Subgroups — C₃₁×#11242 #21468001

Order 7936.

682 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#343	C₂	1	2	yes	maximal
4	C₃₁×#1736	C₂	1	2	yes	maximal
5	C₂×C₃₁×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×#108	—	2	4	no	—
12	C₂×C₃₁×#108	—	2	2	no	—
13	C₃₁×#424	—	2	2	no	—
14	C₃₁×#424	C₄	1	4	yes	complemented
15	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
16	C₃₁×#343	—	2	4	no	—
17	C₃₁×#340	—	2	2	no	—
18	C₃₁×#340	—	2	2	no	—
19	C₃₁×#340	—	2	2	no	—
20	C₂³×C₄×C₃₁	—	2	2	no	—
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₃₁	—	2	1	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	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₃₁	D₄	1	2	yes	—
38	C₂²×C₈×C₃₁	D₄	1	2	yes	—
39	C₃₁×#117	—	2	1	no	—
40	C₃₁×#117	—	2	1	no	—
41	C₃₁×#117	C₂×C₄	1	2	yes	—
42	C₃₁×#117	Q₈	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₃₁	Q₈	1	4	yes	—
51	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄²×C₃₁	—	2	1	no	—
53	C₃₁×#108	—	2	8	no	—
54	C₃₁×C₄⋊C₈	—	2	8	no	—
55	C₂⁴×C₃₁	#41	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	—	2	1	no	—
57	C₂²×C₄×C₃₁	—	2	1	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	—	8	2	no	—
62	C₂²×C₄×C₃₁	—	8	2	no	—
63	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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×D₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	2	yes	—
71	C₂²×C₄×C₃₁	#41	1	2	yes	—
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	M₁₆×C₃₁	—	2	4	no	—
74	M₁₆×C₃₁	—	2	4	no	—
75	M₁₆×C₃₁	—	2	4	no	—
76	M₁₆×C₃₁	—	4	1	no	—
77	M₁₆×C₃₁	—	4	1	no	—
78	M₁₆×C₃₁	—	4	2	no	—
79	M₁₆×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	4	2	no	—
87	C₂×C₈×C₃₁	—	4	2	no	—
88	C₂×C₈×C₃₁	—	4	2	no	—
89	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
90	C₄⋊C₄×C₃₁	—	2	1	no	—
91	C₄⋊C₄×C₃₁	—	2	1	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
97	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
98	C₂²⋊C₄×C₃₁	—	2	2	no	—
99	C₂²⋊C₄×C₃₁	—	2	2	no	—
100	C₄²×C₃₁	—	2	2	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	#11242	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	2	2	no	—
104	C₂³×C₃₁	—	2	1	no	—
105	C₂³×C₃₁	—	2	1	no	—
106	C₂³×C₃₁	#117	1	1	yes	characteristic
107	C₂³×C₃₁	#121	1	1	yes	characteristic
108	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
110	C₂³×C₃₁	#126	1	2	yes	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	1	no	—
115	C₂×C₄×C₃₁	—	4	1	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	8	4	no	—
118	C₂×C₄×C₃₁	—	8	2	no	—
119	C₂×C₄×C₃₁	—	8	2	no	—
120	C₂×C₄×C₃₁	—	8	4	no	—
121	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#121	1	2	yes	—
127	C₂×C₄×C₃₁	#121	1	2	yes	—
128	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#123	1	2	yes	—
134	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#125	1	2	yes	—
136	C₂×C₄×C₃₁	#125	1	2	yes	—
137	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₃₁	—	2	4	no	—
140	C₈×C₃₁	—	4	2	no	—
141	C₈×C₃₁	—	4	4	no	—
142	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic
143	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
144	C₂×#343	C₂×C₃₁	1	2	yes	—
145	#1736	C₂×C₃₁	1	2	yes	—
146	C₂×#340	C₂×C₃₁	1	1	yes	characteristic
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	2	2	no	—
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	2	2	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	#380	1	1	yes	characteristic
156	C₂²×C₃₁	#387	1	2	yes	—
157	C₂²×C₃₁	#395	1	1	yes	characteristic
158	C₂²×C₃₁	#396	1	2	yes	—
159	C₂²×C₃₁	#400	1	1	yes	characteristic
160	C₂²×C₃₁	#471	1	2	yes	—
161	C₂²×C₃₁	#472	1	1	yes	characteristic
162	C₂²×C₃₁	#473	1	1	yes	characteristic
163	C₄×C₃₁	—	4	2	no	—
164	C₄×C₃₁	—	4	2	no	—
165	C₄×C₃₁	—	8	4	no	—
166	C₄×C₃₁	#386	1	1	yes	characteristic
167	C₄×C₃₁	#389	1	2	yes	—
168	C₄×C₃₁	#391	1	1	yes	characteristic
169	C₄×C₃₁	#394	1	1	yes	characteristic
170	C₄×C₃₁	#398	1	2	yes	—
171	C₄×C₃₁	#398	1	1	yes	characteristic
172	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
174	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
176	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
177	C₂×#108	—	2	4	no	—
178	C₂×#108	—	2	2	no	—
179	#424	—	2	2	no	—
180	#424	C₄×C₃₁	1	4	yes	complemented
181	C₂×C₄⋊C₈	—	2	2	no	—
182	#343	—	2	4	no	—
183	#340	—	2	2	no	—
184	#340	—	2	2	no	—
185	#340	—	2	2	no	—
186	C₂×C₃₁	—	2	2	no	—
187	C₂×C₃₁	—	2	1	no	—
188	C₂×C₃₁	—	2	1	no	—
189	C₂×C₃₁	#1441	1	1	yes	characteristic
190	C₂×C₃₁	#1815	1	1	yes	characteristic
191	C₂×C₃₁	#1816	1	1	yes	characteristic
192	C₂×C₃₁	#2063	1	2	yes	—
193	C₂×C₃₁	#2064	1	2	yes	—
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
205	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
211	C₂²×C₈	D₄×C₃₁	1	2	yes	—
212	C₂²×C₈	D₄×C₃₁	1	2	yes	—
213	#117	—	2	1	no	—
214	#117	—	2	1	no	—
215	#117	C₂×C₄×C₃₁	1	2	yes	—
216	#117	Q₈×C₃₁	1	2	yes	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
223	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
224	C₂×C₄⋊C₄	Q₈×C₃₁	1	4	yes	—
225	C₂×C₂²⋊C₄	—	2	1	no	—
226	C₂×C₄²	—	2	1	no	—
227	#108	—	2	8	no	—
228	C₄⋊C₈	—	2	8	no	—
229	C₃₁	#11242	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	#41×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	8	2	no	—
237	C₂²×C₄	—	8	2	no	—
238	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
245	C₂²×C₄	#41×C₃₁	1	2	yes	—
246	C₂²×C₄	#41×C₃₁	1	2	yes	—
247	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	2	4	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	4	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
272	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	C₃₁×#117	1	1	yes	characteristic
281	C₂³	C₃₁×#121	1	1	yes	characteristic
282	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
283	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂³	C₃₁×#126	1	2	yes	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
298	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
299	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
300	C₂×C₄	C₃₁×#121	1	2	yes	—
301	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
302	C₂×C₄	C₃₁×#121	1	2	yes	—
303	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
305	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
306	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
307	C₂×C₄	C₃₁×#123	1	2	yes	—
308	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
309	C₂×C₄	C₃₁×#125	1	2	yes	—
310	C₂×C₄	C₃₁×#125	1	2	yes	—
311	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
312	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
313	C₈	—	2	4	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	C₃₁×#380	1	1	yes	characteristic
325	C₂²	C₃₁×#387	1	2	yes	—
326	C₂²	C₃₁×#395	1	1	yes	characteristic
327	C₂²	C₃₁×#396	1	2	yes	—
328	C₂²	C₃₁×#400	1	1	yes	characteristic
329	C₂²	C₃₁×#471	1	2	yes	—
330	C₂²	C₃₁×#472	1	1	yes	characteristic
331	C₂²	C₃₁×#473	1	1	yes	characteristic
332	C₄	—	4	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	4	no	—
335	C₄	C₃₁×#386	1	1	yes	characteristic
336	C₄	C₃₁×#389	1	2	yes	—
337	C₄	C₃₁×#391	1	1	yes	characteristic
338	C₄	C₃₁×#394	1	1	yes	characteristic
339	C₄	C₃₁×#398	1	2	yes	—
340	C₄	C₃₁×#398	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	2	no	—
344	C₂	C₃₁×#1441	1	1	yes	characteristic
345	C₂	C₃₁×#1815	1	1	yes	characteristic
346	C₂	C₃₁×#1816	1	1	yes	characteristic
347	C₂	C₃₁×#2063	1	2	yes	—
348	C₂	C₃₁×#2064	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₃₁×#424	C₄	semidirect
102. #11242	C₃₁	direct (N × H)
180. #424	C₄×C₃₁	semidirect
229. C₃₁	#11242	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#108	2
C₂×C₃₁×#424	2
C₂×C₃₁×#343	2
C₃₁×#1736	2
C₂×C₃₁×#340	2
#11242	31

Subgroups — C31×#11242 #21468001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11242 #21468001