Subgroups — C₃₁×#11229 #21467988

Order 7936.

654 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	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1740	C₂	1	2	yes	maximal
4	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#117	C₂²	1	2	yes	—
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₃₁×#427	—	2	4	no	—
12	C₂×C₃₁×#107	—	2	1	no	—
13	C₂×C₃₁×#106	—	2	1	no	—
14	C₃₁×#424	—	2	1	no	—
15	C₃₁×#424	—	2	1	no	—
16	C₃₁×#424	—	2	2	no	—
17	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	complemented
18	C₃₁×#342	—	2	4	no	—
19	C₃₁×#340	—	2	2	no	—
20	C₃₁×H(3,ℤ/4)	—	2	4	no	—
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₃₁	—	2	1	no	—
23	C₂×M₁₆×C₃₁	—	2	1	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	D₄	1	2	yes	—
33	C₃₁×#117	—	2	1	no	—
34	C₃₁×#117	—	2	1	no	—
35	C₃₁×#117	—	2	1	no	—
36	C₃₁×#117	—	2	1	no	—
37	C₃₁×#117	—	2	2	no	—
38	C₃₁×#117	—	2	2	no	—
39	C₃₁×#117	—	2	2	no	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	D₄	1	4	yes	—
44	C₃₁×#117	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	Q₈	1	2	yes	—
48	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	—	2	2	no	—
53	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
54	C₃₁×#107	—	2	4	no	—
55	C₃₁×#106	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	4	no	—
57	C₃₁×C₄⋊C₈	—	2	4	no	—
58	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
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	M₁₆×C₃₁	—	4	1	no	—
73	M₁₆×C₃₁	—	4	2	no	—
74	M₁₆×C₃₁	—	4	1	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	1	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	1	no	—
80	C₂×C₈×C₃₁	—	2	1	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	4	2	no	—
90	C₄⋊C₄×C₃₁	—	4	2	no	—
91	C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂²⋊C₄×C₃₁	—	2	4	no	—
94	C₂²⋊C₄×C₃₁	—	2	4	no	—
95	C₂²⋊C₄×C₃₁	—	4	2	no	—
96	C₂²⋊C₄×C₃₁	—	4	2	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	—	2	4	no	—
100	C₄²×C₃₁	—	2	4	no	—
101	C₄²×C₃₁	—	4	2	no	—
102	C₄²×C₃₁	—	4	2	no	—
103	#11229	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	4	2	no	—
105	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₃₁	#121	1	1	yes	characteristic
107	C₂³×C₃₁	#121	1	1	yes	characteristic
108	C₂³×C₃₁	#121	1	2	yes	—
109	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	8	4	no	—
115	C₂×C₄×C₃₁	—	8	4	no	—
116	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#121	1	2	yes	—
122	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#123	1	2	yes	—
128	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#125	1	2	yes	—
130	C₂×C₄×C₃₁	#125	1	2	yes	—
131	C₂×C₄×C₃₁	#126	1	2	yes	—
132	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
134	C₈×C₃₁	—	2	2	no	—
135	C₈×C₃₁	—	2	2	no	—
136	C₈×C₃₁	—	4	2	no	—
137	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
139	#1740	C₂×C₃₁	1	2	yes	—
140	#1735	C₂×C₃₁	1	1	yes	characteristic
141	C₂×H(3,ℤ/4)	C₂×C₃₁	1	2	yes	—
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	—	2	2	no	—
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	4	2	no	—
146	C₂²×C₃₁	—	4	2	no	—
147	C₂²×C₃₁	#380	1	1	yes	characteristic
148	C₂²×C₃₁	#389	1	2	yes	—
149	C₂²×C₃₁	#394	1	1	yes	characteristic
150	C₂²×C₃₁	#396	1	2	yes	—
151	C₂²×C₃₁	#398	1	1	yes	characteristic
152	C₄×C₃₁	—	4	4	no	—
153	C₄×C₃₁	—	8	4	no	—
154	C₄×C₃₁	—	8	4	no	—
155	C₄×C₃₁	#386	1	1	yes	characteristic
156	C₄×C₃₁	#387	1	2	yes	—
157	C₄×C₃₁	#391	1	1	yes	characteristic
158	C₄×C₃₁	#395	1	1	yes	characteristic
159	C₄×C₃₁	#398	1	2	yes	—
160	C₄×C₃₁	#400	1	1	yes	characteristic
161	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
162	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×#117	C₂²×C₃₁	1	2	yes	—
164	C₂×#117	C₂²×C₃₁	1	2	yes	—
165	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
166	#427	—	2	4	no	—
167	C₂×#107	—	2	1	no	—
168	C₂×#106	—	2	1	no	—
169	#424	—	2	2	no	—
170	#424	—	2	1	no	—
171	#424	—	2	1	no	—
172	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	complemented
173	#342	—	2	4	no	—
174	#340	—	2	2	no	—
175	H(3,ℤ/4)	—	2	4	no	—
176	C₂×C₃₁	—	2	1	no	—
177	C₂×C₃₁	—	2	2	no	—
178	C₂×C₃₁	—	2	2	no	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	#1441	1	1	yes	characteristic
181	C₂×C₃₁	#1812	1	2	yes	—
182	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
190	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
191	C₂²×C₈	D₄×C₃₁	1	2	yes	—
192	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	#117	—	2	1	no	—
197	#117	—	2	1	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	D₄×C₃₁	1	4	yes	—
205	#117	Q₈×C₃₁	1	4	yes	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	Q₈×C₃₁	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
215	#107	—	2	4	no	—
216	#106	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	2	4	no	—
219	C₃₁	#11229	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
229	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	#41×C₃₁	1	2	yes	—
233	C₂²×C₄	#41×C₃₁	1	2	yes	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
246	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂³	C₃₁×#121	1	1	yes	characteristic
268	C₂³	C₃₁×#121	1	1	yes	characteristic
269	C₂³	C₃₁×#121	1	2	yes	—
270	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
279	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
280	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
281	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#121	1	2	yes	—
283	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
285	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
286	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
287	C₂×C₄	C₃₁×#123	1	2	yes	—
288	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
289	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
290	C₂×C₄	C₃₁×#125	1	2	yes	—
291	C₂×C₄	C₃₁×#125	1	2	yes	—
292	C₂×C₄	C₃₁×#126	1	2	yes	—
293	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
294	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₃₁×#380	1	1	yes	characteristic
304	C₂²	C₃₁×#389	1	2	yes	—
305	C₂²	C₃₁×#394	1	1	yes	characteristic
306	C₂²	C₃₁×#396	1	2	yes	—
307	C₂²	C₃₁×#398	1	1	yes	characteristic
308	C₄	—	4	4	no	—
309	C₄	—	8	4	no	—
310	C₄	—	8	4	no	—
311	C₄	C₃₁×#386	1	1	yes	characteristic
312	C₄	C₃₁×#387	1	2	yes	—
313	C₄	C₃₁×#391	1	1	yes	characteristic
314	C₄	C₃₁×#395	1	1	yes	characteristic
315	C₄	C₃₁×#398	1	2	yes	—
316	C₄	C₃₁×#400	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	C₃₁×#1441	1	1	yes	characteristic
322	C₂	C₃₁×#1812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#117
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
103. #11229	C₃₁	direct (N × H)
172. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
219. C₃₁	#11229	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×#427	2
C₂×C₃₁×#424	2
C₃₁×#1740	2
C₃₁×#1735	2
C₂×C₃₁×H(3,ℤ/4)	2
#11229	31

Subgroups — C31×#11229 #21467988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11229 #21467988