Subgroups — C₃₁×#8656 #21465415

Order 7936.

494 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1802	C₂	1	2	yes	maximal
4	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1296	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	—	2	1	no	—
7	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#436	—	2	2	no	—
9	C₃₁×#436	C₄	1	4	yes	complemented
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#102	—	2	2	no	—
13	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#384	C₂²	1	2	yes	—
16	C₃₁×#340	—	2	1	no	—
17	C₃₁×#340	—	2	1	no	—
18	C₃₁×#340	—	2	2	no	—
19	C₃₁×#334	—	2	2	no	—
20	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	2	1	no	—
22	C₂×M₁₆×C₃₁	—	2	4	no	—
23	C₂×M₁₆×C₃₁	—	4	4	no	—
24	C₂²×C₈×C₃₁	—	2	1	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₄×D₄×C₃₁	—	2	1	no	—
29	C₄×D₄×C₃₁	—	2	1	no	—
30	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₄×D₄×C₃₁	Q₈	1	2	yes	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₃₁×C₄⋊C₈	—	2	2	no	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₃₁×#102	—	4	4	no	—
43	C₃₁×C₂²⋊C₈	—	2	4	no	—
44	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₃₁	D₄	1	2	yes	—
46	C₂⁴×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
60	M₁₆×C₃₁	—	4	8	no	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	—	4	2	no	—
65	C₂×C₈×C₃₁	—	4	2	no	—
66	C₂×C₈×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
70	C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₄⋊C₄×C₃₁	—	4	2	no	—
72	C₄⋊C₄×C₃₁	—	4	2	no	—
73	C₄⋊C₄×C₃₁	—	4	2	no	—
74	C₄⋊C₄×C₃₁	—	4	2	no	—
75	C₄⋊C₄×C₃₁	—	8	2	no	—
76	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
77	C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₃₁	#41	1	1	yes	characteristic
83	#8656	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	4	2	no	—
86	C₂³×C₃₁	—	4	1	no	—
87	C₂³×C₃₁	—	4	1	no	—
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	#121	1	1	yes	characteristic
90	D₄×C₃₁	—	2	4	no	—
91	D₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	8	2	no	—
99	C₂×C₄×C₃₁	—	8	2	no	—
100	C₂×C₄×C₃₁	—	8	2	no	—
101	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₃₁	#135	1	2	yes	—
108	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₃₁	#137	1	2	yes	—
111	C₈×C₃₁	—	2	2	no	—
112	C₈×C₃₁	—	2	2	no	—
113	C₈×C₃₁	—	4	2	no	—
114	C₈×C₃₁	—	8	2	no	—
115	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
116	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#1802	C₂×C₃₁	1	2	yes	—
118	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#1296	C₂×C₃₁	1	2	yes	complemented
120	C₂²×C₃₁	—	4	2	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	—	4	2	no	—
123	C₂²×C₃₁	—	8	2	no	—
124	C₂²×C₃₁	—	8	2	no	—
125	C₂²×C₃₁	#380	1	1	yes	characteristic
126	C₂²×C₃₁	#463	1	2	yes	—
127	C₂²×C₃₁	#469	1	1	yes	characteristic
128	C₂²×C₃₁	#470	1	1	yes	characteristic
129	C₂²×C₃₁	#471	1	2	yes	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	2	1	no	—
132	C₄×C₃₁	—	4	2	no	—
133	C₄×C₃₁	—	16	2	no	—
134	C₄×C₃₁	#416	1	1	yes	characteristic
135	C₄×C₃₁	#418	1	1	yes	characteristic
136	C₄×C₃₁	#428	1	2	yes	—
137	C₄×C₃₁	#468	1	2	yes	—
138	C₄×C₃₁	#478	1	2	yes	—
139	C₂²×M₁₆	—	2	1	no	—
140	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
141	#436	—	2	2	no	—
142	#436	C₄×C₃₁	1	4	yes	complemented
143	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×#102	—	2	2	no	—
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
148	#384	C₂²×C₃₁	1	2	yes	—
149	#340	—	2	1	no	—
150	#340	—	2	1	no	—
151	#340	—	2	2	no	—
152	#334	—	2	2	no	—
153	C₂×C₃₁	—	8	2	no	—
154	C₂×C₃₁	#1701	1	2	yes	—
155	C₂×C₃₁	#1703	1	2	yes	—
156	C₂×C₃₁	#1804	1	1	yes	characteristic
157	C₂×C₃₁	#1809	1	1	yes	characteristic
158	C₂×C₃₁	#1815	1	1	yes	characteristic
159	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
160	C₂³×C₄	—	2	1	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	4	4	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
170	C₄×D₄	Q₈×C₃₁	1	2	yes	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	—	2	1	no	—
176	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
181	#102	—	4	4	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
184	C₄×C₈	D₄×C₃₁	1	2	yes	—
185	C₃₁	#8656	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
200	M₁₆	—	4	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
209	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	8	2	no	—
216	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
222	C₄²	#41×C₃₁	1	1	yes	characteristic
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₃₁×#121	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
243	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
244	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₃₁×#135	1	2	yes	—
247	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
248	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
249	C₂×C₄	C₃₁×#137	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	8	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	C₃₁×#380	1	1	yes	characteristic
260	C₂²	C₃₁×#463	1	2	yes	—
261	C₂²	C₃₁×#469	1	1	yes	characteristic
262	C₂²	C₃₁×#470	1	1	yes	characteristic
263	C₂²	C₃₁×#471	1	2	yes	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	16	2	no	—
268	C₄	C₃₁×#416	1	1	yes	characteristic
269	C₄	C₃₁×#418	1	1	yes	characteristic
270	C₄	C₃₁×#428	1	2	yes	—
271	C₄	C₃₁×#468	1	2	yes	—
272	C₄	C₃₁×#478	1	2	yes	—
273	C₂	—	8	2	no	—
274	C₂	C₃₁×#1701	1	2	yes	—
275	C₂	C₃₁×#1703	1	2	yes	—
276	C₂	C₃₁×#1804	1	1	yes	characteristic
277	C₂	C₃₁×#1809	1	1	yes	characteristic
278	C₂	C₃₁×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1834	C₂	semidirect
4. C₂×C₃₁×#334	C₂	semidirect
5. C₃₁×#1296	C₂	semidirect
9. C₃₁×#436	C₄	semidirect
83. #8656	C₃₁	direct (N × H)
116. #1834	C₂×C₃₁	semidirect
118. C₂×#334	C₂×C₃₁	semidirect
119. #1296	C₂×C₃₁	semidirect
142. #436	C₄×C₃₁	semidirect
185. C₃₁	#8656	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₃₁×#1834	2
C₃₁×#1802	2
C₂×C₃₁×#334	2
C₃₁×#1296	2
#8656	31

Subgroups — C31×#8656 #21465415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8656 #21465415