Subgroups — C₃₁×#21821 #21478580

Order 7936.

1014 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1985	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1984	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1818	C₂	1	4	yes	maximal
7	C₃₁×#533	—	2	4	no	—
8	C₃₁×#533	—	2	4	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₃₁×#125	C₄	1	2	yes	—
12	C₂×C₃₁×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₂×C₃₁×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#432	—	2	4	no	—
19	C₃₁×#431	—	2	4	no	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#336	C₂²	1	16	yes	—
27	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
28	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₃₁×C₄⋊Q₈	—	2	4	no	—
35	C₃₁×C₄⋊Q₈	—	2	4	no	—
36	C₃₁×#125	—	2	8	no	—
37	C₃₁×C₂²⋊Q₈	—	2	8	no	—
38	C₃₁×C₂²⋊Q₈	—	2	8	no	—
39	C₄×Q₈×C₃₁	—	2	4	no	—
40	C₄×Q₈×C₃₁	—	2	4	no	—
41	C₄×Q₈×C₃₁	—	2	4	no	—
42	C₄×Q₈×C₃₁	—	2	4	no	—
43	C₃₁×#117	—	2	4	no	—
44	C₃₁×#117	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	D₄	1	4	yes	—
55	C₂×C₄²×C₃₁	D₄	1	4	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊C₈	—	2	4	no	—
60	C₃₁×C₄⋊C₈	—	2	4	no	—
61	C₃₁×C₂²⋊C₈	—	2	4	no	—
62	C₃₁×C₂²⋊C₈	—	2	4	no	—
63	C₃₁×#97	—	2	4	no	—
64	C₄×C₈×C₃₁	—	2	4	no	—
65	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×Q₈×C₃₁	—	2	4	no	—
67	C₂×Q₈×C₃₁	—	2	4	no	—
68	C₂×Q₈×C₃₁	—	2	8	no	—
69	C₂×Q₈×C₃₁	—	4	4	no	—
70	C₂²×C₄×C₃₁	—	2	4	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	4	2	no	—
73	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	2	8	no	—
88	C₂×C₈×C₃₁	—	2	8	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	2	8	no	—
92	C₄⋊C₄×C₃₁	—	2	8	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₂²⋊C₄×C₃₁	—	2	8	no	—
96	C₄²×C₃₁	—	2	4	no	—
97	C₄²×C₃₁	—	2	4	no	—
98	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
99	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₃₁	C₂×D₄	1	2	yes	—
101	C₄²×C₃₁	C₂×D₄	1	2	yes	—
102	C₄²×C₃₁	C₂×D₄	1	2	yes	—
103	C₄²×C₃₁	C₂×D₄	1	2	yes	—
104	#21821	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	2	2	no	—
106	C₂³×C₃₁	—	2	2	no	—
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
110	Q₈×C₃₁	—	4	8	no	—
111	Q₈×C₃₁	—	4	8	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	8	no	—
119	C₂×C₄×C₃₁	—	2	4	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	2	8	no	—
122	C₂×C₄×C₃₁	—	2	8	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₃₁	#131	1	4	yes	—
130	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
133	C₈×C₃₁	—	4	4	no	—
134	C₈×C₃₁	—	4	4	no	—
135	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
136	C₂×#432	C₂×C₃₁	1	1	yes	characteristic
137	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
138	#1985	C₂×C₃₁	1	4	yes	complemented
139	#1984	C₂×C₃₁	1	4	yes	complemented
140	#1818	C₂×C₃₁	1	4	yes	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	—	2	2	no	—
144	C₂²×C₃₁	—	4	4	no	—
145	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₃₁	#519	1	2	yes	—
147	C₂²×C₃₁	C₂×#131	1	2	yes	—
148	C₂²×C₃₁	#568	1	2	yes	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	2	2	no	—
152	C₄×C₃₁	—	2	2	no	—
153	C₄×C₃₁	—	4	4	no	—
154	C₄×C₃₁	—	4	4	no	—
155	C₄×C₃₁	#408	1	4	yes	—
156	C₄×C₃₁	#536	1	2	yes	—
157	C₄×C₃₁	#536	1	2	yes	—
158	#533	—	2	4	no	—
159	#533	—	2	4	no	—
160	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
161	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
162	C₂×#125	C₄×C₃₁	1	2	yes	—
163	C₂×C₂²⋊Q₈	C₄×C₃₁	1	2	yes	—
164	C₂×C₂²⋊Q₈	C₄×C₃₁	1	2	yes	—
165	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
167	C₂×#117	C₂²×C₃₁	1	2	yes	—
168	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
169	#432	—	2	4	no	—
170	#431	—	2	4	no	—
171	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
174	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
175	C₂×#97	C₂²×C₃₁	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
177	#336	C₂²×C₃₁	1	16	yes	—
178	C₂×C₃₁	—	4	2	no	—
179	C₂×C₃₁	#2310	1	1	yes	characteristic
180	C₂×C₃₁	C₂×#408	1	1	yes	characteristic
181	C₂×C₃₁	#2879	1	1	yes	characteristic
182	C₂×C₃₁	#2985	1	4	yes	—
183	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
184	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
185	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
189	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	—	2	4	no	—
192	#125	—	2	8	no	—
193	C₂²⋊Q₈	—	2	8	no	—
194	C₂²⋊Q₈	—	2	8	no	—
195	C₄×Q₈	—	2	4	no	—
196	C₄×Q₈	—	2	4	no	—
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	4	no	—
199	#117	—	2	4	no	—
200	#117	—	2	4	no	—
201	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
204	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
205	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
207	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
208	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
209	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₃₁	1	4	yes	—
211	C₂×C₄²	D₄×C₃₁	1	4	yes	—
212	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
213	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	C₂²⋊C₈	—	2	4	no	—
219	#97	—	2	4	no	—
220	C₄×C₈	—	2	4	no	—
221	C₃₁	#21821	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
231	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
234	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	8	no	—
245	C₂×C₈	—	2	8	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
256	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
257	C₄²	C₂×D₄×C₃₁	1	2	yes	—
258	C₄²	C₂×D₄×C₃₁	1	2	yes	—
259	C₄²	C₂×D₄×C₃₁	1	2	yes	—
260	C₄²	C₂×D₄×C₃₁	1	2	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
265	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
266	Q₈	—	4	8	no	—
267	Q₈	—	4	8	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
284	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
285	C₂×C₄	C₃₁×#131	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
289	C₈	—	4	4	no	—
290	C₈	—	4	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	4	no	—
295	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
296	C₂²	C₃₁×#519	1	2	yes	—
297	C₂²	C₂×C₃₁×#131	1	2	yes	—
298	C₂²	C₃₁×#568	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	C₃₁×#408	1	4	yes	—
306	C₄	C₃₁×#536	1	2	yes	—
307	C₄	C₃₁×#536	1	2	yes	—
308	C₂	—	4	2	no	—
309	C₂	C₃₁×#2310	1	1	yes	characteristic
310	C₂	C₂×C₃₁×#408	1	1	yes	characteristic
311	C₂	C₃₁×#2879	1	1	yes	characteristic
312	C₂	C₃₁×#2985	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1985	C₂	semidirect
5. C₃₁×#1984	C₂	semidirect
104. #21821	C₃₁	direct (N × H)
138. #1985	C₂×C₃₁	semidirect
139. #1984	C₂×C₃₁	semidirect
221. C₃₁	#21821	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×#533	2
C₂×C₃₁×#432	2
C₂×C₃₁×#431	2
C₃₁×#1985	2
C₃₁×#1984	2
C₃₁×#1818	2
#21821	31

Subgroups — C31×#21821 #21478580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21821 #21478580