Subgroups — C₃₁×#13004 #21469763

Order 7936.

560 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	—	2	1	no	—
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#121	—	2	1	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
11	C₃₁×#466	—	2	1	no	—
12	C₃₁×#466	—	2	1	no	—
13	C₃₁×#466	C₄	1	2	yes	—
14	C₃₁×#466	C₄	1	2	yes	—
15	C₂×C₃₁×#108	—	2	1	no	—
16	C₂×C₃₁×#108	—	2	4	no	—
17	C₂×C₃₁×#108	C₂²	1	2	yes	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#102	—	2	1	no	—
20	C₃₁×#361	—	2	4	no	—
21	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
22	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
23	C₂²×D₄×C₃₁	—	2	1	no	—
24	C₂²×D₄×C₃₁	—	4	1	no	—
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂×D₈×C₃₁	—	2	2	no	—
27	C₂×D₈×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	—	2	2	no	—
29	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	—	4	2	no	—
33	C₂×M₁₆×C₃₁	—	4	1	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	—	4	1	no	—
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#121	—	2	2	no	—
44	C₃₁×#121	—	2	2	no	—
45	C₃₁×#121	—	4	1	no	—
46	C₃₁×#121	—	4	1	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
49	C₃₁×M₃₂	—	2	2	no	—
50	C₃₁×M₃₂	—	2	2	no	—
51	C₂×C₁₆×C₃₁	—	2	2	no	—
52	C₃₁×#108	—	2	4	no	—
53	C₃₁×#108	—	2	2	no	—
54	C₃₁×#108	D₄	1	4	yes	complemented
55	C₃₁×#107	—	2	1	no	—
56	C₃₁×#107	C₂×C₄	1	2	yes	—
57	C₃₁×#102	—	2	2	no	—
58	C₃₁×#102	—	2	2	no	—
59	C₂⁴×C₃₁	—	4	1	no	—
60	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂×D₄×C₃₁	—	4	2	no	—
64	C₂×D₄×C₃₁	—	4	1	no	—
65	C₂×D₄×C₃₁	—	4	1	no	—
66	C₂×D₄×C₃₁	—	4	1	no	—
67	C₂×D₄×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	4	1	no	—
69	C₂×D₄×C₃₁	—	8	4	no	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	1	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	D₈×C₃₁	—	2	4	no	—
82	D₈×C₃₁	—	4	2	no	—
83	M₁₆×C₃₁	—	2	4	no	—
84	M₁₆×C₃₁	—	4	4	no	—
85	M₁₆×C₃₁	—	4	2	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	4	4	no	—
93	C₂×C₈×C₃₁	—	4	2	no	—
94	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₃₁	#41	1	2	yes	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	4	1	no	—
102	C₂²⋊C₄×C₃₁	—	4	2	no	—
103	C₂²⋊C₄×C₃₁	—	4	2	no	—
104	C₁₆×C₃₁	—	4	2	no	—
105	#13004	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	2	2	no	—
107	C₂³×C₃₁	—	2	1	no	—
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	—	4	1	no	—
110	C₂³×C₃₁	—	4	2	no	—
111	C₂³×C₃₁	—	8	2	no	—
112	C₂³×C₃₁	—	8	2	no	—
113	C₂³×C₃₁	—	8	1	no	—
114	C₂³×C₃₁	—	8	1	no	—
115	C₂³×C₃₁	#102	1	1	yes	characteristic
116	C₂³×C₃₁	#102	1	1	yes	characteristic
117	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	8	4	no	—
121	D₄×C₃₁	—	8	4	no	—
122	D₄×C₃₁	—	8	1	no	—
123	D₄×C₃₁	—	8	1	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	8	2	no	—
136	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
137	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
140	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
144	C₈×C₃₁	—	2	1	no	—
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	4	4	no	—
147	C₈×C₃₁	C₄×D₄	1	2	yes	—
148	C₈×C₃₁	#121	1	2	yes	—
149	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
150	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂×#361	C₂×C₃₁	1	2	yes	complemented
152	#2112	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1387	C₂×C₃₁	1	2	yes	complemented
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	2	1	no	—
160	C₂²×C₃₁	—	2	1	no	—
161	C₂²×C₃₁	—	4	2	no	—
162	C₂²×C₃₁	—	4	2	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	16	2	no	—
167	C₂²×C₃₁	—	16	2	no	—
168	C₂²×C₃₁	#361	1	2	yes	complemented
169	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
170	C₂²×C₃₁	#448	1	1	yes	characteristic
171	C₂²×C₃₁	#481	1	1	yes	characteristic
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	8	2	no	—
175	C₄×C₃₁	#386	1	1	yes	characteristic
176	C₄×C₃₁	#417	1	1	yes	characteristic
177	C₄×C₃₁	#447	1	1	yes	characteristic
178	C₄×C₃₁	#480	1	1	yes	characteristic
179	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
180	C₂²×M₁₆	—	2	1	no	—
181	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×#121	—	2	1	no	—
183	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
184	#466	—	2	1	no	—
185	#466	—	2	1	no	—
186	#466	C₄×C₃₁	1	2	yes	—
187	#466	C₄×C₃₁	1	2	yes	—
188	C₂×#108	—	2	1	no	—
189	C₂×#108	—	2	4	no	—
190	C₂×#108	C₂²×C₃₁	1	2	yes	—
191	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×#102	—	2	1	no	—
193	#361	—	2	4	no	—
194	C₂²⋊C₁₆	—	2	1	no	—
195	C₂²⋊C₁₆	—	2	1	no	—
196	C₂×C₃₁	—	2	1	no	—
197	C₂×C₃₁	—	2	1	no	—
198	C₂×C₃₁	—	2	2	no	—
199	C₂×C₃₁	—	4	1	no	—
200	C₂×C₃₁	—	16	2	no	—
201	C₂×C₃₁	#1874	1	1	yes	characteristic
202	C₂×C₃₁	C₂×#361	1	1	yes	characteristic
203	C₂×C₃₁	#2148	1	1	yes	characteristic
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	—	4	1	no	—
206	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
207	C₂×D₈	—	2	2	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
211	C₂×M₁₆	—	2	4	no	—
212	C₂×M₁₆	—	2	2	no	—
213	C₂×M₁₆	—	4	2	no	—
214	C₂×M₁₆	—	4	1	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	4	1	no	—
218	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	#121	—	2	2	no	—
225	#121	—	2	2	no	—
226	#121	—	4	1	no	—
227	#121	—	4	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₂²⋊C₄	—	4	1	no	—
230	M₃₂	—	2	2	no	—
231	M₃₂	—	2	2	no	—
232	C₂×C₁₆	—	2	2	no	—
233	#108	—	2	4	no	—
234	#108	—	2	2	no	—
235	#108	D₄×C₃₁	1	4	yes	complemented
236	#107	—	2	1	no	—
237	#107	C₂×C₄×C₃₁	1	2	yes	—
238	#102	—	2	2	no	—
239	#102	—	2	2	no	—
240	C₃₁	#13004	1	1	yes	characteristic, direct product factor, Sylow
241	C₂⁴	—	4	1	no	—
242	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	8	4	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
259	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
260	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
261	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	M₁₆	—	2	4	no	—
266	M₁₆	—	4	4	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
278	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₈	#41×C₃₁	1	2	yes	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₁₆	—	4	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	1	no	—
295	C₂³	—	8	1	no	—
296	C₂³	C₃₁×#102	1	1	yes	characteristic
297	C₂³	C₃₁×#102	1	1	yes	characteristic
298	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	8	1	no	—
304	D₄	—	8	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
318	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
320	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
321	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
324	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	4	4	no	—
328	C₈	C₄×D₄×C₃₁	1	2	yes	—
329	C₈	C₃₁×#121	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	16	2	no	—
343	C₂²	—	16	2	no	—
344	C₂²	C₃₁×#361	1	2	yes	complemented
345	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
346	C₂²	C₃₁×#448	1	1	yes	characteristic
347	C₂²	C₃₁×#481	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	8	2	no	—
351	C₄	C₃₁×#386	1	1	yes	characteristic
352	C₄	C₃₁×#417	1	1	yes	characteristic
353	C₄	C₃₁×#447	1	1	yes	characteristic
354	C₄	C₃₁×#480	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	2	no	—
358	C₂	—	4	1	no	—
359	C₂	—	16	2	no	—
360	C₂	C₃₁×#1874	1	1	yes	characteristic
361	C₂	C₂×C₃₁×#361	1	1	yes	characteristic
362	C₂	C₃₁×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#108	C₂	semidirect
3. C₂×C₃₁×#361	C₂	semidirect
4. C₃₁×#2112	C₂	semidirect
5. C₃₁×#1387	C₂	semidirect
54. C₃₁×#108	D₄	semidirect
105. #13004	C₃₁	direct (N × H)
150. C₂²×#108	C₂×C₃₁	semidirect
151. C₂×#361	C₂×C₃₁	semidirect
152. #2112	C₂×C₃₁	semidirect
153. #1387	C₂×C₃₁	semidirect
168. C₂²×C₃₁	#361	semidirect
235. #108	D₄×C₃₁	semidirect
240. C₃₁	#13004	direct (N × H)
344. C₂²	C₃₁×#361	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#466	2
C₂²×C₃₁×#108	2
C₂×C₃₁×#361	2
C₃₁×#2112	2
C₃₁×#1387	2
#13004	31

Subgroups — C31×#13004 #21469763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13004 #21469763