Subgroups — C₃₁×#33782 #21490541

Order 7936.

588 conjugacy classes of subgroups.

The subgroup lattice has 342 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	C₃₁×#3147	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×#2209	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2207	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2189	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	—	2	1	no	—
10	C₂×C₃₁×#135	C₂²	1	2	yes	—
11	C₃₁×#533	—	2	1	no	—
12	C₃₁×#532	—	2	1	no	—
13	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	—
14	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#495	—	2	1	no	—
17	C₃₁×#495	C₂²	1	2	yes	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#493	C₂²	1	1	yes	characteristic
21	C₃₁×#492	—	2	1	no	—
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	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₃₁×#358	C₂²	1	4	yes	—
30	C₃₁×#357	C₂²	1	4	yes	—
31	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₃₁×#346	C₂²	1	2	yes	—
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₃₁×#135	—	4	4	no	—
35	C₃₁×#135	—	4	4	no	—
36	C₂×Q₁₆×C₃₁	—	2	1	no	—
37	C₂×Q₁₆×C₃₁	—	2	1	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×D₈×C₃₁	—	2	1	no	—
43	C₂×D₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	—	2	1	no	—
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	—	2	1	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#125	—	4	1	no	—
51	C₃₁×#124	—	2	2	no	—
52	C₃₁×C₂²⋊Q₈	—	2	2	no	—
53	C₃₁×C₂²⋊Q₈	—	4	1	no	—
54	C₃₁×#121	—	4	1	no	—
55	C₃₁×#121	—	4	1	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	—	4	1	no	—
58	C₄×D₄×C₃₁	—	4	1	no	—
59	C₄×D₄×C₃₁	—	4	1	no	—
60	C₃₁×#117	—	4	1	no	—
61	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
62	C₃₁×Q₃₂	—	4	4	no	—
63	C₃₁×SD₃₂	—	4	4	no	—
64	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
65	C₃₁×#107	—	2	1	no	—
66	C₃₁×#107	C₂³	1	2	yes	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	#41×C₃₁	—	4	4	no	—
74	#41×C₃₁	—	4	4	no	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	Q₁₆×C₃₁	—	2	2	no	—
87	Q₁₆×C₃₁	—	2	2	no	—
88	Q₁₆×C₃₁	—	2	4	no	—
89	Q₁₆×C₃₁	—	4	2	no	—
90	Q₁₆×C₃₁	—	4	2	no	—
91	SD₁₆×C₃₁	—	4	4	no	—
92	SD₁₆×C₃₁	—	4	4	no	—
93	D₈×C₃₁	—	2	4	no	—
94	D₈×C₃₁	—	4	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	4	1	no	—
103	C₄⋊C₄×C₃₁	—	4	1	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	C₄²×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	D₈	1	1	yes	characteristic
114	C₄²×C₃₁	D₈	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₃₁	—	2	4	no	—
118	#33782	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₃₁	—	4	2	no	—
122	Q₈×C₃₁	—	4	2	no	—
123	Q₈×C₃₁	—	4	2	no	—
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	8	2	no	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	8	4	no	—
130	D₄×C₃₁	—	8	2	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	4	1	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	8	4	no	—
139	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	2	1	no	—
147	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
148	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
149	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#3147	C₂×C₃₁	1	1	yes	characteristic
151	#2251	C₂×C₃₁	1	2	yes	complemented
152	#2243	C₂×C₃₁	1	2	yes	complemented
153	#2209	C₂×C₃₁	1	2	yes	complemented
154	#2207	C₂×C₃₁	1	2	yes	complemented
155	#2189	C₂×C₃₁	1	4	yes	complemented
156	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	2	1	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	2	1	no	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	#510	1	2	yes	—
172	C₄×C₃₁	#535	1	1	yes	characteristic
173	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×#135	—	2	1	no	—
175	C₂×#135	C₂²×C₃₁	1	2	yes	—
176	#533	—	2	1	no	—
177	#532	—	2	1	no	—
178	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	—
179	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
180	#500	C₂²×C₃₁	1	1	yes	characteristic
181	#495	—	2	1	no	—
182	#495	C₂²×C₃₁	1	2	yes	—
183	#494	C₂²×C₃₁	1	1	yes	characteristic
184	#494	C₂²×C₃₁	1	1	yes	characteristic
185	#493	C₂²×C₃₁	1	1	yes	characteristic
186	#492	—	2	1	no	—
187	#487	—	2	1	no	—
188	#467	—	2	1	no	—
189	#467	C₂²×C₃₁	1	2	yes	—
190	C₄×Q₁₆	—	2	1	no	—
191	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
192	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
193	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
194	#358	C₂²×C₃₁	1	4	yes	—
195	#357	C₂²×C₃₁	1	4	yes	—
196	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
197	#346	C₂²×C₃₁	1	2	yes	—
198	C₂×C₃₁	—	4	1	no	—
199	C₂×C₃₁	—	16	2	no	—
200	C₂×C₃₁	#3205	1	1	yes	characteristic
201	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
202	C₂×C₃₁	#3414	1	1	yes	characteristic
203	C₂×#41	—	2	2	no	—
204	#135	—	4	4	no	—
205	#135	—	4	4	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
209	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	C₂³×C₃₁	1	2	yes	—
214	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	#127	—	2	1	no	—
220	#125	—	4	1	no	—
221	#124	—	2	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	#121	—	4	1	no	—
225	#121	—	4	1	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	4	1	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	4	1	no	—
230	#117	—	4	1	no	—
231	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
232	Q₃₂	—	4	4	no	—
233	SD₃₂	—	4	4	no	—
234	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
235	#107	—	2	1	no	—
236	#107	C₂³×C₃₁	1	2	yes	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
240	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
243	C₃₁	#33782	1	1	yes	characteristic, direct product factor, Sylow
244	#41	—	4	4	no	—
245	#41	—	4	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	2	2	no	—
259	Q₁₆	—	2	4	no	—
260	Q₁₆	—	4	2	no	—
261	Q₁₆	—	4	2	no	—
262	SD₁₆	—	4	4	no	—
263	SD₁₆	—	4	4	no	—
264	D₈	—	2	4	no	—
265	D₈	—	4	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	D₈×C₃₁	1	1	yes	characteristic
285	C₄²	D₈×C₃₁	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₁₆	—	2	4	no	—
289	C₂³	—	4	2	no	—
290	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	4	no	—
300	D₄	—	8	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
311	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
312	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
313	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
318	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	C₃₁×#510	1	2	yes	—
334	C₄	C₃₁×#535	1	1	yes	characteristic
335	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₃₁×#3205	1	1	yes	characteristic
339	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
340	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	C₃₁×#3147	C₂
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×#2209	C₂	semidirect
6. C₃₁×#2207	C₂	semidirect
7. C₃₁×#2189	C₂	semidirect
8. C₃₁×#2163	C₂	semidirect
118. #33782	C₃₁	direct (N × H)
149. #3170	C₂×C₃₁	semidirect
151. #2251	C₂×C₃₁	semidirect
152. #2243	C₂×C₃₁	semidirect
153. #2209	C₂×C₃₁	semidirect
154. #2207	C₂×C₃₁	semidirect
155. #2189	C₂×C₃₁	semidirect
156. #2163	C₂×C₃₁	semidirect
243. C₃₁	#33782	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
C₃₁×#3147	2
C₃₁×#2251	2
C₃₁×#2243	2
C₃₁×#2209	2
C₃₁×#2207	2
C₃₁×#2189	2
C₃₁×#2163	2
#33782	31

Subgroups — C31×#33782 #21490541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33782 #21490541