Subgroups — C₃₁×#20660 #21477419

Order 7936.

1518 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#427	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2280	C₂	1	2	yes	maximal
4	C₃₁×#1809	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1735	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	—	2	8	no	—
8	C₂²×M₁₆×C₃₁	C₄	1	8	yes	complemented
9	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#117	C₂²	1	4	yes	—
13	C₃₁×#427	—	2	8	no	—
14	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
15	C₂×C₃₁×#106	C₂²	1	4	yes	complemented
16	C₃₁×#381	—	2	2	no	—
17	C₃₁×#380	C₂²	1	8	yes	—
18	C₃₁×#379	—	2	2	no	—
19	C₄×C₂²⋊C₄×C₃₁	—	2	4	no	—
20	C₃₁×#340	C₂²	1	8	yes	complemented
21	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	—	2	4	no	—
23	C₂³×C₄×C₃₁	—	2	4	no	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₃₁	D₄	1	2	yes	—
27	C₂³×C₄×C₃₁	Q₈	1	4	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₃₁	—	2	32	no	—
31	C₂×M₁₆×C₃₁	—	2	8	no	—
32	C₂×M₁₆×C₃₁	—	2	8	no	—
33	C₂×M₁₆×C₃₁	C₂×C₄	1	16	yes	—
34	C₂²×C₈×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₃₁	D₄	1	8	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₃₁×#117	—	2	8	no	—
41	C₃₁×#117	—	2	8	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
45	C₂×C₄²×C₃₁	—	2	4	no	—
46	C₃₁×#107	—	2	8	no	—
47	C₃₁×#106	—	2	8	no	—
48	C₃₁×#95	—	2	8	no	—
49	C₂⁴×C₃₁	—	2	4	no	—
50	C₂⁴×C₃₁	—	4	4	no	—
51	C₂⁴×C₃₁	C₄⋊C₄	1	4	yes	—
52	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
53	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	2	8	no	—
56	C₂²×C₄×C₃₁	—	2	8	no	—
57	C₂²×C₄×C₃₁	—	2	8	no	—
58	C₂²×C₄×C₃₁	—	2	16	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	8	no	—
62	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₃₁	#41	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₃₁	—	2	32	no	—
74	C₂×C₈×C₃₁	—	2	8	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	C₂²⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
81	C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₂²⋊C₄×C₃₁	—	4	16	no	—
84	C₂²⋊C₄×C₃₁	—	4	8	no	—
85	C₄²×C₃₁	—	4	8	no	—
86	#20660	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	2	4	no	—
88	C₂³×C₃₁	—	2	4	no	—
89	C₂³×C₃₁	—	2	8	no	—
90	C₂³×C₃₁	—	2	8	no	—
91	C₂³×C₃₁	—	4	8	no	—
92	C₂³×C₃₁	—	4	16	no	—
93	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
94	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₃₁	#121	1	4	yes	—
96	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	8	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	8	no	—
102	C₂×C₄×C₃₁	—	2	16	no	—
103	C₂×C₄×C₃₁	—	4	8	no	—
104	C₂×C₄×C₃₁	—	4	16	no	—
105	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₃₁	#117	1	4	yes	—
109	C₂×C₄×C₃₁	#121	1	4	yes	—
110	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
111	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
115	C₈×C₃₁	—	2	4	no	—
116	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
117	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic
118	C₂×#427	C₂×C₃₁	1	2	yes	complemented
119	#2280	C₂×C₃₁	1	2	yes	—
120	#1809	C₂×C₃₁	1	8	yes	complemented
121	#1735	C₂×C₃₁	1	2	yes	complemented
122	C₂²×C₃₁	—	2	4	no	—
123	C₂²×C₃₁	—	2	8	no	—
124	C₂²×C₃₁	—	2	8	no	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	4	16	no	—
127	C₂²×C₃₁	—	4	8	no	—
128	C₂²×C₃₁	#380	1	4	yes	—
129	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₃₁	C₂×#121	1	2	yes	—
131	C₂²×C₃₁	#575	1	4	yes	—
132	C₄×C₃₁	—	2	4	no	—
133	C₄×C₃₁	—	8	8	no	—
134	C₄×C₃₁	#380	1	4	yes	—
135	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
137	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
138	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
139	C₂²×M₁₆	—	2	8	no	—
140	C₂²×M₁₆	C₄×C₃₁	1	8	yes	complemented
141	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
142	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
143	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×#117	C₂²×C₃₁	1	4	yes	—
145	#427	—	2	8	no	—
146	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
147	C₂×#106	C₂²×C₃₁	1	4	yes	complemented
148	#381	—	2	2	no	—
149	#380	C₂²×C₃₁	1	8	yes	—
150	#379	—	2	2	no	—
151	C₄×C₂²⋊C₄	—	2	4	no	—
152	#340	C₂²×C₃₁	1	8	yes	complemented
153	C₂×C₃₁	—	2	4	no	—
154	C₂×C₃₁	—	4	4	no	—
155	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
156	C₂×C₃₁	#2911	1	2	yes	—
157	C₂×C₃₁	#3054	1	4	yes	—
158	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
159	C₂³×C₄	—	2	4	no	—
160	C₂³×C₄	—	2	4	no	—
161	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
162	C₂³×C₄	D₄×C₃₁	1	2	yes	—
163	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
164	C₂³×C₄	Q₈×C₃₁	1	4	yes	—
165	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
166	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	32	no	—
168	C₂×M₁₆	—	2	8	no	—
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	C₂×C₄×C₃₁	1	16	yes	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
173	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
174	C₂²×C₈	D₄×C₃₁	1	8	yes	—
175	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
176	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
177	#117	—	2	8	no	—
178	#117	—	2	8	no	—
179	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
180	C₂×C₂²⋊C₄	—	2	4	no	—
181	C₂×C₂²⋊C₄	—	2	8	no	—
182	C₂×C₄²	—	2	4	no	—
183	#107	—	2	8	no	—
184	#106	—	2	8	no	—
185	#95	—	2	8	no	—
186	C₃₁	#20660	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	4	4	no	—
189	C₂⁴	C₄⋊C₄×C₃₁	1	4	yes	—
190	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
191	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	16	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
201	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
202	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
206	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
208	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
209	C₂²×C₄	#41×C₃₁	1	4	yes	—
210	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
211	M₁₆	—	2	32	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂²⋊C₄×C₃₁	1	16	yes	complemented
215	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
216	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
218	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₂²⋊C₄	—	4	16	no	—
222	C₂²⋊C₄	—	4	8	no	—
223	C₄²	—	4	8	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	2	8	no	—
228	C₂³	—	4	8	no	—
229	C₂³	—	4	16	no	—
230	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
231	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
232	C₂³	C₃₁×#121	1	4	yes	—
233	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	16	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
244	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
245	C₂×C₄	C₃₁×#117	1	4	yes	—
246	C₂×C₄	C₃₁×#121	1	4	yes	—
247	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
248	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
250	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
252	C₈	—	2	4	no	—
253	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	16	no	—
259	C₂²	—	4	8	no	—
260	C₂²	C₃₁×#380	1	4	yes	—
261	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂²	C₂×C₃₁×#121	1	2	yes	—
263	C₂²	C₃₁×#575	1	4	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	8	8	no	—
266	C₄	C₃₁×#380	1	4	yes	—
267	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
268	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
269	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	—	4	4	no	—
272	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
273	C₂	C₃₁×#2911	1	2	yes	—
274	C₂	C₃₁×#3054	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂⁵×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#427	C₂	semidirect
4. C₃₁×#1809	C₂	semidirect
5. C₃₁×#1735	C₂	semidirect
8. C₂²×M₁₆×C₃₁	C₄	semidirect
14. C₂×C₃₁×#107	C₂²	semidirect
15. C₂×C₃₁×#106	C₂²	semidirect
20. C₃₁×#340	C₂²	semidirect
76. C₂×C₈×C₃₁	C₂²⋊C₄	semidirect
86. #20660	C₃₁	direct (N × H)
118. C₂×#427	C₂×C₃₁	semidirect
120. #1809	C₂×C₃₁	semidirect
121. #1735	C₂×C₃₁	semidirect
140. C₂²×M₁₆	C₄×C₃₁	semidirect
146. C₂×#107	C₂²×C₃₁	semidirect
147. C₂×#106	C₂²×C₃₁	semidirect
152. #340	C₂²×C₃₁	semidirect
186. C₃₁	#20660	direct (N × H)
214. C₂×C₈	C₂²⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₃₁	2
C₂×C₃₁×#427	2
C₃₁×#2280	2
C₃₁×#1809	2
C₃₁×#1735	2
#20660	31

Subgroups — C31×#20660 #21477419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20660 #21477419