Subgroups — C₃₁×#19629 #21476388

Order 7936.

1050 conjugacy classes of subgroups.

The subgroup lattice has 334 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	2	yes	maximal, complemented
2	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3151	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1721	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1712	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
10	C₂²×D₈×C₃₁	—	2	2	no	—
11	C₃₁×#532	C₂²	1	2	yes	—
12	C₂×C₃₁×#127	C₂²	1	2	yes	—
13	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#496	—	2	4	no	—
15	C₃₁×#493	—	2	2	no	—
16	C₃₁×#493	C₂²	1	2	yes	—
17	C₃₁×#493	C₂²	1	2	yes	—
18	C₃₁×#493	C₂²	1	4	yes	complemented
19	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#493	C₂²	1	4	yes	complemented
21	C₃₁×#492	C₂²	1	4	yes	complemented
22	C₃₁×#486	C₂²	1	2	yes	complemented
23	C₃₁×#486	C₂²	1	4	yes	complemented
24	C₃₁×#466	—	2	2	no	—
25	C₃₁×#459	—	2	4	no	—
26	C₃₁×#447	—	2	4	no	—
27	C₄×D₈×C₃₁	—	2	2	no	—
28	C₃₁×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#322	C₂²	1	2	yes	complemented
32	C₈²×C₃₁	C₂²	1	2	yes	complemented
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₂²×D₄×C₃₁	—	2	2	no	—
35	C₂²×D₄×C₃₁	—	2	2	no	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₃₁×AGL(1,ℤ/8)	—	4	16	no	—
38	C₂×SD₁₆×C₃₁	—	2	8	no	—
39	C₂×D₈×C₃₁	—	2	2	no	—
40	C₂×D₈×C₃₁	—	2	4	no	—
41	C₂×D₈×C₃₁	—	2	4	no	—
42	C₂×D₈×C₃₁	—	2	8	no	—
43	C₂×D₈×C₃₁	—	2	8	no	—
44	C₂×D₈×C₃₁	—	4	8	no	—
45	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
46	C₂²×C₈×C₃₁	D₄	1	2	yes	—
47	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₃₁×#127	—	2	2	no	—
49	C₃₁×#127	—	2	2	no	—
50	C₃₁×#127	—	2	2	no	—
51	C₃₁×#127	C₂³	1	4	yes	—
52	C₃₁×#127	C₂³	1	2	yes	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×#121	—	2	2	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#107	—	2	2	no	—
60	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
61	C₃₁×#102	—	2	4	no	—
62	C₃₁×#102	—	2	8	no	—
63	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
64	C₃₁×#97	D₄	1	2	yes	—
65	C₄×C₈×C₃₁	D₄	1	2	yes	—
66	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₃₁	C₂³	1	2	yes	—
69	C₄×C₈×C₃₁	C₂³	1	2	yes	—
70	C₂⁴×C₃₁	—	4	2	no	—
71	#41×C₃₁	—	4	8	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	—	2	4	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	4	no	—
78	C₂×D₄×C₃₁	—	2	4	no	—
79	C₂×D₄×C₃₁	—	4	8	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂×D₄×C₃₁	—	4	2	no	—
82	C₂×D₄×C₃₁	—	4	2	no	—
83	C₂×D₄×C₃₁	—	4	2	no	—
84	C₂×D₄×C₃₁	—	4	8	no	—
85	C₂×D₄×C₃₁	—	8	4	no	—
86	C₂²×C₄×C₃₁	—	4	2	no	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	SD₁₆×C₃₁	—	4	16	no	—
91	D₈×C₃₁	—	4	8	no	—
92	D₈×C₃₁	—	4	16	no	—
93	D₈×C₃₁	—	4	4	no	—
94	D₈×C₃₁	—	4	8	no	—
95	D₈×C₃₁	—	8	8	no	—
96	M₁₆×C₃₁	D₈	1	8	yes	complemented
97	C₂×C₈×C₃₁	—	2	4	no	—
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
110	#19629	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	—	4	4	no	—
113	C₂³×C₃₁	—	8	4	no	—
114	C₂³×C₃₁	—	8	4	no	—
115	C₂³×C₃₁	—	8	4	no	—
116	C₂³×C₃₁	#127	1	1	yes	characteristic
117	Q₈×C₃₁	—	4	4	no	—
118	D₄×C₃₁	—	4	4	no	—
119	D₄×C₃₁	—	4	8	no	—
120	D₄×C₃₁	—	4	8	no	—
121	D₄×C₃₁	—	4	4	no	—
122	D₄×C₃₁	—	4	4	no	—
123	D₄×C₃₁	—	4	8	no	—
124	D₄×C₃₁	—	8	4	no	—
125	D₄×C₃₁	—	8	4	no	—
126	D₄×C₃₁	—	8	4	no	—
127	D₄×C₃₁	—	8	4	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	8	4	no	—
132	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
136	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₃₁	—	2	2	no	—
140	C₈×C₃₁	—	2	4	no	—
141	C₈×C₃₁	C₂×D₈	1	8	yes	complemented
142	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
143	#3205	C₂×C₃₁	1	2	yes	complemented
144	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#3151	C₂×C₃₁	1	2	yes	complemented
146	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1721	C₂×C₃₁	1	4	yes	complemented
148	#1712	C₂×C₃₁	1	2	yes	complemented
149	#1706	C₂×C₃₁	1	2	yes	complemented
150	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	8	2	no	—
154	C₂²×C₃₁	—	8	4	no	—
155	C₂²×C₃₁	—	8	4	no	—
156	C₂²×C₃₁	—	8	2	no	—
157	C₂²×C₃₁	—	8	4	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	16	8	no	—
160	C₂²×C₃₁	#493	1	2	yes	—
161	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	#493	1	2	yes	—
166	C₄×C₃₁	C₂²×D₈	1	2	yes	—
167	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
168	C₄×C₃₁	#576	1	2	yes	—
169	C₂×AGL(1,ℤ/8)	—	2	4	no	—
170	C₂²×D₈	—	2	2	no	—
171	#532	C₂²×C₃₁	1	2	yes	—
172	C₂×#127	C₂²×C₃₁	1	2	yes	—
173	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
174	#496	—	2	4	no	—
175	#493	—	2	2	no	—
176	#493	C₂²×C₃₁	1	2	yes	—
177	#493	C₂²×C₃₁	1	2	yes	—
178	#493	C₂²×C₃₁	1	4	yes	complemented
179	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
180	#493	C₂²×C₃₁	1	4	yes	complemented
181	#492	C₂²×C₃₁	1	4	yes	complemented
182	#486	C₂²×C₃₁	1	2	yes	complemented
183	#486	C₂²×C₃₁	1	4	yes	complemented
184	#466	—	2	2	no	—
185	#459	—	2	4	no	—
186	#447	—	2	4	no	—
187	C₄×D₈	—	2	2	no	—
188	#431	C₂²×C₃₁	1	1	yes	characteristic
189	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
191	#322	C₂²×C₃₁	1	2	yes	complemented
192	C₈²	C₂²×C₃₁	1	2	yes	complemented
193	C₂×C₃₁	—	2	2	no	—
194	C₂×C₃₁	—	16	2	no	—
195	C₂×C₃₁	—	16	4	no	—
196	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
197	C₂×C₃₁	#3151	1	1	yes	characteristic
198	C₂×C₃₁	#3155	1	1	yes	characteristic
199	C₂×#41	—	2	2	no	—
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	—	2	2	no	—
203	AGL(1,ℤ/8)	—	4	16	no	—
204	C₂×SD₁₆	—	2	8	no	—
205	C₂×D₈	—	2	4	no	—
206	C₂×D₈	—	2	4	no	—
207	C₂×D₈	—	2	8	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	2	8	no	—
210	C₂×D₈	—	4	8	no	—
211	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
212	C₂²×C₈	D₄×C₃₁	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
214	#127	—	2	2	no	—
215	#127	—	2	2	no	—
216	#127	—	2	2	no	—
217	#127	C₂³×C₃₁	1	4	yes	—
218	#127	C₂³×C₃₁	1	2	yes	—
219	#124	—	2	2	no	—
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
225	#107	—	2	2	no	—
226	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
227	#102	—	2	4	no	—
228	#102	—	2	8	no	—
229	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
230	#97	D₄×C₃₁	1	2	yes	—
231	C₄×C₈	D₄×C₃₁	1	2	yes	—
232	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₃₁	1	2	yes	—
234	C₄×C₈	C₂³×C₃₁	1	2	yes	—
235	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
236	C₃₁	#19629	1	1	yes	characteristic, direct product factor, Sylow
237	C₂⁴	—	4	2	no	—
238	#41	—	4	8	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	4	8	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	8	no	—
252	C₂×D₄	—	8	4	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	SD₁₆	—	4	16	no	—
258	D₈	—	4	8	no	—
259	D₈	—	4	16	no	—
260	D₈	—	4	4	no	—
261	D₈	—	4	8	no	—
262	D₈	—	8	8	no	—
263	M₁₆	D₈×C₃₁	1	8	yes	complemented
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
267	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
268	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
277	C₂³	—	4	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	8	4	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	8	4	no	—
282	C₂³	C₃₁×#127	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	8	no	—
286	D₄	—	4	8	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	8	no	—
290	D₄	—	8	4	no	—
291	D₄	—	8	4	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
299	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
300	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	C₂×D₈×C₃₁	1	8	yes	complemented
308	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	16	8	no	—
318	C₂²	C₃₁×#493	1	2	yes	—
319	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	C₃₁×#493	1	2	yes	—
324	C₄	C₂²×D₈×C₃₁	1	2	yes	—
325	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
326	C₄	C₃₁×#576	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	16	4	no	—
329	C₂	—	16	2	no	—
330	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
331	C₂	C₃₁×#3151	1	1	yes	characteristic
332	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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
1. C₃₁×#3205	C₂	semidirect
2. C₃₁×#3169	C₂	semidirect
3. C₃₁×#3151	C₂	semidirect
4. C₂×C₃₁×#493	C₂	semidirect
5. C₃₁×#1721	C₂	semidirect
6. C₃₁×#1712	C₂	semidirect
7. C₃₁×#1706	C₂	semidirect
8. C₈×M₁₆×C₃₁	C₂	semidirect
13. C₃₁×#500	C₂²	semidirect
18. C₃₁×#493	C₂²	semidirect
19. C₃₁×#493	C₂²	semidirect
20. C₃₁×#493	C₂²	semidirect
21. C₃₁×#492	C₂²	semidirect
22. C₃₁×#486	C₂²	semidirect
23. C₃₁×#486	C₂²	semidirect
31. C₃₁×#322	C₂²	semidirect
32. C₈²×C₃₁	C₂²	semidirect
96. M₁₆×C₃₁	D₈	semidirect
110. #19629	C₃₁	direct (N × H)
141. C₈×C₃₁	C₂×D₈	semidirect
142. C₈×C₃₁	AGL(1,ℤ/8)	semidirect
143. #3205	C₂×C₃₁	semidirect
144. #3169	C₂×C₃₁	semidirect
145. #3151	C₂×C₃₁	semidirect
146. C₂×#493	C₂×C₃₁	semidirect
147. #1721	C₂×C₃₁	semidirect
148. #1712	C₂×C₃₁	semidirect
149. #1706	C₂×C₃₁	semidirect
150. C₈×M₁₆	C₂×C₃₁	semidirect
173. #500	C₂²×C₃₁	semidirect
178. #493	C₂²×C₃₁	semidirect
179. #493	C₂²×C₃₁	semidirect
180. #493	C₂²×C₃₁	semidirect
181. #492	C₂²×C₃₁	semidirect
182. #486	C₂²×C₃₁	semidirect
183. #486	C₂²×C₃₁	semidirect
191. #322	C₂²×C₃₁	semidirect
192. C₈²	C₂²×C₃₁	semidirect
236. C₃₁	#19629	direct (N × H)
263. M₁₆	D₈×C₃₁	semidirect
307. C₈	C₂×D₈×C₃₁	semidirect
308. C₈	C₃₁×AGL(1,ℤ/8)	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×#3169	2
C₃₁×#3151	2
C₂×C₃₁×#493	2
C₃₁×#1721	2
C₃₁×#1712	2
C₃₁×#1706	2
C₈×M₁₆×C₃₁	2
#19629	31

Subgroups — C31×#19629 #21476388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19629 #21476388