Subgroups — C₃₁×#8666 #21465425

Order 7936.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1806	C₂	1	2	yes	maximal
3	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1303	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
8	C₈×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
9	C₈×D₄×C₃₁	—	2	1	no	—
10	C₈×D₄×C₃₁	—	2	1	no	—
11	C₈×D₄×C₃₁	C₄	1	1	yes	characteristic, complemented
12	C₈×D₄×C₃₁	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#431	—	2	1	no	—
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#106	—	2	1	no	—
16	C₃₁×#423	C₂²	1	2	yes	—
17	C₂×C₃₁×#104	—	2	2	no	—
18	C₂×C₄×C₈×C₃₁	—	2	1	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#340	—	2	1	no	—
22	C₃₁×#339	—	2	2	no	—
23	C₃₁×#334	—	2	1	no	—
24	C₃₁×#334	—	2	1	no	—
25	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	—	2	1	no	—
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₃₁	Q₈	1	1	yes	characteristic
33	C₄×D₄×C₃₁	—	2	1	no	—
34	C₄×D₄×C₃₁	—	2	1	no	—
35	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₃₁	Q₈	1	1	yes	characteristic
37	C₃₁×#117	—	2	1	no	—
38	C₃₁×#117	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄²×C₃₁	—	2	1	no	—
41	C₂×C₄²×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#106	—	2	4	no	—
44	C₃₁×C₄⋊C₈	—	2	1	no	—
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	—	2	2	no	—
48	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₃₁×#104	—	4	4	no	—
51	C₃₁×C₂²⋊C₈	—	2	1	no	—
52	C₃₁×C₂²⋊C₈	—	2	1	no	—
53	C₃₁×C₂²⋊C₈	—	2	1	no	—
54	C₄×C₈×C₃₁	—	2	1	no	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₃₁×#95	—	2	2	no	—
62	#41×C₃₁	—	2	2	no	—
63	#41×C₃₁	SD₁₆	1	4	yes	—
64	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₄×C₃₁	—	2	1	no	—
66	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	—	2	1	no	—
68	C₂²×C₄×C₃₁	—	2	1	no	—
69	C₂²×C₄×C₃₁	—	2	1	no	—
70	C₂²×C₄×C₃₁	—	4	2	no	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
74	M₁₆×C₃₁	—	4	2	no	—
75	M₁₆×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	1	no	—
79	C₂×C₈×C₃₁	—	4	1	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	4	2	no	—
83	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₄⋊C₄×C₃₁	—	2	1	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	4	2	no	—
91	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
93	C₂²⋊C₄×C₃₁	—	2	1	no	—
94	C₂²⋊C₄×C₃₁	—	2	1	no	—
95	C₂²⋊C₄×C₃₁	—	2	1	no	—
96	C₄²×C₃₁	—	2	1	no	—
97	C₄²×C₃₁	—	2	1	no	—
98	C₄²×C₃₁	—	2	1	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	2	no	—
101	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₃₁	#41	1	1	yes	characteristic
105	#8666	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	#121	1	1	yes	characteristic
108	Q₈×C₃₁	—	2	1	no	—
109	Q₈×C₃₁	#106	1	2	yes	complemented
110	D₄×C₃₁	—	2	4	no	—
111	D₄×C₃₁	—	2	1	no	—
112	D₄×C₃₁	#106	1	2	yes	complemented
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	4	1	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	1	no	—
122	C₂×C₄×C₃₁	—	4	1	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	8	2	no	—
125	C₂×C₄×C₃₁	—	8	2	no	—
126	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₃₁	#135	1	2	yes	—
134	C₈×C₃₁	—	2	1	no	—
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	—	2	1	no	—
137	C₈×C₃₁	—	4	1	no	—
138	C₈×C₃₁	—	4	1	no	—
139	C₈×C₃₁	—	8	2	no	—
140	C₈×C₃₁	#104	1	2	yes	complemented
141	C₈×#41	C₂×C₃₁	1	1	yes	characteristic
142	#1806	C₂×C₃₁	1	2	yes	—
143	C₄×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#1303	C₂×C₃₁	1	2	yes	complemented
146	C₂²×C₃₁	—	2	1	no	—
147	C₂²×C₃₁	—	2	1	no	—
148	C₂²×C₃₁	—	2	2	no	—
149	C₂²×C₃₁	—	4	1	no	—
150	C₂²×C₃₁	—	4	1	no	—
151	C₂²×C₃₁	#380	1	1	yes	characteristic
152	C₂²×C₃₁	#465	1	2	yes	—
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	4	1	no	—
158	C₄×C₃₁	—	4	1	no	—
159	C₄×C₃₁	—	8	4	no	—
160	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
161	C₄×C₃₁	#421	1	1	yes	characteristic
162	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
163	C₄×C₃₁	#427	1	1	yes	characteristic
164	C₄×C₃₁	#465	1	2	yes	—
165	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
166	C₈×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
167	C₈×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
168	C₈×D₄	—	2	1	no	—
169	C₈×D₄	—	2	1	no	—
170	C₈×D₄	C₄×C₃₁	1	1	yes	characteristic, complemented
171	C₈×D₄	C₄×C₃₁	1	1	yes	characteristic, complemented
172	#431	—	2	1	no	—
173	#431	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×#106	—	2	1	no	—
175	#423	C₂²×C₃₁	1	2	yes	—
176	C₂×#104	—	2	2	no	—
177	C₂×C₄×C₈	—	2	1	no	—
178	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
179	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
180	#340	—	2	1	no	—
181	#339	—	2	2	no	—
182	#334	—	2	1	no	—
183	#334	—	2	1	no	—
184	C₂×C₃₁	—	2	2	no	—
185	C₂×C₃₁	—	4	2	no	—
186	C₂×C₃₁	#1806	1	1	yes	characteristic
187	C₂×C₃₁	#1807	1	1	yes	characteristic
188	C₂×C₃₁	#1814	1	1	yes	characteristic
189	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
195	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
196	C₄×Q₈	Q₈×C₃₁	1	1	yes	characteristic
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₄×D₄	Q₈×C₃₁	1	1	yes	characteristic
201	#117	—	2	1	no	—
202	#117	D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
207	#106	—	2	4	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
213	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
214	#104	—	4	4	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
222	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
223	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
224	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
225	#95	—	2	2	no	—
226	C₃₁	#8666	1	1	yes	characteristic, direct product factor, Sylow
227	#41	—	2	2	no	—
228	#41	SD₁₆×C₃₁	1	4	yes	—
229	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
257	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
267	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
269	C₄²	#41×C₃₁	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	C₃₁×#121	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	C₃₁×#106	1	2	yes	complemented
274	D₄	—	2	4	no	—
275	D₄	—	2	1	no	—
276	D₄	C₃₁×#106	1	2	yes	complemented
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
293	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
294	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
295	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
296	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
297	C₂×C₄	C₃₁×#135	1	2	yes	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₈	—	8	2	no	—
304	C₈	C₃₁×#104	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	C₃₁×#380	1	1	yes	characteristic
311	C₂²	C₃₁×#465	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	8	4	no	—
319	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
320	C₄	C₃₁×#421	1	1	yes	characteristic
321	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
322	C₄	C₃₁×#427	1	1	yes	characteristic
323	C₄	C₃₁×#465	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	C₃₁×#1806	1	1	yes	characteristic
327	C₂	C₃₁×#1807	1	1	yes	characteristic
328	C₂	C₃₁×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₃₁×#106	C₂	semidirect
4. C₃₁×#1566	C₂	semidirect
5. C₃₁×#1303	C₂	semidirect
7. C₈×Q₈×C₃₁	C₄	semidirect
8. C₈×Q₈×C₃₁	C₄	semidirect
11. C₈×D₄×C₃₁	C₄	semidirect
12. C₈×D₄×C₃₁	C₄	semidirect
105. #8666	C₃₁	direct (N × H)
109. Q₈×C₃₁	#106	semidirect
112. D₄×C₃₁	#106	semidirect
140. C₈×C₃₁	#104	semidirect
143. C₄×#106	C₂×C₃₁	semidirect
144. #1566	C₂×C₃₁	semidirect
145. #1303	C₂×C₃₁	semidirect
166. C₈×Q₈	C₄×C₃₁	semidirect
167. C₈×Q₈	C₄×C₃₁	semidirect
170. C₈×D₄	C₄×C₃₁	semidirect
171. C₈×D₄	C₄×C₃₁	semidirect
226. C₃₁	#8666	direct (N × H)
273. Q₈	C₃₁×#106	semidirect
276. D₄	C₃₁×#106	semidirect
304. C₈	C₃₁×#104	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₃₁	2
C₃₁×#1806	2
C₄×C₃₁×#106	2
C₃₁×#1566	2
C₃₁×#1303	2
#8666	31

Subgroups — C31×#8666 #21465425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8666 #21465425