Subgroups — C₃₁×#11223 #21467982

Order 7936.

706 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	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#344	C₂	1	2	yes	maximal
4	C₃₁×#1742	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1728	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	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	C₃₁×#424	—	2	4	no	—
12	C₃₁×#423	—	2	2	no	—
13	C₃₁×#389	—	2	2	no	—
14	C₃₁×#385	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
16	C₃₁×#344	—	2	4	no	—
17	C₃₁×#343	—	2	2	no	—
18	C₃₁×#336	—	2	4	no	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
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	—	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	—	2	2	no	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	D₄	1	4	yes	—
35	C₃₁×#117	—	2	2	no	—
36	C₃₁×#117	—	2	1	no	—
37	C₃₁×#117	—	2	1	no	—
38	C₃₁×#117	—	2	2	no	—
39	C₃₁×#117	D₄	1	2	yes	—
40	C₃₁×#117	Q₈	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	—	2	1	no	—
50	C₂×C₄²×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	D₄	1	2	yes	—
52	C₃₁×C₄⋊C₈	—	2	8	no	—
53	C₃₁×#95	—	2	4	no	—
54	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	#41	1	2	yes	—
69	C₂²×C₄×C₃₁	#41	1	2	yes	—
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	M₁₆×C₃₁	—	4	2	no	—
76	M₁₆×C₃₁	—	4	2	no	—
77	M₁₆×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	4	2	no	—
83	C₂×C₈×C₃₁	—	4	2	no	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	4	8	no	—
88	C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂²⋊C₄×C₃₁	—	2	4	no	—
90	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
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	—	4	1	no	—
96	C₄²×C₃₁	—	4	2	no	—
97	C₄²×C₃₁	—	4	1	no	—
98	#11223	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	—	4	2	no	—
100	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₃₁	#117	1	1	yes	characteristic
103	C₂³×C₃₁	C₄×D₄	1	2	yes	—
104	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	8	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#121	1	2	yes	—
116	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₃₁	#121	1	2	yes	—
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#123	1	2	yes	—
125	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#125	1	2	yes	—
128	C₂×C₄×C₃₁	#125	1	2	yes	—
129	C₂×C₄×C₃₁	#126	1	2	yes	—
130	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
132	C₈×C₃₁	—	4	4	no	—
133	C₈×C₃₁	—	4	4	no	—
134	#2905	C₂×C₃₁	1	1	yes	characteristic
135	#2296	C₂×C₃₁	1	1	yes	characteristic
136	C₂×#344	C₂×C₃₁	1	2	yes	—
137	#1742	C₂×C₃₁	1	1	yes	characteristic
138	#1728	C₂×C₃₁	1	2	yes	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	4	no	—
141	C₂²×C₃₁	—	4	1	no	—
142	C₂²×C₃₁	—	4	1	no	—
143	C₂²×C₃₁	—	4	2	no	—
144	C₂²×C₃₁	#380	1	1	yes	characteristic
145	C₂²×C₃₁	#386	1	1	yes	characteristic
146	C₂²×C₃₁	#387	1	2	yes	—
147	C₂²×C₃₁	#389	1	2	yes	—
148	C₂²×C₃₁	#391	1	1	yes	characteristic
149	C₄×C₃₁	—	4	4	no	—
150	C₄×C₃₁	—	4	4	no	—
151	C₄×C₃₁	—	4	4	no	—
152	C₄×C₃₁	#394	1	1	yes	characteristic
153	C₄×C₃₁	#395	1	1	yes	characteristic
154	C₄×C₃₁	#396	1	2	yes	—
155	C₄×C₃₁	#398	1	1	yes	characteristic
156	C₄×C₃₁	#398	1	2	yes	—
157	C₄×C₃₁	#400	1	1	yes	characteristic
158	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
159	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
160	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
163	#424	—	2	4	no	—
164	#423	—	2	2	no	—
165	#389	—	2	2	no	—
166	#385	C₄×C₃₁	1	4	yes	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	#344	—	2	4	no	—
169	#343	—	2	2	no	—
170	#336	—	2	4	no	—
171	C₂×C₃₁	—	2	1	no	—
172	C₂×C₃₁	—	2	2	no	—
173	C₂×C₃₁	—	2	2	no	—
174	C₂×C₃₁	—	2	1	no	—
175	C₂×C₃₁	#1441	1	1	yes	characteristic
176	C₂×C₃₁	#1891	1	2	yes	—
177	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
178	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	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	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	D₄×C₃₁	1	4	yes	—
193	#117	—	2	2	no	—
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	#117	—	2	2	no	—
197	#117	D₄×C₃₁	1	2	yes	—
198	#117	Q₈×C₃₁	1	2	yes	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
202	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
203	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	D₄×C₃₁	1	2	yes	—
210	C₄⋊C₈	—	2	8	no	—
211	#95	—	2	4	no	—
212	C₃₁	#11223	1	1	yes	characteristic, direct product factor, Sylow
213	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
219	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
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	1	yes	characteristic
223	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	#41×C₃₁	1	2	yes	—
228	C₂²×C₄	#41×C₃₁	1	2	yes	—
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	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	8	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	2	no	—
256	C₄²	—	4	1	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
259	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂³	C₃₁×#117	1	1	yes	characteristic
261	C₂³	C₄×D₄×C₃₁	1	2	yes	—
262	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
274	C₂×C₄	C₃₁×#121	1	2	yes	—
275	C₂×C₄	C₃₁×#121	1	2	yes	—
276	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
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×C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
283	C₂×C₄	C₃₁×#123	1	2	yes	—
284	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
285	C₂×C₄	C₃₁×#125	1	2	yes	—
286	C₂×C₄	C₃₁×#125	1	2	yes	—
287	C₂×C₄	C₃₁×#126	1	2	yes	—
288	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
289	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
290	C₈	—	4	4	no	—
291	C₈	—	4	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₃₁×#380	1	1	yes	characteristic
298	C₂²	C₃₁×#386	1	1	yes	characteristic
299	C₂²	C₃₁×#387	1	2	yes	—
300	C₂²	C₃₁×#389	1	2	yes	—
301	C₂²	C₃₁×#391	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	C₃₁×#394	1	1	yes	characteristic
306	C₄	C₃₁×#395	1	1	yes	characteristic
307	C₄	C₃₁×#396	1	2	yes	—
308	C₄	C₃₁×#398	1	2	yes	—
309	C₄	C₃₁×#398	1	1	yes	characteristic
310	C₄	C₃₁×#400	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	C₃₁×#1441	1	1	yes	characteristic
316	C₂	C₃₁×#1891	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₂⁴	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#386
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
98. #11223	C₃₁	direct (N × H)
212. C₃₁	#11223	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2905	2
C₃₁×#2296	2
C₂×C₃₁×#344	2
C₃₁×#1742	2
C₃₁×#1728	2
#11223	31

Subgroups — C31×#11223 #21467982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11223 #21467982