Subgroups — C₃₁×#11239 #21467998

Order 7936.

682 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#107	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	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1736	C₂	1	2	yes	maximal
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×#107	—	2	2	no	—
12	C₂×C₃₁×#107	—	2	4	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×#341	—	2	2	no	—
18	C₃₁×#340	—	2	4	no	—
19	C₂³×C₄×C₃₁	—	2	2	no	—
20	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	1	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	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
31	C₂²×C₈×C₃₁	—	2	2	no	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₃₁	D₄	1	2	yes	—
37	C₂²×C₈×C₃₁	D₄	1	2	yes	—
38	C₃₁×#117	—	2	1	no	—
39	C₃₁×#117	—	2	1	no	—
40	C₃₁×#117	C₂×C₄	1	2	yes	—
41	C₃₁×#117	Q₈	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
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	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₄²×C₃₁	—	2	1	no	—
51	C₃₁×#107	—	2	8	no	—
52	C₃₁×C₄⋊C₈	—	2	8	no	—
53	C₂⁴×C₃₁	#41	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	—	4	2	no	—
58	C₂²×C₄×C₃₁	—	8	4	no	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₃₁	#41	1	2	yes	—
67	C₂²×C₄×C₃₁	#41	1	2	yes	—
68	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	2	4	no	—
72	M₁₆×C₃₁	—	4	2	no	—
73	M₁₆×C₃₁	—	4	2	no	—
74	M₁₆×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	4	1	no	—
82	C₂×C₈×C₃₁	—	4	2	no	—
83	C₂×C₈×C₃₁	—	4	2	no	—
84	C₂×C₈×C₃₁	—	4	1	no	—
85	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
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	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	#11239	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	2	2	no	—
98	C₂³×C₃₁	—	2	2	no	—
99	C₂³×C₃₁	#117	1	1	yes	characteristic
100	C₂³×C₃₁	#121	1	1	yes	characteristic
101	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂³×C₃₁	#123	1	2	yes	—
103	C₂³×C₃₁	#125	1	2	yes	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	8	4	no	—
109	C₂×C₄×C₃₁	—	8	4	no	—
110	C₂×C₄×C₃₁	—	8	4	no	—
111	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#121	1	2	yes	—
117	C₂×C₄×C₃₁	#121	1	2	yes	—
118	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#125	1	2	yes	—
126	C₂×C₄×C₃₁	#126	1	2	yes	—
127	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
129	C₈×C₃₁	—	2	4	no	—
130	C₈×C₃₁	—	4	2	no	—
131	C₈×C₃₁	—	4	4	no	—
132	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic
133	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
134	C₂×#343	C₂×C₃₁	1	2	yes	—
135	C₂×#341	C₂×C₃₁	1	1	yes	characteristic
136	#1736	C₂×C₃₁	1	2	yes	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	2	4	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	#380	1	1	yes	characteristic
143	C₂²×C₃₁	#387	1	2	yes	—
144	C₂²×C₃₁	#394	1	1	yes	characteristic
145	C₂²×C₃₁	#398	1	2	yes	—
146	C₂²×C₃₁	#398	1	1	yes	characteristic
147	C₂²×C₃₁	#472	1	2	yes	—
148	C₂²×C₃₁	#473	1	2	yes	—
149	C₄×C₃₁	—	4	4	no	—
150	C₄×C₃₁	—	8	4	no	—
151	C₄×C₃₁	#386	1	1	yes	characteristic
152	C₄×C₃₁	#389	1	2	yes	—
153	C₄×C₃₁	#391	1	1	yes	characteristic
154	C₄×C₃₁	#395	1	1	yes	characteristic
155	C₄×C₃₁	#396	1	2	yes	—
156	C₄×C₃₁	#400	1	1	yes	characteristic
157	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
158	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
159	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
162	C₂×#107	—	2	2	no	—
163	C₂×#107	—	2	4	no	—
164	#424	—	2	2	no	—
165	#424	C₄×C₃₁	1	4	yes	complemented
166	C₂×C₄⋊C₈	—	2	2	no	—
167	#343	—	2	4	no	—
168	#341	—	2	2	no	—
169	#340	—	2	4	no	—
170	C₂×C₃₁	—	2	2	no	—
171	C₂×C₃₁	—	2	2	no	—
172	C₂×C₃₁	#1441	1	1	yes	characteristic
173	C₂×C₃₁	#1816	1	2	yes	—
174	C₂×C₃₁	#2045	1	1	yes	characteristic
175	C₂×C₃₁	#2047	1	1	yes	characteristic
176	C₂×C₃₁	#2089	1	2	yes	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
188	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
194	C₂²×C₈	D₄×C₃₁	1	2	yes	—
195	C₂²×C₈	D₄×C₃₁	1	2	yes	—
196	#117	—	2	1	no	—
197	#117	—	2	1	no	—
198	#117	C₂×C₄×C₃₁	1	2	yes	—
199	#117	Q₈×C₃₁	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
206	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	#107	—	2	8	no	—
210	C₄⋊C₈	—	2	8	no	—
211	C₃₁	#11239	1	1	yes	characteristic, direct product factor, Sylow
212	C₂⁴	#41×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	8	4	no	—
218	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
225	C₂²×C₄	#41×C₃₁	1	2	yes	—
226	C₂²×C₄	#41×C₃₁	1	2	yes	—
227	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₃₁×#117	1	1	yes	characteristic
258	C₂³	C₃₁×#121	1	1	yes	characteristic
259	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
260	C₂³	C₃₁×#123	1	2	yes	—
261	C₂³	C₃₁×#125	1	2	yes	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
272	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
273	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
274	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
275	C₂×C₄	C₃₁×#121	1	2	yes	—
276	C₂×C₄	C₃₁×#121	1	2	yes	—
277	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
278	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
280	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
283	C₂×C₄	C₃₁×#125	1	2	yes	—
284	C₂×C₄	C₃₁×#126	1	2	yes	—
285	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
286	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₈	—	4	4	no	—
289	C₈	—	4	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	C₃₁×#380	1	1	yes	characteristic
296	C₂²	C₃₁×#387	1	2	yes	—
297	C₂²	C₃₁×#394	1	1	yes	characteristic
298	C₂²	C₃₁×#398	1	2	yes	—
299	C₂²	C₃₁×#398	1	1	yes	characteristic
300	C₂²	C₃₁×#472	1	2	yes	—
301	C₂²	C₃₁×#473	1	2	yes	—
302	C₄	—	4	4	no	—
303	C₄	—	8	4	no	—
304	C₄	C₃₁×#386	1	1	yes	characteristic
305	C₄	C₃₁×#389	1	2	yes	—
306	C₄	C₃₁×#391	1	1	yes	characteristic
307	C₄	C₃₁×#395	1	1	yes	characteristic
308	C₄	C₃₁×#396	1	2	yes	—
309	C₄	C₃₁×#400	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	C₃₁×#1441	1	1	yes	characteristic
313	C₂	C₃₁×#1816	1	2	yes	—
314	C₂	C₃₁×#2045	1	1	yes	characteristic
315	C₂	C₃₁×#2047	1	1	yes	characteristic
316	C₂	C₃₁×#2089	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
96. #11239	C₃₁	direct (N × H)
165. #424	C₄×C₃₁	semidirect
211. C₃₁	#11239	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×#107	2
C₂×C₃₁×#424	2
C₂×C₃₁×#343	2
C₂×C₃₁×#341	2
C₃₁×#1736	2
#11239	31

Subgroups — C31×#11239 #21467998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11239 #21467998