Subgroups — C₃₁×#13002 #21469761

Order 7936.

528 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#360	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1386	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#525	—	2	1	no	—
9	C₂×C₃₁×#117	—	2	1	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
11	C₃₁×#467	C₄	1	2	yes	complemented
12	C₃₁×#466	C₄	1	2	yes	complemented
13	C₃₁×#465	—	2	2	no	—
14	C₃₁×#427	—	2	4	no	—
15	C₃₁×#427	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#107	—	2	1	no	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₃₁×#416	—	2	1	no	—
19	C₃₁×#360	—	2	4	no	—
20	C₃₁×C₂²⋊C₁₆	—	2	2	no	—
21	C₂×#41×C₃₁	—	2	1	no	—
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₃₁×#135	—	2	2	no	—
24	C₃₁×#135	—	2	2	no	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	D₄	1	2	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×#123	—	4	1	no	—
35	C₃₁×C₂²⋊Q₈	—	2	2	no	—
36	C₃₁×#121	—	2	2	no	—
37	C₃₁×#120	—	4	1	no	—
38	C₄×D₄×C₃₁	—	4	2	no	—
39	C₃₁×#117	—	2	4	no	—
40	C₃₁×#117	—	2	1	no	—
41	C₃₁×#117	—	4	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
44	C₂×C₄²×C₃₁	—	4	1	no	—
45	C₃₁×M₃₂	—	2	2	no	—
46	C₃₁×M₃₂	—	2	2	no	—
47	C₂×C₁₆×C₃₁	—	2	2	no	—
48	C₃₁×#107	—	2	4	no	—
49	C₃₁×#107	C₂×C₄	1	2	yes	—
50	C₃₁×#106	—	2	2	no	—
51	C₃₁×#106	—	2	1	no	—
52	C₃₁×#106	D₄	1	4	yes	complemented
53	C₃₁×#103	—	2	2	no	—
54	C₃₁×#102	—	2	2	no	—
55	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
56	#41×C₃₁	—	4	1	no	—
57	#41×C₃₁	—	4	1	no	—
58	#41×C₃₁	—	4	1	no	—
59	#41×C₃₁	—	4	1	no	—
60	C₂×Q₈×C₃₁	—	2	1	no	—
61	C₂×D₄×C₃₁	—	2	1	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂×D₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	—	2	1	no	—
65	C₂²×C₄×C₃₁	—	2	1	no	—
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	—	2	1	no	—
68	C₂²×C₄×C₃₁	—	4	2	no	—
69	C₂²×C₄×C₃₁	—	4	1	no	—
70	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₃₁	—	2	2	no	—
75	SD₁₆×C₃₁	—	4	2	no	—
76	D₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	1	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	#41	1	2	yes	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	4	2	no	—
89	C₄⋊C₄×C₃₁	—	4	2	no	—
90	C₂²⋊C₄×C₃₁	—	4	4	no	—
91	C₂²⋊C₄×C₃₁	—	4	2	no	—
92	C₂²⋊C₄×C₃₁	—	4	1	no	—
93	C₂²⋊C₄×C₃₁	—	4	2	no	—
94	C₄²×C₃₁	—	4	4	no	—
95	C₄²×C₃₁	—	4	1	no	—
96	C₁₆×C₃₁	—	4	2	no	—
97	#13002	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	2	1	no	—
99	C₂³×C₃₁	—	2	1	no	—
100	C₂³×C₃₁	—	4	2	no	—
101	C₂³×C₃₁	—	4	1	no	—
102	C₂³×C₃₁	#102	1	2	yes	—
103	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
104	Q₈×C₃₁	—	4	1	no	—
105	Q₈×C₃₁	—	4	1	no	—
106	D₄×C₃₁	—	4	1	no	—
107	D₄×C₃₁	—	4	1	no	—
108	D₄×C₃₁	—	8	4	no	—
109	D₄×C₃₁	—	8	1	no	—
110	D₄×C₃₁	—	8	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	1	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	8	4	no	—
121	C₂×C₄×C₃₁	—	8	1	no	—
122	C₂×C₄×C₃₁	—	8	1	no	—
123	C₂×C₄×C₃₁	#102	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
130	C₈×C₃₁	—	2	2	no	—
131	C₈×C₃₁	C₄×D₄	1	2	yes	—
132	C₈×C₃₁	#121	1	2	yes	—
133	#3050	C₂×C₃₁	1	1	yes	characteristic
134	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂×#360	C₂×C₃₁	1	2	yes	complemented
136	#2112	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#1386	C₂×C₃₁	1	2	yes	complemented
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	4	2	no	—
143	C₂²×C₃₁	—	4	2	no	—
144	C₂²×C₃₁	—	8	1	no	—
145	C₂²×C₃₁	—	8	2	no	—
146	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
147	C₂²×C₃₁	#447	1	1	yes	characteristic
148	C₂²×C₃₁	#480	1	1	yes	characteristic
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	8	4	no	—
151	C₄×C₃₁	—	8	1	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	#386	1	1	yes	characteristic
154	C₄×C₃₁	#417	1	1	yes	characteristic
155	C₄×C₃₁	#448	1	1	yes	characteristic
156	C₄×C₃₁	#481	1	1	yes	characteristic
157	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
158	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	#525	—	2	1	no	—
160	C₂×#117	—	2	1	no	—
161	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
162	#467	C₄×C₃₁	1	2	yes	complemented
163	#466	C₄×C₃₁	1	2	yes	complemented
164	#465	—	2	2	no	—
165	#427	—	2	4	no	—
166	#427	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×#107	—	2	1	no	—
168	C₂×#106	C₂²×C₃₁	1	2	yes	—
169	#416	—	2	1	no	—
170	#360	—	2	4	no	—
171	C₂²⋊C₁₆	—	2	2	no	—
172	C₂×C₃₁	—	2	1	no	—
173	C₂×C₃₁	—	2	1	no	—
174	C₂×C₃₁	—	4	2	no	—
175	C₂×C₃₁	—	16	1	no	—
176	C₂×C₃₁	#1874	1	1	yes	characteristic
177	C₂×C₃₁	#2145	1	2	yes	—
178	C₂×#41	—	2	1	no	—
179	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
180	#135	—	2	2	no	—
181	#135	—	2	2	no	—
182	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂×SD₁₆	—	2	1	no	—
184	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
188	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₃₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
191	#123	—	4	1	no	—
192	C₂²⋊Q₈	—	2	2	no	—
193	#121	—	2	2	no	—
194	#120	—	4	1	no	—
195	C₄×D₄	—	4	2	no	—
196	#117	—	2	4	no	—
197	#117	—	2	1	no	—
198	#117	—	4	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	4	1	no	—
201	C₂×C₄²	—	4	1	no	—
202	M₃₂	—	2	2	no	—
203	M₃₂	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	#107	—	2	4	no	—
206	#107	C₂×C₄×C₃₁	1	2	yes	—
207	#106	—	2	2	no	—
208	#106	—	2	1	no	—
209	#106	D₄×C₃₁	1	4	yes	complemented
210	#103	—	2	2	no	—
211	#102	—	2	2	no	—
212	C₃₁	#13002	1	1	yes	characteristic, direct product factor, Sylow
213	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
214	#41	—	4	1	no	—
215	#41	—	4	1	no	—
216	#41	—	4	1	no	—
217	#41	—	4	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
229	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	SD₁₆	—	4	2	no	—
234	D₈	—	2	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
242	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₈	#41×C₃₁	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₄²	—	4	4	no	—
253	C₄²	—	4	1	no	—
254	C₁₆	—	4	2	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	1	no	—
259	C₂³	C₃₁×#102	1	2	yes	—
260	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
261	Q₈	—	4	1	no	—
262	Q₈	—	4	1	no	—
263	D₄	—	4	1	no	—
264	D₄	—	4	1	no	—
265	D₄	—	8	4	no	—
266	D₄	—	8	1	no	—
267	D₄	—	8	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	C₃₁×#102	1	2	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
283	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	C₄×D₄×C₃₁	1	2	yes	—
289	C₈	C₃₁×#121	1	2	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	8	1	no	—
297	C₂²	—	8	2	no	—
298	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
299	C₂²	C₃₁×#447	1	1	yes	characteristic
300	C₂²	C₃₁×#480	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	8	4	no	—
303	C₄	—	8	1	no	—
304	C₄	—	8	2	no	—
305	C₄	C₃₁×#386	1	1	yes	characteristic
306	C₄	C₃₁×#417	1	1	yes	characteristic
307	C₄	C₃₁×#448	1	1	yes	characteristic
308	C₄	C₃₁×#481	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	4	2	no	—
312	C₂	—	16	1	no	—
313	C₂	C₃₁×#1874	1	1	yes	characteristic
314	C₂	C₃₁×#2145	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₃₁×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#427	C₂	semidirect
3. C₂×C₃₁×#360	C₂	semidirect
4. C₃₁×#2112	C₂	semidirect
5. C₃₁×#1386	C₂	semidirect
11. C₃₁×#467	C₄	semidirect
12. C₃₁×#466	C₄	semidirect
52. C₃₁×#106	D₄	semidirect
97. #13002	C₃₁	direct (N × H)
134. C₂×#427	C₂×C₃₁	semidirect
135. C₂×#360	C₂×C₃₁	semidirect
136. #2112	C₂×C₃₁	semidirect
137. #1386	C₂×C₃₁	semidirect
162. #467	C₄×C₃₁	semidirect
163. #466	C₄×C₃₁	semidirect
209. #106	D₄×C₃₁	semidirect
212. C₃₁	#13002	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×#3050	2
C₂×C₃₁×#427	2
C₂×C₃₁×#360	2
C₃₁×#2112	2
C₃₁×#1386	2
#13002	31

Subgroups — C31×#13002 #21469761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13002 #21469761