Subgroups — C₃₁×#20368 #21477127

Order 7936.

1042 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#405	C₂	1	2	yes	maximal
2	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1746	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1745	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1744	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1729	C₂	1	2	yes	maximal
7	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#405	C₄	1	16	yes	complemented
15	C₂×C₃₁×#97	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
19	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
20	C₃₁×#376	C₂²	1	2	yes	—
21	C₄³×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#336	C₂²	1	1	yes	characteristic
23	C₃₁×#336	C₂²	1	1	yes	characteristic
24	C₃₁×#336	C₂²	1	2	yes	—
25	C₃₁×#336	C₂²	1	2	yes	—
26	C₃₁×#336	C₂²	1	2	yes	—
27	C₃₁×#336	C₂²	1	4	yes	—
28	C₃₁×#336	C₂²	1	4	yes	—
29	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₃₁	—	2	4	no	—
32	C₂×M₁₆×C₃₁	—	2	4	no	—
33	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₃₁×#117	C₂×C₄	1	8	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₃₁	—	2	4	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#97	C₂×C₄	1	16	yes	complemented
57	C₄×C₈×C₃₁	C₂×C₄	1	16	yes	complemented
58	C₃₁×#95	C₂×C₄	1	4	yes	—
59	C₃₁×#95	C₂×C₄	1	4	yes	—
60	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₃₁	#41	1	4	yes	—
72	C₂²×C₄×C₃₁	#41	1	2	yes	—
73	C₂²×C₄×C₃₁	#41	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
75	M₁₆×C₃₁	—	2	16	no	—
76	C₂×C₈×C₃₁	—	2	16	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
85	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
88	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
89	C₂²⋊C₄×C₃₁	—	2	4	no	—
90	C₂²⋊C₄×C₃₁	C₄²	1	8	yes	—
91	C₄²×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	2	8	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	4	4	no	—
98	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
99	#20368	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	2	no	—
101	C₂³×C₃₁	—	2	2	no	—
102	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
103	C₂³×C₃₁	#117	1	2	yes	—
104	C₂³×C₃₁	#117	1	2	yes	—
105	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	4	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	4	8	no	—
117	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
118	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
119	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₃₁	#117	1	8	yes	—
121	C₂×C₄×C₃₁	#117	1	8	yes	—
122	C₂×C₄×C₃₁	#117	1	2	yes	—
123	C₂×C₄×C₃₁	#117	1	2	yes	—
124	C₂×C₄×C₃₁	#131	1	4	yes	—
125	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
126	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
127	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
129	C₈×C₃₁	—	2	4	no	—
130	C₈×C₃₁	—	2	4	no	—
131	C₈×C₃₁	—	2	8	no	—
132	C₂×#405	C₂×C₃₁	1	2	yes	—
133	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
134	#1746	C₂×C₃₁	1	4	yes	complemented
135	#1745	C₂×C₃₁	1	2	yes	complemented
136	#1744	C₂×C₃₁	1	2	yes	complemented
137	#1729	C₂×C₃₁	1	2	yes	—
138	#1728	C₂×C₃₁	1	1	yes	characteristic
139	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	2	4	no	—
143	C₂²×C₃₁	—	2	4	no	—
144	C₂²×C₃₁	#376	1	4	yes	—
145	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
147	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
148	C₂²×C₃₁	C₂×#131	1	2	yes	—
149	C₂²×C₃₁	#568	1	2	yes	—
150	C₄×C₃₁	—	2	4	no	—
151	C₄×C₃₁	—	2	4	no	—
152	C₄×C₃₁	—	2	4	no	—
153	C₄×C₃₁	—	4	2	no	—
154	C₄×C₃₁	—	4	4	no	—
155	C₄×C₃₁	#376	1	4	yes	—
156	C₄×C₃₁	C₂×#117	1	2	yes	—
157	C₄×C₃₁	C₂×#117	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
159	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
160	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
163	#405	C₄×C₃₁	1	16	yes	complemented
164	C₂×#97	C₂²×C₃₁	1	4	yes	—
165	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
166	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
167	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
168	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
169	#376	C₂²×C₃₁	1	2	yes	—
170	C₄³	C₂²×C₃₁	1	2	yes	—
171	#336	C₂²×C₃₁	1	1	yes	characteristic
172	#336	C₂²×C₃₁	1	1	yes	characteristic
173	#336	C₂²×C₃₁	1	2	yes	—
174	#336	C₂²×C₃₁	1	2	yes	—
175	#336	C₂²×C₃₁	1	2	yes	—
176	#336	C₂²×C₃₁	1	4	yes	—
177	#336	C₂²×C₃₁	1	4	yes	—
178	C₂×C₃₁	—	2	4	no	—
179	C₂×C₃₁	C₂×#376	1	1	yes	characteristic
180	C₂×C₃₁	C₄×#131	1	1	yes	characteristic
181	C₂×C₃₁	#2875	1	1	yes	characteristic
182	C₂×C₃₁	#2924	1	2	yes	—
183	C₂×C₃₁	#2925	1	2	yes	—
184	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
185	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
192	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
193	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	C₂×C₄×C₃₁	1	8	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
202	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
205	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
206	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
207	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
208	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
211	#97	C₂×C₄×C₃₁	1	16	yes	complemented
212	C₄×C₈	C₂×C₄×C₃₁	1	16	yes	complemented
213	#95	C₂×C₄×C₃₁	1	4	yes	—
214	#95	C₂×C₄×C₃₁	1	4	yes	—
215	C₃₁	#20368	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
227	C₂²×C₄	#41×C₃₁	1	4	yes	—
228	C₂²×C₄	#41×C₃₁	1	2	yes	—
229	C₂²×C₄	#41×C₃₁	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
231	M₁₆	—	2	16	no	—
232	C₂×C₈	—	2	16	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
241	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
244	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	C₄²×C₃₁	1	8	yes	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	8	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
258	C₂³	C₃₁×#117	1	2	yes	—
259	C₂³	C₃₁×#117	1	2	yes	—
260	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
273	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
274	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
275	C₂×C₄	C₃₁×#117	1	8	yes	—
276	C₂×C₄	C₃₁×#117	1	8	yes	—
277	C₂×C₄	C₃₁×#117	1	2	yes	—
278	C₂×C₄	C₃₁×#117	1	2	yes	—
279	C₂×C₄	C₃₁×#131	1	4	yes	—
280	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
281	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
282	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	4	no	—
286	C₈	—	2	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₃₁×#376	1	4	yes	—
292	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
293	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
294	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
295	C₂²	C₂×C₃₁×#131	1	2	yes	—
296	C₂²	C₃₁×#568	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	4	no	—
302	C₄	C₃₁×#376	1	4	yes	—
303	C₄	C₂×C₃₁×#117	1	2	yes	—
304	C₄	C₂×C₃₁×#117	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₂×C₃₁×#376	1	1	yes	characteristic
307	C₂	C₄×C₃₁×#131	1	1	yes	characteristic
308	C₂	C₃₁×#2875	1	1	yes	characteristic
309	C₂	C₃₁×#2924	1	2	yes	—
310	C₂	C₃₁×#2925	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1746	C₂	semidirect
4. C₃₁×#1745	C₂	semidirect
5. C₃₁×#1744	C₂	semidirect
14. C₃₁×#405	C₄	semidirect
56. C₃₁×#97	C₂×C₄	semidirect
57. C₄×C₈×C₃₁	C₂×C₄	semidirect
99. #20368	C₃₁	direct (N × H)
134. #1746	C₂×C₃₁	semidirect
135. #1745	C₂×C₃₁	semidirect
136. #1744	C₂×C₃₁	semidirect
163. #405	C₄×C₃₁	semidirect
211. #97	C₂×C₄×C₃₁	semidirect
212. C₄×C₈	C₂×C₄×C₃₁	semidirect
215. C₃₁	#20368	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×#405	2
C₄×C₃₁×#117	2
C₃₁×#1746	2
C₃₁×#1745	2
C₃₁×#1744	2
C₃₁×#1729	2
C₃₁×#1728	2
C₂×C₃₁×#336	2
#20368	31

Subgroups — C31×#20368 #21477127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20368 #21477127