Subgroups — C₃₁×#10152 #21466911

Order 7936.

642 conjugacy classes of subgroups.

The subgroup lattice has 370 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	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1549	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	C₂²×SD₁₆×C₃₁	—	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×#493	—	2	1	no	—
13	C₃₁×#492	—	2	2	no	—
14	C₃₁×#492	—	2	2	no	—
15	C₃₁×#489	—	2	2	no	—
16	C₃₁×#487	—	2	1	no	—
17	C₃₁×#486	—	2	1	no	—
18	C₃₁×#486	—	2	1	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	C₃₁×#417	—	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	C₂×SD₁₆×C₃₁	—	2	8	no	—
35	C₂×SD₁₆×C₃₁	—	4	8	no	—
36	C₂×D₈×C₃₁	—	2	4	no	—
37	C₂×M₁₆×C₃₁	—	2	1	no	—
38	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
39	C₂²×C₈×C₃₁	—	2	1	no	—
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×C₄⋊Q₈	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
45	C₃₁×#127	—	2	1	no	—
46	C₃₁×#127	—	2	1	no	—
47	C₃₁×#127	—	2	1	no	—
48	C₃₁×#127	—	2	1	no	—
49	C₃₁×#127	D₄	1	1	yes	characteristic
50	C₃₁×#124	—	2	1	no	—
51	C₃₁×#121	—	2	1	no	—
52	C₃₁×#121	—	2	1	no	—
53	C₄×D₄×C₃₁	—	2	1	no	—
54	C₄×D₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×C₄⋊C₈	—	2	1	no	—
59	C₃₁×C₄⋊C₈	—	2	1	no	—
60	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
61	C₃₁×#103	—	2	4	no	—
62	C₃₁×#102	—	2	4	no	—
63	C₃₁×#102	—	2	4	no	—
64	C₃₁×#100	—	4	4	no	—
65	C₃₁×C₂²⋊C₈	—	2	1	no	—
66	C₃₁×#97	—	2	1	no	—
67	C₄×C₈×C₃₁	—	2	1	no	—
68	C₄×C₈×C₃₁	—	2	1	no	—
69	C₄×C₈×C₃₁	—	2	1	no	—
70	C₄×C₈×C₃₁	D₄	1	2	yes	—
71	C₂⁴×C₃₁	—	2	2	no	—
72	#41×C₃₁	—	4	2	no	—
73	#41×C₃₁	—	4	2	no	—
74	C₂×Q₈×C₃₁	—	2	1	no	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	4	4	no	—
77	C₂×Q₈×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	2	1	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	2	1	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	4	4	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	—	4	1	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	2	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	8	2	no	—
92	C₂²×C₄×C₃₁	—	4	1	no	—
93	C₂²×C₄×C₃₁	—	4	1	no	—
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
97	SD₁₆×C₃₁	—	4	16	no	—
98	D₈×C₃₁	—	8	4	no	—
99	M₁₆×C₃₁	—	2	2	no	—
100	M₁₆×C₃₁	—	2	2	no	—
101	M₁₆×C₃₁	—	4	2	no	—
102	C₂×C₈×C₃₁	—	2	4	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	4	1	no	—
107	C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	1	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₄⋊C₄×C₃₁	—	4	1	no	—
113	C₂²⋊C₄×C₃₁	—	4	1	no	—
114	C₂²⋊C₄×C₃₁	—	4	1	no	—
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
119	#10152	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	4	1	no	—
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	—	4	4	no	—
123	C₂³×C₃₁	—	8	2	no	—
124	C₂³×C₃₁	—	8	2	no	—
125	C₂³×C₃₁	#121	1	1	yes	characteristic
126	Q₈×C₃₁	—	4	4	no	—
127	Q₈×C₃₁	—	4	2	no	—
128	Q₈×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	4	4	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	4	4	no	—
133	D₄×C₃₁	—	4	2	no	—
134	D₄×C₃₁	—	8	2	no	—
135	D₄×C₃₁	—	8	2	no	—
136	D₄×C₃₁	—	8	2	no	—
137	D₄×C₃₁	—	8	2	no	—
138	D₄×C₃₁	—	8	1	no	—
139	D₄×C₃₁	—	8	1	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	8	2	no	—
145	C₂×C₄×C₃₁	—	8	1	no	—
146	C₂×C₄×C₃₁	—	8	1	no	—
147	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
148	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
149	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
150	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
151	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
152	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
153	C₂×C₄×C₃₁	#135	1	2	yes	—
154	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
155	C₈×C₃₁	—	2	4	no	—
156	C₈×C₃₁	—	4	2	no	—
157	C₈×C₃₁	—	4	2	no	—
158	C₈×C₃₁	—	8	1	no	—
159	#3205	C₂×C₃₁	1	1	yes	characteristic
160	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#3141	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#1492	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#1278	C₂×C₃₁	1	1	yes	characteristic, complemented
166	C₂²×C₃₁	—	2	1	no	—
167	C₂²×C₃₁	—	2	1	no	—
168	C₂²×C₃₁	—	8	2	no	—
169	C₂²×C₃₁	—	8	1	no	—
170	C₂²×C₃₁	—	8	1	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	—	8	1	no	—
173	C₂²×C₃₁	—	8	2	no	—
174	C₂²×C₃₁	—	16	4	no	—
175	C₂²×C₃₁	#392	1	1	yes	characteristic
176	C₂²×C₃₁	#463	1	2	yes	—
177	C₄×C₃₁	—	2	1	no	—
178	C₄×C₃₁	—	2	1	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	—	8	1	no	—
181	C₄×C₃₁	#449	1	2	yes	—
182	C₄×C₃₁	#453	1	2	yes	—
183	C₄×C₃₁	#471	1	2	yes	—
184	C₄×C₃₁	#495	1	1	yes	characteristic
185	C₄×C₃₁	#496	1	1	yes	characteristic
186	C₂²×SD₁₆	—	2	2	no	—
187	#532	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
190	#493	—	2	1	no	—
191	#492	—	2	2	no	—
192	#492	—	2	2	no	—
193	#489	—	2	2	no	—
194	#487	—	2	1	no	—
195	#486	—	2	1	no	—
196	#486	—	2	1	no	—
197	#459	—	2	2	no	—
198	#447	—	2	2	no	—
199	#431	C₂²×C₃₁	1	1	yes	characteristic
200	#423	C₂²×C₃₁	1	1	yes	characteristic
201	#417	—	2	2	no	—
202	C₂×#100	—	2	2	no	—
203	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
205	#332	—	2	2	no	—
206	D₄⋊C₈	—	2	2	no	—
207	C₂×C₃₁	—	2	2	no	—
208	C₂×C₃₁	—	16	1	no	—
209	C₂×C₃₁	—	16	2	no	—
210	C₂×C₃₁	#2002	1	1	yes	characteristic
211	C₂×C₃₁	#2012	1	1	yes	characteristic
212	C₂×C₃₁	#2019	1	1	yes	characteristic
213	C₂×#41	—	2	1	no	—
214	C₂²×Q₈	—	2	1	no	—
215	C₂²×D₄	—	2	1	no	—
216	C₂²×D₄	—	2	1	no	—
217	C₂²×D₄	D₄×C₃₁	1	2	yes	—
218	C₂×SD₁₆	—	2	8	no	—
219	C₂×SD₁₆	—	4	8	no	—
220	C₂×D₈	—	2	4	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
223	C₂²×C₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
228	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
229	#127	—	2	1	no	—
230	#127	—	2	1	no	—
231	#127	—	2	1	no	—
232	#127	—	2	1	no	—
233	#127	D₄×C₃₁	1	1	yes	characteristic
234	#124	—	2	1	no	—
235	#121	—	2	1	no	—
236	#121	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
245	#103	—	2	4	no	—
246	#102	—	2	4	no	—
247	#102	—	2	4	no	—
248	#100	—	4	4	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	#97	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	D₄×C₃₁	1	2	yes	—
255	C₃₁	#10152	1	1	yes	characteristic, direct product factor, Sylow
256	C₂⁴	—	2	2	no	—
257	#41	—	4	2	no	—
258	#41	—	4	2	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	4	4	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	4	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	8	2	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
282	SD₁₆	—	4	16	no	—
283	D₈	—	8	4	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	4	2	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	4	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
304	C₂³	—	4	1	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	4	no	—
307	C₂³	—	8	2	no	—
308	C₂³	—	8	2	no	—
309	C₂³	C₃₁×#121	1	1	yes	characteristic
310	Q₈	—	4	4	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	2	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	2	no	—
320	D₄	—	8	2	no	—
321	D₄	—	8	2	no	—
322	D₄	—	8	1	no	—
323	D₄	—	8	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	8	1	no	—
331	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
332	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
333	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
334	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
335	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
336	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
337	C₂×C₄	C₃₁×#135	1	2	yes	—
338	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
339	C₈	—	2	4	no	—
340	C₈	—	4	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	8	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	1	no	—
347	C₂²	—	8	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	16	4	no	—
352	C₂²	C₃₁×#392	1	1	yes	characteristic
353	C₂²	C₃₁×#463	1	2	yes	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	1	no	—
358	C₄	C₃₁×#449	1	2	yes	—
359	C₄	C₃₁×#453	1	2	yes	—
360	C₄	C₃₁×#471	1	2	yes	—
361	C₄	C₃₁×#495	1	1	yes	characteristic
362	C₄	C₃₁×#496	1	1	yes	characteristic
363	C₂	—	2	2	no	—
364	C₂	—	16	2	no	—
365	C₂	—	16	1	no	—
366	C₂	C₃₁×#2002	1	1	yes	characteristic
367	C₂	C₃₁×#2012	1	1	yes	characteristic
368	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. C₂×C₃₁×#492	C₂	semidirect
3. C₃₁×#3141	C₂	semidirect
4. C₃₁×#1965	C₂	semidirect
5. C₃₁×#1549	C₂	semidirect
6. C₃₁×#1492	C₂	semidirect
7. C₃₁×#1278	C₂	semidirect
119. #10152	C₃₁	direct (N × H)
160. C₂×#492	C₂×C₃₁	semidirect
161. #3141	C₂×C₃₁	semidirect
162. #1965	C₂×C₃₁	semidirect
163. #1549	C₂×C₃₁	semidirect
164. #1492	C₂×C₃₁	semidirect
165. #1278	C₂×C₃₁	semidirect
255. C₃₁	#10152	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×#3205	2
C₂×C₃₁×#492	2
C₃₁×#3141	2
C₃₁×#1965	2
C₃₁×#1549	2
C₃₁×#1492	2
C₃₁×#1278	2
#10152	31

Subgroups — C31×#10152 #21466911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10152 #21466911