Subgroups — C₃₁×#13017 #21469776

Order 7936.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2140	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#518	—	2	1	no	—
11	C₂×C₃₁×D₁₆	—	2	2	no	—
12	C₂×C₃₁×D₁₆	—	2	2	no	—
13	C₂×C₃₁×D₁₆	C₄	1	4	yes	complemented
14	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#442	—	2	2	no	—
17	C₄×D₈×C₃₁	—	2	2	no	—
18	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
19	C₃₁×#427	C₂²	1	1	yes	characteristic
20	C₃₁×#418	—	2	1	no	—
21	C₃₁×#418	—	2	1	no	—
22	C₂×C₃₁×#100	—	2	1	no	—
23	C₃₁×#405	C₂²	1	1	yes	characteristic
24	C₃₁×#361	—	2	2	no	—
25	C₃₁×#357	—	2	2	no	—
26	C₂²×D₄×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	—	2	1	no	—
29	C₂×D₈×C₃₁	—	2	1	no	—
30	C₂×D₈×C₃₁	—	2	1	no	—
31	C₂×D₈×C₃₁	—	2	1	no	—
32	C₂×D₈×C₃₁	—	2	1	no	—
33	C₂×D₈×C₃₁	—	2	1	no	—
34	C₂×D₈×C₃₁	—	2	1	no	—
35	C₂×D₈×C₃₁	—	2	1	no	—
36	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	complemented
38	C₂×D₈×C₃₁	D₄	1	2	yes	—
39	C₂×D₈×C₃₁	D₄	1	2	yes	—
40	C₂×M₁₆×C₃₁	—	2	1	no	—
41	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₃₁	—	2	4	no	—
44	C₄×D₄×C₃₁	—	4	2	no	—
45	C₃₁×#117	—	2	1	no	—
46	C₃₁×#117	D₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
48	C₃₁×D₁₆	—	2	8	no	—
49	C₃₁×M₃₂	—	2	1	no	—
50	C₃₁×M₃₂	—	2	1	no	—
51	C₂×C₁₆×C₃₁	—	2	1	no	—
52	C₂×C₁₆×C₃₁	—	2	1	no	—
53	C₂×C₁₆×C₃₁	—	2	1	no	—
54	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
55	C₃₁×#108	—	2	1	no	—
56	C₃₁×#108	—	2	1	no	—
57	C₃₁×#107	D₄	1	2	yes	—
58	C₃₁×#106	—	2	1	no	—
59	C₃₁×#102	—	2	4	no	—
60	C₃₁×#102	—	4	2	no	—
61	C₃₁×#100	—	4	2	no	—
62	C₃₁×#97	—	2	1	no	—
63	C₄×C₈×C₃₁	—	2	1	no	—
64	C₂⁴×C₃₁	—	4	1	no	—
65	C₂⁴×C₃₁	—	4	1	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	2	2	no	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂×D₄×C₃₁	—	4	1	no	—
72	C₂×D₄×C₃₁	—	4	1	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂×D₄×C₃₁	—	4	1	no	—
76	C₂×D₄×C₃₁	—	4	1	no	—
77	C₂×D₄×C₃₁	—	4	1	no	—
78	C₂²×C₄×C₃₁	—	4	4	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	D₈×C₃₁	—	2	2	no	—
81	D₈×C₃₁	—	2	2	no	—
82	D₈×C₃₁	—	2	2	no	—
83	D₈×C₃₁	—	2	2	no	—
84	D₈×C₃₁	—	2	2	no	—
85	D₈×C₃₁	—	4	1	no	—
86	D₈×C₃₁	—	4	1	no	—
87	D₈×C₃₁	C₂²⋊C₄	1	4	yes	—
88	M₁₆×C₃₁	—	2	1	no	—
89	M₁₆×C₃₁	—	2	1	no	—
90	M₁₆×C₃₁	—	4	1	no	—
91	M₁₆×C₃₁	—	4	1	no	—
92	C₂×C₈×C₃₁	—	2	1	no	—
93	C₂×C₈×C₃₁	—	4	1	no	—
94	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
99	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
102	C₂²⋊C₄×C₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₃₁	—	4	4	no	—
104	C₂²⋊C₄×C₃₁	—	4	1	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	4	1	no	—
107	C₁₆×C₃₁	—	2	2	no	—
108	C₁₆×C₃₁	—	4	1	no	—
109	#13017	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₃₁	—	4	2	no	—
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	—	4	2	no	—
113	C₂³×C₃₁	—	4	2	no	—
114	C₂³×C₃₁	—	8	1	no	—
115	C₂³×C₃₁	—	8	1	no	—
116	C₂³×C₃₁	—	16	2	no	—
117	C₂³×C₃₁	—	16	2	no	—
118	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
119	D₄×C₃₁	—	2	4	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	4	2	no	—
123	D₄×C₃₁	—	4	2	no	—
124	D₄×C₃₁	—	4	2	no	—
125	D₄×C₃₁	—	8	1	no	—
126	D₄×C₃₁	—	8	1	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	16	4	no	—
130	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
135	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₈×C₃₁	—	4	1	no	—
137	C₈×C₃₁	—	8	1	no	—
138	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
140	C₈×C₃₁	#121	1	1	yes	characteristic
141	C₈×C₃₁	#123	1	1	yes	characteristic
142	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic
143	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#2140	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1382	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#1379	C₂×C₃₁	1	1	yes	characteristic, complemented
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	4	4	no	—
151	C₂²×C₃₁	—	8	4	no	—
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	16	2	no	—
154	C₂²×C₃₁	—	16	2	no	—
155	C₂²×C₃₁	—	16	2	no	—
156	C₂²×C₃₁	—	16	2	no	—
157	C₂²×C₃₁	—	16	2	no	—
158	C₂²×C₃₁	#442	1	1	yes	characteristic
159	C₂²×C₃₁	#447	1	1	yes	characteristic
160	C₂²×C₃₁	#463	1	1	yes	characteristic
161	C₄×C₃₁	—	4	2	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	#386	1	1	yes	characteristic
164	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
165	C₄×C₃₁	#449	1	1	yes	characteristic
166	C₄×C₃₁	#459	1	1	yes	characteristic
167	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
168	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
169	#518	—	2	1	no	—
170	C₂×D₁₆	—	2	2	no	—
171	C₂×D₁₆	—	2	2	no	—
172	C₂×D₁₆	C₄×C₃₁	1	4	yes	complemented
173	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
174	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
175	#442	—	2	2	no	—
176	C₄×D₈	—	2	2	no	—
177	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
178	#427	C₂²×C₃₁	1	1	yes	characteristic
179	#418	—	2	1	no	—
180	#418	—	2	1	no	—
181	C₂×#100	—	2	1	no	—
182	#405	C₂²×C₃₁	1	1	yes	characteristic
183	#361	—	2	2	no	—
184	#357	—	2	2	no	—
185	C₂×C₃₁	—	2	1	no	—
186	C₂×C₃₁	—	2	1	no	—
187	C₂×C₃₁	—	8	4	no	—
188	C₂×C₃₁	—	16	2	no	—
189	C₂×C₃₁	#1879	1	1	yes	characteristic
190	C₂×C₃₁	#2192	1	2	yes	—
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	—	2	1	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
202	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	complemented
203	C₂×D₈	D₄×C₃₁	1	2	yes	—
204	C₂×D₈	D₄×C₃₁	1	2	yes	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₄×D₄	—	2	4	no	—
209	C₄×D₄	—	4	2	no	—
210	#117	—	2	1	no	—
211	#117	D₄×C₃₁	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	4	2	no	—
213	D₁₆	—	2	8	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	complemented
220	#108	—	2	1	no	—
221	#108	—	2	1	no	—
222	#107	D₄×C₃₁	1	2	yes	—
223	#106	—	2	1	no	—
224	#102	—	2	4	no	—
225	#102	—	4	2	no	—
226	#100	—	4	2	no	—
227	#97	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₃₁	#13017	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	—	4	1	no	—
231	C₂⁴	—	4	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	4	1	no	—
252	D₈	—	4	1	no	—
253	D₈	C₂²⋊C₄×C₃₁	1	4	yes	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
265	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	4	1	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	—	4	1	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	8	1	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	16	2	no	—
282	C₂³	—	16	2	no	—
283	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
284	D₄	—	2	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	1	no	—
291	D₄	—	8	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	4	no	—
295	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
297	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
298	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
299	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
300	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₈	—	4	1	no	—
302	C₈	—	8	1	no	—
303	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
304	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
305	C₈	C₃₁×#121	1	1	yes	characteristic
306	C₈	C₃₁×#123	1	1	yes	characteristic
307	C₂²	—	2	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	4	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	16	2	no	—
312	C₂²	—	16	2	no	—
313	C₂²	—	16	2	no	—
314	C₂²	—	16	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	C₃₁×#442	1	1	yes	characteristic
317	C₂²	C₃₁×#447	1	1	yes	characteristic
318	C₂²	C₃₁×#463	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₃₁×#386	1	1	yes	characteristic
322	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
323	C₄	C₃₁×#449	1	1	yes	characteristic
324	C₄	C₃₁×#459	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	8	4	no	—
328	C₂	—	16	2	no	—
329	C₂	C₃₁×#1879	1	1	yes	characteristic
330	C₂	C₃₁×#2192	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#447
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×C₃₁×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2948	C₂	semidirect
3. C₂×C₃₁×#361	C₂	semidirect
4. C₃₁×#2140	C₂	semidirect
5. C₃₁×#1978	C₂	semidirect
6. C₃₁×#1382	C₂	semidirect
7. C₃₁×#1379	C₂	semidirect
13. C₂×C₃₁×D₁₆	C₄	semidirect
37. C₂×D₈×C₃₁	C₂×C₄	semidirect
54. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
109. #13017	C₃₁	direct (N × H)
143. #2948	C₂×C₃₁	semidirect
144. C₂×#361	C₂×C₃₁	semidirect
145. #2140	C₂×C₃₁	semidirect
146. #1978	C₂×C₃₁	semidirect
147. #1382	C₂×C₃₁	semidirect
148. #1379	C₂×C₃₁	semidirect
172. C₂×D₁₆	C₄×C₃₁	semidirect
202. C₂×D₈	C₂×C₄×C₃₁	semidirect
219. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
229. C₃₁	#13017	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×D₁₆	2
C₃₁×#2948	2
C₂×C₃₁×#361	2
C₃₁×#2140	2
C₃₁×#1978	2
C₃₁×#1382	2
C₃₁×#1379	2
#13017	31

Subgroups — C31×#13017 #21469776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13017 #21469776