Subgroups — C₃₁×#21963 #21478722

Order 7936.

830 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1949	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1948	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1947	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1809	C₂	1	4	yes	maximal
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#117	—	2	1	no	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#107	C₂²	1	4	yes	—
12	C₂×C₃₁×#106	C₂²	1	4	yes	—
13	C₃₁×#423	C₄	1	8	yes	complemented
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₃₁×#384	C₂²	1	2	yes	—
17	C₃₁×#383	—	2	1	no	—
18	C₃₁×#382	—	2	1	no	—
19	C₃₁×#382	C₂²	1	2	yes	—
20	C₃₁×#381	—	2	1	no	—
21	C₃₁×#380	—	2	1	no	—
22	C₃₁×#380	C₂²	1	2	yes	—
23	C₃₁×#380	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
29	C₃₁×#376	—	2	1	no	—
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₃₁×#340	C₂²	1	4	yes	—
32	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	—	4	4	no	—
36	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
38	C₃₁×#117	—	4	4	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	—	2	1	no	—
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#107	—	4	4	no	—
56	C₃₁×#106	—	4	4	no	—
57	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
58	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
59	C₃₁×#95	—	2	2	no	—
60	C₃₁×#95	—	2	2	no	—
61	C₃₁×#95	—	2	2	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	1	no	—
65	C₃₁×#95	—	2	1	no	—
66	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	4	no	—
74	C₂²×C₄×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₃₁	—	4	8	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	4	4	no	—
96	C₄⋊C₄×C₃₁	—	4	4	no	—
97	C₂²⋊C₄×C₃₁	—	4	4	no	—
98	C₂²⋊C₄×C₃₁	—	4	4	no	—
99	C₄²×C₃₁	—	4	4	no	—
100	C₄²×C₃₁	—	4	2	no	—
101	C₄²×C₃₁	—	4	2	no	—
102	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
103	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
104	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
105	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
106	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
107	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
108	#21963	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₃₁	—	2	2	no	—
110	C₂³×C₃₁	—	2	1	no	—
111	C₂³×C₃₁	—	2	1	no	—
112	C₂³×C₃₁	—	2	2	no	—
113	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
114	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	2	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	4	no	—
123	C₂×C₄×C₃₁	—	2	8	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	—	4	2	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	4	no	—
134	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₃₁	#137	1	2	yes	—
141	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
144	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
145	C₈×C₃₁	—	2	8	no	—
146	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
147	#2285	C₂×C₃₁	1	1	yes	characteristic
148	#2283	C₂×C₃₁	1	1	yes	characteristic
149	#1949	C₂×C₃₁	1	4	yes	complemented
150	#1948	C₂×C₃₁	1	2	yes	complemented
151	#1947	C₂×C₃₁	1	2	yes	complemented
152	#1809	C₂×C₃₁	1	4	yes	—
153	C₂²×C₃₁	—	2	2	no	—
154	C₂²×C₃₁	—	2	2	no	—
155	C₂²×C₃₁	—	2	2	no	—
156	C₂²×C₃₁	—	4	2	no	—
157	C₂²×C₃₁	—	4	2	no	—
158	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₃₁	#537	1	2	yes	—
160	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
162	C₂²×C₃₁	#575	1	2	yes	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	2	2	no	—
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	—	2	2	no	—
167	C₄×C₃₁	—	8	4	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	#428	1	4	yes	—
171	C₄×C₃₁	#518	1	1	yes	characteristic
172	C₄×C₃₁	#519	1	1	yes	characteristic
173	C₄×C₃₁	#544	1	2	yes	—
174	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
175	C₂×#117	—	2	1	no	—
176	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×#107	C₂²×C₃₁	1	4	yes	—
178	C₂×#106	C₂²×C₃₁	1	4	yes	—
179	#423	C₄×C₃₁	1	8	yes	complemented
180	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
181	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
182	#384	C₂²×C₃₁	1	2	yes	—
183	#383	—	2	1	no	—
184	#382	—	2	1	no	—
185	#382	C₂²×C₃₁	1	2	yes	—
186	#381	—	2	1	no	—
187	#380	—	2	1	no	—
188	#380	C₂²×C₃₁	1	2	yes	—
189	#380	C₂²×C₃₁	1	2	yes	—
190	C₄×C₄⋊C₄	—	2	1	no	—
191	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
192	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
193	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
194	C₄×C₂²⋊C₄	—	2	1	no	—
195	#376	—	2	1	no	—
196	#340	C₂²×C₃₁	1	4	yes	—
197	#340	C₂²×C₃₁	1	4	yes	—
198	C₂×C₃₁	—	4	2	no	—
199	C₂×C₃₁	#2316	1	1	yes	characteristic
200	C₂×C₃₁	C₂×#428	1	1	yes	characteristic
201	C₂×C₃₁	#2911	1	1	yes	characteristic
202	C₂×C₃₁	#3251	1	2	yes	—
203	C₂×C₃₁	#3252	1	2	yes	—
204	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
205	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
206	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
207	C₂×M₁₆	—	4	4	no	—
208	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
209	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
210	#117	—	4	4	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
222	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
223	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
224	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
225	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
227	#107	—	4	4	no	—
228	#106	—	4	4	no	—
229	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
230	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	C₃₁	#21963	1	1	yes	characteristic, direct product factor, Sylow
239	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
249	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
250	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
258	M₁₆	—	4	8	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₂²⋊C₄	—	4	4	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₄²	—	4	4	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
276	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
277	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
278	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
279	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
280	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
281	C₂³	—	2	2	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
286	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
308	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
309	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
311	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
312	C₂×C₄	C₃₁×#137	1	2	yes	—
313	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
315	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
316	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
317	C₈	—	2	8	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
324	C₂²	C₃₁×#537	1	2	yes	—
325	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
326	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
327	C₂²	C₃₁×#575	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	4	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	C₃₁×#428	1	4	yes	—
336	C₄	C₃₁×#518	1	1	yes	characteristic
337	C₄	C₃₁×#519	1	1	yes	characteristic
338	C₄	C₃₁×#544	1	2	yes	—
339	C₂	—	4	2	no	—
340	C₂	C₃₁×#2316	1	1	yes	characteristic
341	C₂	C₂×C₃₁×#428	1	1	yes	characteristic
342	C₂	C₃₁×#2911	1	1	yes	characteristic
343	C₂	C₃₁×#3251	1	2	yes	—
344	C₂	C₃₁×#3252	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1949	C₂	semidirect
5. C₃₁×#1948	C₂	semidirect
6. C₃₁×#1947	C₂	semidirect
13. C₃₁×#423	C₄	semidirect
108. #21963	C₃₁	direct (N × H)
149. #1949	C₂×C₃₁	semidirect
150. #1948	C₂×C₃₁	semidirect
151. #1947	C₂×C₃₁	semidirect
179. #423	C₄×C₃₁	semidirect
238. C₃₁	#21963	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#423	2
C₃₁×#2285	2
C₃₁×#2283	2
C₃₁×#1949	2
C₃₁×#1948	2
C₃₁×#1947	2
C₃₁×#1809	2
#21963	31

Subgroups — C31×#21963 #21478722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21963 #21478722