Subgroups — C₃₁×#11224 #21467983

Order 7936.

706 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2316	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#343	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1733	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	1	yes	characteristic
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#389	—	2	2	no	—
12	C₃₁×#389	—	2	4	no	—
13	C₃₁×#385	—	2	2	no	—
14	C₃₁×#384	C₄	1	2	yes	complemented
15	C₃₁×#382	C₄	1	2	yes	complemented
16	C₃₁×#380	—	2	2	no	—
17	C₃₁×#343	—	2	2	no	—
18	C₃₁×#339	—	2	4	no	—
19	C₃₁×H(3,ℤ/4)	—	2	4	no	—
20	C₂³×C₄×C₃₁	D₄	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
27	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₃₁×#117	—	2	1	no	—
30	C₃₁×#117	—	2	1	no	—
31	C₃₁×#117	—	2	2	no	—
32	C₃₁×#117	—	2	2	no	—
33	C₃₁×#117	—	2	2	no	—
34	C₃₁×#117	—	2	2	no	—
35	C₃₁×#117	—	2	2	no	—
36	C₃₁×#117	—	2	2	no	—
37	C₃₁×#117	D₄	1	2	yes	—
38	C₃₁×#117	Q₈	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄²×C₃₁	—	2	1	no	—
53	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₃₁	D₄	1	4	yes	—
55	C₃₁×#95	—	2	4	no	—
56	C₃₁×#95	—	2	4	no	—
57	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	—	2	2	no	—
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	1	no	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	—	8	2	no	—
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	#41	1	2	yes	—
75	C₂²×C₄×C₃₁	#41	1	2	yes	—
76	M₁₆×C₃₁	—	2	4	no	—
77	M₁₆×C₃₁	—	2	4	no	—
78	M₁₆×C₃₁	—	4	2	no	—
79	M₁₆×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	2	4	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	4	2	no	—
90	C₄⋊C₄×C₃₁	—	4	2	no	—
91	C₄⋊C₄×C₃₁	—	8	4	no	—
92	C₄⋊C₄×C₃₁	—	8	4	no	—
93	C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂²⋊C₄×C₃₁	—	2	2	no	—
95	C₂²⋊C₄×C₃₁	—	4	1	no	—
96	C₂²⋊C₄×C₃₁	—	4	2	no	—
97	C₂²⋊C₄×C₃₁	—	4	1	no	—
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₄²×C₃₁	—	2	2	no	—
100	C₄²×C₃₁	—	2	4	no	—
101	C₄²×C₃₁	—	2	4	no	—
102	C₄²×C₃₁	—	4	2	no	—
103	C₄²×C₃₁	C₄⋊C₄	1	4	yes	complemented
104	#11224	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	4	1	no	—
106	C₂³×C₃₁	—	4	1	no	—
107	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂³×C₃₁	#120	1	1	yes	characteristic
109	C₂³×C₃₁	#121	1	1	yes	characteristic
110	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂³×C₃₁	#123	1	1	yes	characteristic
112	C₂³×C₃₁	#124	1	1	yes	characteristic
113	C₂³×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₃₁	—	2	4	no	—
115	C₂×C₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	4	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	1	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	8	2	no	—
128	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₃₁	#121	1	2	yes	—
133	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#121	1	2	yes	—
135	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	#123	1	2	yes	—
140	C₂×C₄×C₃₁	#125	1	2	yes	—
141	C₂×C₄×C₃₁	#125	1	2	yes	—
142	C₂×C₄×C₃₁	#126	1	2	yes	—
143	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
144	C₈×C₃₁	—	4	4	no	—
145	#2317	C₂×C₃₁	1	1	yes	characteristic
146	#2316	C₂×C₃₁	1	1	yes	characteristic
147	C₂×#343	C₂×C₃₁	1	1	yes	characteristic
148	#1733	C₂×C₃₁	1	2	yes	—
149	C₂×H(3,ℤ/4)	C₂×C₃₁	1	2	yes	—
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	4	1	no	—
157	C₂²×C₃₁	—	4	1	no	—
158	C₂²×C₃₁	—	4	1	no	—
159	C₂²×C₃₁	—	4	1	no	—
160	C₂²×C₃₁	#380	1	1	yes	characteristic
161	C₂²×C₃₁	#386	1	1	yes	characteristic
162	C₂²×C₃₁	#391	1	1	yes	characteristic
163	C₂²×C₃₁	#394	1	1	yes	characteristic
164	C₂²×C₃₁	#395	1	1	yes	characteristic
165	C₂²×C₃₁	#398	1	1	yes	characteristic
166	C₂²×C₃₁	#400	1	1	yes	characteristic
167	C₄×C₃₁	—	4	4	no	—
168	C₄×C₃₁	—	4	4	no	—
169	C₄×C₃₁	—	8	4	no	—
170	C₄×C₃₁	—	8	1	no	—
171	C₄×C₃₁	—	8	1	no	—
172	C₄×C₃₁	#387	1	2	yes	—
173	C₄×C₃₁	#389	1	2	yes	—
174	C₄×C₃₁	#396	1	2	yes	—
175	C₄×C₃₁	#398	1	2	yes	—
176	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
177	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×#117	C₂²×C₃₁	1	2	yes	—
180	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
181	#389	—	2	2	no	—
182	#389	—	2	4	no	—
183	#385	—	2	2	no	—
184	#384	C₄×C₃₁	1	2	yes	complemented
185	#382	C₄×C₃₁	1	2	yes	complemented
186	#380	—	2	2	no	—
187	#343	—	2	2	no	—
188	#339	—	2	4	no	—
189	H(3,ℤ/4)	—	2	4	no	—
190	C₂×C₃₁	—	2	1	no	—
191	C₂×C₃₁	—	2	1	no	—
192	C₂×C₃₁	—	2	1	no	—
193	C₂×C₃₁	—	2	1	no	—
194	C₂×C₃₁	—	2	1	no	—
195	C₂×C₃₁	—	2	1	no	—
196	C₂×C₃₁	#1441	1	1	yes	characteristic
197	C₂×C₃₁	#1911	1	1	yes	characteristic
198	C₂×C₃₁	#1912	1	1	yes	characteristic
199	C₂³×C₄	D₄×C₃₁	1	2	yes	—
200	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
207	C₂²×C₈	—	2	2	no	—
208	#117	—	2	1	no	—
209	#117	—	2	1	no	—
210	#117	—	2	2	no	—
211	#117	—	2	2	no	—
212	#117	—	2	2	no	—
213	#117	—	2	2	no	—
214	#117	—	2	2	no	—
215	#117	—	2	2	no	—
216	#117	D₄×C₃₁	1	2	yes	—
217	#117	Q₈×C₃₁	1	2	yes	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	4	no	—
223	C₂×C₄⋊C₄	—	2	4	no	—
224	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
228	C₂×C₂²⋊C₄	—	2	1	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	1	no	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
233	C₂×C₄²	D₄×C₃₁	1	4	yes	—
234	#95	—	2	4	no	—
235	#95	—	2	4	no	—
236	C₃₁	#11224	1	1	yes	characteristic, direct product factor, Sylow
237	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	8	2	no	—
245	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
246	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
252	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
254	C₂²×C₄	#41×C₃₁	1	2	yes	—
255	C₂²×C₄	#41×C₃₁	1	2	yes	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	2	4	no	—
258	M₁₆	—	4	2	no	—
259	M₁₆	—	4	2	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	8	4	no	—
272	C₄⋊C₄	—	8	4	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	4	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	4	2	no	—
283	C₄²	C₄⋊C₄×C₃₁	1	4	yes	complemented
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
287	C₂³	C₃₁×#120	1	1	yes	characteristic
288	C₂³	C₃₁×#121	1	1	yes	characteristic
289	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
290	C₂³	C₃₁×#123	1	1	yes	characteristic
291	C₂³	C₃₁×#124	1	1	yes	characteristic
292	C₂³	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
309	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
310	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
311	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
312	C₂×C₄	C₃₁×#121	1	2	yes	—
313	C₂×C₄	C₃₁×#121	1	2	yes	—
314	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
315	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
316	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
317	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
318	C₂×C₄	C₃₁×#123	1	2	yes	—
319	C₂×C₄	C₃₁×#125	1	2	yes	—
320	C₂×C₄	C₃₁×#125	1	2	yes	—
321	C₂×C₄	C₃₁×#126	1	2	yes	—
322	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
323	C₈	—	4	4	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	C₃₁×#380	1	1	yes	characteristic
335	C₂²	C₃₁×#386	1	1	yes	characteristic
336	C₂²	C₃₁×#391	1	1	yes	characteristic
337	C₂²	C₃₁×#394	1	1	yes	characteristic
338	C₂²	C₃₁×#395	1	1	yes	characteristic
339	C₂²	C₃₁×#398	1	1	yes	characteristic
340	C₂²	C₃₁×#400	1	1	yes	characteristic
341	C₄	—	4	4	no	—
342	C₄	—	4	4	no	—
343	C₄	—	8	4	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	C₃₁×#387	1	2	yes	—
347	C₄	C₃₁×#389	1	2	yes	—
348	C₄	C₃₁×#396	1	2	yes	—
349	C₄	C₃₁×#398	1	2	yes	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	C₃₁×#1441	1	1	yes	characteristic
357	C₂	C₃₁×#1911	1	1	yes	characteristic
358	C₂	C₃₁×#1912	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#386
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₃₁×#384	C₄	semidirect
15. C₃₁×#382	C₄	semidirect
103. C₄²×C₃₁	C₄⋊C₄	semidirect
104. #11224	C₃₁	direct (N × H)
184. #384	C₄×C₃₁	semidirect
185. #382	C₄×C₃₁	semidirect
236. C₃₁	#11224	direct (N × H)
283. C₄²	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2317	2
C₃₁×#2316	2
C₂×C₃₁×#343	2
C₃₁×#1733	2
C₂×C₃₁×H(3,ℤ/4)	2
#11224	31

Subgroups — C31×#11224 #21467983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11224 #21467983