Subgroups — C₃₁×#33637 #21490396

Order 7936.

524 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2231	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2230	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2155	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2145	C₂	1	2	yes	maximal
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#131	—	2	1	no	—
10	C₃₁×#533	—	2	1	no	—
11	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
12	C₃₁×#500	—	2	1	no	—
13	C₃₁×#499	—	2	1	no	—
14	C₃₁×#499	C₂²	1	2	yes	complemented
15	C₃₁×#498	C₂²	1	2	yes	complemented
16	C₃₁×#443	—	2	1	no	—
17	C₃₁×#443	—	2	1	no	—
18	C₃₁×#443	C₂²	1	2	yes	—
19	C₃₁×#443	C₂²	1	2	yes	—
20	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
21	C₃₁×#427	—	2	1	no	—
22	C₃₁×#427	C₂²	1	2	yes	—
23	C₂×C₃₁×#104	—	2	1	no	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#365	C₂²	1	8	yes	complemented
26	C₃₁×#364	C₂²	1	2	yes	complemented
27	C₃₁×#364	C₂²	1	2	yes	complemented
28	C₃₁×#360	C₂²	1	8	yes	—
29	C₂×#41×C₃₁	—	2	1	no	—
30	C₃₁×#135	—	2	1	no	—
31	C₃₁×#135	—	2	1	no	—
32	C₃₁×#135	Q₈	1	2	yes	complemented
33	C₃₁×#135	C₂³	1	2	yes	—
34	C₂×Q₁₆×C₃₁	Q₈	1	1	yes	characteristic, complemented
35	C₂×SD₁₆×C₃₁	—	2	1	no	—
36	C₂×D₈×C₃₁	Q₈	1	1	yes	characteristic, complemented
37	C₃₁×#131	—	2	2	no	—
38	C₃₁×#131	—	2	2	no	—
39	C₃₁×#131	—	4	1	no	—
40	C₃₁×#131	—	4	1	no	—
41	C₂×M₁₆×C₃₁	—	2	1	no	—
42	C₂×M₁₆×C₃₁	—	4	1	no	—
43	C₂²×C₈×C₃₁	—	4	1	no	—
44	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×#125	—	2	2	no	—
48	C₃₁×C₂²⋊Q₈	—	4	2	no	—
49	C₄×Q₈×C₃₁	—	4	2	no	—
50	C₃₁×#117	—	2	2	no	—
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₃₁×M₃₂	—	2	2	no	—
53	C₃₁×M₃₂	C₂³	1	4	yes	—
54	C₂×C₁₆×C₃₁	—	2	2	no	—
55	C₃₁×#108	—	2	1	no	—
56	C₃₁×#108	C₂³	1	2	yes	—
57	C₃₁×#107	—	2	2	no	—
58	C₃₁×#107	—	2	2	no	—
59	C₃₁×#106	—	2	2	no	—
60	C₃₁×#106	C₂³	1	4	yes	—
61	C₃₁×#104	—	2	2	no	—
62	C₃₁×#104	—	2	2	no	—
63	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
65	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	#41×C₃₁	—	2	2	no	—
68	#41×C₃₁	—	2	2	no	—
69	#41×C₃₁	—	4	1	no	—
70	#41×C₃₁	—	4	1	no	—
71	C₂×Q₈×C₃₁	—	2	1	no	—
72	C₂×Q₈×C₃₁	—	4	2	no	—
73	C₂×D₄×C₃₁	—	2	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₃₁	—	2	1	no	—
79	Q₁₆×C₃₁	C₂×Q₈	1	2	yes	—
80	SD₁₆×C₃₁	—	2	2	no	—
81	SD₁₆×C₃₁	—	2	2	no	—
82	D₈×C₃₁	—	2	1	no	—
83	D₈×C₃₁	C₂×Q₈	1	2	yes	—
84	M₁₆×C₃₁	—	2	2	no	—
85	M₁₆×C₃₁	—	4	1	no	—
86	M₁₆×C₃₁	—	4	2	no	—
87	M₁₆×C₃₁	—	4	2	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	4	2	no	—
90	C₂×C₈×C₃₁	—	4	2	no	—
91	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
92	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
93	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₃₁	SD₁₆	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	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
101	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₄⋊C₄×C₃₁	—	4	4	no	—
105	C₄⋊C₄×C₃₁	—	4	2	no	—
106	C₂²⋊C₄×C₃₁	—	4	2	no	—
107	C₄²×C₃₁	—	2	1	no	—
108	C₄²×C₃₁	—	4	2	no	—
109	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₁₆×C₃₁	—	2	4	no	—
112	#33637	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	—	4	1	no	—
114	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
115	Q₈×C₃₁	—	2	2	no	—
116	Q₈×C₃₁	—	4	1	no	—
117	Q₈×C₃₁	—	8	4	no	—
118	D₄×C₃₁	—	2	2	no	—
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₃₁	—	4	2	no	—
136	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
139	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
140	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
141	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
142	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
144	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
146	#2231	C₂×C₃₁	1	4	yes	complemented
147	#2230	C₂×C₃₁	1	4	yes	complemented
148	#2155	C₂×C₃₁	1	2	yes	complemented
149	C₂×#364	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2145	C₂×C₃₁	1	2	yes	—
151	C₂²×C₃₁	—	4	1	no	—
152	C₂²×C₃₁	—	4	2	no	—
153	C₂²×C₃₁	—	8	2	no	—
154	C₂²×C₃₁	#476	1	1	yes	characteristic
155	C₂²×C₃₁	#476	1	1	yes	characteristic
156	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₄×C₃₁	—	2	2	no	—
158	C₄×C₃₁	—	4	1	no	—
159	C₄×C₃₁	—	4	2	no	—
160	C₄×C₃₁	—	8	4	no	—
161	C₄×C₃₁	#476	1	1	yes	characteristic
162	C₄×C₃₁	#476	1	1	yes	characteristic
163	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
164	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
165	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#131	—	2	1	no	—
167	#533	—	2	1	no	—
168	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
169	#500	—	2	1	no	—
170	#499	—	2	1	no	—
171	#499	C₂²×C₃₁	1	2	yes	complemented
172	#498	C₂²×C₃₁	1	2	yes	complemented
173	#443	—	2	1	no	—
174	#443	—	2	1	no	—
175	#443	C₂²×C₃₁	1	2	yes	—
176	#443	C₂²×C₃₁	1	2	yes	—
177	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
178	#427	—	2	1	no	—
179	#427	C₂²×C₃₁	1	2	yes	—
180	C₂×#104	—	2	1	no	—
181	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
182	#365	C₂²×C₃₁	1	8	yes	complemented
183	#364	C₂²×C₃₁	1	2	yes	complemented
184	#364	C₂²×C₃₁	1	2	yes	complemented
185	#360	C₂²×C₃₁	1	8	yes	—
186	C₂×C₃₁	—	2	1	no	—
187	C₂×C₃₁	—	2	1	no	—
188	C₂×C₃₁	—	2	1	no	—
189	C₂×C₃₁	—	8	2	no	—
190	C₂×C₃₁	C₂×#476	1	1	yes	characteristic
191	C₂×#41	—	2	1	no	—
192	#135	—	2	1	no	—
193	#135	—	2	1	no	—
194	#135	Q₈×C₃₁	1	2	yes	complemented
195	#135	C₂³×C₃₁	1	2	yes	—
196	C₂×Q₁₆	Q₈×C₃₁	1	1	yes	characteristic, complemented
197	C₂×SD₁₆	—	2	1	no	—
198	C₂×D₈	Q₈×C₃₁	1	1	yes	characteristic, complemented
199	#131	—	2	2	no	—
200	#131	—	2	2	no	—
201	#131	—	4	1	no	—
202	#131	—	4	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	4	1	no	—
205	C₂²×C₈	—	4	1	no	—
206	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	2	no	—
209	#125	—	2	2	no	—
210	C₂²⋊Q₈	—	4	2	no	—
211	C₄×Q₈	—	4	2	no	—
212	#117	—	2	2	no	—
213	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
214	M₃₂	—	2	2	no	—
215	M₃₂	C₂³×C₃₁	1	4	yes	—
216	C₂×C₁₆	—	2	2	no	—
217	#108	—	2	1	no	—
218	#108	C₂³×C₃₁	1	2	yes	—
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	#106	—	2	2	no	—
222	#106	C₂³×C₃₁	1	4	yes	—
223	#104	—	2	2	no	—
224	#104	—	2	2	no	—
225	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
226	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
229	C₃₁	#33637	1	1	yes	characteristic, direct product factor, Sylow
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	#41	—	4	1	no	—
233	#41	—	4	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	Q₁₆	—	2	1	no	—
242	Q₁₆	C₂×Q₈×C₃₁	1	2	yes	—
243	SD₁₆	—	2	2	no	—
244	SD₁₆	—	2	2	no	—
245	D₈	—	2	1	no	—
246	D₈	C₂×Q₈×C₃₁	1	2	yes	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
255	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
256	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
257	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
262	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
263	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
264	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₁₆	—	2	4	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	8	4	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
292	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
293	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
299	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
300	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
301	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
302	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
303	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
304	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
305	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₃₁×#476	1	1	yes	characteristic
310	C₂²	C₃₁×#476	1	1	yes	characteristic
311	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	8	4	no	—
316	C₄	C₃₁×#476	1	1	yes	characteristic
317	C₄	C₃₁×#476	1	1	yes	characteristic
318	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
319	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×C₃₁×#476	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×#476
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3160	C₂	semidirect
3. C₃₁×#2231	C₂	semidirect
4. C₃₁×#2230	C₂	semidirect
5. C₃₁×#2155	C₂	semidirect
6. C₂×C₃₁×#364	C₂	semidirect
14. C₃₁×#499	C₂²	semidirect
15. C₃₁×#498	C₂²	semidirect
25. C₃₁×#365	C₂²	semidirect
26. C₃₁×#364	C₂²	semidirect
27. C₃₁×#364	C₂²	semidirect
32. C₃₁×#135	Q₈	semidirect
34. C₂×Q₁₆×C₃₁	Q₈	semidirect
36. C₂×D₈×C₃₁	Q₈	semidirect
112. #33637	C₃₁	direct (N × H)
144. #3160	C₂×C₃₁	semidirect
146. #2231	C₂×C₃₁	semidirect
147. #2230	C₂×C₃₁	semidirect
148. #2155	C₂×C₃₁	semidirect
149. C₂×#364	C₂×C₃₁	semidirect
171. #499	C₂²×C₃₁	semidirect
172. #498	C₂²×C₃₁	semidirect
182. #365	C₂²×C₃₁	semidirect
183. #364	C₂²×C₃₁	semidirect
184. #364	C₂²×C₃₁	semidirect
194. #135	Q₈×C₃₁	semidirect
196. C₂×Q₁₆	Q₈×C₃₁	semidirect
198. C₂×D₈	Q₈×C₃₁	semidirect
229. C₃₁	#33637	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×#3160	2
C₂×C₃₁×#443	2
C₃₁×#2231	2
C₃₁×#2230	2
C₃₁×#2155	2
C₂×C₃₁×#364	2
C₃₁×#2145	2
#33637	31

Subgroups — C31×#33637 #21490396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33637 #21490396