Subgroups — C₃₁×#13231 #21469990

Order 7936.

468 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3058	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2128	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2122	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1812	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1757	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
7	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
8	C₃₁×#535	C₂²	1	2	yes	—
9	C₃₁×#469	—	2	1	no	—
10	C₃₁×#469	—	2	1	no	—
11	C₃₁×#466	—	2	1	no	—
12	C₃₁×#466	—	2	1	no	—
13	C₃₁×#427	—	2	1	no	—
14	C₃₁×#427	C₄	1	1	yes	characteristic, complemented
15	C₃₁×#418	C₄	1	2	yes	complemented
16	C₃₁×#413	C₂²	1	1	yes	characteristic
17	C₃₁×#410	C₂²	1	2	yes	—
18	C₃₁×#410	C₂²	1	1	yes	characteristic
19	C₃₁×#405	—	2	1	no	—
20	C₃₁×#353	—	2	4	no	—
21	C₃₁×AΣL(1,𝔽₁₆)	—	2	4	no	—
22	C₃₁×#344	—	2	1	no	—
23	C₃₁×H(3,ℤ/4)	—	2	1	no	—
24	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×D₈×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	—	4	1	no	—
29	C₂×D₈×C₃₁	—	4	1	no	—
30	C₃₁×#131	—	4	1	no	—
31	C₃₁×#131	—	4	1	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₃₁×#127	—	2	2	no	—
40	C₃₁×#124	—	2	2	no	—
41	C₃₁×#121	—	2	2	no	—
42	C₃₁×#121	—	2	2	no	—
43	C₃₁×#120	—	2	2	no	—
44	C₃₁×#120	—	2	2	no	—
45	C₃₁×#117	—	2	1	no	—
46	C₃₁×#117	—	2	2	no	—
47	C₃₁×#117	C₂×C₄	1	2	yes	—
48	C₃₁×#107	—	2	1	no	—
49	C₃₁×#107	—	4	1	no	—
50	C₃₁×#106	—	2	1	no	—
51	C₃₁×#106	—	4	1	no	—
52	C₃₁×#102	—	2	2	no	—
53	C₃₁×#102	—	2	2	no	—
54	C₃₁×#101	—	2	1	no	—
55	C₃₁×#100	D₄	1	2	yes	—
56	C₃₁×#99	—	2	1	no	—
57	C₃₁×#99	—	2	2	no	—
58	C₃₁×#99	—	2	2	no	—
59	C₃₁×#99	D₄	1	2	yes	—
60	C₃₁×#97	—	4	1	no	—
61	C₄×C₈×C₃₁	—	4	1	no	—
62	C₂⁴×C₃₁	—	2	2	no	—
63	#41×C₃₁	—	4	1	no	—
64	#41×C₃₁	—	4	1	no	—
65	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	4	2	no	—
69	C₂×D₄×C₃₁	—	4	2	no	—
70	C₂×D₄×C₃₁	—	4	2	no	—
71	C₂×D₄×C₃₁	—	4	2	no	—
72	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	D₈×C₃₁	—	4	2	no	—
79	D₈×C₃₁	—	4	1	no	—
80	D₈×C₃₁	—	4	1	no	—
81	M₁₆×C₃₁	—	2	2	no	—
82	M₁₆×C₃₁	—	4	1	no	—
83	M₁₆×C₃₁	—	4	1	no	—
84	M₁₆×C₃₁	—	4	1	no	—
85	M₁₆×C₃₁	—	4	1	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	2	1	no	—
89	C₂×C₈×C₃₁	—	2	1	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	—	2	1	no	—
92	C₂×C₈×C₃₁	—	4	1	no	—
93	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₂²⋊C₄×C₃₁	—	2	2	no	—
101	C₂²⋊C₄×C₃₁	—	2	2	no	—
102	C₂²⋊C₄×C₃₁	—	4	2	no	—
103	C₂²⋊C₄×C₃₁	—	4	2	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	4	1	no	—
107	C₄²×C₃₁	—	4	2	no	—
108	#13231	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₃₁	—	4	2	no	—
110	C₂³×C₃₁	—	8	2	no	—
111	C₂³×C₃₁	—	16	2	no	—
112	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
113	C₂³×C₃₁	#121	1	1	yes	characteristic
114	C₂³×C₃₁	#127	1	1	yes	characteristic
115	Q₈×C₃₁	—	4	1	no	—
116	D₄×C₃₁	—	4	1	no	—
117	D₄×C₃₁	—	4	1	no	—
118	D₄×C₃₁	—	4	1	no	—
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	8	2	no	—
122	D₄×C₃₁	—	16	2	no	—
123	D₄×C₃₁	—	16	2	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	8	2	no	—
133	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₈×C₃₁	—	2	1	no	—
140	C₈×C₃₁	—	4	2	no	—
141	C₈×C₃₁	#99	1	1	yes	characteristic
142	C₈×C₃₁	#99	1	1	yes	characteristic
143	#3058	C₂×C₃₁	1	1	yes	characteristic
144	#2128	C₂×C₃₁	1	2	yes	complemented
145	#2122	C₂×C₃₁	1	2	yes	complemented
146	#1812	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1757	C₂×C₃₁	1	1	yes	characteristic, complemented
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	4	1	no	—
152	C₂²×C₃₁	—	4	1	no	—
153	C₂²×C₃₁	—	4	1	no	—
154	C₂²×C₃₁	—	8	2	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	8	2	no	—
157	C₂²×C₃₁	—	16	4	no	—
158	C₂²×C₃₁	#390	1	1	yes	characteristic
159	C₄×C₃₁	—	2	1	no	—
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	—	16	2	no	—
165	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
166	C₄×C₃₁	#410	1	1	yes	characteristic
167	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
168	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
169	#535	C₂²×C₃₁	1	2	yes	—
170	#469	—	2	1	no	—
171	#469	—	2	1	no	—
172	#466	—	2	1	no	—
173	#466	—	2	1	no	—
174	#427	—	2	1	no	—
175	#427	C₄×C₃₁	1	1	yes	characteristic, complemented
176	#418	C₄×C₃₁	1	2	yes	complemented
177	#413	C₂²×C₃₁	1	1	yes	characteristic
178	#410	C₂²×C₃₁	1	2	yes	—
179	#410	C₂²×C₃₁	1	1	yes	characteristic
180	#405	—	2	1	no	—
181	#353	—	2	4	no	—
182	AΣL(1,𝔽₁₆)	—	2	4	no	—
183	#344	—	2	1	no	—
184	H(3,ℤ/4)	—	2	1	no	—
185	C₂×C₃₁	—	2	1	no	—
186	C₂×C₃₁	—	4	1	no	—
187	C₂×C₃₁	—	4	1	no	—
188	C₂×C₃₁	—	4	1	no	—
189	C₂×C₃₁	—	16	2	no	—
190	C₂×C₃₁	#1964	1	1	yes	characteristic
191	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
192	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	4	1	no	—
196	C₂×D₈	—	4	1	no	—
197	#131	—	4	1	no	—
198	#131	—	4	1	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
201	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
202	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
206	#127	—	2	2	no	—
207	#124	—	2	2	no	—
208	#121	—	2	2	no	—
209	#121	—	2	2	no	—
210	#120	—	2	2	no	—
211	#120	—	2	2	no	—
212	#117	—	2	1	no	—
213	#117	—	2	2	no	—
214	#117	C₂×C₄×C₃₁	1	2	yes	—
215	#107	—	2	1	no	—
216	#107	—	4	1	no	—
217	#106	—	2	1	no	—
218	#106	—	4	1	no	—
219	#102	—	2	2	no	—
220	#102	—	2	2	no	—
221	#101	—	2	1	no	—
222	#100	D₄×C₃₁	1	2	yes	—
223	#99	—	2	1	no	—
224	#99	—	2	2	no	—
225	#99	—	2	2	no	—
226	#99	D₄×C₃₁	1	2	yes	—
227	#97	—	4	1	no	—
228	C₄×C₈	—	4	1	no	—
229	C₃₁	#13231	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	—	2	2	no	—
231	#41	—	4	1	no	—
232	#41	—	4	1	no	—
233	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
246	D₈	—	4	2	no	—
247	D₈	—	4	1	no	—
248	D₈	—	4	1	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	8	2	no	—
278	C₂³	—	16	2	no	—
279	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
280	C₂³	C₃₁×#121	1	1	yes	characteristic
281	C₂³	C₃₁×#127	1	1	yes	characteristic
282	Q₈	—	4	1	no	—
283	D₄	—	4	1	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	16	2	no	—
290	D₄	—	16	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
303	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	4	2	no	—
308	C₈	C₃₁×#99	1	1	yes	characteristic
309	C₈	C₃₁×#99	1	1	yes	characteristic
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	16	4	no	—
320	C₂²	C₃₁×#390	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	16	2	no	—
327	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
328	C₄	C₃₁×#410	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	4	1	no	—
332	C₂	—	4	1	no	—
333	C₂	—	16	2	no	—
334	C₂	C₃₁×#1964	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#1964
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×#41	C₂³×C₃₁
prime radical	C₃₁×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2128	C₂	semidirect
3. C₃₁×#2122	C₂	semidirect
4. C₃₁×#1812	C₂	semidirect
5. C₃₁×#1757	C₂	semidirect
6. C₂²×D₈×C₃₁	C₄	semidirect
14. C₃₁×#427	C₄	semidirect
15. C₃₁×#418	C₄	semidirect
108. #13231	C₃₁	direct (N × H)
144. #2128	C₂×C₃₁	semidirect
145. #2122	C₂×C₃₁	semidirect
146. #1812	C₂×C₃₁	semidirect
147. #1757	C₂×C₃₁	semidirect
167. C₂²×D₈	C₄×C₃₁	semidirect
175. #427	C₄×C₃₁	semidirect
176. #418	C₄×C₃₁	semidirect
229. C₃₁	#13231	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×#3058	2
C₃₁×#2128	2
C₃₁×#2122	2
C₃₁×#1812	2
C₃₁×#1757	2
#13231	31

Subgroups — C31×#13231 #21469990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13231 #21469990