Subgroups — C₃₁×#19965 #21476724

Order 7936.

590 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×#3400	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3365	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1686	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1682	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1648	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1620	C₂	1	2	yes	maximal
8	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#559	C₂²	1	2	yes	—
11	C₃₁×#557	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#497	—	2	2	no	—
14	C₃₁×#493	C₂²	1	1	yes	characteristic
15	C₃₁×#492	C₂²	1	2	yes	—
16	C₃₁×#488	—	2	2	no	—
17	C₃₁×#486	C₂²	1	2	yes	—
18	C₃₁×#478	—	2	2	no	—
19	C₃₁×#475	C₂²	1	2	yes	—
20	C₃₁×#464	—	2	2	no	—
21	C₃₁×#464	—	2	2	no	—
22	C₃₁×#462	C₂²	1	2	yes	—
23	C₃₁×#461	—	2	2	no	—
24	C₃₁×#460	C₂²	1	2	yes	—
25	C₃₁×#460	C₂²	1	2	yes	—
26	C₃₁×#446	C₂²	1	2	yes	—
27	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#440	—	2	2	no	—
29	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆×C₃₁	—	2	2	no	—
32	C₃₁×#335	C₂²	1	1	yes	characteristic
33	C₃₁×#335	C₂²	1	1	yes	characteristic
34	C₃₁×#332	C₂²	1	2	yes	—
35	C₃₁×#331	C₂²	1	2	yes	—
36	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
41	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
42	C₂×SD₁₆×C₃₁	—	2	2	no	—
43	C₂×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×SD₁₆×C₃₁	—	2	2	no	—
45	C₂×SD₁₆×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	4	1	no	—
47	C₃₁×C₄⋊Q₈	—	2	2	no	—
48	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₃₁×#127	C₂³	1	2	yes	—
50	C₃₁×#125	—	2	2	no	—
51	C₃₁×#125	—	4	2	no	—
52	C₃₁×#124	—	2	2	no	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×#123	—	4	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
58	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₃₁×#107	—	4	1	no	—
62	C₃₁×#107	D₄	1	1	yes	characteristic
63	C₃₁×#107	D₄	1	1	yes	characteristic
64	C₃₁×#106	—	2	2	no	—
65	C₃₁×C₄⋊C₈	—	2	2	no	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#103	D₄	1	2	yes	—
72	C₃₁×#103	D₄	1	2	yes	—
73	C₃₁×#102	—	2	2	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₃₁×#97	—	2	2	no	—
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
84	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	Q₁₆×C₃₁	—	2	2	no	—
90	Q₁₆×C₃₁	—	4	1	no	—
91	Q₁₆×C₃₁	—	8	2	no	—
92	SD₁₆×C₃₁	—	4	4	no	—
93	SD₁₆×C₃₁	—	4	4	no	—
94	SD₁₆×C₃₁	—	8	2	no	—
95	SD₁₆×C₃₁	—	8	2	no	—
96	D₈×C₃₁	—	8	2	no	—
97	C₂×C₈×C₃₁	—	2	1	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₂²⋊C₄×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	—	2	2	no	—
117	C₄²×C₃₁	—	2	2	no	—
118	C₄²×C₃₁	—	4	2	no	—
119	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
120	#19965	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	2	no	—
122	Q₈×C₃₁	—	2	2	no	—
123	Q₈×C₃₁	—	2	2	no	—
124	Q₈×C₃₁	—	2	2	no	—
125	Q₈×C₃₁	—	2	2	no	—
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	4	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	8	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	16	2	no	—
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
142	C₈×C₃₁	—	2	2	no	—
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	—	2	1	no	—
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	4	2	no	—
147	C₈×C₃₁	—	4	2	no	—
148	C₈×C₃₁	#135	1	2	yes	—
149	#3400	C₂×C₃₁	1	2	yes	complemented
150	#3365	C₂×C₃₁	1	2	yes	complemented
151	#1706	C₂×C₃₁	1	1	yes	characteristic
152	#1686	C₂×C₃₁	1	2	yes	complemented
153	#1682	C₂×C₃₁	1	2	yes	complemented
154	#1648	C₂×C₃₁	1	2	yes	complemented
155	#1620	C₂×C₃₁	1	2	yes	—
156	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
157	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	D₄²	1	1	yes	characteristic
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	4	2	no	—
164	C₄×C₃₁	—	4	2	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	—	4	2	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
169	C₄×C₃₁	#575	1	1	yes	characteristic
170	C₄×C₃₁	#576	1	1	yes	characteristic
171	C₄×C₃₁	#578	1	1	yes	characteristic
172	#559	C₂²×C₃₁	1	2	yes	—
173	#557	C₂²×C₃₁	1	2	yes	—
174	#500	C₂²×C₃₁	1	1	yes	characteristic
175	#497	—	2	2	no	—
176	#493	C₂²×C₃₁	1	1	yes	characteristic
177	#492	C₂²×C₃₁	1	2	yes	—
178	#488	—	2	2	no	—
179	#486	C₂²×C₃₁	1	2	yes	—
180	#478	—	2	2	no	—
181	#475	C₂²×C₃₁	1	2	yes	—
182	#464	—	2	2	no	—
183	#464	—	2	2	no	—
184	#462	C₂²×C₃₁	1	2	yes	—
185	#461	—	2	2	no	—
186	#460	C₂²×C₃₁	1	2	yes	—
187	#460	C₂²×C₃₁	1	2	yes	—
188	#446	C₂²×C₃₁	1	2	yes	—
189	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
190	#440	—	2	2	no	—
191	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
192	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
193	C₄×SD₁₆	—	2	2	no	—
194	#335	C₂²×C₃₁	1	1	yes	characteristic
195	#335	C₂²×C₃₁	1	1	yes	characteristic
196	#332	C₂²×C₃₁	1	2	yes	—
197	#331	C₂²×C₃₁	1	2	yes	—
198	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
199	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
200	C₈²	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×C₃₁	—	16	2	no	—
202	C₂×C₃₁	#3277	1	1	yes	characteristic
203	C₂×C₃₁	#3287	1	1	yes	characteristic
204	C₂×C₃₁	#3291	1	1	yes	characteristic
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
207	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
208	C₂×SD₁₆	—	2	2	no	—
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×D₈	—	4	1	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
215	#127	C₂³×C₃₁	1	2	yes	—
216	#125	—	2	2	no	—
217	#125	—	4	2	no	—
218	#124	—	2	2	no	—
219	#124	—	2	2	no	—
220	#123	—	4	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
224	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	—	2	2	no	—
227	#107	—	4	1	no	—
228	#107	D₄×C₃₁	1	1	yes	characteristic
229	#107	D₄×C₃₁	1	1	yes	characteristic
230	#106	—	2	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
234	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	D₄×C₃₁	1	2	yes	—
238	#103	D₄×C₃₁	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#97	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₃₁	#19965	1	1	yes	characteristic, direct product factor, Sylow
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
251	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	4	1	no	—
258	Q₁₆	—	8	2	no	—
259	SD₁₆	—	4	4	no	—
260	SD₁₆	—	4	4	no	—
261	SD₁₆	—	8	2	no	—
262	SD₁₆	—	8	2	no	—
263	D₈	—	8	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
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	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
279	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	4	2	no	—
286	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
287	C₂³	—	4	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	2	2	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	2	no	—
314	C₈	C₃₁×#135	1	2	yes	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	D₄²×C₃₁	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
326	C₄	C₃₁×#575	1	1	yes	characteristic
327	C₄	C₃₁×#576	1	1	yes	characteristic
328	C₄	C₃₁×#578	1	1	yes	characteristic
329	C₂	—	16	2	no	—
330	C₂	C₃₁×#3277	1	1	yes	characteristic
331	C₂	C₃₁×#3287	1	1	yes	characteristic
332	C₂	C₃₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#493	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3400	C₂	semidirect
2. C₃₁×#3365	C₂	semidirect
4. C₃₁×#1686	C₂	semidirect
5. C₃₁×#1682	C₂	semidirect
6. C₃₁×#1648	C₂	semidirect
8. C₃₁×#1591	C₂	semidirect
9. C₈×Q₁₆×C₃₁	C₂	semidirect
120. #19965	C₃₁	direct (N × H)
149. #3400	C₂×C₃₁	semidirect
150. #3365	C₂×C₃₁	semidirect
152. #1686	C₂×C₃₁	semidirect
153. #1682	C₂×C₃₁	semidirect
154. #1648	C₂×C₃₁	semidirect
156. #1591	C₂×C₃₁	semidirect
157. C₈×Q₁₆	C₂×C₃₁	semidirect
248. C₃₁	#19965	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×#3400	2
C₃₁×#3365	2
C₃₁×#1706	2
C₃₁×#1686	2
C₃₁×#1682	2
C₃₁×#1648	2
C₃₁×#1620	2
C₃₁×#1591	2
C₈×Q₁₆×C₃₁	2
#19965	31

Subgroups — C31×#19965 #21476724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19965 #21476724