314	C₄	C₃₁×#536	1	1	yes	characteristic
315	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
316	C₂	—	4	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₃₁×#3226	1	1	yes	characteristic
319	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
320	C₂	C₂×C₃₁×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3169	C₂
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
2. C₃₁×#3160	C₂	semidirect
3. C₃₁×#2254	C₂	semidirect
4. C₃₁×#2252	C₂	semidirect
5. C₃₁×#2240	C₂	semidirect
6. C₃₁×#2240	C₂	semidirect
7. C₃₁×#2234	C₂	semidirect
8. C₃₁×#2228	C₂	semidirect
9. C₃₁×#2162	C₂	semidirect
110. #33794	C₃₁	direct (N × H)
140. #3160	C₂×C₃₁	semidirect
141. #2254	C₂×C₃₁	semidirect
142. #2252	C₂×C₃₁	semidirect
143. #2240	C₂×C₃₁	semidirect
144. #2240	C₂×C₃₁	semidirect
145. #2234	C₂×C₃₁	semidirect
146. #2228	C₂×C₃₁	semidirect
147. #2162	C₂×C₃₁	semidirect
232. C₃₁	#33794	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×#3169	2
C₃₁×#3160	2
C₃₁×#2254	2
C₃₁×#2252	2
C₃₁×#2240	2
C₃₁×#2234	2
C₃₁×#2228	2
C₃₁×#2162	2
#33794	31

Subgroups — C31×#33794 #21490553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33794 #21490553