Subgroups — C₃₁×#13068 #21469827

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2140	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1814	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
11	C₃₁×#504	—	2	2	no	—
12	C₃₁×#504	—	2	2	no	—
13	C₂×C₃₁×M₃₂	—	2	2	no	—
14	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
16	C₃₁×#472	—	2	2	no	—
17	C₃₁×#472	C₄	1	4	yes	complemented
18	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
20	C₃₁×#427	C₂²	1	1	yes	characteristic
21	C₃₁×#421	—	2	1	no	—
22	C₂×C₃₁×#100	—	2	1	no	—
23	C₃₁×#361	—	2	2	no	—
24	C₃₁×#357	—	2	2	no	—
25	C₃₁×#344	—	2	1	no	—
26	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
29	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
30	C₂×SD₁₆×C₃₁	—	4	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	—	4	1	no	—
37	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₃₁×#131	—	2	1	no	—
39	C₃₁×#131	—	2	1	no	—
40	C₃₁×#131	C₂×C₄	1	2	yes	—
41	C₃₁×#131	D₄	1	2	yes	—
42	C₂×M₁₆×C₃₁	—	2	1	no	—
43	C₂×M₁₆×C₃₁	—	2	1	no	—
44	C₂×M₁₆×C₃₁	—	2	1	no	—
45	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	—	2	1	no	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#117	—	2	1	no	—
49	C₃₁×M₃₂	—	2	4	no	—
50	C₃₁×M₃₂	—	2	1	no	—
51	C₃₁×M₃₂	—	2	1	no	—
52	C₃₁×M₃₂	—	2	1	no	—
53	C₃₁×M₃₂	D₄	1	2	yes	—
54	C₂×C₁₆×C₃₁	—	2	4	no	—
55	C₂×C₁₆×C₃₁	—	2	1	no	—
56	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
57	C₃₁×#108	—	2	1	no	—
58	C₃₁×#108	—	2	1	no	—
59	C₃₁×#108	—	2	1	no	—
60	C₃₁×#108	C₂×C₄	1	2	yes	—
61	C₃₁×#107	—	2	1	no	—
62	C₃₁×#106	—	2	1	no	—
63	C₃₁×#104	—	4	2	no	—
64	C₃₁×#100	—	2	4	no	—
65	C₂⁴×C₃₁	—	4	1	no	—
66	#41×C₃₁	—	2	1	no	—
67	#41×C₃₁	—	2	1	no	—
68	#41×C₃₁	C₂²⋊C₄	1	2	yes	—
69	#41×C₃₁	D₈	1	2	yes	—
70	C₂×Q₈×C₃₁	SD₁₆	1	1	yes	characteristic
71	C₂×D₄×C₃₁	—	2	1	no	—
72	C₂×D₄×C₃₁	—	2	2	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	C₂×D₄×C₃₁	SD₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	—	2	1	no	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	SD₁₆×C₃₁	—	4	4	no	—
82	D₈×C₃₁	—	2	2	no	—
83	D₈×C₃₁	—	2	2	no	—
84	D₈×C₃₁	—	4	4	no	—
85	D₈×C₃₁	—	4	1	no	—
86	D₈×C₃₁	—	4	1	no	—
87	M₁₆×C₃₁	—	2	1	no	—
88	M₁₆×C₃₁	—	2	2	no	—
89	M₁₆×C₃₁	—	2	2	no	—
90	M₁₆×C₃₁	—	2	2	no	—
91	M₁₆×C₃₁	—	4	1	no	—
92	M₁₆×C₃₁	—	4	1	no	—
93	M₁₆×C₃₁	—	4	1	no	—
94	M₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₃₁	—	2	1	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	4	1	no	—
99	C₂×C₈×C₃₁	—	4	1	no	—
100	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
105	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	4	1	no	—
107	C₂²⋊C₄×C₃₁	—	4	1	no	—
108	C₄²×C₃₁	—	4	1	no	—
109	C₁₆×C₃₁	—	2	2	no	—
110	C₁₆×C₃₁	—	2	2	no	—
111	#13068	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₃₁	—	2	1	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	#135	1	1	yes	characteristic
118	Q₈×C₃₁	—	2	1	no	—
119	Q₈×C₃₁	#102	1	2	yes	—
120	D₄×C₃₁	—	2	2	no	—
121	D₄×C₃₁	—	2	2	no	—
122	D₄×C₃₁	—	2	1	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	D₄×C₃₁	#102	1	2	yes	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	8	1	no	—
132	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
138	C₈×C₃₁	—	2	2	no	—
139	C₈×C₃₁	—	4	2	no	—
140	C₈×C₃₁	—	8	1	no	—
141	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
142	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
143	C₈×C₃₁	#120	1	1	yes	characteristic
144	C₈×C₃₁	#123	1	1	yes	characteristic
145	#3407	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₂×#472	C₂×C₃₁	1	1	yes	characteristic
147	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#2140	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#1814	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1389	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1385	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂²×C₃₁	—	2	2	no	—
153	C₂²×C₃₁	—	4	1	no	—
154	C₂²×C₃₁	—	8	1	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	16	2	no	—
157	C₂²×C₃₁	—	16	2	no	—
158	C₂²×C₃₁	—	16	2	no	—
159	C₂²×C₃₁	#416	1	1	yes	characteristic
160	C₂²×C₃₁	#465	1	1	yes	characteristic
161	C₂²×C₃₁	#466	1	1	yes	characteristic
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	16	1	no	—
164	C₄×C₃₁	#386	1	1	yes	characteristic
165	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
166	C₄×C₃₁	#465	1	1	yes	characteristic
167	C₄×C₃₁	#466	1	1	yes	characteristic
168	C₂×AGL(1,ℤ/8)	—	2	1	no	—
169	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
171	#504	—	2	2	no	—
172	#504	—	2	2	no	—
173	C₂×M₃₂	—	2	2	no	—
174	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
176	#472	—	2	2	no	—
177	#472	C₄×C₃₁	1	4	yes	complemented
178	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
180	#427	C₂²×C₃₁	1	1	yes	characteristic
181	#421	—	2	1	no	—
182	C₂×#100	—	2	1	no	—
183	#361	—	2	2	no	—
184	#357	—	2	2	no	—
185	#344	—	2	1	no	—
186	C₂×C₃₁	—	2	1	no	—
187	C₂×C₃₁	—	2	1	no	—
188	C₂×C₃₁	—	2	1	no	—
189	C₂×C₃₁	—	4	2	no	—
190	C₂×C₃₁	—	16	2	no	—
191	C₂×C₃₁	#1893	1	1	yes	characteristic
192	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
193	C₂²×D₄	—	2	1	no	—
194	AGL(1,ℤ/8)	—	2	4	no	—
195	AGL(1,ℤ/8)	—	4	2	no	—
196	C₂×SD₁₆	—	4	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	—	2	1	no	—
202	C₂×D₈	—	4	1	no	—
203	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
204	#131	—	2	1	no	—
205	#131	—	2	1	no	—
206	#131	C₂×C₄×C₃₁	1	2	yes	—
207	#131	D₄×C₃₁	1	2	yes	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
214	#117	—	2	1	no	—
215	M₃₂	—	2	4	no	—
216	M₃₂	—	2	1	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	D₄×C₃₁	1	2	yes	—
220	C₂×C₁₆	—	2	4	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
223	#108	—	2	1	no	—
224	#108	—	2	1	no	—
225	#108	—	2	1	no	—
226	#108	C₂×C₄×C₃₁	1	2	yes	—
227	#107	—	2	1	no	—
228	#106	—	2	1	no	—
229	#104	—	4	2	no	—
230	#100	—	2	4	no	—
231	C₃₁	#13068	1	1	yes	characteristic, direct product factor, Sylow
232	C₂⁴	—	4	1	no	—
233	#41	—	2	1	no	—
234	#41	—	2	1	no	—
235	#41	C₂²⋊C₄×C₃₁	1	2	yes	—
236	#41	D₈×C₃₁	1	2	yes	—
237	C₂×Q₈	SD₁₆×C₃₁	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	2	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	C₂×D₄	—	4	1	no	—
245	C₂×D₄	SD₁₆×C₃₁	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
248	SD₁₆	—	4	4	no	—
249	D₈	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	4	4	no	—
252	D₈	—	4	1	no	—
253	D₈	—	4	1	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
272	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
273	C₄⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₁₆	—	2	2	no	—
277	C₁₆	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	16	2	no	—
283	C₂³	C₃₁×#135	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	C₃₁×#102	1	2	yes	—
286	D₄	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	1	no	—
292	D₄	—	8	1	no	—
293	D₄	C₃₁×#102	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
300	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	8	1	no	—
307	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
308	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
309	C₈	C₃₁×#120	1	1	yes	characteristic
310	C₈	C₃₁×#123	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	—	16	2	no	—
317	C₂²	—	16	2	no	—
318	C₂²	C₃₁×#416	1	1	yes	characteristic
319	C₂²	C₃₁×#465	1	1	yes	characteristic
320	C₂²	C₃₁×#466	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	16	1	no	—
323	C₄	C₃₁×#386	1	1	yes	characteristic
324	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
325	C₄	C₃₁×#465	1	1	yes	characteristic
326	C₄	C₃₁×#466	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	4	2	no	—
331	C₂	—	16	2	no	—
332	C₂	C₃₁×#1893	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#1893
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3407	C₂	semidirect
3. C₂×C₃₁×#361	C₂	semidirect
4. C₃₁×#2140	C₂	semidirect
5. C₃₁×#1814	C₂	semidirect
6. C₃₁×#1389	C₂	semidirect
7. C₃₁×#1385	C₂	semidirect
17. C₃₁×#472	C₄	semidirect
111. #13068	C₃₁	direct (N × H)
145. #3407	C₂×C₃₁	semidirect
147. C₂×#361	C₂×C₃₁	semidirect
148. #2140	C₂×C₃₁	semidirect
149. #1814	C₂×C₃₁	semidirect
150. #1389	C₂×C₃₁	semidirect
151. #1385	C₂×C₃₁	semidirect
177. #472	C₄×C₃₁	semidirect
231. C₃₁	#13068	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×#3407	2
C₂×C₃₁×#472	2
C₂×C₃₁×#361	2
C₃₁×#2140	2
C₃₁×#1814	2
C₃₁×#1389	2
C₃₁×#1385	2
#13068	31

Subgroups — C31×#13068 #21469827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13068 #21469827