Subgroups — C₃₁×#33692 #21490451

Order 7936.

536 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×#3176	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2955	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2238	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2231	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2213	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2190	C₂	1	2	yes	maximal, complemented
7	C₃₁×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2166	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#575	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#131	—	2	1	no	—
11	C₃₁×#540	—	2	1	no	—
12	C₃₁×#509	C₂²	1	4	yes	—
13	C₃₁×#499	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#490	—	2	1	no	—
16	C₃₁×#469	C₂²	1	2	yes	—
17	C₃₁×#466	—	2	1	no	—
18	C₃₁×#444	—	2	1	no	—
19	C₃₁×#444	C₂²	1	2	yes	—
20	C₃₁×#443	—	2	1	no	—
21	C₃₁×#443	C₂²	1	2	yes	—
22	C₃₁×#442	C₂²	1	2	yes	complemented
23	C₄×D₈×C₃₁	—	2	1	no	—
24	C₃₁×#430	C₂²	1	1	yes	characteristic
25	C₃₁×#421	—	2	1	no	—
26	C₃₁×#405	C₂²	1	1	yes	characteristic
27	C₃₁×#365	C₂²	1	4	yes	complemented
28	C₃₁×#364	C₂²	1	2	yes	complemented
29	C₃₁×#361	C₂²	1	4	yes	complemented
30	C₃₁×#360	C₂²	1	4	yes	—
31	C₃₁×#349	C₂²	1	2	yes	—
32	C₃₁×#347	C₂²	1	2	yes	complemented
33	C₂×#41×C₃₁	—	2	1	no	—
34	C₃₁×#137	—	2	2	no	—
35	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
36	C₃₁×#135	—	2	1	no	—
37	C₃₁×#135	C₂³	1	2	yes	—
38	C₂×D₈×C₃₁	—	2	1	no	—
39	C₂×D₈×C₃₁	C₂³	1	2	yes	—
40	C₃₁×#131	—	2	2	no	—
41	C₃₁×#131	—	2	2	no	—
42	C₃₁×#131	—	4	1	no	—
43	C₃₁×#131	—	4	1	no	—
44	C₂×M₁₆×C₃₁	—	2	1	no	—
45	C₂×M₁₆×C₃₁	—	4	1	no	—
46	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	—	4	1	no	—
48	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#125	—	2	1	no	—
51	C₃₁×#123	—	4	2	no	—
52	C₃₁×#121	—	2	2	no	—
53	C₃₁×#121	—	4	2	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₃₁×#117	D₄	1	1	yes	characteristic
56	C₃₁×SD₃₂	—	2	4	no	—
57	C₃₁×D₁₆	—	2	4	no	—
58	C₃₁×M₃₂	C₂³	1	4	yes	—
59	C₃₁×#108	—	2	1	no	—
60	C₃₁×#108	C₂³	1	2	yes	—
61	C₃₁×#107	—	2	1	no	—
62	C₃₁×#106	C₂³	1	2	yes	—
63	C₃₁×#104	—	2	2	no	—
64	C₃₁×#104	—	2	2	no	—
65	C₃₁×#102	—	2	2	no	—
66	C₃₁×#102	—	2	2	no	—
67	C₃₁×#97	D₄	1	1	yes	characteristic
68	C₃₁×#97	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	#41×C₃₁	—	2	2	no	—
72	#41×C₃₁	—	2	2	no	—
73	#41×C₃₁	—	4	1	no	—
74	#41×C₃₁	—	4	1	no	—
75	C₂×Q₈×C₃₁	—	2	1	no	—
76	C₂×D₄×C₃₁	—	2	1	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	4	2	no	—
79	C₂×D₄×C₃₁	—	4	1	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₃₁	—	2	1	no	—
85	Q₁₆×C₃₁	#41	1	2	yes	—
86	SD₁₆×C₃₁	—	2	2	no	—
87	SD₁₆×C₃₁	—	2	2	no	—
88	D₈×C₃₁	—	2	1	no	—
89	D₈×C₃₁	—	2	4	no	—
90	D₈×C₃₁	—	4	1	no	—
91	D₈×C₃₁	—	4	1	no	—
92	D₈×C₃₁	#41	1	2	yes	—
93	M₁₆×C₃₁	—	2	2	no	—
94	M₁₆×C₃₁	—	2	2	no	—
95	M₁₆×C₃₁	—	4	2	no	—
96	M₁₆×C₃₁	—	4	1	no	—
97	M₁₆×C₃₁	—	4	2	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	4	2	no	—
101	C₂×C₈×C₃₁	—	4	1	no	—
102	C₂×C₈×C₃₁	—	4	2	no	—
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	2	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₃₁	—	2	4	no	—
116	#33692	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	4	1	no	—
118	C₂³×C₃₁	—	4	2	no	—
119	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₃₁	—	2	2	no	—
121	Q₈×C₃₁	—	4	1	no	—
122	D₄×C₃₁	—	2	2	no	—
123	D₄×C₃₁	—	4	1	no	—
124	D₄×C₃₁	—	4	2	no	—
125	D₄×C₃₁	—	4	2	no	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	16	2	no	—
129	D₄×C₃₁	—	16	2	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	16	2	no	—
139	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	—	4	2	no	—
144	C₈×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₈×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₈×C₃₁	C₂×#41	1	2	yes	—
147	#3176	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#2955	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#2238	C₂×C₃₁	1	2	yes	complemented
150	#2231	C₂×C₃₁	1	2	yes	complemented
151	#2213	C₂×C₃₁	1	2	yes	complemented
152	#2190	C₂×C₃₁	1	2	yes	complemented
153	AGL(1,ℤ/16)	C₂×C₃₁	1	4	yes	complemented
154	#2166	C₂×C₃₁	1	1	yes	characteristic, complemented
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	4	1	no	—
157	C₂²×C₃₁	—	4	2	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	4	1	no	—
165	C₄×C₃₁	—	4	1	no	—
166	C₄×C₃₁	—	4	2	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	#525	1	1	yes	characteristic
169	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	#575	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×#131	—	2	1	no	—
172	#540	—	2	1	no	—
173	#509	C₂²×C₃₁	1	4	yes	—
174	#499	C₂²×C₃₁	1	1	yes	characteristic, complemented
175	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
176	#490	—	2	1	no	—
177	#469	C₂²×C₃₁	1	2	yes	—
178	#466	—	2	1	no	—
179	#444	—	2	1	no	—
180	#444	C₂²×C₃₁	1	2	yes	—
181	#443	—	2	1	no	—
182	#443	C₂²×C₃₁	1	2	yes	—
183	#442	C₂²×C₃₁	1	2	yes	complemented
184	C₄×D₈	—	2	1	no	—
185	#430	C₂²×C₃₁	1	1	yes	characteristic
186	#421	—	2	1	no	—
187	#405	C₂²×C₃₁	1	1	yes	characteristic
188	#365	C₂²×C₃₁	1	4	yes	complemented
189	#364	C₂²×C₃₁	1	2	yes	complemented
190	#361	C₂²×C₃₁	1	4	yes	complemented
191	#360	C₂²×C₃₁	1	4	yes	—
192	#349	C₂²×C₃₁	1	2	yes	—
193	#347	C₂²×C₃₁	1	2	yes	complemented
194	C₂×C₃₁	—	2	1	no	—
195	C₂×C₃₁	—	4	1	no	—
196	C₂×C₃₁	—	8	2	no	—
197	C₂×C₃₁	—	16	2	no	—
198	C₂×C₃₁	#3109	1	1	yes	characteristic
199	C₂×#41	—	2	1	no	—
200	#137	—	2	2	no	—
201	AGL(1,ℤ/8)	—	2	2	no	—
202	#135	—	2	1	no	—
203	#135	C₂³×C₃₁	1	2	yes	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	C₂³×C₃₁	1	2	yes	—
206	#131	—	2	2	no	—
207	#131	—	2	2	no	—
208	#131	—	4	1	no	—
209	#131	—	4	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	4	1	no	—
212	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
213	C₂²×C₈	—	4	1	no	—
214	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
215	#127	—	2	1	no	—
216	#125	—	2	1	no	—
217	#123	—	4	2	no	—
218	#121	—	2	2	no	—
219	#121	—	4	2	no	—
220	C₄×D₄	—	2	2	no	—
221	#117	D₄×C₃₁	1	1	yes	characteristic
222	SD₃₂	—	2	4	no	—
223	D₁₆	—	2	4	no	—
224	M₃₂	C₂³×C₃₁	1	4	yes	—
225	#108	—	2	1	no	—
226	#108	C₂³×C₃₁	1	2	yes	—
227	#107	—	2	1	no	—
228	#106	C₂³×C₃₁	1	2	yes	—
229	#104	—	2	2	no	—
230	#104	—	2	2	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#97	D₄×C₃₁	1	1	yes	characteristic
234	#97	C₂³×C₃₁	1	1	yes	characteristic
235	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
237	C₃₁	#33692	1	1	yes	characteristic, direct product factor, Sylow
238	#41	—	2	2	no	—
239	#41	—	2	2	no	—
240	#41	—	4	1	no	—
241	#41	—	4	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	Q₁₆	—	2	1	no	—
252	Q₁₆	#41×C₃₁	1	2	yes	—
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	2	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	4	no	—
257	D₈	—	4	1	no	—
258	D₈	—	4	1	no	—
259	D₈	#41×C₃₁	1	2	yes	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	2	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₁₆	—	2	4	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	16	2	no	—
295	D₄	—	16	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₈	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₈	C₂×#41×C₃₁	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₃₁×#525	1	1	yes	characteristic
327	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	4	1	no	—
330	C₂	—	8	2	no	—
331	C₂	—	16	2	no	—
332	C₂	C₃₁×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3176	C₂	semidirect
2. C₃₁×#2955	C₂	semidirect
3. C₃₁×#2238	C₂	semidirect
4. C₃₁×#2231	C₂	semidirect
5. C₃₁×#2213	C₂	semidirect
6. C₃₁×#2190	C₂	semidirect
7. C₃₁×AGL(1,ℤ/16)	C₂	semidirect
8. C₃₁×#2166	C₂	semidirect
13. C₃₁×#499	C₂²	semidirect
14. C₃₁×#493	C₂²	semidirect
22. C₃₁×#442	C₂²	semidirect
27. C₃₁×#365	C₂²	semidirect
28. C₃₁×#364	C₂²	semidirect
29. C₃₁×#361	C₂²	semidirect
32. C₃₁×#347	C₂²	semidirect
116. #33692	C₃₁	direct (N × H)
147. #3176	C₂×C₃₁	semidirect
148. #2955	C₂×C₃₁	semidirect
149. #2238	C₂×C₃₁	semidirect
150. #2231	C₂×C₃₁	semidirect
151. #2213	C₂×C₃₁	semidirect
152. #2190	C₂×C₃₁	semidirect
153. AGL(1,ℤ/16)	C₂×C₃₁	semidirect
154. #2166	C₂×C₃₁	semidirect
174. #499	C₂²×C₃₁	semidirect
175. #493	C₂²×C₃₁	semidirect
183. #442	C₂²×C₃₁	semidirect
188. #365	C₂²×C₃₁	semidirect
189. #364	C₂²×C₃₁	semidirect
190. #361	C₂²×C₃₁	semidirect
193. #347	C₂²×C₃₁	semidirect
237. C₃₁	#33692	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×#3176	2
C₃₁×#2955	2
C₃₁×#2238	2
C₃₁×#2231	2
C₃₁×#2213	2
C₃₁×#2190	2
C₃₁×AGL(1,ℤ/16)	2
C₃₁×#2166	2
#33692	31

Subgroups — C31×#33692 #21490451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33692 #21490451