Subgroups — C₃₁×#19286 #21476045

Order 7936.

690 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3037	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1702	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1694	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1688	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1547	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#137	—	2	2	no	—
10	C₃₁×#532	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#493	C₂²	1	1	yes	characteristic
14	C₃₁×#486	C₂²	1	2	yes	—
15	C₃₁×#466	—	2	1	no	—
16	C₃₁×#466	—	2	1	no	—
17	C₃₁×#464	—	2	4	no	—
18	C₃₁×#462	C₂²	1	4	yes	—
19	C₃₁×#460	C₂²	1	4	yes	—
20	C₃₁×#446	—	2	1	no	—
21	C₃₁×#446	—	2	1	no	—
22	C₃₁×#446	C₂²	1	2	yes	—
23	C₃₁×#446	C₂²	1	2	yes	—
24	C₄×D₈×C₃₁	—	2	1	no	—
25	C₄×D₈×C₃₁	—	2	1	no	—
26	C₃₁×#423	C₂²	1	2	yes	—
27	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₃₁×#417	—	2	2	no	—
30	C₂×C₃₁×#97	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#334	C₂²	1	2	yes	—
33	C₃₁×#334	C₂²	1	2	yes	—
34	C₃₁×#332	C₂²	1	4	yes	—
35	C₃₁×#331	C₂²	1	4	yes	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
38	C₃₁×#137	—	4	8	no	—
39	C₂×Q₁₆×C₃₁	—	2	4	no	—
40	C₂×SD₁₆×C₃₁	—	2	4	no	—
41	C₂×D₈×C₃₁	—	2	1	no	—
42	C₂×D₈×C₃₁	—	2	1	no	—
43	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	—	2	1	no	—
45	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₃₁×#127	C₂³	1	2	yes	—
49	C₃₁×#124	—	2	2	no	—
50	C₃₁×#121	—	2	2	no	—
51	C₃₁×#121	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	2	1	no	—
53	C₄×Q₈×C₃₁	—	2	1	no	—
54	C₄×Q₈×C₃₁	D₄	1	2	yes	—
55	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	—	2	1	no	—
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#107	—	2	1	no	—
63	C₃₁×#107	—	2	1	no	—
64	C₃₁×C₄⋊C₈	—	2	2	no	—
65	C₃₁×C₄⋊C₈	—	2	2	no	—
66	C₃₁×C₄⋊C₈	—	2	1	no	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
70	C₃₁×#103	—	2	4	no	—
71	C₃₁×#102	—	2	2	no	—
72	C₃₁×#102	—	2	2	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×#97	—	2	2	no	—
75	C₃₁×#97	—	2	2	no	—
76	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	#41×C₃₁	—	4	4	no	—
81	#41×C₃₁	—	4	4	no	—
82	C₂×Q₈×C₃₁	—	2	1	no	—
83	C₂×Q₈×C₃₁	—	2	4	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	4	2	no	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
96	Q₁₆×C₃₁	—	4	8	no	—
97	SD₁₆×C₃₁	—	4	8	no	—
98	D₈×C₃₁	—	4	2	no	—
99	D₈×C₃₁	—	4	2	no	—
100	M₁₆×C₃₁	—	4	4	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₂×C₈×C₃₁	—	4	1	no	—
105	C₂×C₈×C₃₁	—	4	2	no	—
106	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²⋊C₄×C₃₁	—	4	2	no	—
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₄²×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₃₁	#41	1	1	yes	characteristic
123	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
124	#19286	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂³×C₃₁	—	4	2	no	—
126	C₂³×C₃₁	#121	1	1	yes	characteristic
127	Q₈×C₃₁	—	2	4	no	—
128	Q₈×C₃₁	—	4	2	no	—
129	Q₈×C₃₁	—	4	2	no	—
130	Q₈×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	4	2	no	—
134	D₄×C₃₁	—	4	2	no	—
135	D₄×C₃₁	—	8	2	no	—
136	D₄×C₃₁	—	8	4	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	8	4	no	—
146	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
147	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
148	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
149	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
150	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
153	C₈×C₃₁	—	2	1	no	—
154	C₈×C₃₁	—	2	1	no	—
155	C₈×C₃₁	—	2	1	no	—
156	C₈×C₃₁	—	4	4	no	—
157	C₈×C₃₁	—	4	2	no	—
158	C₈×C₃₁	#137	1	2	yes	complemented
159	#3169	C₂×C₃₁	1	1	yes	characteristic
160	#3037	C₂×C₃₁	1	2	yes	complemented
161	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#1702	C₂×C₃₁	1	4	yes	complemented
163	#1694	C₂×C₃₁	1	2	yes	complemented
164	#1688	C₂×C₃₁	1	2	yes	complemented
165	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#1547	C₂×C₃₁	1	2	yes	complemented
167	C₂²×C₃₁	—	2	1	no	—
168	C₂²×C₃₁	—	2	1	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	#469	1	2	yes	—
173	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	4	2	no	—
178	C₄×C₃₁	—	4	2	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	#469	1	2	yes	—
181	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
182	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
183	C₄×C₃₁	#575	1	1	yes	characteristic
184	C₂×#137	—	2	2	no	—
185	#532	C₂²×C₃₁	1	2	yes	—
186	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
187	#500	C₂²×C₃₁	1	1	yes	characteristic
188	#493	C₂²×C₃₁	1	1	yes	characteristic
189	#486	C₂²×C₃₁	1	2	yes	—
190	#466	—	2	1	no	—
191	#466	—	2	1	no	—
192	#464	—	2	4	no	—
193	#462	C₂²×C₃₁	1	4	yes	—
194	#460	C₂²×C₃₁	1	4	yes	—
195	#446	—	2	1	no	—
196	#446	—	2	1	no	—
197	#446	C₂²×C₃₁	1	2	yes	—
198	#446	C₂²×C₃₁	1	2	yes	—
199	C₄×D₈	—	2	1	no	—
200	C₄×D₈	—	2	1	no	—
201	#423	C₂²×C₃₁	1	2	yes	—
202	C₂×C₄⋊C₈	—	2	1	no	—
203	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
204	#417	—	2	2	no	—
205	C₂×#97	—	2	1	no	—
206	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
207	#334	C₂²×C₃₁	1	2	yes	—
208	#334	C₂²×C₃₁	1	2	yes	—
209	#332	C₂²×C₃₁	1	4	yes	—
210	#331	C₂²×C₃₁	1	4	yes	—
211	C₂×C₃₁	—	2	2	no	—
212	C₂×C₃₁	—	16	2	no	—
213	C₂×C₃₁	#3037	1	1	yes	characteristic
214	C₂×C₃₁	C₂×#469	1	1	yes	characteristic
215	C₂×C₃₁	#3067	1	1	yes	characteristic
216	C₂×#41	—	2	2	no	—
217	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
218	#137	—	4	8	no	—
219	C₂×Q₁₆	—	2	4	no	—
220	C₂×SD₁₆	—	2	4	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
227	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
228	#127	C₂³×C₃₁	1	2	yes	—
229	#124	—	2	2	no	—
230	#121	—	2	2	no	—
231	#121	—	2	2	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	D₄×C₃₁	1	2	yes	—
235	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
242	#107	—	2	1	no	—
243	#107	—	2	1	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	2	no	—
248	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
249	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
250	#103	—	2	4	no	—
251	#102	—	2	2	no	—
252	#102	—	2	2	no	—
253	#102	—	2	4	no	—
254	#97	—	2	2	no	—
255	#97	—	2	2	no	—
256	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
257	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₃₁	#19286	1	1	yes	characteristic, direct product factor, Sylow
261	#41	—	4	4	no	—
262	#41	—	4	4	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
277	Q₁₆	—	4	8	no	—
278	SD₁₆	—	4	8	no	—
279	D₈	—	4	2	no	—
280	D₈	—	4	2	no	—
281	M₁₆	—	4	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₄²	#41×C₃₁	1	1	yes	characteristic
304	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
305	C₂³	—	4	2	no	—
306	C₂³	C₃₁×#121	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	2	no	—
316	D₄	—	8	4	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
327	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
328	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
329	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
330	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	4	4	no	—
337	C₈	—	4	2	no	—
338	C₈	C₃₁×#137	1	2	yes	complemented
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	C₃₁×#469	1	2	yes	—
345	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	C₃₁×#469	1	2	yes	—
353	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
354	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
355	C₄	C₃₁×#575	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	16	2	no	—
358	C₂	C₃₁×#3037	1	1	yes	characteristic
359	C₂	C₂×C₃₁×#469	1	1	yes	characteristic
360	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3169	C₂
LCS	G ▷ 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×#3037	C₂	semidirect
3. C₂×C₃₁×#446	C₂	semidirect
4. C₃₁×#1702	C₂	semidirect
5. C₃₁×#1694	C₂	semidirect
6. C₃₁×#1688	C₂	semidirect
7. C₃₁×#1566	C₂	semidirect
8. C₃₁×#1547	C₂	semidirect
124. #19286	C₃₁	direct (N × H)
158. C₈×C₃₁	#137	semidirect
160. #3037	C₂×C₃₁	semidirect
161. C₂×#446	C₂×C₃₁	semidirect
162. #1702	C₂×C₃₁	semidirect
163. #1694	C₂×C₃₁	semidirect
164. #1688	C₂×C₃₁	semidirect
165. #1566	C₂×C₃₁	semidirect
166. #1547	C₂×C₃₁	semidirect
260. C₃₁	#19286	direct (N × H)
338. C₈	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3169	2
C₃₁×#3037	2
C₂×C₃₁×#446	2
C₃₁×#1702	2
C₃₁×#1694	2
C₃₁×#1688	2
C₃₁×#1566	2
C₃₁×#1547	2
#19286	31

Subgroups — C31×#19286 #21476045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19286 #21476045