Subgroups — C₃₁×#10168 #21466927

Order 7936.

642 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#137	—	2	2	no	—
9	C₃₁×#532	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₃₁×#497	—	2	2	no	—
13	C₃₁×#494	—	2	1	no	—
14	C₃₁×#493	—	2	1	no	—
15	C₃₁×#492	—	2	1	no	—
16	C₃₁×#486	—	2	2	no	—
17	C₃₁×#486	—	2	1	no	—
18	C₃₁×#486	—	2	2	no	—
19	C₃₁×#459	—	2	2	no	—
20	C₃₁×#447	—	2	2	no	—
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₃₁×#423	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#102	—	2	2	no	—
24	C₂×C₃₁×#100	—	2	2	no	—
25	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#332	—	2	2	no	—
28	C₃₁×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₃₁	—	2	1	no	—
30	C₂²×Q₈×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	—	2	1	no	—
33	C₂²×D₄×C₃₁	D₄	1	2	yes	—
34	C₃₁×#137	—	4	4	no	—
35	C₃₁×#137	—	4	4	no	—
36	C₂×Q₁₆×C₃₁	—	2	4	no	—
37	C₂×SD₁₆×C₃₁	—	2	4	no	—
38	C₂×D₈×C₃₁	—	2	4	no	—
39	C₂×M₁₆×C₃₁	—	2	1	no	—
40	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
41	C₂²×C₈×C₃₁	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×C₄⋊Q₈	—	2	1	no	—
45	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
47	C₃₁×#127	—	2	1	no	—
48	C₃₁×#127	—	2	1	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#127	—	2	1	no	—
51	C₃₁×#127	D₄	1	1	yes	characteristic
52	C₃₁×#124	—	2	1	no	—
53	C₃₁×#121	—	2	1	no	—
54	C₃₁×#121	—	2	1	no	—
55	C₄×D₄×C₃₁	—	2	1	no	—
56	C₄×D₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊C₈	—	2	1	no	—
61	C₃₁×C₄⋊C₈	—	2	1	no	—
62	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
63	C₃₁×#102	—	2	8	no	—
64	C₃₁×#102	—	2	4	no	—
65	C₃₁×#100	—	4	4	no	—
66	C₃₁×C₂²⋊C₈	—	2	1	no	—
67	C₃₁×#97	—	2	1	no	—
68	C₄×C₈×C₃₁	—	2	1	no	—
69	C₄×C₈×C₃₁	—	2	1	no	—
70	C₄×C₈×C₃₁	—	2	1	no	—
71	C₄×C₈×C₃₁	D₄	1	2	yes	—
72	C₂⁴×C₃₁	—	2	2	no	—
73	#41×C₃₁	—	4	2	no	—
74	#41×C₃₁	—	4	2	no	—
75	C₂×Q₈×C₃₁	—	2	1	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	4	1	no	—
78	C₂×Q₈×C₃₁	—	4	2	no	—
79	C₂×Q₈×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	2	1	no	—
85	C₂×D₄×C₃₁	—	4	4	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	1	no	—
89	C₂×D₄×C₃₁	—	4	1	no	—
90	C₂×D₄×C₃₁	—	4	1	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	8	2	no	—
94	C₂²×C₄×C₃₁	—	4	1	no	—
95	C₂²×C₄×C₃₁	—	4	1	no	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
99	Q₁₆×C₃₁	—	4	8	no	—
100	SD₁₆×C₃₁	—	4	8	no	—
101	D₈×C₃₁	—	8	4	no	—
102	M₁₆×C₃₁	—	2	2	no	—
103	M₁₆×C₃₁	—	4	2	no	—
104	M₁₆×C₃₁	D₈	1	4	yes	complemented
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	4	2	no	—
109	C₂×C₈×C₃₁	—	4	1	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	1	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	2	1	no	—
114	C₄⋊C₄×C₃₁	—	4	1	no	—
115	C₄⋊C₄×C₃₁	—	4	1	no	—
116	C₂²⋊C₄×C₃₁	—	4	1	no	—
117	C₂²⋊C₄×C₃₁	—	4	1	no	—
118	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
122	#10168	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₃₁	—	4	1	no	—
124	C₂³×C₃₁	—	4	2	no	—
125	C₂³×C₃₁	—	4	4	no	—
126	C₂³×C₃₁	—	8	2	no	—
127	C₂³×C₃₁	—	8	2	no	—
128	C₂³×C₃₁	#121	1	1	yes	characteristic
129	Q₈×C₃₁	—	4	4	no	—
130	Q₈×C₃₁	—	4	2	no	—
131	Q₈×C₃₁	—	8	1	no	—
132	Q₈×C₃₁	—	8	1	no	—
133	D₄×C₃₁	—	4	4	no	—
134	D₄×C₃₁	—	4	2	no	—
135	D₄×C₃₁	—	4	2	no	—
136	D₄×C₃₁	—	4	4	no	—
137	D₄×C₃₁	—	4	2	no	—
138	D₄×C₃₁	—	8	2	no	—
139	D₄×C₃₁	—	8	2	no	—
140	D₄×C₃₁	—	8	2	no	—
141	D₄×C₃₁	—	8	2	no	—
142	D₄×C₃₁	—	8	1	no	—
143	D₄×C₃₁	—	8	1	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	4	1	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	8	2	no	—
149	C₂×C₄×C₃₁	—	8	1	no	—
150	C₂×C₄×C₃₁	—	8	1	no	—
151	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
152	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
153	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
154	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
155	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
156	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
158	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
159	C₈×C₃₁	—	2	4	no	—
160	C₈×C₃₁	—	4	2	no	—
161	C₈×C₃₁	—	4	2	no	—
162	C₈×C₃₁	—	8	1	no	—
163	#3205	C₂×C₃₁	1	1	yes	characteristic
164	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
165	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
167	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#1492	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1278	C₂×C₃₁	1	1	yes	characteristic, complemented
170	C₂²×C₃₁	—	2	1	no	—
171	C₂²×C₃₁	—	2	1	no	—
172	C₂²×C₃₁	—	8	2	no	—
173	C₂²×C₃₁	—	8	1	no	—
174	C₂²×C₃₁	—	8	1	no	—
175	C₂²×C₃₁	—	8	2	no	—
176	C₂²×C₃₁	—	8	1	no	—
177	C₂²×C₃₁	—	8	2	no	—
178	C₂²×C₃₁	—	16	4	no	—
179	C₂²×C₃₁	#392	1	1	yes	characteristic
180	C₂²×C₃₁	#459	1	2	yes	—
181	C₄×C₃₁	—	2	1	no	—
182	C₄×C₃₁	—	2	1	no	—
183	C₄×C₃₁	—	8	1	no	—
184	C₄×C₃₁	—	8	2	no	—
185	C₄×C₃₁	#447	1	2	yes	—
186	C₄×C₃₁	#455	1	2	yes	—
187	C₄×C₃₁	#471	1	2	yes	—
188	C₄×C₃₁	#493	1	1	yes	characteristic
189	C₄×C₃₁	#496	1	1	yes	characteristic
190	C₂×#137	—	2	2	no	—
191	#532	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
194	#497	—	2	2	no	—
195	#494	—	2	1	no	—
196	#493	—	2	1	no	—
197	#492	—	2	1	no	—
198	#486	—	2	2	no	—
199	#486	—	2	1	no	—
200	#486	—	2	2	no	—
201	#459	—	2	2	no	—
202	#447	—	2	2	no	—
203	#431	C₂²×C₃₁	1	1	yes	characteristic
204	#423	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×#102	—	2	2	no	—
206	C₂×#100	—	2	2	no	—
207	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
209	#332	—	2	2	no	—
210	D₄⋊C₈	—	2	2	no	—
211	C₂×C₃₁	—	2	2	no	—
212	C₂×C₃₁	—	16	1	no	—
213	C₂×C₃₁	—	16	2	no	—
214	C₂×C₃₁	#2004	1	1	yes	characteristic
215	C₂×C₃₁	#2011	1	1	yes	characteristic
216	C₂×C₃₁	#2019	1	1	yes	characteristic
217	C₂×#41	—	2	1	no	—
218	C₂²×Q₈	—	2	1	no	—
219	C₂²×D₄	—	2	1	no	—
220	C₂²×D₄	—	2	1	no	—
221	C₂²×D₄	D₄×C₃₁	1	2	yes	—
222	#137	—	4	4	no	—
223	#137	—	4	4	no	—
224	C₂×Q₁₆	—	2	4	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×D₈	—	2	4	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
229	C₂²×C₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
234	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
235	#127	—	2	1	no	—
236	#127	—	2	1	no	—
237	#127	—	2	1	no	—
238	#127	—	2	1	no	—
239	#127	D₄×C₃₁	1	1	yes	characteristic
240	#124	—	2	1	no	—
241	#121	—	2	1	no	—
242	#121	—	2	1	no	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
251	#102	—	2	8	no	—
252	#102	—	2	4	no	—
253	#100	—	4	4	no	—
254	C₂²⋊C₈	—	2	1	no	—
255	#97	—	2	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	D₄×C₃₁	1	2	yes	—
260	C₃₁	#10168	1	1	yes	characteristic, direct product factor, Sylow
261	C₂⁴	—	2	2	no	—
262	#41	—	4	2	no	—
263	#41	—	4	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	4	4	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	—	8	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
288	Q₁₆	—	4	8	no	—
289	SD₁₆	—	4	8	no	—
290	D₈	—	8	4	no	—
291	M₁₆	—	2	2	no	—
292	M₁₆	—	4	2	no	—
293	M₁₆	D₈×C₃₁	1	4	yes	complemented
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	4	2	no	—
298	C₂×C₈	—	4	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂³	—	4	1	no	—
312	C₂³	—	4	2	no	—
313	C₂³	—	4	4	no	—
314	C₂³	—	8	2	no	—
315	C₂³	—	8	2	no	—
316	C₂³	C₃₁×#121	1	1	yes	characteristic
317	Q₈	—	4	4	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	8	1	no	—
320	Q₈	—	8	1	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	4	no	—
325	D₄	—	4	2	no	—
326	D₄	—	8	2	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	2	no	—
330	D₄	—	8	1	no	—
331	D₄	—	8	1	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
340	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
341	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
342	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
343	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
344	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
345	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
346	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
347	C₈	—	2	4	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	16	4	no	—
360	C₂²	C₃₁×#392	1	1	yes	characteristic
361	C₂²	C₃₁×#459	1	2	yes	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	2	no	—
366	C₄	C₃₁×#447	1	2	yes	—
367	C₄	C₃₁×#455	1	2	yes	—
368	C₄	C₃₁×#471	1	2	yes	—
369	C₄	C₃₁×#493	1	1	yes	characteristic
370	C₄	C₃₁×#496	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	16	2	no	—
373	C₂	—	16	1	no	—
374	C₂	C₃₁×#2004	1	1	yes	characteristic
375	C₂	C₃₁×#2011	1	1	yes	characteristic
376	C₂	C₃₁×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#3205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3152	C₂	semidirect
3. C₂×C₃₁×#486	C₂	semidirect
4. C₃₁×#1965	C₂	semidirect
5. C₃₁×#1546	C₂	semidirect
6. C₃₁×#1492	C₂	semidirect
7. C₃₁×#1278	C₂	semidirect
104. M₁₆×C₃₁	D₈	semidirect
122. #10168	C₃₁	direct (N × H)
164. #3152	C₂×C₃₁	semidirect
165. C₂×#486	C₂×C₃₁	semidirect
166. #1965	C₂×C₃₁	semidirect
167. #1546	C₂×C₃₁	semidirect
168. #1492	C₂×C₃₁	semidirect
169. #1278	C₂×C₃₁	semidirect
260. C₃₁	#10168	direct (N × H)
293. M₁₆	D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3205	2
C₃₁×#3152	2
C₂×C₃₁×#486	2
C₃₁×#1965	2
C₃₁×#1546	2
C₃₁×#1492	2
C₃₁×#1278	2
#10168	31

Subgroups — C31×#10168 #21466927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10168 #21466927