Subgroups — C₃₁×#33801 #21490560

Order 7936.

500 conjugacy classes of subgroups.

The subgroup lattice has 314 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×#2255	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2244	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2234	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2226	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2224	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#533	—	2	1	no	—
10	C₃₁×#532	—	2	1	no	—
11	C₃₁×#500	C₂²	1	1	yes	characteristic
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#499	—	2	1	no	—
15	C₃₁×#499	C₂²	1	2	yes	—
16	C₃₁×#498	—	2	1	no	—
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#486	—	2	1	no	—
19	C₃₁×#466	—	2	1	no	—
20	C₃₁×#466	C₂²	1	2	yes	—
21	C₄×D₈×C₃₁	—	2	1	no	—
22	C₄×D₈×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#427	—	2	1	no	—
24	C₃₁×#427	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#367	C₂²	1	4	yes	—
27	C₃₁×#366	C₂²	1	4	yes	—
28	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₃₁×#357	C₂²	1	4	yes	—
30	C₃₁×#357	C₂²	1	4	yes	—
31	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₃₁×#346	C₂²	1	2	yes	—
33	C₂×#41×C₃₁	—	4	1	no	—
34	C₂×D₈×C₃₁	—	2	1	no	—
35	C₂×D₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×C₄⋊Q₈	—	2	1	no	—
39	C₃₁×C₄⋊Q₈	—	2	1	no	—
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×#127	—	2	1	no	—
42	C₃₁×#125	—	2	2	no	—
43	C₃₁×#124	—	4	1	no	—
44	C₃₁×C₂²⋊Q₈	—	4	1	no	—
45	C₃₁×C₂²⋊Q₈	—	4	1	no	—
46	C₃₁×#121	—	2	2	no	—
47	C₃₁×#121	—	4	1	no	—
48	C₄×Q₈×C₃₁	—	4	1	no	—
49	C₄×Q₈×C₃₁	—	4	1	no	—
50	C₄×D₄×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	—	4	1	no	—
52	C₃₁×#117	—	2	2	no	—
53	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
54	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
55	C₃₁×#107	—	2	1	no	—
56	C₃₁×#107	—	2	1	no	—
57	C₃₁×#107	—	2	1	no	—
58	C₃₁×#107	C₂³	1	2	yes	—
59	C₃₁×#107	C₂³	1	2	yes	—
60	C₃₁×#107	C₂³	1	2	yes	—
61	C₃₁×#106	—	2	2	no	—
62	C₃₁×#106	—	2	2	no	—
63	C₃₁×#102	—	2	2	no	—
64	C₃₁×#102	—	2	2	no	—
65	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	#41×C₃₁	—	8	4	no	—
70	C₂×Q₈×C₃₁	—	4	1	no	—
71	C₂×Q₈×C₃₁	—	4	1	no	—
72	C₂×Q₈×C₃₁	—	4	1	no	—
73	C₂×D₄×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	—	4	2	no	—
75	C₂×D₄×C₃₁	—	4	1	no	—
76	C₂×D₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	D₈×C₃₁	—	2	4	no	—
82	D₈×C₃₁	—	4	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	4	1	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	1	no	—
96	C₄⋊C₄×C₃₁	—	4	1	no	—
97	C₂²⋊C₄×C₃₁	—	4	2	no	—
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	2	no	—
101	C₄²×C₃₁	D₈	1	1	yes	characteristic
102	C₄²×C₃₁	D₈	1	1	yes	characteristic
103	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₁₆×C₃₁	—	2	4	no	—
106	#33801	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	4	2	no	—
108	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
109	Q₈×C₃₁	—	8	2	no	—
110	Q₈×C₃₁	—	8	2	no	—
111	Q₈×C₃₁	—	8	2	no	—
112	D₄×C₃₁	—	4	2	no	—
113	D₄×C₃₁	—	4	2	no	—
114	D₄×C₃₁	—	8	2	no	—
115	D₄×C₃₁	—	8	2	no	—
116	D₄×C₃₁	—	8	4	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	4	1	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	—	8	4	no	—
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	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	—	2	1	no	—
133	C₈×C₃₁	#137	1	2	yes	—
134	C₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
135	#3169	C₂×C₃₁	1	1	yes	characteristic
136	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#2255	C₂×C₃₁	1	2	yes	complemented
138	#2244	C₂×C₃₁	1	2	yes	complemented
139	#2234	C₂×C₃₁	1	4	yes	complemented
140	#2226	C₂×C₃₁	1	2	yes	complemented
141	#2224	C₂×C₃₁	1	2	yes	complemented
142	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	2	1	no	—
146	C₂²×C₃₁	—	8	2	no	—
147	C₂²×C₃₁	—	8	2	no	—
148	C₂²×C₃₁	—	8	2	no	—
149	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₃₁	—	2	1	no	—
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	#509	1	2	yes	—
158	C₄×C₃₁	#536	1	1	yes	characteristic
159	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
160	#533	—	2	1	no	—
161	#532	—	2	1	no	—
162	#500	C₂²×C₃₁	1	1	yes	characteristic
163	#500	C₂²×C₃₁	1	1	yes	characteristic
164	#500	C₂²×C₃₁	1	1	yes	characteristic
165	#499	—	2	1	no	—
166	#499	C₂²×C₃₁	1	2	yes	—
167	#498	—	2	1	no	—
168	#493	C₂²×C₃₁	1	1	yes	characteristic
169	#486	—	2	1	no	—
170	#466	—	2	1	no	—
171	#466	C₂²×C₃₁	1	2	yes	—
172	C₄×D₈	—	2	1	no	—
173	C₄×D₈	C₂²×C₃₁	1	2	yes	—
174	#427	—	2	1	no	—
175	#427	C₂²×C₃₁	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
177	#367	C₂²×C₃₁	1	4	yes	—
178	#366	C₂²×C₃₁	1	4	yes	—
179	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
180	#357	C₂²×C₃₁	1	4	yes	—
181	#357	C₂²×C₃₁	1	4	yes	—
182	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
183	#346	C₂²×C₃₁	1	2	yes	—
184	C₂×C₃₁	—	4	1	no	—
185	C₂×C₃₁	—	16	2	no	—
186	C₂×C₃₁	#3226	1	1	yes	characteristic
187	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
188	C₂×C₃₁	#3414	1	1	yes	characteristic
189	C₂×#41	—	4	1	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	C₂³×C₃₁	1	2	yes	—
192	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	#127	—	2	1	no	—
198	#125	—	2	2	no	—
199	#124	—	4	1	no	—
200	C₂²⋊Q₈	—	4	1	no	—
201	C₂²⋊Q₈	—	4	1	no	—
202	#121	—	2	2	no	—
203	#121	—	4	1	no	—
204	C₄×Q₈	—	4	1	no	—
205	C₄×Q₈	—	4	1	no	—
206	C₄×D₄	—	2	2	no	—
207	C₄×D₄	—	4	1	no	—
208	#117	—	2	2	no	—
209	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	2	1	no	—
214	#107	C₂³×C₃₁	1	2	yes	—
215	#107	C₂³×C₃₁	1	2	yes	—
216	#107	C₂³×C₃₁	1	2	yes	—
217	#106	—	2	2	no	—
218	#106	—	2	2	no	—
219	#102	—	2	2	no	—
220	#102	—	2	2	no	—
221	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
222	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
225	C₃₁	#33801	1	1	yes	characteristic, direct product factor, Sylow
226	#41	—	8	4	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
238	D₈	—	2	4	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	D₈×C₃₁	1	1	yes	characteristic
259	C₄²	D₈×C₃₁	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₁₆	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
265	Q₈	—	8	2	no	—
266	Q₈	—	8	2	no	—
267	Q₈	—	8	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₃₁×#137	1	2	yes	—
290	C₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₃₁×#509	1	2	yes	—
306	C₄	C₃₁×#536	1	1	yes	characteristic
307	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
308	C₂	—	4	1	no	—
309	C₂	—	16	2	no	—
310	C₂	C₃₁×#3226	1	1	yes	characteristic
311	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
312	C₂	C₃₁×#3414	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×#2255	C₂	semidirect
4. C₃₁×#2244	C₂	semidirect
5. C₃₁×#2234	C₂	semidirect
6. C₃₁×#2226	C₂	semidirect
7. C₃₁×#2224	C₂	semidirect
8. C₃₁×#2163	C₂	semidirect
106. #33801	C₃₁	direct (N × H)
136. #3160	C₂×C₃₁	semidirect
137. #2255	C₂×C₃₁	semidirect
138. #2244	C₂×C₃₁	semidirect
139. #2234	C₂×C₃₁	semidirect
140. #2226	C₂×C₃₁	semidirect
141. #2224	C₂×C₃₁	semidirect
142. #2163	C₂×C₃₁	semidirect
225. C₃₁	#33801	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×#2255	2
C₃₁×#2244	2
C₃₁×#2234	2
C₃₁×#2226	2
C₃₁×#2224	2
C₃₁×#2163	2
#33801	31

Subgroups — C31×#33801 #21490560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33801 #21490560