Subgroups — C₃₁×#20465 #21477224

Order 7936.

930 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	2	yes	maximal
2	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1768	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1767	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1750	C₂	1	2	yes	maximal
6	C₃₁×#1749	C₂	1	2	yes	maximal
7	C₃₁×#1746	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#433	C₄	1	16	yes	complemented
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₂×C₃₁×#97	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₃₁	C₄	1	4	yes	—
23	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
24	C₃₁×#376	C₂²	1	1	yes	characteristic
25	C₃₁×#376	C₂²	1	1	yes	characteristic
26	C₄³×C₃₁	C₂²	1	1	yes	characteristic
27	C₄³×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#336	C₂²	1	4	yes	—
29	C₃₁×#336	C₂²	1	4	yes	—
30	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₃₁×#117	—	2	4	no	—
35	C₃₁×#117	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₁	—	2	4	no	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×C₄⋊C₈	C₂×C₄	1	16	yes	complemented
51	C₃₁×C₂²⋊C₈	C₂×C₄	1	16	yes	—
52	C₃₁×#97	—	2	4	no	—
53	C₃₁×#97	—	2	4	no	—
54	C₃₁×#97	—	2	4	no	—
55	C₃₁×#97	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₃₁	—	2	4	no	—
57	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
58	C₃₁×#95	C₂×C₄	1	4	yes	—
59	C₃₁×#95	C₂×C₄	1	4	yes	—
60	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	—	4	4	no	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
79	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	C₄²	1	8	yes	—
82	C₂²⋊C₄×C₃₁	—	2	4	no	—
83	C₂²⋊C₄×C₃₁	C₄²	1	8	yes	—
84	C₄²×C₃₁	—	2	4	no	—
85	C₄²×C₃₁	—	2	2	no	—
86	C₄²×C₃₁	—	2	2	no	—
87	C₄²×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	—	4	4	no	—
89	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
90	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
91	C₄²×C₃₁	#41	1	4	yes	—
92	C₄²×C₃₁	#41	1	4	yes	—
93	#20465	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	1	no	—
95	C₂³×C₃₁	—	2	1	no	—
96	C₂³×C₃₁	—	2	4	no	—
97	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
98	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	8	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	4	8	no	—
111	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
112	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₃₁	#117	1	4	yes	—
115	C₂×C₄×C₃₁	#117	1	4	yes	—
116	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
119	C₈×C₃₁	—	2	8	no	—
120	C₈×C₃₁	—	2	8	no	—
121	C₂×#433	C₂×C₃₁	1	2	yes	—
122	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
123	#1768	C₂×C₃₁	1	2	yes	complemented
124	#1767	C₂×C₃₁	1	2	yes	complemented
125	#1750	C₂×C₃₁	1	2	yes	—
126	#1749	C₂×C₃₁	1	2	yes	—
127	#1746	C₂×C₃₁	1	2	yes	complemented
128	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
129	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂²×C₃₁	—	2	4	no	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	4	2	no	—
133	C₂²×C₃₁	—	4	2	no	—
134	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
135	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
137	C₂²×C₃₁	#568	1	4	yes	—
138	C₄×C₃₁	—	2	4	no	—
139	C₄×C₃₁	—	2	4	no	—
140	C₄×C₃₁	—	2	4	no	—
141	C₄×C₃₁	—	2	4	no	—
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₄×C₃₁	—	4	2	no	—
145	C₄×C₃₁	C₄×#41	1	4	yes	—
146	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
148	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
149	#433	C₄×C₃₁	1	16	yes	complemented
150	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
151	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
152	C₂×#97	C₂²×C₃₁	1	2	yes	—
153	C₂×#97	C₂²×C₃₁	1	2	yes	—
154	C₂×#97	C₂²×C₃₁	1	2	yes	—
155	C₂×#97	C₂²×C₃₁	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
158	C₄×C₄⋊C₄	C₄×C₃₁	1	4	yes	—
159	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
160	#376	C₂²×C₃₁	1	1	yes	characteristic
161	#376	C₂²×C₃₁	1	1	yes	characteristic
162	C₄³	C₂²×C₃₁	1	1	yes	characteristic
163	C₄³	C₂²×C₃₁	1	1	yes	characteristic
164	#336	C₂²×C₃₁	1	4	yes	—
165	#336	C₂²×C₃₁	1	4	yes	—
166	C₂×C₃₁	—	4	2	no	—
167	C₂×C₃₁	C₄×#117	1	1	yes	characteristic
168	C₂×C₃₁	#2875	1	2	yes	—
169	C₂×C₃₁	#2921	1	2	yes	—
170	C₂×C₃₁	#2923	1	2	yes	—
171	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
172	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
174	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
175	#117	—	2	4	no	—
176	#117	C₂×C₄×C₃₁	1	8	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
184	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
185	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
191	C₄⋊C₈	C₂×C₄×C₃₁	1	16	yes	complemented
192	C₂²⋊C₈	C₂×C₄×C₃₁	1	16	yes	—
193	#97	—	2	4	no	—
194	#97	—	2	4	no	—
195	#97	—	2	4	no	—
196	#97	C₂×C₄×C₃₁	1	8	yes	complemented
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
199	#95	C₂×C₄×C₃₁	1	4	yes	—
200	#95	C₂×C₄×C₃₁	1	4	yes	—
201	C₃₁	#20465	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
215	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
221	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	C₄²×C₃₁	1	8	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	C₄²×C₃₁	1	8	yes	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	4	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
232	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
233	C₄²	#41×C₃₁	1	4	yes	—
234	C₄²	#41×C₃₁	1	4	yes	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
239	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
253	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
254	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
255	C₂×C₄	C₃₁×#117	1	4	yes	—
256	C₂×C₄	C₃₁×#117	1	4	yes	—
257	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
258	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
259	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
260	C₈	—	2	8	no	—
261	C₈	—	2	8	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
267	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
268	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
269	C₂²	C₃₁×#568	1	4	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	C₄×#41×C₃₁	1	4	yes	—
278	C₂	—	4	2	no	—
279	C₂	C₄×C₃₁×#117	1	1	yes	characteristic
280	C₂	C₃₁×#2875	1	2	yes	—
281	C₂	C₃₁×#2921	1	2	yes	—
282	C₂	C₃₁×#2923	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1768	C₂	semidirect
4. C₃₁×#1767	C₂	semidirect
7. C₃₁×#1746	C₂	semidirect
8. C₄×C₃₁×#97	C₂	semidirect
9. C₄×C₃₁×#97	C₂	semidirect
13. C₃₁×#433	C₄	semidirect
50. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
55. C₃₁×#97	C₂×C₄	semidirect
57. C₄×C₈×C₃₁	C₂×C₄	semidirect
93. #20465	C₃₁	direct (N × H)
123. #1768	C₂×C₃₁	semidirect
124. #1767	C₂×C₃₁	semidirect
127. #1746	C₂×C₃₁	semidirect
128. C₄×#97	C₂×C₃₁	semidirect
129. C₄×#97	C₂×C₃₁	semidirect
149. #433	C₄×C₃₁	semidirect
191. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
196. #97	C₂×C₄×C₃₁	semidirect
198. C₄×C₈	C₂×C₄×C₃₁	semidirect
201. C₃₁	#20465	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×#433	2
C₄×C₃₁×#117	2
C₃₁×#1768	2
C₃₁×#1767	2
C₃₁×#1750	2
C₃₁×#1749	2
C₃₁×#1746	2
C₄×C₃₁×#97	2
#20465	31

Subgroups — C31×#20465 #21477224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20465 #21477224