Subgroups — C₃₁×#12976 #21469735

Order 7936.

578 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1374	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
9	C₂²×C₁₆×C₃₁	—	2	2	no	—
10	C₂²×C₁₆×C₃₁	—	2	4	no	—
11	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
13	C₈×D₄×C₃₁	—	2	2	no	—
14	C₈×D₄×C₃₁	—	2	2	no	—
15	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×C₂²⋊C₁₆	—	2	2	no	—
20	C₃₁×C₂²⋊C₁₆	—	2	4	no	—
21	C₂²×D₄×C₃₁	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	—	2	1	no	—
27	C₂²×C₈×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	D₄	1	2	yes	—
34	C₂²×C₈×C₃₁	D₄	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₃₁	—	2	2	no	—
37	C₄×D₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₃₁	C₈	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₃₁	C₈	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₃₁	—	2	8	no	—
43	C₂×C₁₆×C₃₁	—	2	2	no	—
44	C₂×C₁₆×C₃₁	—	2	2	no	—
45	C₂×C₁₆×C₃₁	—	2	2	no	—
46	C₂×C₁₆×C₃₁	—	2	2	no	—
47	C₂×C₁₆×C₃₁	—	2	2	no	—
48	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
49	C₃₁×C₄⋊C₈	—	2	2	no	—
50	C₃₁×C₂²⋊C₈	—	2	2	no	—
51	C₃₁×C₂²⋊C₈	—	2	2	no	—
52	C₄×C₈×C₃₁	—	2	1	no	—
53	C₄×C₈×C₃₁	—	2	1	no	—
54	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
55	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	C₁₆	1	4	yes	complemented
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	2	1	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	1	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	4	1	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	4	1	no	—
83	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	#41	1	2	yes	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₄²×C₃₁	—	2	1	no	—
92	C₄²×C₃₁	—	2	1	no	—
93	C₁₆×C₃₁	—	2	4	no	—
94	C₁₆×C₃₁	—	4	2	no	—
95	#12976	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	—	2	2	no	—
97	C₂³×C₃₁	—	2	2	no	—
98	C₂³×C₃₁	—	2	1	no	—
99	C₂³×C₃₁	—	2	2	no	—
100	C₂³×C₃₁	—	2	1	no	—
101	C₂³×C₃₁	—	2	2	no	—
102	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
103	C₂³×C₃₁	C₂×C₁₆	1	2	yes	—
104	C₂³×C₃₁	C₂×C₁₆	1	2	yes	complemented
105	C₂³×C₃₁	M₃₂	1	2	yes	—
106	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
107	D₄×C₃₁	—	2	8	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	4	1	no	—
119	C₂×C₄×C₃₁	—	4	1	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×C₁₆	1	2	yes	complemented
123	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#131	1	2	yes	—
127	C₈×C₃₁	—	2	2	no	—
128	C₈×C₃₁	—	4	1	no	—
129	C₈×C₃₁	—	4	1	no	—
130	C₈×C₃₁	#120	1	1	yes	characteristic
131	C₈×C₃₁	#121	1	2	yes	—
132	C₈×C₃₁	#123	1	1	yes	characteristic
133	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
134	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
135	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	2	yes	complemented
136	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#1374	C₂×C₃₁	1	2	yes	complemented
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	4	no	—
141	C₂²×C₃₁	—	2	4	no	—
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	—	2	4	no	—
144	C₂²×C₃₁	—	4	1	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	4	2	no	—
147	C₂²×C₃₁	C₂²⋊C₁₆	1	4	yes	complemented
148	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₃₁	C₈×D₄	1	2	yes	—
150	C₂²×C₃₁	C₂²×C₁₆	1	1	yes	characteristic
151	C₂²×C₃₁	C₂×M₃₂	1	1	yes	characteristic
152	C₂²×C₃₁	#504	1	2	yes	—
153	C₄×C₃₁	—	2	2	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	4	1	no	—
156	C₄×C₃₁	—	4	1	no	—
157	C₄×C₃₁	#386	1	1	yes	characteristic
158	C₄×C₃₁	#408	1	1	yes	characteristic
159	C₄×C₃₁	#435	1	2	yes	—
160	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
161	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
163	C₂²×C₁₆	—	2	2	no	—
164	C₂²×C₁₆	—	2	4	no	—
165	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
166	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
167	C₈×D₄	—	2	2	no	—
168	C₈×D₄	—	2	2	no	—
169	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
170	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic
171	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²⋊C₁₆	—	2	2	no	—
174	C₂²⋊C₁₆	—	2	4	no	—
175	C₂×C₃₁	—	2	4	no	—
176	C₂×C₃₁	—	4	2	no	—
177	C₂×C₃₁	#1869	1	1	yes	characteristic
178	C₂×C₃₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
179	C₂×C₃₁	#2113	1	1	yes	characteristic
180	C₂×C₃₁	D₄×C₁₆	1	2	yes	—
181	C₂×C₃₁	#2168	1	2	yes	—
182	C₂²×D₄	C₈×C₃₁	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₈	D₄×C₃₁	1	2	yes	—
195	C₂²×C₈	D₄×C₃₁	1	2	yes	—
196	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
200	C₂×C₂²⋊C₄	C₈×C₃₁	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	C₈×C₃₁	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₁₆	—	2	8	no	—
204	C₂×C₁₆	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	—	2	2	no	—
209	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
210	C₄⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	2	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₃₁	#12976	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
217	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	C₁₆×C₃₁	1	4	yes	complemented
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₈	#41×C₃₁	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₁₆	—	2	4	no	—
256	C₁₆	—	4	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	2	no	—
263	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
264	C₂³	C₂×C₁₆×C₃₁	1	2	yes	—
265	C₂³	C₂×C₁₆×C₃₁	1	2	yes	complemented
266	C₂³	C₃₁×M₃₂	1	2	yes	—
267	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
268	D₄	—	2	8	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
283	C₂×C₄	C₂×C₁₆×C₃₁	1	2	yes	complemented
284	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
286	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₃₁×#131	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	4	1	no	—
290	C₈	—	4	1	no	—
291	C₈	C₃₁×#120	1	1	yes	characteristic
292	C₈	C₃₁×#121	1	2	yes	—
293	C₈	C₃₁×#123	1	1	yes	characteristic
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₃₁×C₂²⋊C₁₆	1	4	yes	complemented
304	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
305	C₂²	C₈×D₄×C₃₁	1	2	yes	—
306	C₂²	C₂²×C₁₆×C₃₁	1	1	yes	characteristic
307	C₂²	C₂×C₃₁×M₃₂	1	1	yes	characteristic
308	C₂²	C₃₁×#504	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	C₃₁×#386	1	1	yes	characteristic
314	C₄	C₃₁×#408	1	1	yes	characteristic
315	C₄	C₃₁×#435	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	4	2	no	—
318	C₂	C₃₁×#1869	1	1	yes	characteristic
319	C₂	C₂×C₃₁×C₂²⋊C₁₆	1	1	yes	characteristic
320	C₂	C₃₁×#2113	1	1	yes	characteristic
321	C₂	D₄×C₁₆×C₃₁	1	2	yes	—
322	C₂	C₃₁×#2168	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₃₁	C₂³
commutator	C₂²	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂³×C₃₁	C₂²×C₈
prime radical	C₂²×D₄×C₃₁	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₆×C₃₁	C₂	semidirect
3. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
4. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
5. C₃₁×#1374	C₂	semidirect
48. C₂×C₁₆×C₃₁	D₄	semidirect
58. C₂×D₄×C₃₁	C₁₆	semidirect
95. #12976	C₃₁	direct (N × H)
104. C₂³×C₃₁	C₂×C₁₆	semidirect
122. C₂×C₄×C₃₁	C₂×C₁₆	semidirect
133. C₂³×C₁₆	C₂×C₃₁	semidirect
135. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
136. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
137. #1374	C₂×C₃₁	semidirect
147. C₂²×C₃₁	C₂²⋊C₁₆	semidirect
209. C₂×C₁₆	D₄×C₃₁	semidirect
215. C₃₁	#12976	direct (N × H)
220. C₂×D₄	C₁₆×C₃₁	semidirect
265. C₂³	C₂×C₁₆×C₃₁	semidirect
283. C₂×C₄	C₂×C₁₆×C₃₁	semidirect
303. C₂²	C₃₁×C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₃₁	2
C₂×C₈×D₄×C₃₁	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₃₁×#1374	2
#12976	31

Subgroups — C31×#12976 #21469735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12976 #21469735