Subgroups — C₃₁×#33781 #21490540

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2251	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2243	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2211	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2188	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2187	C₂	1	2	yes	maximal
8	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₃₁	—	2	1	no	—
11	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#533	—	2	1	no	—
13	C₂×C₃₁×#124	—	2	1	no	—
14	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	complemented
15	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#495	—	2	1	no	—
18	C₃₁×#495	—	2	1	no	—
19	C₃₁×#495	C₂²	1	2	yes	—
20	C₃₁×#495	C₂²	1	2	yes	—
21	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#487	—	2	1	no	—
23	C₃₁×#467	—	2	1	no	—
24	C₃₁×#467	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₃₁	—	2	1	no	—
26	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	C₃₁×#358	C₂²	1	4	yes	complemented
30	C₃₁×#357	C₂²	1	4	yes	—
31	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₃₁×#346	C₂²	1	2	yes	complemented
33	C₂²×Q₈×C₃₁	—	2	1	no	—
34	C₂²×D₄×C₃₁	—	2	1	no	—
35	C₂×Q₁₆×C₃₁	—	2	1	no	—
36	C₂×Q₁₆×C₃₁	—	2	1	no	—
37	C₂×Q₁₆×C₃₁	—	4	2	no	—
38	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
39	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₃₁	—	2	2	no	—
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₂×SD₁₆×C₃₁	—	4	4	no	—
43	C₂×D₈×C₃₁	—	2	1	no	—
44	C₂×D₈×C₃₁	—	4	2	no	—
45	C₂×D₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	—	2	1	no	—
49	C₃₁×C₄⋊Q₈	—	2	1	no	—
50	C₃₁×#125	—	4	1	no	—
51	C₃₁×#124	—	2	2	no	—
52	C₃₁×#124	—	2	2	no	—
53	C₃₁×#124	—	4	1	no	—
54	C₃₁×#124	—	4	1	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	4	1	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	4	1	no	—
59	C₃₁×#117	—	4	1	no	—
60	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
62	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
63	C₃₁×Q₃₂	—	4	4	no	—
64	C₃₁×SD₃₂	—	4	4	no	—
65	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	C₂³	1	2	yes	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#103	—	2	2	no	—
70	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂⁴×C₃₁	—	4	1	no	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	4	2	no	—
78	C₂×Q₈×C₃₁	—	4	2	no	—
79	C₂×Q₈×C₃₁	—	4	1	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	4	2	no	—
83	C₂×D₄×C₃₁	—	4	2	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂²×C₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₃₁	—	2	4	no	—
90	Q₁₆×C₃₁	—	2	4	no	—
91	Q₁₆×C₃₁	—	4	2	no	—
92	Q₁₆×C₃₁	—	4	2	no	—
93	SD₁₆×C₃₁	—	4	4	no	—
94	SD₁₆×C₃₁	—	4	4	no	—
95	D₈×C₃₁	—	2	4	no	—
96	D₈×C₃₁	—	4	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	4	1	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₄⋊C₄×C₃₁	—	4	2	no	—
108	C₄⋊C₄×C₃₁	—	4	1	no	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	2	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	D₈	1	1	yes	characteristic
116	C₄²×C₃₁	D₈	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₃₁	—	2	4	no	—
120	#33781	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	—	8	2	no	—
123	C₂³×C₃₁	—	8	2	no	—
124	C₂³×C₃₁	—	8	2	no	—
125	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	4	2	no	—
128	Q₈×C₃₁	—	4	2	no	—
129	Q₈×C₃₁	—	4	2	no	—
130	Q₈×C₃₁	—	8	2	no	—
131	Q₈×C₃₁	—	8	2	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	4	2	no	—
134	D₄×C₃₁	—	8	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	4	1	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	8	2	no	—
143	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₃₁	—	2	1	no	—
150	C₈×C₃₁	—	2	1	no	—
151	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
152	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
153	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
154	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
155	#2251	C₂×C₃₁	1	2	yes	complemented
156	#2243	C₂×C₃₁	1	2	yes	complemented
157	#2211	C₂×C₃₁	1	4	yes	complemented
158	#2188	C₂×C₃₁	1	2	yes	complemented
159	#2187	C₂×C₃₁	1	2	yes	—
160	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
161	C₂²×C₃₁	—	2	1	no	—
162	C₂²×C₃₁	—	2	1	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	16	4	no	—
167	C₂²×C₃₁	#510	1	2	yes	—
168	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₃₁	—	2	1	no	—
170	C₄×C₃₁	—	2	1	no	—
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	8	2	no	—
175	C₄×C₃₁	—	8	2	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	#535	1	1	yes	characteristic
178	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
180	C₂²×SD₁₆	—	2	1	no	—
181	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
182	#533	—	2	1	no	—
183	C₂×#124	—	2	1	no	—
184	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	complemented
185	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
186	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
187	#495	—	2	1	no	—
188	#495	—	2	1	no	—
189	#495	C₂²×C₃₁	1	2	yes	—
190	#495	C₂²×C₃₁	1	2	yes	—
191	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
192	#487	—	2	1	no	—
193	#467	—	2	1	no	—
194	#467	C₂²×C₃₁	1	2	yes	—
195	C₄×Q₁₆	—	2	1	no	—
196	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
197	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
198	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
199	#358	C₂²×C₃₁	1	4	yes	complemented
200	#357	C₂²×C₃₁	1	4	yes	—
201	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
202	#346	C₂²×C₃₁	1	2	yes	complemented
203	C₂×C₃₁	—	2	2	no	—
204	C₂×C₃₁	—	16	2	no	—
205	C₂×C₃₁	#3205	1	1	yes	characteristic
206	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
207	C₂×C₃₁	#3414	1	1	yes	characteristic
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	—	2	1	no	—
212	C₂×Q₁₆	—	4	2	no	—
213	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
214	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	4	4	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	—	4	2	no	—
220	C₂×D₈	C₂³×C₃₁	1	2	yes	—
221	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	#125	—	4	1	no	—
226	#124	—	2	2	no	—
227	#124	—	2	2	no	—
228	#124	—	4	1	no	—
229	#124	—	4	1	no	—
230	C₂²⋊Q₈	—	2	2	no	—
231	C₂²⋊Q₈	—	4	1	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	4	1	no	—
234	#117	—	4	1	no	—
235	C₂×C₂²⋊C₄	—	4	1	no	—
236	C₂×C₂²⋊C₄	—	4	1	no	—
237	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
238	Q₃₂	—	4	4	no	—
239	SD₃₂	—	4	4	no	—
240	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
241	#107	—	2	1	no	—
242	#107	C₂³×C₃₁	1	2	yes	—
243	#103	—	2	2	no	—
244	#103	—	2	2	no	—
245	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
246	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₃₁	#33781	1	1	yes	characteristic, direct product factor, Sylow
250	C₂⁴	—	4	1	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
263	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	2	4	no	—
267	Q₁₆	—	4	2	no	—
268	Q₁₆	—	4	2	no	—
269	SD₁₆	—	4	4	no	—
270	SD₁₆	—	4	4	no	—
271	D₈	—	2	4	no	—
272	D₈	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	D₈×C₃₁	1	1	yes	characteristic
292	C₄²	D₈×C₃₁	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
295	C₁₆	—	2	4	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	—	8	2	no	—
299	C₂³	—	8	2	no	—
300	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	8	2	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
319	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
320	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
321	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
327	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	C₃₁×#510	1	2	yes	—
335	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₃₁×#535	1	1	yes	characteristic
345	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₃₁×#3205	1	1	yes	characteristic
349	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
350	C₂	C₃₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3170	C₂	semidirect
3. C₃₁×#2251	C₂	semidirect
4. C₃₁×#2243	C₂	semidirect
5. C₃₁×#2211	C₂	semidirect
6. C₃₁×#2188	C₂	semidirect
8. C₃₁×#2163	C₂	semidirect
14. C₂×C₃₁×Q₃₂	C₂²	semidirect
16. C₃₁×#500	C₂²	semidirect
21. C₃₁×#494	C₂²	semidirect
26. C₄×Q₁₆×C₃₁	C₂²	semidirect
28. C₃₁×C₄⋊C₁₆	C₂²	semidirect
29. C₃₁×#358	C₂²	semidirect
32. C₃₁×#346	C₂²	semidirect
120. #33781	C₃₁	direct (N × H)
153. #3170	C₂×C₃₁	semidirect
155. #2251	C₂×C₃₁	semidirect
156. #2243	C₂×C₃₁	semidirect
157. #2211	C₂×C₃₁	semidirect
158. #2188	C₂×C₃₁	semidirect
160. #2163	C₂×C₃₁	semidirect
184. C₂×Q₃₂	C₂²×C₃₁	semidirect
186. #500	C₂²×C₃₁	semidirect
191. #494	C₂²×C₃₁	semidirect
196. C₄×Q₁₆	C₂²×C₃₁	semidirect
198. C₄⋊C₁₆	C₂²×C₃₁	semidirect
199. #358	C₂²×C₃₁	semidirect
202. #346	C₂²×C₃₁	semidirect
249. C₃₁	#33781	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×#3170	2
C₂×C₃₁×#495	2
C₃₁×#2251	2
C₃₁×#2243	2
C₃₁×#2211	2
C₃₁×#2188	2
C₃₁×#2187	2
C₃₁×#2163	2
#33781	31

Subgroups — C31×#33781 #21490540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33781 #21490540