Subgroups — C₃₁×#9747 #21466506

Order 7936.

510 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×#486	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1987	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₃₁×#486	—	2	2	no	—
11	C₃₁×#486	C₄	1	4	yes	complemented
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#102	—	2	1	no	—
16	C₂×C₃₁×#102	—	2	1	no	—
17	C₂×C₃₁×#102	—	2	2	no	—
18	C₂×C₃₁×#97	—	2	1	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#340	—	2	2	no	—
22	C₃₁×#336	—	2	1	no	—
23	C₃₁×#335	—	2	2	no	—
24	C₃₁×#333	—	2	2	no	—
25	C₃₁×#331	—	2	2	no	—
26	C₂²×Q₈×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	—	2	1	no	—
29	C₂²×D₄×C₃₁	—	2	1	no	—
30	C₂²×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₃₁×C₄⋊Q₈	—	2	1	no	—
36	C₃₁×C₄⋊Q₈	—	2	1	no	—
37	C₃₁×C₄⋊Q₈	—	2	1	no	—
38	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₃₁×#127	—	2	1	no	—
40	C₃₁×#127	—	2	1	no	—
41	C₃₁×#127	—	2	1	no	—
42	C₃₁×#127	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊C₈	—	2	1	no	—
49	C₃₁×C₄⋊C₈	—	2	1	no	—
50	C₃₁×C₄⋊C₈	—	2	1	no	—
51	C₃₁×C₄⋊C₈	—	2	1	no	—
52	C₃₁×C₄⋊C₈	—	2	1	no	—
53	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
54	C₃₁×#102	—	2	4	no	—
55	C₃₁×#102	—	2	4	no	—
56	C₃₁×#102	—	4	4	no	—
57	C₃₁×#97	—	4	2	no	—
58	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₃₁	D₄	1	2	yes	—
60	C₃₁×#95	—	2	1	no	—
61	C₂⁴×C₃₁	—	4	1	no	—
62	C₂×Q₈×C₃₁	—	2	2	no	—
63	C₂×Q₈×C₃₁	—	4	1	no	—
64	C₂×Q₈×C₃₁	—	8	2	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	4	1	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂×D₄×C₃₁	—	4	1	no	—
72	C₂×D₄×C₃₁	—	4	1	no	—
73	C₂×D₄×C₃₁	—	4	2	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂×D₄×C₃₁	—	4	1	no	—
76	C₂×D₄×C₃₁	—	4	2	no	—
77	C₂×D₄×C₃₁	—	8	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	4	1	no	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	—	2	1	no	—
85	C₂×C₈×C₃₁	—	2	1	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	4	1	no	—
90	C₂×C₈×C₃₁	—	4	1	no	—
91	C₂×C₈×C₃₁	—	4	1	no	—
92	C₂×C₈×C₃₁	—	4	2	no	—
93	C₂×C₈×C₃₁	—	4	2	no	—
94	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	4	1	no	—
101	C₄⋊C₄×C₃₁	—	4	1	no	—
102	C₄²×C₃₁	—	4	1	no	—
103	C₄²×C₃₁	—	4	1	no	—
104	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	#41	1	1	yes	characteristic
108	#9747	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₃₁	—	4	2	no	—
110	C₂³×C₃₁	—	8	1	no	—
111	C₂³×C₃₁	—	8	1	no	—
112	C₂³×C₃₁	—	8	2	no	—
113	C₂³×C₃₁	—	8	2	no	—
114	C₂³×C₃₁	#121	1	1	yes	characteristic
115	Q₈×C₃₁	—	8	2	no	—
116	Q₈×C₃₁	—	8	2	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	4	4	no	—
120	D₄×C₃₁	—	8	2	no	—
121	D₄×C₃₁	—	8	2	no	—
122	D₄×C₃₁	—	8	1	no	—
123	D₄×C₃₁	—	8	1	no	—
124	D₄×C₃₁	—	8	1	no	—
125	D₄×C₃₁	—	8	1	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	8	2	no	—
132	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
137	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
138	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
139	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
140	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#137	1	2	yes	—
144	C₈×C₃₁	—	2	2	no	—
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	—	4	2	no	—
147	C₈×C₃₁	—	8	1	no	—
148	C₈×C₃₁	—	8	1	no	—
149	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
150	#1987	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1982	C₂×C₃₁	1	1	yes	characteristic
152	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
154	C₂×#333	C₂×C₃₁	1	1	yes	characteristic, complemented
155	C₂×#331	C₂×C₃₁	1	1	yes	characteristic
156	C₂²×C₃₁	—	8	2	no	—
157	C₂²×C₃₁	—	8	2	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	16	2	no	—
160	C₂²×C₃₁	—	16	2	no	—
161	C₂²×C₃₁	#390	1	1	yes	characteristic
162	C₂²×C₃₁	#463	1	1	yes	characteristic
163	C₂²×C₃₁	#464	1	1	yes	characteristic
164	C₂²×C₃₁	#468	1	2	yes	—
165	C₂²×C₃₁	#472	1	2	yes	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	#416	1	1	yes	characteristic
171	C₄×C₃₁	#417	1	1	yes	characteristic
172	C₄×C₃₁	#440	1	2	yes	—
173	C₄×C₃₁	#463	1	1	yes	characteristic
174	C₄×C₃₁	#464	1	1	yes	characteristic
175	C₄×C₃₁	#496	1	1	yes	characteristic
176	C₄×C₃₁	#497	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
179	#486	—	2	2	no	—
180	#486	C₄×C₃₁	1	4	yes	complemented
181	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×#102	—	2	1	no	—
185	C₂×#102	—	2	1	no	—
186	C₂×#102	—	2	2	no	—
187	C₂×#97	—	2	1	no	—
188	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
189	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
190	#340	—	2	2	no	—
191	#336	—	2	1	no	—
192	#335	—	2	2	no	—
193	#333	—	2	2	no	—
194	#331	—	2	2	no	—
195	C₂×C₃₁	—	16	2	no	—
196	C₂×C₃₁	#1697	1	2	yes	—
197	C₂×C₃₁	#1698	1	2	yes	—
198	C₂×C₃₁	#1973	1	1	yes	characteristic
199	C₂×C₃₁	#1978	1	1	yes	characteristic
200	C₂×C₃₁	#1983	1	1	yes	characteristic
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
214	#127	—	2	1	no	—
215	#127	—	2	1	no	—
216	#127	—	2	1	no	—
217	#127	C₂×C₄×C₃₁	1	2	yes	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	#102	—	4	4	no	—
232	#97	—	4	2	no	—
233	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
234	C₄×C₈	D₄×C₃₁	1	2	yes	—
235	#95	—	2	1	no	—
236	C₃₁	#9747	1	1	yes	characteristic, direct product factor, Sylow
237	C₂⁴	—	4	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
271	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₄²	#41×C₃₁	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	C₃₁×#121	1	1	yes	characteristic
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	1	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	8	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
311	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
312	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
313	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
314	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
315	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
317	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
318	C₂×C₄	C₃₁×#137	1	2	yes	—
319	C₈	—	2	2	no	—
320	C₈	—	2	2	no	—
321	C₈	—	4	2	no	—
322	C₈	—	8	1	no	—
323	C₈	—	8	1	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	2	no	—
328	C₂²	—	16	2	no	—
329	C₂²	C₃₁×#390	1	1	yes	characteristic
330	C₂²	C₃₁×#463	1	1	yes	characteristic
331	C₂²	C₃₁×#464	1	1	yes	characteristic
332	C₂²	C₃₁×#468	1	2	yes	—
333	C₂²	C₃₁×#472	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₃₁×#416	1	1	yes	characteristic
339	C₄	C₃₁×#417	1	1	yes	characteristic
340	C₄	C₃₁×#440	1	2	yes	—
341	C₄	C₃₁×#463	1	1	yes	characteristic
342	C₄	C₃₁×#464	1	1	yes	characteristic
343	C₄	C₃₁×#496	1	1	yes	characteristic
344	C₄	C₃₁×#497	1	1	yes	characteristic
345	C₂	—	16	2	no	—
346	C₂	C₃₁×#1697	1	2	yes	—
347	C₂	C₃₁×#1698	1	2	yes	—
348	C₂	C₃₁×#1973	1	1	yes	characteristic
349	C₂	C₃₁×#1978	1	1	yes	characteristic
350	C₂	C₃₁×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1987	C₂	semidirect
4. C₃₁×#1770	C₂	semidirect
5. C₂×C₃₁×#335	C₂	semidirect
6. C₂×C₃₁×#333	C₂	semidirect
11. C₃₁×#486	C₄	semidirect
108. #9747	C₃₁	direct (N × H)
150. #1987	C₂×C₃₁	semidirect
152. #1770	C₂×C₃₁	semidirect
153. C₂×#335	C₂×C₃₁	semidirect
154. C₂×#333	C₂×C₃₁	semidirect
180. #486	C₄×C₃₁	semidirect
236. C₃₁	#9747	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×#486	2
C₃₁×#1987	2
C₃₁×#1982	2
C₃₁×#1770	2
C₂×C₃₁×#335	2
C₂×C₃₁×#333	2
C₂×C₃₁×#331	2
#9747	31

Subgroups — C31×#9747 #21466506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9747 #21466506