Subgroups — C₃₁×#19264 #21476023

Order 7936.

650 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×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3112	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1678	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1676	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1668	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1551	C₂	1	2	yes	maximal, complemented
9	C₃₁×#532	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#500	C₂²	1	1	yes	characteristic
12	C₃₁×#493	C₂²	1	1	yes	characteristic
13	C₃₁×#486	C₂²	1	2	yes	—
14	C₃₁×#480	—	2	2	no	—
15	C₃₁×#477	C₂²	1	4	yes	—
16	C₃₁×#466	—	2	1	no	—
17	C₃₁×#466	—	2	1	no	—
18	C₃₁×#460	C₂²	1	4	yes	—
19	C₃₁×#448	—	2	2	no	—
20	C₈×Q₈×C₃₁	—	2	1	no	—
21	C₈×Q₈×C₃₁	—	2	1	no	—
22	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
23	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#440	—	2	4	no	—
25	C₄×D₈×C₃₁	—	2	1	no	—
26	C₄×D₈×C₃₁	—	2	1	no	—
27	C₃₁×#432	C₂²	1	2	yes	—
28	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₃₁	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#335	C₂²	1	2	yes	—
33	C₃₁×#334	C₂²	1	2	yes	—
34	C₃₁×#333	C₂²	1	4	yes	—
35	C₃₁×#331	C₂²	1	4	yes	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂×SD₁₆×C₃₁	—	2	4	no	—
39	C₂×D₈×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	—	2	1	no	—
41	C₂²×C₈×C₃₁	—	2	1	no	—
42	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₃₁×#127	C₂³	1	2	yes	—
46	C₃₁×#124	—	2	2	no	—
47	C₃₁×#121	—	2	2	no	—
48	C₃₁×#121	—	2	2	no	—
49	C₄×Q₈×C₃₁	—	2	1	no	—
50	C₄×Q₈×C₃₁	—	2	1	no	—
51	C₄×Q₈×C₃₁	D₄	1	2	yes	—
52	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	—	2	1	no	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#107	—	2	1	no	—
60	C₃₁×#107	—	2	1	no	—
61	C₃₁×#107	—	2	4	no	—
62	C₃₁×C₄⋊C₈	—	2	2	no	—
63	C₃₁×C₄⋊C₈	—	2	2	no	—
64	C₃₁×C₄⋊C₈	—	2	1	no	—
65	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
67	C₃₁×#103	—	2	4	no	—
68	C₃₁×#102	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	4	no	—
71	C₃₁×C₂²⋊C₈	—	2	2	no	—
72	C₃₁×#97	—	2	2	no	—
73	C₄×C₈×C₃₁	—	2	2	no	—
74	C₄×C₈×C₃₁	—	2	2	no	—
75	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
76	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	#41×C₃₁	—	8	4	no	—
80	C₂×Q₈×C₃₁	—	2	4	no	—
81	C₂×Q₈×C₃₁	—	2	1	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	—	2	1	no	—
89	C₂²×C₄×C₃₁	—	2	1	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
94	SD₁₆×C₃₁	—	4	8	no	—
95	D₈×C₃₁	—	4	2	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	—	2	4	no	—
100	C₂×C₈×C₃₁	—	4	1	no	—
101	C₂×C₈×C₃₁	—	4	2	no	—
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	2	2	no	—
115	C₄²×C₃₁	—	2	2	no	—
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	#41	1	1	yes	characteristic
119	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
120	#19264	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	#121	1	1	yes	characteristic
123	Q₈×C₃₁	—	2	4	no	—
124	Q₈×C₃₁	—	2	2	no	—
125	Q₈×C₃₁	—	8	2	no	—
126	Q₈×C₃₁	#135	1	4	yes	—
127	D₄×C₃₁	—	4	4	no	—
128	D₄×C₃₁	—	4	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	8	2	no	—
131	D₄×C₃₁	—	8	4	no	—
132	C₂×C₄×C₃₁	—	2	4	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	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	8	4	no	—
142	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₈×C₃₁	—	2	1	no	—
149	C₈×C₃₁	—	2	1	no	—
150	C₈×C₃₁	—	2	1	no	—
151	C₈×C₃₁	—	4	4	no	—
152	C₈×C₃₁	—	4	2	no	—
153	C₈×C₃₁	#137	1	2	yes	—
154	#3169	C₂×C₃₁	1	1	yes	characteristic
155	#3112	C₂×C₃₁	1	2	yes	complemented
156	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#1678	C₂×C₃₁	1	4	yes	complemented
158	#1676	C₂×C₃₁	1	2	yes	complemented
159	#1668	C₂×C₃₁	1	2	yes	complemented
160	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#1551	C₂×C₃₁	1	2	yes	complemented
162	C₂²×C₃₁	—	2	1	no	—
163	C₂²×C₃₁	—	2	1	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	#472	1	2	yes	—
168	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
169	C₄×C₃₁	—	2	1	no	—
170	C₄×C₃₁	—	2	4	no	—
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	4	2	no	—
175	C₄×C₃₁	—	8	2	no	—
176	C₄×C₃₁	C₂×#135	1	2	yes	—
177	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
178	C₄×C₃₁	#575	1	1	yes	characteristic
179	#532	C₂²×C₃₁	1	2	yes	—
180	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
181	#500	C₂²×C₃₁	1	1	yes	characteristic
182	#493	C₂²×C₃₁	1	1	yes	characteristic
183	#486	C₂²×C₃₁	1	2	yes	—
184	#480	—	2	2	no	—
185	#477	C₂²×C₃₁	1	4	yes	—
186	#466	—	2	1	no	—
187	#466	—	2	1	no	—
188	#460	C₂²×C₃₁	1	4	yes	—
189	#448	—	2	2	no	—
190	C₈×Q₈	—	2	1	no	—
191	C₈×Q₈	—	2	1	no	—
192	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
193	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
194	#440	—	2	4	no	—
195	C₄×D₈	—	2	1	no	—
196	C₄×D₈	—	2	1	no	—
197	#432	C₂²×C₃₁	1	2	yes	—
198	C₂×C₄⋊C₈	—	2	1	no	—
199	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
200	C₂×C₄×C₈	—	2	1	no	—
201	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
202	#335	C₂²×C₃₁	1	2	yes	—
203	#334	C₂²×C₃₁	1	2	yes	—
204	#333	C₂²×C₃₁	1	4	yes	—
205	#331	C₂²×C₃₁	1	4	yes	—
206	C₂×C₃₁	—	2	2	no	—
207	C₂×C₃₁	—	16	2	no	—
208	C₂×C₃₁	#3037	1	1	yes	characteristic
209	C₂×C₃₁	#3050	1	1	yes	characteristic
210	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
211	C₂×#41	—	2	2	no	—
212	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
213	C₂×SD₁₆	—	2	4	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
219	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
220	#127	C₂³×C₃₁	1	2	yes	—
221	#124	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	D₄×C₃₁	1	2	yes	—
227	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
234	#107	—	2	1	no	—
235	#107	—	2	1	no	—
236	#107	—	2	4	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
241	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
242	#103	—	2	4	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	4	no	—
246	C₂²⋊C₈	—	2	2	no	—
247	#97	—	2	2	no	—
248	C₄×C₈	—	2	2	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
251	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
254	C₃₁	#19264	1	1	yes	characteristic, direct product factor, Sylow
255	#41	—	8	4	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
270	SD₁₆	—	4	8	no	—
271	D₈	—	4	2	no	—
272	D₈	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
294	C₄²	#41×C₃₁	1	1	yes	characteristic
295	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
296	C₂³	—	4	2	no	—
297	C₂³	C₃₁×#121	1	1	yes	characteristic
298	Q₈	—	2	4	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	8	2	no	—
301	Q₈	C₃₁×#135	1	4	yes	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
318	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
319	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	4	4	no	—
327	C₈	—	4	2	no	—
328	C₈	C₃₁×#137	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	C₃₁×#472	1	2	yes	—
335	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×C₃₁×#135	1	2	yes	—
344	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
345	C₄	C₃₁×#575	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₃₁×#3037	1	1	yes	characteristic
349	C₂	C₃₁×#3050	1	1	yes	characteristic
350	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3112	C₂	semidirect
3. C₂×C₈×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#1678	C₂	semidirect
5. C₃₁×#1676	C₂	semidirect
6. C₃₁×#1668	C₂	semidirect
7. C₃₁×#1568	C₂	semidirect
8. C₃₁×#1551	C₂	semidirect
120. #19264	C₃₁	direct (N × H)
155. #3112	C₂×C₃₁	semidirect
156. C₂×C₈×Q₈	C₂×C₃₁	semidirect
157. #1678	C₂×C₃₁	semidirect
158. #1676	C₂×C₃₁	semidirect
159. #1668	C₂×C₃₁	semidirect
160. #1568	C₂×C₃₁	semidirect
161. #1551	C₂×C₃₁	semidirect
254. C₃₁	#19264	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×#3169	2
C₃₁×#3112	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1678	2
C₃₁×#1676	2
C₃₁×#1668	2
C₃₁×#1568	2
C₃₁×#1551	2
#19264	31

Subgroups — C31×#19264 #21476023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19264 #21476023