Subgroups — C₃₁×#33280 #21490039

Order 7936.

798 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×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1936	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1934	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#124	—	2	2	no	—
13	C₂×C₃₁×#102	C₂²	1	4	yes	—
14	C₂×C₃₁×#102	C₂²	1	4	yes	—
15	C₂×C₃₁×#97	C₂²	1	2	yes	—
16	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#390	—	2	2	no	—
20	C₃₁×#390	C₂²	1	2	yes	—
21	C₃₁×#390	C₂²	1	2	yes	—
22	C₃₁×#389	C₂²	1	2	yes	—
23	C₃₁×#389	C₂²	1	2	yes	—
24	C₃₁×#387	—	2	2	no	—
25	C₃₁×#387	—	2	2	no	—
26	C₃₁×#382	C₂²	1	1	yes	characteristic
27	C₃₁×#382	C₂²	1	1	yes	characteristic
28	C₄³×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#340	C₂²	1	4	yes	—
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₂²×Q₈×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	—	2	1	no	—
33	C₂²×D₄×C₃₁	—	2	1	no	—
34	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	—	2	2	no	—
39	C₃₁×#127	—	2	2	no	—
40	C₃₁×#127	—	2	2	no	—
41	C₃₁×#124	—	4	8	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₂×C₄²×C₃₁	D₄	1	2	yes	—
51	C₂×C₄²×C₃₁	D₄	1	2	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#102	—	2	8	no	—
56	C₃₁×#102	—	2	8	no	—
57	C₃₁×#97	—	2	4	no	—
58	C₃₁×#97	—	2	2	no	—
59	C₃₁×#97	—	2	2	no	—
60	C₄×C₈×C₃₁	—	2	4	no	—
61	C₃₁×#95	—	2	1	no	—
62	C₃₁×#95	—	2	1	no	—
63	C₃₁×#95	—	2	1	no	—
64	C₃₁×#95	—	2	1	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	—	2	2	no	—
67	C₂⁴×C₃₁	—	2	2	no	—
68	C₂×Q₈×C₃₁	—	4	4	no	—
69	C₂×Q₈×C₃₁	—	4	2	no	—
70	C₂×D₄×C₃₁	—	2	4	no	—
71	C₂×D₄×C₃₁	—	2	4	no	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	—	4	4	no	—
75	C₂×D₄×C₃₁	—	4	4	no	—
76	C₂×D₄×C₃₁	—	4	2	no	—
77	C₂×D₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	#41	1	4	yes	—
91	C₂×C₈×C₃₁	#41	1	4	yes	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	4	4	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	4	no	—
99	C₂²⋊C₄×C₃₁	—	4	8	no	—
100	C₂²⋊C₄×C₃₁	—	4	8	no	—
101	C₄²×C₃₁	—	2	4	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	4	2	no	—
105	C₄²×C₃₁	#41	1	2	yes	—
106	C₄²×C₃₁	#41	1	2	yes	—
107	#33280	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	4	2	no	—
109	C₂³×C₃₁	—	4	2	no	—
110	C₂³×C₃₁	—	4	2	no	—
111	C₂³×C₃₁	—	4	4	no	—
112	C₂³×C₃₁	—	4	4	no	—
113	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₃₁	—	8	4	no	—
115	D₄×C₃₁	—	4	4	no	—
116	D₄×C₃₁	—	4	4	no	—
117	D₄×C₃₁	—	8	4	no	—
118	D₄×C₃₁	—	8	4	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	#124	1	2	yes	—
135	C₂×C₄×C₃₁	#124	1	2	yes	—
136	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
140	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
141	C₈×C₃₁	—	2	4	no	—
142	C₈×C₃₁	—	2	4	no	—
143	#2864	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#2858	C₂×C₃₁	1	1	yes	characteristic
145	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1961	C₂×C₃₁	1	1	yes	characteristic
148	#1961	C₂×C₃₁	1	1	yes	characteristic
149	#1936	C₂×C₃₁	1	4	yes	complemented
150	#1934	C₂×C₃₁	1	4	yes	complemented
151	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	8	2	no	—
154	C₂²×C₃₁	—	8	2	no	—
155	C₂²×C₃₁	—	8	4	no	—
156	C₂²×C₃₁	—	8	4	no	—
157	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
158	C₂²×C₃₁	#532	1	1	yes	characteristic
159	C₂²×C₃₁	#532	1	1	yes	characteristic
160	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
161	C₂²×C₃₁	#575	1	2	yes	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	2	4	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	#529	1	2	yes	—
170	C₄×C₃₁	#529	1	2	yes	—
171	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
172	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×#124	—	2	2	no	—
174	C₂×#102	C₂²×C₃₁	1	4	yes	—
175	C₂×#102	C₂²×C₃₁	1	4	yes	—
176	C₂×#97	C₂²×C₃₁	1	2	yes	—
177	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
180	#390	—	2	2	no	—
181	#390	C₂²×C₃₁	1	2	yes	—
182	#390	C₂²×C₃₁	1	2	yes	—
183	#389	C₂²×C₃₁	1	2	yes	—
184	#389	C₂²×C₃₁	1	2	yes	—
185	#387	—	2	2	no	—
186	#387	—	2	2	no	—
187	#382	C₂²×C₃₁	1	1	yes	characteristic
188	#382	C₂²×C₃₁	1	1	yes	characteristic
189	C₄³	C₂²×C₃₁	1	1	yes	characteristic
190	#340	C₂²×C₃₁	1	4	yes	—
191	#340	C₂²×C₃₁	1	4	yes	—
192	C₂×C₃₁	—	16	2	no	—
193	C₂×C₃₁	#2861	1	1	yes	characteristic
194	C₂×C₃₁	#3138	1	1	yes	characteristic
195	C₂×C₃₁	#3140	1	1	yes	characteristic
196	C₂×C₃₁	#3171	1	2	yes	—
197	C₂×C₃₁	#3180	1	2	yes	—
198	C₂²×Q₈	—	2	1	no	—
199	C₂²×D₄	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
203	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	#127	—	2	2	no	—
207	#127	—	2	2	no	—
208	#124	—	4	8	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	4	no	—
216	C₂×C₂²⋊C₄	—	2	4	no	—
217	C₂×C₄²	D₄×C₃₁	1	2	yes	—
218	C₂×C₄²	D₄×C₃₁	1	2	yes	—
219	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
222	#102	—	2	8	no	—
223	#102	—	2	8	no	—
224	#97	—	2	4	no	—
225	#97	—	2	2	no	—
226	#97	—	2	2	no	—
227	C₄×C₈	—	2	4	no	—
228	#95	—	2	1	no	—
229	#95	—	2	1	no	—
230	#95	—	2	1	no	—
231	#95	—	2	1	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	C₃₁	#33280	1	1	yes	characteristic, direct product factor, Sylow
235	C₂⁴	—	2	2	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	#41×C₃₁	1	4	yes	—
259	C₂×C₈	#41×C₃₁	1	4	yes	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	4	8	no	—
268	C₂²⋊C₄	—	4	8	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	#41×C₃₁	1	2	yes	—
274	C₄²	#41×C₃₁	1	2	yes	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	4	no	—
280	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
281	Q₈	—	8	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	8	4	no	—
285	D₄	—	8	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₃₁×#124	1	2	yes	—
302	C₂×C₄	C₃₁×#124	1	2	yes	—
303	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
304	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
307	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
308	C₈	—	2	4	no	—
309	C₈	—	2	4	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
316	C₂²	C₃₁×#532	1	1	yes	characteristic
317	C₂²	C₃₁×#532	1	1	yes	characteristic
318	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
319	C₂²	C₃₁×#575	1	2	yes	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	C₃₁×#529	1	2	yes	—
328	C₄	C₃₁×#529	1	2	yes	—
329	C₂	—	16	2	no	—
330	C₂	C₃₁×#2861	1	1	yes	characteristic
331	C₂	C₃₁×#3138	1	1	yes	characteristic
332	C₂	C₃₁×#3140	1	1	yes	characteristic
333	C₂	C₃₁×#3171	1	2	yes	—
334	C₂	C₃₁×#3180	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2864	C₂	semidirect
3. C₃₁×#1993	C₂	semidirect
4. C₃₁×#1993	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1934	C₂	semidirect
9. C₄×C₃₁×#97	C₂	semidirect
107. #33280	C₃₁	direct (N × H)
143. #2864	C₂×C₃₁	semidirect
145. #1993	C₂×C₃₁	semidirect
146. #1993	C₂×C₃₁	semidirect
149. #1936	C₂×C₃₁	semidirect
150. #1934	C₂×C₃₁	semidirect
151. C₄×#97	C₂×C₃₁	semidirect
234. C₃₁	#33280	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×#2864	2
C₃₁×#2858	2
C₃₁×#1993	2
C₃₁×#1961	2
C₃₁×#1936	2
C₃₁×#1934	2
C₄×C₃₁×#97	2
#33280	31

Subgroups — C31×#33280 #21490039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33280 #21490039