Subgroups — C₃₁×#12992 #21469751

Order 7936.

582 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	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#121	—	2	1	no	—
9	C₂×C₃₁×#117	—	2	1	no	—
10	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#466	—	2	2	no	—
12	C₃₁×#466	C₄	1	4	yes	complemented
13	C₃₁×#427	—	2	4	no	—
14	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#106	—	2	1	no	—
17	C₂×C₃₁×#102	—	2	1	no	—
18	C₃₁×#357	—	2	4	no	—
19	C₃₁×C₂²⋊C₁₆	—	2	2	no	—
20	C₂²×D₄×C₃₁	—	2	1	no	—
21	C₂²×D₄×C₃₁	—	4	1	no	—
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂×D₈×C₃₁	—	2	2	no	—
24	C₂×D₈×C₃₁	—	2	1	no	—
25	C₂×D₈×C₃₁	—	2	2	no	—
26	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×#121	—	2	4	no	—
34	C₃₁×#121	—	4	2	no	—
35	C₃₁×#117	—	2	4	no	—
36	C₃₁×#117	—	4	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
40	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
41	C₂×C₄²×C₃₁	—	4	1	no	—
42	C₂×C₁₆×C₃₁	—	2	2	no	—
43	C₂×C₁₆×C₃₁	—	2	2	no	—
44	C₂×C₁₆×C₃₁	—	2	2	no	—
45	C₃₁×#107	—	2	2	no	—
46	C₃₁×#107	—	2	1	no	—
47	C₃₁×#107	C₂×C₄	1	2	yes	—
48	C₃₁×#107	D₄	1	4	yes	complemented
49	C₃₁×#106	—	2	4	no	—
50	C₃₁×#102	—	2	4	no	—
51	C₂⁴×C₃₁	—	4	1	no	—
52	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
53	C₂×D₄×C₃₁	—	2	2	no	—
54	C₂×D₄×C₃₁	—	4	4	no	—
55	C₂×D₄×C₃₁	—	4	1	no	—
56	C₂×D₄×C₃₁	—	4	1	no	—
57	C₂×D₄×C₃₁	—	4	1	no	—
58	C₂×D₄×C₃₁	—	4	1	no	—
59	C₂×D₄×C₃₁	—	4	1	no	—
60	C₂×D₄×C₃₁	—	8	4	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×C₄×C₃₁	—	2	1	no	—
64	C₂²×C₄×C₃₁	—	2	1	no	—
65	C₂²×C₄×C₃₁	—	4	2	no	—
66	C₂²×C₄×C₃₁	—	4	1	no	—
67	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	D₈×C₃₁	—	2	4	no	—
72	D₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	2	1	no	—
74	C₂×C₈×C₃₁	—	2	1	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	#41	1	2	yes	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₄⋊C₄×C₃₁	—	4	2	no	—
86	C₂²⋊C₄×C₃₁	—	4	4	no	—
87	C₂²⋊C₄×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	—	4	4	no	—
89	C₁₆×C₃₁	—	4	2	no	—
90	#12992	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	1	no	—
92	C₂³×C₃₁	—	2	1	no	—
93	C₂³×C₃₁	—	2	2	no	—
94	C₂³×C₃₁	—	2	2	no	—
95	C₂³×C₃₁	—	4	2	no	—
96	C₂³×C₃₁	—	8	1	no	—
97	C₂³×C₃₁	—	8	1	no	—
98	C₂³×C₃₁	—	8	2	no	—
99	C₂³×C₃₁	—	8	2	no	—
100	C₂³×C₃₁	#102	1	2	yes	—
101	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
102	D₄×C₃₁	—	4	2	no	—
103	D₄×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	8	8	no	—
105	D₄×C₃₁	—	8	1	no	—
106	D₄×C₃₁	—	8	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	4	no	—
115	C₂×C₄×C₃₁	—	8	2	no	—
116	C₂×C₄×C₃₁	—	8	2	no	—
117	C₂×C₄×C₃₁	—	8	2	no	—
118	C₂×C₄×C₃₁	#102	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
125	C₈×C₃₁	—	2	2	no	—
126	C₈×C₃₁	C₄×D₄	1	2	yes	—
127	C₈×C₃₁	#121	1	2	yes	—
128	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
129	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂×#357	C₂×C₃₁	1	2	yes	complemented
131	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1380	C₂×C₃₁	1	2	yes	complemented
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	2	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	4	2	no	—
138	C₂²×C₃₁	—	8	2	no	—
139	C₂²×C₃₁	—	8	2	no	—
140	C₂²×C₃₁	—	8	2	no	—
141	C₂²×C₃₁	—	16	2	no	—
142	C₂²×C₃₁	—	16	2	no	—
143	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
144	C₂²×C₃₁	#447	1	1	yes	characteristic
145	C₂²×C₃₁	#480	1	1	yes	characteristic
146	C₂²×C₃₁	#508	1	2	yes	—
147	C₂²×C₃₁	#509	1	2	yes	—
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	8	2	no	—
150	C₄×C₃₁	—	8	4	no	—
151	C₄×C₃₁	#386	1	1	yes	characteristic
152	C₄×C₃₁	#417	1	1	yes	characteristic
153	C₄×C₃₁	#448	1	1	yes	characteristic
154	C₄×C₃₁	#481	1	1	yes	characteristic
155	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
156	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#121	—	2	1	no	—
158	C₂×#117	—	2	1	no	—
159	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
160	#466	—	2	2	no	—
161	#466	C₄×C₃₁	1	4	yes	complemented
162	#427	—	2	4	no	—
163	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#107	C₂²×C₃₁	1	2	yes	—
165	C₂×#106	—	2	1	no	—
166	C₂×#102	—	2	1	no	—
167	#357	—	2	4	no	—
168	C₂²⋊C₁₆	—	2	2	no	—
169	C₂×C₃₁	—	4	2	no	—
170	C₂×C₃₁	—	16	2	no	—
171	C₂×C₃₁	#1874	1	1	yes	characteristic
172	C₂×C₃₁	#2138	1	1	yes	characteristic
173	C₂×C₃₁	#2140	1	1	yes	characteristic
174	C₂×C₃₁	#2186	1	2	yes	—
175	C₂×C₃₁	#2234	1	2	yes	—
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	4	1	no	—
178	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
179	C₂×D₈	—	2	2	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
186	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₃₁	1	2	yes	—
188	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
189	#121	—	2	4	no	—
190	#121	—	4	2	no	—
191	#117	—	2	4	no	—
192	#117	—	4	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	—	4	1	no	—
196	C₂×C₂²⋊C₄	—	4	1	no	—
197	C₂×C₄²	—	4	1	no	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	#107	—	2	2	no	—
202	#107	—	2	1	no	—
203	#107	C₂×C₄×C₃₁	1	2	yes	—
204	#107	D₄×C₃₁	1	4	yes	complemented
205	#106	—	2	4	no	—
206	#102	—	2	4	no	—
207	C₃₁	#12992	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	—	4	1	no	—
209	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	8	4	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
225	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	D₈	—	2	4	no	—
229	D₈	—	4	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₈	#41×C₃₁	1	2	yes	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₄²	—	4	4	no	—
246	C₁₆	—	4	2	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	8	1	no	—
253	C₂³	—	8	1	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	2	no	—
256	C₂³	C₃₁×#102	1	2	yes	—
257	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	8	no	—
261	D₄	—	8	1	no	—
262	D₄	—	8	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	C₃₁×#102	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
277	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
278	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	C₄×D₄×C₃₁	1	2	yes	—
283	C₈	C₃₁×#121	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	16	2	no	—
293	C₂²	—	16	2	no	—
294	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
295	C₂²	C₃₁×#447	1	1	yes	characteristic
296	C₂²	C₃₁×#480	1	1	yes	characteristic
297	C₂²	C₃₁×#508	1	2	yes	—
298	C₂²	C₃₁×#509	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₃₁×#386	1	1	yes	characteristic
303	C₄	C₃₁×#417	1	1	yes	characteristic
304	C₄	C₃₁×#448	1	1	yes	characteristic
305	C₄	C₃₁×#481	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₃₁×#1874	1	1	yes	characteristic
309	C₂	C₃₁×#2138	1	1	yes	characteristic
310	C₂	C₃₁×#2140	1	1	yes	characteristic
311	C₂	C₃₁×#2186	1	2	yes	—
312	C₂	C₃₁×#2234	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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×#357	C₂	semidirect
4. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
5. C₃₁×#1380	C₂	semidirect
12. C₃₁×#466	C₄	semidirect
48. C₃₁×#107	D₄	semidirect
90. #12992	C₃₁	direct (N × H)
129. C₂×#427	C₂×C₃₁	semidirect
130. C₂×#357	C₂×C₃₁	semidirect
131. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
132. #1380	C₂×C₃₁	semidirect
161. #466	C₄×C₃₁	semidirect
204. #107	D₄×C₃₁	semidirect
207. C₃₁	#12992	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#466	2
C₂×C₃₁×#427	2
C₂×C₃₁×#357	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₃₁×#1380	2
#12992	31

Subgroups — C31×#12992 #21469751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12992 #21469751