Subgroups — C₃₁×#11357 #21468116

Order 7936.

886 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#340	C₂	1	4	yes	maximal
3	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
4	C₃₁×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#384	—	2	2	no	—
12	C₃₁×#384	C₄	1	4	yes	complemented
13	C₂×C₃₁×#95	—	2	1	no	—
14	C₂×C₃₁×#95	—	2	1	no	—
15	C₃₁×#340	—	2	8	no	—
16	C₃₁×#336	—	2	2	no	—
17	C₂³×C₄×C₃₁	—	2	2	no	—
18	C₂³×C₄×C₃₁	—	2	2	no	—
19	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	—	2	4	no	—
24	C₂²×C₈×C₃₁	—	2	4	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
27	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
36	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
40	C₂×C₄²×C₃₁	—	2	4	no	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₃₁×#95	—	4	2	no	—
44	C₃₁×#95	—	4	2	no	—
45	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	—	4	4	no	—
53	C₂²×C₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	—	4	2	no	—
58	C₂²×C₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	—	4	4	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	D₈	1	2	yes	—
65	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₃₁	#41	1	4	yes	—
73	C₂²×C₄×C₃₁	#41	1	2	yes	—
74	C₂²×C₄×C₃₁	#41	1	2	yes	—
75	C₂×C₈×C₃₁	—	2	8	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	4	4	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	2	8	no	—
81	C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₄⋊C₄×C₃₁	—	4	8	no	—
83	C₄⋊C₄×C₃₁	—	4	8	no	—
84	C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₄⋊C₄×C₃₁	—	4	4	no	—
86	C₄⋊C₄×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	—	2	4	no	—
88	C₄²×C₃₁	—	2	4	no	—
89	#11357	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₃₁	#121	1	2	yes	—
92	C₂³×C₃₁	#123	1	2	yes	—
93	C₂³×C₃₁	#124	1	1	yes	characteristic
94	C₂³×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
95	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
96	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
97	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
98	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
99	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
100	C₂³×C₃₁	#137	1	2	yes	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	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	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	—	4	4	no	—
115	C₂×C₄×C₃₁	—	8	4	no	—
116	C₂×C₄×C₃₁	—	8	4	no	—
117	C₂×C₄×C₃₁	—	8	2	no	—
118	C₂×C₄×C₃₁	—	8	2	no	—
119	C₂×C₄×C₃₁	—	8	2	no	—
120	C₂×C₄×C₃₁	—	8	2	no	—
121	C₂×C₄×C₃₁	#102	1	4	yes	—
122	C₂×C₄×C₃₁	#103	1	4	yes	—
123	C₂×C₄×C₃₁	#117	1	2	yes	—
124	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₃₁	#123	1	4	yes	—
129	C₂×C₄×C₃₁	#123	1	2	yes	—
130	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
131	C₂×C₄×C₃₁	#125	1	4	yes	—
132	C₂×C₄×C₃₁	#126	1	4	yes	—
133	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
134	C₈×C₃₁	—	4	4	no	—
135	C₂×#384	C₂×C₃₁	1	1	yes	characteristic
136	C₂×#340	C₂×C₃₁	1	4	yes	—
137	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
138	C₄⋊#95	C₂×C₃₁	1	1	yes	characteristic
139	C₂²×C₃₁	#381	1	1	yes	characteristic
140	C₂²×C₃₁	#390	1	1	yes	characteristic
141	C₂²×C₃₁	#391	1	1	yes	characteristic
142	C₂²×C₃₁	#397	1	2	yes	—
143	C₂²×C₃₁	#398	1	2	yes	—
144	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
145	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
146	C₂²×C₃₁	#418	1	1	yes	characteristic
147	C₂²×C₃₁	#419	1	1	yes	characteristic
148	C₂²×C₃₁	#459	1	2	yes	—
149	C₂²×C₃₁	#460	1	2	yes	—
150	C₂²×C₃₁	#461	1	2	yes	—
151	C₂²×C₃₁	#462	1	2	yes	—
152	C₂²×C₃₁	#480	1	2	yes	—
153	C₂²×C₃₁	#481	1	2	yes	—
154	C₂²×C₃₁	#483	1	2	yes	—
155	C₂²×C₃₁	#484	1	2	yes	—
156	C₂²×C₃₁	#486	1	1	yes	characteristic
157	C₂²×C₃₁	#487	1	1	yes	characteristic
158	C₂²×C₃₁	#489	1	2	yes	—
159	C₂²×C₃₁	#498	1	1	yes	characteristic
160	C₂²×C₃₁	#500	1	1	yes	characteristic
161	C₂²×C₃₁	#501	1	2	yes	—
162	C₄×C₃₁	—	2	4	no	—
163	C₄×C₃₁	—	2	4	no	—
164	C₄×C₃₁	—	8	4	no	—
165	C₄×C₃₁	—	8	4	no	—
166	C₄×C₃₁	#387	1	4	yes	—
167	C₄×C₃₁	#399	1	2	yes	—
168	C₄×C₃₁	#400	1	2	yes	—
169	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
170	C₂²×C₄⋊C₄	—	2	1	no	—
171	C₂²×C₄⋊C₄	—	2	1	no	—
172	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
173	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
174	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
175	#384	—	2	2	no	—
176	#384	C₄×C₃₁	1	4	yes	complemented
177	C₂×#95	—	2	1	no	—
178	C₂×#95	—	2	1	no	—
179	#340	—	2	8	no	—
180	#336	—	2	2	no	—
181	C₂×C₃₁	#1445	1	1	yes	characteristic
182	C₂×C₃₁	#1823	1	1	yes	characteristic
183	C₂×C₃₁	#1824	1	1	yes	characteristic
184	C₂×C₃₁	#1981	1	1	yes	characteristic
185	C₂×C₃₁	#1982	1	1	yes	characteristic
186	C₂×C₃₁	#1990	1	1	yes	characteristic
187	C₂×C₃₁	#1993	1	1	yes	characteristic
188	C₂×C₃₁	#2071	1	2	yes	—
189	C₂×C₃₁	#2072	1	2	yes	—
190	C₂×C₃₁	#2083	1	2	yes	—
191	C₂×C₃₁	#2084	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
195	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
196	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	4	4	no	—
211	C₂×C₄⋊C₄	—	4	2	no	—
212	C₂×C₄⋊C₄	—	4	2	no	—
213	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
214	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
215	C₂×C₄²	—	2	4	no	—
216	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
217	C₂×C₄²	D₄×C₃₁	1	2	yes	—
218	#95	—	4	2	no	—
219	#95	—	4	2	no	—
220	C₃₁	#11357	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
239	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
240	C₂²×C₄	D₈×C₃₁	1	2	yes	—
241	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
242	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
243	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
248	C₂²×C₄	#41×C₃₁	1	4	yes	—
249	C₂²×C₄	#41×C₃₁	1	2	yes	—
250	C₂²×C₄	#41×C₃₁	1	2	yes	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	4	no	—
255	C₂×C₈	—	4	2	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	4	8	no	—
259	C₄⋊C₄	—	4	8	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	4	no	—
265	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
266	C₂³	C₃₁×#121	1	2	yes	—
267	C₂³	C₃₁×#123	1	2	yes	—
268	C₂³	C₃₁×#124	1	1	yes	characteristic
269	C₂³	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
270	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
271	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
272	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
273	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
274	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
275	C₂³	C₃₁×#137	1	2	yes	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	C₃₁×#102	1	4	yes	—
297	C₂×C₄	C₃₁×#103	1	4	yes	—
298	C₂×C₄	C₃₁×#117	1	2	yes	—
299	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
300	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
301	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
302	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
303	C₂×C₄	C₃₁×#123	1	2	yes	—
304	C₂×C₄	C₃₁×#123	1	4	yes	—
305	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
306	C₂×C₄	C₃₁×#125	1	4	yes	—
307	C₂×C₄	C₃₁×#126	1	4	yes	—
308	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
309	C₈	—	4	4	no	—
310	C₂²	C₃₁×#381	1	1	yes	characteristic
311	C₂²	C₃₁×#390	1	1	yes	characteristic
312	C₂²	C₃₁×#391	1	1	yes	characteristic
313	C₂²	C₃₁×#397	1	2	yes	—
314	C₂²	C₃₁×#398	1	2	yes	—
315	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
316	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
317	C₂²	C₃₁×#418	1	1	yes	characteristic
318	C₂²	C₃₁×#419	1	1	yes	characteristic
319	C₂²	C₃₁×#459	1	2	yes	—
320	C₂²	C₃₁×#460	1	2	yes	—
321	C₂²	C₃₁×#461	1	2	yes	—
322	C₂²	C₃₁×#462	1	2	yes	—
323	C₂²	C₃₁×#480	1	2	yes	—
324	C₂²	C₃₁×#481	1	2	yes	—
325	C₂²	C₃₁×#483	1	2	yes	—
326	C₂²	C₃₁×#484	1	2	yes	—
327	C₂²	C₃₁×#486	1	1	yes	characteristic
328	C₂²	C₃₁×#487	1	1	yes	characteristic
329	C₂²	C₃₁×#489	1	2	yes	—
330	C₂²	C₃₁×#498	1	1	yes	characteristic
331	C₂²	C₃₁×#500	1	1	yes	characteristic
332	C₂²	C₃₁×#501	1	2	yes	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	8	4	no	—
336	C₄	—	8	4	no	—
337	C₄	C₃₁×#387	1	4	yes	—
338	C₄	C₃₁×#399	1	2	yes	—
339	C₄	C₃₁×#400	1	2	yes	—
340	C₂	C₃₁×#1445	1	1	yes	characteristic
341	C₂	C₃₁×#1823	1	1	yes	characteristic
342	C₂	C₃₁×#1824	1	1	yes	characteristic
343	C₂	C₃₁×#1981	1	1	yes	characteristic
344	C₂	C₃₁×#1982	1	1	yes	characteristic
345	C₂	C₃₁×#1990	1	1	yes	characteristic
346	C₂	C₃₁×#1993	1	1	yes	characteristic
347	C₂	C₃₁×#2071	1	2	yes	—
348	C₂	C₃₁×#2072	1	2	yes	—
349	C₂	C₃₁×#2083	1	2	yes	—
350	C₂	C₃₁×#2084	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₁×#384	C₄	semidirect
89. #11357	C₃₁	direct (N × H)
176. #384	C₄×C₃₁	semidirect
220. C₃₁	#11357	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×#384	2
C₂×C₃₁×#340	2
C₂×C₃₁×#336	2
C₃₁×C₄⋊#95	2
#11357	31

Subgroups — C31×#11357 #21468116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11357 #21468116