Subgroups — C₃₁×#2440 #16842373

Order 3968.

608 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal
3	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#398	C₂	1	2	yes	maximal, complemented
5	C₃₁×#395	C₂	1	2	yes	maximal, complemented
6	C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#385	C₂	1	2	yes	maximal
9	C₃₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#380	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊Q₈	—	2	4	no	—
16	C₃₁×C₂²⋊Q₈	—	2	8	no	—
17	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#95	C₂²	1	1	yes	characteristic
34	C₃₁×#95	C₂²	1	2	yes	—
35	C₃₁×#95	C₂²	1	2	yes	—
36	C₃₁×#95	C₂²	1	1	yes	characteristic
37	C₃₁×#95	C₂²	1	2	yes	—
38	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₃₁	—	2	4	no	—
40	C₂×Q₈×C₃₁	—	2	4	no	—
41	C₂×Q₈×C₃₁	—	4	2	no	—
42	C₂²×C₄×C₃₁	—	2	2	no	—
43	C₂²×C₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	—	2	4	no	—
46	C₂²×C₄×C₃₁	—	4	2	no	—
47	C₂²×C₄×C₃₁	Q₈	1	4	yes	complemented
48	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₃₁	—	2	4	no	—
59	C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₄⋊C₄×C₃₁	—	2	4	no	—
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₄⋊C₄×C₃₁	D₄	1	4	yes	—
68	C₂²⋊C₄×C₃₁	—	2	4	no	—
69	C₂²⋊C₄×C₃₁	D₄	1	4	yes	—
70	C₂²⋊C₄×C₃₁	Q₈	1	4	yes	complemented
71	C₄²×C₃₁	—	2	2	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	2	1	no	—
74	C₂³×C₃₁	—	2	1	no	—
75	C₂³×C₃₁	—	2	1	no	—
76	C₂³×C₃₁	—	2	1	no	—
77	C₂³×C₃₁	C₂×D₄	1	2	yes	—
78	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
79	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
80	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
81	Q₈×C₃₁	—	4	8	no	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	1	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	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
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	—	4	8	no	—
105	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
108	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
110	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
112	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
113	C₂×C₄×C₃₁	#41	1	2	yes	—
114	#2440	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	2	2	no	—
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	2	2	no	—
121	C₂²×C₃₁	C₄⋊Q₈	1	4	yes	complemented
122	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
126	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
127	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
129	C₄×C₃₁	—	2	2	no	—
130	C₄×C₃₁	—	2	4	no	—
131	C₄×C₃₁	—	2	2	no	—
132	C₄×C₃₁	—	2	2	no	—
133	C₄×C₃₁	—	2	2	no	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	4	2	no	—
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	—	4	2	no	—
138	C₄×C₃₁	C₂²⋊Q₈	1	4	yes	complemented
139	C₂×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
140	C₂×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	—
141	C₂²×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
142	#398	C₂×C₃₁	1	2	yes	complemented
143	#395	C₂×C₃₁	1	2	yes	complemented
144	#389	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#389	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#385	C₂×C₃₁	1	2	yes	—
147	#384	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#380	C₂×C₃₁	1	1	yes	characteristic
149	C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
150	C₂×C₃₁	—	2	4	no	—
151	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₂×C₃₁	#544	1	1	yes	characteristic
154	C₂×C₃₁	D₄²	1	1	yes	characteristic
155	C₂×C₃₁	#547	1	1	yes	characteristic
156	C₂×C₃₁	D₄×Q₈	1	1	yes	characteristic
157	C₂×C₃₁	#554	1	1	yes	characteristic
158	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
159	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
160	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
161	C₄⋊Q₈	—	2	4	no	—
162	C₂²⋊Q₈	—	2	8	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
171	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
173	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
179	#95	C₂²×C₃₁	1	1	yes	characteristic
180	#95	C₂²×C₃₁	1	1	yes	characteristic
181	#95	C₂²×C₃₁	1	2	yes	—
182	#95	C₂²×C₃₁	1	2	yes	—
183	#95	C₂²×C₃₁	1	2	yes	—
184	C₃₁	#2440	1	1	yes	characteristic, direct product factor, Sylow
185	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	Q₈×C₃₁	1	4	yes	complemented
195	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
196	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
198	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
201	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	D₄×C₃₁	1	4	yes	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	D₄×C₃₁	1	4	yes	—
217	C₂²⋊C₄	Q₈×C₃₁	1	4	yes	complemented
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	C₂×D₄×C₃₁	1	2	yes	—
225	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
226	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
227	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
253	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
255	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
256	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
257	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
258	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
259	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
260	C₂×C₄	#41×C₃₁	1	2	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	C₃₁×C₄⋊Q₈	1	4	yes	complemented
268	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
269	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
270	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
271	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
272	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
273	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	C₃₁×C₂²⋊Q₈	1	4	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
287	C₂	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
288	C₂	C₃₁×#544	1	1	yes	characteristic
289	C₂	D₄²×C₃₁	1	1	yes	characteristic
290	C₂	C₃₁×#547	1	1	yes	characteristic
291	C₂	D₄×Q₈×C₃₁	1	1	yes	characteristic
292	C₂	C₃₁×#554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂³
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
1. C₂×C₃₁×C₄⋊Q₈	C₂	semidirect
4. C₃₁×#398	C₂	semidirect
5. C₃₁×#395	C₂	semidirect
6. C₃₁×#389	C₂	semidirect
7. C₃₁×#389	C₂	semidirect
9. C₃₁×#384	C₂	semidirect
47. C₂²×C₄×C₃₁	Q₈	semidirect
70. C₂²⋊C₄×C₃₁	Q₈	semidirect
114. #2440	C₃₁	direct (N × H)
121. C₂²×C₃₁	C₄⋊Q₈	semidirect
138. C₄×C₃₁	C₂²⋊Q₈	semidirect
139. C₂×C₄⋊Q₈	C₂×C₃₁	semidirect
142. #398	C₂×C₃₁	semidirect
143. #395	C₂×C₃₁	semidirect
144. #389	C₂×C₃₁	semidirect
145. #389	C₂×C₃₁	semidirect
147. #384	C₂×C₃₁	semidirect
184. C₃₁	#2440	direct (N × H)
194. C₂²×C₄	Q₈×C₃₁	semidirect
217. C₂²⋊C₄	Q₈×C₃₁	semidirect
267. C₂²	C₃₁×C₄⋊Q₈	semidirect
284. C₄	C₃₁×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₄⋊Q₈	2
C₂×C₃₁×C₂²⋊Q₈	2
C₂²×C₄⋊C₄×C₃₁	2
C₃₁×#398	2
C₃₁×#395	2
C₃₁×#389	2
C₃₁×#385	2
C₃₁×#384	2
C₃₁×#380	2
C₄×C₂²⋊C₄×C₃₁	2
#2440	31

Subgroups — C31×#2440 #16842373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2440 #16842373