Subgroups — C₃₁×#8172 #21464931

Order 7936.

730 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₃₁×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₃₁	—	2	1	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃₁×#97	—	2	2	no	—
11	C₂×C₃₁×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
14	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
16	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
17	C₄³×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#336	—	2	1	no	—
19	C₃₁×D₄⋊C₈	C₄	1	8	yes	complemented
20	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	2	2	no	—
22	C₂³×C₄×C₃₁	—	2	1	no	—
23	C₂²×C₈×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₄×D₄×C₃₁	—	2	4	no	—
26	C₄×D₄×C₃₁	—	2	4	no	—
27	C₄×D₄×C₃₁	—	2	2	no	—
28	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
33	C₂×C₄²×C₃₁	—	2	2	no	—
34	C₂×C₄²×C₃₁	—	2	1	no	—
35	C₂×C₄²×C₃₁	—	4	4	no	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
43	C₃₁×#97	—	2	8	no	—
44	C₃₁×#97	—	4	2	no	—
45	C₄×C₈×C₃₁	—	2	2	no	—
46	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
47	C₃₁×#95	—	2	2	no	—
48	C₃₁×#95	—	2	1	no	—
49	C₂⁴×C₃₁	—	2	1	no	—
50	C₂×D₄×C₃₁	—	2	1	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	C₄²	1	2	yes	—
53	C₂×D₄×C₃₁	M₁₆	1	4	yes	—
54	C₂²×C₄×C₃₁	—	2	8	no	—
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	—	4	4	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	—	2	8	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₄⋊C₄×C₃₁	C₄²	1	2	yes	—
76	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	—
77	C₂²⋊C₄×C₃₁	—	2	8	no	—
78	C₂²⋊C₄×C₃₁	—	2	4	no	—
79	C₄²×C₃₁	—	2	4	no	—
80	C₄²×C₃₁	—	2	2	no	—
81	C₄²×C₃₁	—	2	2	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₄²×C₃₁	—	2	2	no	—
84	C₄²×C₃₁	—	2	1	no	—
85	C₄²×C₃₁	—	4	4	no	—
86	C₄²×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
88	C₄²×C₃₁	#41	1	2	yes	—
89	#8172	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	—	2	4	no	—
92	C₂³×C₃₁	—	4	1	no	—
93	C₂³×C₃₁	—	4	1	no	—
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	D₄×C₃₁	—	2	2	no	—
97	D₄×C₃₁	—	2	4	no	—
98	D₄×C₃₁	#97	1	4	yes	complemented
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	4	16	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	1	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	#104	1	4	yes	—
115	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
124	C₈×C₃₁	—	2	8	no	—
125	C₈×C₃₁	—	4	4	no	—
126	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
127	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
129	C₂×D₄⋊C₈	C₂×C₃₁	1	2	yes	—
130	#1279	C₂×C₃₁	1	2	yes	complemented
131	C₂²×C₃₁	—	2	4	no	—
132	C₂²×C₃₁	—	4	4	no	—
133	C₂²×C₃₁	—	4	4	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃₁	#407	1	2	yes	—
138	C₂²×C₃₁	#417	1	1	yes	characteristic
139	C₂²×C₃₁	#418	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×#104	1	2	yes	—
141	C₄×C₃₁	—	2	4	no	—
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₄×C₃₁	—	4	4	no	—
145	C₄×C₃₁	—	4	2	no	—
146	C₄×C₃₁	C₂×#97	1	1	yes	characteristic
147	C₄×C₃₁	C₄×M₁₆	1	1	yes	characteristic
148	C₄×C₃₁	#432	1	2	yes	—
149	C₄×C₃₁	#440	1	2	yes	—
150	C₄×C₃₁	#442	1	2	yes	—
151	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
152	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×C₄²	—	2	1	no	—
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
155	C₂×#97	—	2	2	no	—
156	C₂×#97	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
158	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
159	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
160	C₄×C₂²⋊C₄	—	2	1	no	—
161	C₄×C₂²⋊C₄	—	2	2	no	—
162	C₄³	C₂²×C₃₁	1	1	yes	characteristic
163	#336	—	2	1	no	—
164	D₄⋊C₈	C₄×C₃₁	1	8	yes	complemented
165	C₂×C₃₁	—	4	4	no	—
166	C₂×C₃₁	#1489	1	2	yes	—
167	C₂×C₃₁	#1490	1	2	yes	—
168	C₂×C₃₁	#1749	1	1	yes	characteristic
169	C₂×C₃₁	C₄×#104	1	1	yes	characteristic
170	C₂×C₃₁	#1762	1	1	yes	characteristic
171	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	2	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₄×D₄	—	2	4	no	—
177	C₄×D₄	—	2	4	no	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
180	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	—	2	1	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	—	4	4	no	—
187	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
188	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₃₁	1	2	yes	—
190	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
194	#97	—	2	8	no	—
195	#97	—	4	2	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
198	#95	—	2	2	no	—
199	#95	—	2	1	no	—
200	C₃₁	#8172	1	1	yes	characteristic, direct product factor, Sylow
201	C₂⁴	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	C₄²×C₃₁	1	2	yes	—
205	C₂×D₄	M₁₆×C₃₁	1	4	yes	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	C₄²×C₃₁	1	2	yes	—
228	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	4	no	—
238	C₄²	—	4	2	no	—
239	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
240	C₄²	#41×C₃₁	1	2	yes	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
247	D₄	—	2	2	no	—
248	D₄	—	2	4	no	—
249	D₄	C₃₁×#97	1	4	yes	complemented
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₃₁×#104	1	4	yes	—
266	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
268	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
269	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
270	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
271	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
272	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
273	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
275	C₈	—	2	8	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
283	C₂²	C₃₁×#407	1	2	yes	—
284	C₂²	C₃₁×#417	1	1	yes	characteristic
285	C₂²	C₃₁×#418	1	1	yes	characteristic
286	C₂²	C₂×C₃₁×#104	1	2	yes	—
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	2	no	—
292	C₄	C₂×C₃₁×#97	1	1	yes	characteristic
293	C₄	C₄×M₁₆×C₃₁	1	1	yes	characteristic
294	C₄	C₃₁×#432	1	2	yes	—
295	C₄	C₃₁×#440	1	2	yes	—
296	C₄	C₃₁×#442	1	2	yes	—
297	C₂	—	4	4	no	—
298	C₂	C₃₁×#1489	1	2	yes	—
299	C₂	C₃₁×#1490	1	2	yes	—
300	C₂	C₃₁×#1749	1	1	yes	characteristic
301	C₂	C₄×C₃₁×#104	1	1	yes	characteristic
302	C₂	C₃₁×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1767	C₂	semidirect
3. C₄×C₃₁×#97	C₂	semidirect
5. C₃₁×#1279	C₂	semidirect
19. C₃₁×D₄⋊C₈	C₄	semidirect
42. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
46. C₄×C₈×C₃₁	C₂×C₄	semidirect
89. #8172	C₃₁	direct (N × H)
98. D₄×C₃₁	#97	semidirect
127. #1767	C₂×C₃₁	semidirect
128. C₄×#97	C₂×C₃₁	semidirect
130. #1279	C₂×C₃₁	semidirect
164. D₄⋊C₈	C₄×C₃₁	semidirect
193. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
197. C₄×C₈	C₂×C₄×C₃₁	semidirect
200. C₃₁	#8172	direct (N × H)
249. D₄	C₃₁×#97	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₃₁	2
C₃₁×#1767	2
C₄×C₃₁×#97	2
C₂×C₃₁×D₄⋊C₈	2
C₃₁×#1279	2
#8172	31

Subgroups — C31×#8172 #21464931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8172 #21464931