Subgroups — C₃₁×#2405 #16842338

Order 3968.

1116 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#123	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#394	C₂	1	4	yes	maximal, complemented
5	C₃₁×#386	C₂	1	2	yes	maximal, complemented
6	C₃₁×#385	C₂	1	2	yes	maximal, complemented
7	C₃₁×#379	C₂	1	2	yes	maximal, complemented
8	C₃₁×#379	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
10	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₃₁	—	2	4	no	—
12	C₂²×D₄×C₃₁	—	2	8	no	—
13	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#123	—	2	8	no	—
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
26	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
28	C₃₁×#95	C₂²	1	4	yes	complemented
29	C₂⁴×C₃₁	—	2	4	no	—
30	C₂⁴×C₃₁	—	2	4	no	—
31	C₂⁴×C₃₁	—	2	2	no	—
32	C₂⁴×C₃₁	—	2	4	no	—
33	C₂⁴×C₃₁	—	2	4	no	—
34	C₂⁴×C₃₁	—	4	2	no	—
35	C₂⁴×C₃₁	—	4	2	no	—
36	C₂⁴×C₃₁	D₄	1	4	yes	—
37	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
38	C₂⁴×C₃₁	C₂³	1	2	yes	—
39	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
40	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₃₁	—	2	32	no	—
42	C₂×D₄×C₃₁	—	2	4	no	—
43	C₂×D₄×C₃₁	—	2	4	no	—
44	C₂×D₄×C₃₁	—	2	4	no	—
45	C₂×D₄×C₃₁	—	2	4	no	—
46	C₂×D₄×C₃₁	—	2	4	no	—
47	C₂×D₄×C₃₁	D₄	1	8	yes	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
56	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
58	C₄⋊C₄×C₃₁	—	2	8	no	—
59	C₂²⋊C₄×C₃₁	—	2	4	no	—
60	C₂²⋊C₄×C₃₁	—	2	4	no	—
61	C₂²⋊C₄×C₃₁	—	2	4	no	—
62	C₂²⋊C₄×C₃₁	—	2	8	no	—
63	C₂²⋊C₄×C₃₁	—	2	4	no	—
64	C₂²⋊C₄×C₃₁	—	4	2	no	—
65	C₂²⋊C₄×C₃₁	—	4	4	no	—
66	C₂³×C₃₁	—	2	2	no	—
67	C₂³×C₃₁	—	2	2	no	—
68	C₂³×C₃₁	—	2	2	no	—
69	C₂³×C₃₁	—	2	2	no	—
70	C₂³×C₃₁	—	2	8	no	—
71	C₂³×C₃₁	—	2	8	no	—
72	C₂³×C₃₁	—	2	8	no	—
73	C₂³×C₃₁	—	2	2	no	—
74	C₂³×C₃₁	—	2	2	no	—
75	C₂³×C₃₁	—	2	2	no	—
76	C₂³×C₃₁	—	2	2	no	—
77	C₂³×C₃₁	—	2	4	no	—
78	C₂³×C₃₁	—	2	2	no	—
79	C₂³×C₃₁	—	2	4	no	—
80	C₂³×C₃₁	—	4	4	no	—
81	C₂³×C₃₁	—	4	16	no	—
82	C₂³×C₃₁	—	4	16	no	—
83	C₂³×C₃₁	—	4	4	no	—
84	C₂³×C₃₁	—	4	4	no	—
85	C₂³×C₃₁	C₂×D₄	1	2	yes	—
86	C₂³×C₃₁	C₂×D₄	1	4	yes	—
87	C₂³×C₃₁	C₂×D₄	1	4	yes	—
88	C₂³×C₃₁	C₂×D₄	1	2	yes	—
89	C₂³×C₃₁	C₂×D₄	1	2	yes	—
90	C₂³×C₃₁	#41	1	4	yes	—
91	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
92	D₄×C₃₁	—	2	16	no	—
93	D₄×C₃₁	—	2	16	no	—
94	C₂×C₄×C₃₁	—	2	8	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	4	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	—	4	4	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
107	#2405	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	2	2	no	—
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	2	8	no	—
113	C₂²×C₃₁	—	2	8	no	—
114	C₂²×C₃₁	—	2	2	no	—
115	C₂²×C₃₁	—	2	4	no	—
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	2	8	no	—
119	C₂²×C₃₁	—	4	16	no	—
120	C₂²×C₃₁	—	4	16	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	—	4	2	no	—
123	C₂²×C₃₁	—	4	4	no	—
124	C₂²×C₃₁	—	4	4	no	—
125	C₂²×C₃₁	#120	1	4	yes	—
126	C₂²×C₃₁	#123	1	4	yes	complemented
127	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
129	C₂²×C₃₁	C₂×#41	1	2	yes	—
130	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
131	C₄×C₃₁	—	2	4	no	—
132	C₄×C₃₁	—	4	2	no	—
133	C₄×C₃₁	—	4	2	no	—
134	C₄×C₃₁	—	4	4	no	—
135	C₂³×D₄	C₂×C₃₁	1	1	yes	characteristic
136	C₂×#123	C₂×C₃₁	1	2	yes	complemented
137	C₂²×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#394	C₂×C₃₁	1	4	yes	complemented
139	#386	C₂×C₃₁	1	2	yes	complemented
140	#385	C₂×C₃₁	1	2	yes	complemented
141	#379	C₂×C₃₁	1	2	yes	complemented
142	#379	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂×C₃₁	—	2	4	no	—
144	C₂×C₃₁	—	2	4	no	—
145	C₂×C₃₁	—	4	2	no	—
146	C₂×C₃₁	—	4	4	no	—
147	C₂×C₃₁	C₂×#120	1	1	yes	characteristic
148	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
149	C₂×C₃₁	#534	1	1	yes	characteristic
150	C₂×C₃₁	#546	1	4	yes	—
151	C₂⁵	C₂²×C₃₁	1	1	yes	characteristic
152	C₂⁵	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×D₄	—	2	4	no	—
154	C₂²×D₄	—	2	8	no	—
155	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
156	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
157	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
159	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
160	#123	—	2	8	no	—
161	C₂×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
162	C₂×C₂²⋊C₄	—	2	4	no	—
163	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
166	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
167	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
168	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
169	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
170	#95	C₂²×C₃₁	1	4	yes	complemented
171	C₃₁	#2405	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	—	2	4	no	—
173	C₂⁴	—	2	4	no	—
174	C₂⁴	—	2	2	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	—	2	4	no	—
177	C₂⁴	—	4	2	no	—
178	C₂⁴	—	4	2	no	—
179	C₂⁴	D₄×C₃₁	1	4	yes	—
180	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
181	C₂⁴	C₂³×C₃₁	1	2	yes	—
182	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
183	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
184	C₂×D₄	—	2	32	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	D₄×C₃₁	1	8	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
198	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
199	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
200	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
201	C₄⋊C₄	—	2	8	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	8	no	—
214	C₂³	—	2	8	no	—
215	C₂³	—	2	8	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	16	no	—
225	C₂³	—	4	16	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	4	no	—
228	C₂³	C₂×D₄×C₃₁	1	2	yes	—
229	C₂³	C₂×D₄×C₃₁	1	4	yes	—
230	C₂³	C₂×D₄×C₃₁	1	4	yes	—
231	C₂³	C₂×D₄×C₃₁	1	2	yes	—
232	C₂³	C₂×D₄×C₃₁	1	2	yes	—
233	C₂³	#41×C₃₁	1	4	yes	—
234	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
235	D₄	—	2	16	no	—
236	D₄	—	2	16	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	16	no	—
262	C₂²	—	4	16	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	4	no	—
266	C₂²	—	4	4	no	—
267	C₂²	C₃₁×#120	1	4	yes	—
268	C₂²	C₃₁×#123	1	4	yes	complemented
269	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
270	C₂²	C₂²×D₄×C₃₁	1	2	yes	—
271	C₂²	C₂×#41×C₃₁	1	2	yes	—
272	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	4	no	—
277	C₂	—	2	4	no	—
278	C₂	—	2	4	no	—
279	C₂	—	4	2	no	—
280	C₂	—	4	4	no	—
281	C₂	C₂×C₃₁×#120	1	1	yes	characteristic
282	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
283	C₂	C₃₁×#534	1	1	yes	characteristic
284	C₂	C₃₁×#546	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#123	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#394	C₂	semidirect
5. C₃₁×#386	C₂	semidirect
6. C₃₁×#385	C₂	semidirect
7. C₃₁×#379	C₂	semidirect
8. C₃₁×#379	C₂	semidirect
16. C₂³×C₄×C₃₁	C₂²	semidirect
17. C₂³×C₄×C₃₁	C₂²	semidirect
19. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
23. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
24. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
25. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
27. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
28. C₃₁×#95	C₂²	semidirect
107. #2405	C₃₁	direct (N × H)
126. C₂²×C₃₁	#123	semidirect
136. C₂×#123	C₂×C₃₁	semidirect
137. C₂²×C₂²⋊C₄	C₂×C₃₁	semidirect
138. #394	C₂×C₃₁	semidirect
139. #386	C₂×C₃₁	semidirect
140. #385	C₂×C₃₁	semidirect
141. #379	C₂×C₃₁	semidirect
142. #379	C₂×C₃₁	semidirect
158. C₂³×C₄	C₂²×C₃₁	semidirect
159. C₂³×C₄	C₂²×C₃₁	semidirect
161. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
165. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
166. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
167. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
168. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
169. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
170. #95	C₂²×C₃₁	semidirect
171. C₃₁	#2405	direct (N × H)
268. C₂²	C₃₁×#123	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₃₁	2
C₂×C₃₁×#123	2
C₂²×C₂²⋊C₄×C₃₁	2
C₃₁×#394	2
C₃₁×#386	2
C₃₁×#385	2
C₃₁×#379	2
#2405	31

Subgroups — C31×#2405 #16842338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2405 #16842338