Subgroups — C₃₁×#10816 #21467575

Order 7936.

722 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1742	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1740	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#339	C₂	1	2	yes	maximal
5	C₂×C₃₁×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×#117	C₂²	1	2	yes	—
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
12	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
13	C₃₁×#376	C₄	1	2	yes	complemented
14	C₄³×C₃₁	C₄	1	2	yes	complemented
15	C₃₁×#343	—	2	2	no	—
16	C₃₁×#342	—	2	2	no	—
17	C₃₁×#339	—	2	4	no	—
18	C₃₁×H(3,ℤ/4)	C₄	1	8	yes	complemented
19	C₂³×C₄×C₃₁	—	2	1	no	—
20	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₃₁×#117	—	2	2	no	—
29	C₃₁×#117	—	2	2	no	—
30	C₃₁×#117	—	2	2	no	—
31	C₃₁×#117	—	2	2	no	—
32	C₃₁×#117	—	2	2	no	—
33	C₃₁×#117	—	2	2	no	—
34	C₃₁×#117	C₂×C₄	1	4	yes	—
35	C₃₁×#117	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
42	C₂×C₄²×C₃₁	—	2	1	no	—
43	C₂×C₄²×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	—	2	4	no	—
47	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	D₄	1	2	yes	—
52	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
53	C₃₁×#95	—	2	4	no	—
54	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	—	2	1	no	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	4	4	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	#41	1	2	yes	—
71	C₂²×C₄×C₃₁	#41	1	2	yes	—
72	C₂²×C₄×C₃₁	#41	1	2	yes	—
73	M₁₆×C₃₁	—	2	4	no	—
74	M₁₆×C₃₁	—	2	4	no	—
75	M₁₆×C₃₁	—	4	2	no	—
76	M₁₆×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₂²⋊C₄×C₃₁	—	2	4	no	—
85	C₂²⋊C₄×C₃₁	—	2	4	no	—
86	C₂²⋊C₄×C₃₁	—	2	4	no	—
87	C₂²⋊C₄×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	—	2	1	no	—
89	C₄²×C₃₁	—	2	1	no	—
90	C₄²×C₃₁	—	2	2	no	—
91	C₄²×C₃₁	—	2	2	no	—
92	C₄²×C₃₁	—	2	2	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	2	4	no	—
96	C₄²×C₃₁	—	4	8	no	—
97	C₄²×C₃₁	—	4	2	no	—
98	C₄²×C₃₁	C₄²	1	4	yes	—
99	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
100	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
101	#10816	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₃₁	—	4	2	no	—
103	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₃₁	C₄×D₄	1	2	yes	—
105	C₂³×C₃₁	#121	1	1	yes	characteristic
106	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₃₁	#123	1	2	yes	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	4	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	—	8	4	no	—
120	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#117	1	2	yes	—
124	C₂×C₄×C₃₁	#117	1	2	yes	—
125	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#123	1	2	yes	—
134	C₂×C₄×C₃₁	#124	1	2	yes	—
135	C₂×C₄×C₃₁	#125	1	2	yes	—
136	C₂×C₄×C₃₁	#126	1	2	yes	—
137	C₂×C₄×C₃₁	#126	1	2	yes	—
138	C₈×C₃₁	—	4	4	no	—
139	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
140	#1742	C₂×C₃₁	1	1	yes	characteristic
141	#1740	C₂×C₃₁	1	1	yes	characteristic
142	C₂×#339	C₂×C₃₁	1	2	yes	—
143	C₂×H(3,ℤ/4)	C₂×C₃₁	1	2	yes	—
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	4	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	—	4	2	no	—
149	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₃₁	#380	1	1	yes	characteristic
151	C₂²×C₃₁	#381	1	1	yes	characteristic
152	C₂²×C₃₁	#387	1	2	yes	—
153	C₂²×C₃₁	#388	1	2	yes	—
154	C₄×C₃₁	—	2	4	no	—
155	C₄×C₃₁	—	4	4	no	—
156	C₄×C₃₁	—	4	2	no	—
157	C₄×C₃₁	—	4	4	no	—
158	C₄×C₃₁	—	8	4	no	—
159	C₄×C₃₁	#376	1	1	yes	characteristic
160	C₄×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₃₁	#382	1	1	yes	characteristic
162	C₄×C₃₁	#383	1	1	yes	characteristic
163	C₄×C₃₁	#387	1	2	yes	—
164	C₄×C₃₁	#388	1	2	yes	—
165	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
166	C₂×#117	C₂²×C₃₁	1	2	yes	—
167	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
169	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
170	C₄×C₄⋊C₄	—	2	2	no	—
171	C₄×C₂²⋊C₄	—	2	2	no	—
172	#376	C₄×C₃₁	1	2	yes	complemented
173	C₄³	C₄×C₃₁	1	2	yes	complemented
174	#343	—	2	2	no	—
175	#342	—	2	2	no	—
176	#339	—	2	4	no	—
177	H(3,ℤ/4)	C₄×C₃₁	1	8	yes	complemented
178	C₂×C₃₁	—	2	1	no	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	—	2	2	no	—
181	C₂×C₃₁	—	2	2	no	—
182	C₂×C₃₁	#1434	1	1	yes	characteristic
183	C₂×C₃₁	#1827	1	2	yes	—
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	C₂×C₄×C₃₁	1	4	yes	—
200	#117	C₂×C₄×C₃₁	1	4	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
213	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
214	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
215	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄²	D₄×C₃₁	1	2	yes	—
217	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
218	#95	—	2	4	no	—
219	C₃₁	#10816	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	#41×C₃₁	1	2	yes	—
237	C₂²×C₄	#41×C₃₁	1	2	yes	—
238	C₂²×C₄	#41×C₃₁	1	2	yes	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	4	8	no	—
263	C₄²	—	4	2	no	—
264	C₄²	C₄²×C₃₁	1	4	yes	—
265	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
266	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
267	C₂³	—	4	2	no	—
268	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
269	C₂³	C₄×D₄×C₃₁	1	2	yes	—
270	C₂³	C₃₁×#121	1	1	yes	characteristic
271	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
272	C₂³	C₃₁×#123	1	2	yes	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
288	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
289	C₂×C₄	C₃₁×#117	1	2	yes	—
290	C₂×C₄	C₃₁×#117	1	2	yes	—
291	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
292	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
293	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
294	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
295	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
296	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
297	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
298	C₂×C₄	C₃₁×#123	1	2	yes	—
299	C₂×C₄	C₃₁×#124	1	2	yes	—
300	C₂×C₄	C₃₁×#125	1	2	yes	—
301	C₂×C₄	C₃₁×#126	1	2	yes	—
302	C₂×C₄	C₃₁×#126	1	2	yes	—
303	C₈	—	4	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
310	C₂²	C₃₁×#380	1	1	yes	characteristic
311	C₂²	C₃₁×#381	1	1	yes	characteristic
312	C₂²	C₃₁×#387	1	2	yes	—
313	C₂²	C₃₁×#388	1	2	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	4	no	—
319	C₄	C₃₁×#376	1	1	yes	characteristic
320	C₄	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
321	C₄	C₃₁×#382	1	1	yes	characteristic
322	C₄	C₃₁×#383	1	1	yes	characteristic
323	C₄	C₃₁×#387	1	2	yes	—
324	C₄	C₃₁×#388	1	2	yes	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	C₃₁×#1434	1	1	yes	characteristic
330	C₂	C₃₁×#1827	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#117
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#381
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
13. C₃₁×#376	C₄	semidirect
14. C₄³×C₃₁	C₄	semidirect
18. C₃₁×H(3,ℤ/4)	C₄	semidirect
101. #10816	C₃₁	direct (N × H)
172. #376	C₄×C₃₁	semidirect
173. C₄³	C₄×C₃₁	semidirect
177. H(3,ℤ/4)	C₄×C₃₁	semidirect
219. C₃₁	#10816	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#117	2
C₃₁×#1742	2
C₃₁×#1740	2
C₂×C₃₁×#339	2
C₂×C₃₁×H(3,ℤ/4)	2
#10816	31

Subgroups — C31×#10816 #21467575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10816 #21467575