Subgroups — C₃₁×#17854 #21474613

Order 7936.

782 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2972	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1580	C₂	1	4	yes	maximal, complemented
4	C₈×D₈×C₃₁	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1478	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic
9	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
10	C₄×D₈×C₃₁	C₄	1	2	yes	—
11	C₄×D₈×C₃₁	C₄	1	2	yes	—
12	C₄×D₈×C₃₁	C₂²	1	2	yes	—
13	C₄×D₈×C₃₁	C₂²	1	2	yes	—
14	C₈×D₄×C₃₁	—	2	4	no	—
15	C₈×D₄×C₃₁	C₂²	1	8	yes	complemented
16	C₃₁×#431	C₂²	1	2	yes	—
17	C₃₁×#431	C₂²	1	1	yes	characteristic
18	C₃₁×#431	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
20	C₂×C₃₁×#102	C₄	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	—	2	4	no	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#335	C₂²	1	2	yes	complemented
24	C₃₁×#334	C₂²	1	2	yes	complemented
25	C₃₁×D₄⋊C₈	C₂²	1	8	yes	complemented
26	C₃₁×#322	C₂²	1	2	yes	complemented
27	C₈²×C₃₁	C₂²	1	2	yes	complemented
28	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₃₁	—	2	2	no	—
30	C₂×D₈×C₃₁	C₈	1	8	yes	complemented
31	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₃₁	—	2	8	no	—
34	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₃₁	—	2	2	no	—
37	C₄×D₄×C₃₁	—	2	2	no	—
38	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#107	C₂×C₄	1	2	yes	—
44	C₃₁×#107	C₂×C₄	1	2	yes	—
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
47	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
48	C₃₁×#102	C₂×C₄	1	4	yes	—
49	C₃₁×#102	C₂×C₄	1	4	yes	—
50	C₃₁×C₂²⋊C₈	—	2	2	no	—
51	C₃₁×C₂²⋊C₈	—	2	1	no	—
52	C₃₁×C₂²⋊C₈	—	2	8	no	—
53	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
54	C₄×C₈×C₃₁	—	2	2	no	—
55	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₃₁	C₂³	1	2	yes	—
58	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂⁴×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	—	2	4	no	—
64	C₂×D₄×C₃₁	C₂×C₈	1	8	yes	complemented
65	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	—	2	4	no	—
69	C₂²×C₄×C₃₁	—	8	4	no	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	D₈×C₃₁	C₂×C₈	1	8	yes	complemented
74	D₈×C₃₁	C₂×C₈	1	8	yes	complemented
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	8	8	no	—
79	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	complemented
80	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₃₁	—	2	4	no	—
88	C₂²⋊C₄×C₃₁	—	2	4	no	—
89	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₃₁	#41	1	1	yes	characteristic
92	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
93	#17854	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	4	no	—
95	C₂³×C₃₁	—	2	8	no	—
96	C₂³×C₃₁	—	8	2	no	—
97	C₂³×C₃₁	—	8	2	no	—
98	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
99	D₄×C₃₁	—	2	4	no	—
100	D₄×C₃₁	—	2	8	no	—
101	D₄×C₃₁	C₂²×C₈	1	8	yes	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	4	8	no	—
106	C₂×C₄×C₃₁	—	8	4	no	—
107	C₂×C₄×C₃₁	—	8	2	no	—
108	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₃₁	#131	1	2	yes	—
113	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
116	C₈×C₃₁	—	2	1	no	—
117	C₈×C₃₁	—	2	1	no	—
118	C₈×C₃₁	—	2	4	no	—
119	C₈×C₃₁	—	4	2	no	—
120	C₈×C₃₁	—	4	4	no	—
121	C₈×C₃₁	C₂²×C₈	1	2	yes	—
122	C₈×C₃₁	C₂²×C₈	1	2	yes	—
123	#2972	C₂×C₃₁	1	2	yes	complemented
124	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
125	#1580	C₂×C₃₁	1	4	yes	complemented
126	C₈×D₈	C₂×C₃₁	1	4	yes	complemented
127	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#1478	C₂×C₃₁	1	2	yes	complemented
129	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂²×C₃₁	—	2	1	no	—
131	C₂²×C₃₁	—	2	1	no	—
132	C₂²×C₃₁	—	2	8	no	—
133	C₂²×C₃₁	—	4	4	no	—
134	C₂²×C₃₁	—	4	4	no	—
135	C₂²×C₃₁	—	8	8	no	—
136	C₂²×C₃₁	C₈×D₄	1	2	yes	—
137	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	4	4	no	—
142	C₄×C₃₁	C₈×D₄	1	2	yes	—
143	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
145	C₄×C₃₁	#576	1	2	yes	—
146	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic
147	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
148	C₄×D₈	C₄×C₃₁	1	2	yes	—
149	C₄×D₈	C₄×C₃₁	1	2	yes	—
150	C₄×D₈	C₂²×C₃₁	1	2	yes	—
151	C₄×D₈	C₂²×C₃₁	1	2	yes	—
152	C₈×D₄	—	2	4	no	—
153	C₈×D₄	C₂²×C₃₁	1	8	yes	complemented
154	#431	C₂²×C₃₁	1	2	yes	—
155	#431	C₂²×C₃₁	1	1	yes	characteristic
156	#431	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
158	C₂×#102	C₄×C₃₁	1	2	yes	—
159	C₂×C₂²⋊C₈	—	2	4	no	—
160	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
161	#335	C₂²×C₃₁	1	2	yes	complemented
162	#334	C₂²×C₃₁	1	2	yes	complemented
163	D₄⋊C₈	C₂²×C₃₁	1	8	yes	complemented
164	#322	C₂²×C₃₁	1	2	yes	complemented
165	C₈²	C₂²×C₃₁	1	2	yes	complemented
166	C₂×C₃₁	—	2	2	no	—
167	C₂×C₃₁	—	4	8	no	—
168	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
169	C₂×C₃₁	#2948	1	1	yes	characteristic
170	C₂×C₃₁	#2957	1	1	yes	characteristic
171	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
172	C₂³×C₄	—	2	2	no	—
173	C₂×D₈	C₈×C₃₁	1	8	yes	complemented
174	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
175	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
176	C₂²×C₈	—	2	8	no	—
177	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
182	C₄×D₄	C₂³×C₃₁	1	4	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
186	#107	C₂×C₄×C₃₁	1	2	yes	—
187	#107	C₂×C₄×C₃₁	1	2	yes	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
190	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
191	#102	C₂×C₄×C₃₁	1	4	yes	—
192	#102	C₂×C₄×C₃₁	1	4	yes	—
193	C₂²⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₂²⋊C₈	—	2	8	no	—
196	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
199	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₄×C₈	C₂³×C₃₁	1	2	yes	—
201	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₃₁	1	2	yes	—
204	C₃₁	#17854	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	C₂×C₈×C₃₁	1	8	yes	complemented
209	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	8	4	no	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
217	D₈	C₂×C₈×C₃₁	1	8	yes	complemented
218	D₈	C₂×C₈×C₃₁	1	8	yes	complemented
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	8	8	no	—
223	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	complemented
224	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₄²	#41×C₃₁	1	1	yes	characteristic
236	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
242	D₄	—	2	8	no	—
243	D₄	—	2	4	no	—
244	D₄	C₂²×C₈×C₃₁	1	8	yes	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
255	C₂×C₄	C₃₁×#131	1	2	yes	—
256	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	4	no	—
264	C₈	C₂²×C₈×C₃₁	1	2	yes	—
265	C₈	C₂²×C₈×C₃₁	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	8	no	—
272	C₂²	C₈×D₄×C₃₁	1	2	yes	—
273	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	4	no	—
278	C₄	C₈×D₄×C₃₁	1	2	yes	—
279	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
280	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
281	C₄	C₃₁×#576	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	4	8	no	—
284	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic
285	C₂	C₃₁×#2948	1	1	yes	characteristic
286	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂²×D₈×C₃₁	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2972	C₂	semidirect
3. C₃₁×#1580	C₂	semidirect
4. C₈×D₈×C₃₁	C₂	semidirect
5. C₃₁×#1564	C₂	semidirect
6. C₃₁×#1478	C₂	semidirect
7. C₃₁×#1450	C₂	semidirect
15. C₈×D₄×C₃₁	C₂²	semidirect
23. C₃₁×#335	C₂²	semidirect
24. C₃₁×#334	C₂²	semidirect
25. C₃₁×D₄⋊C₈	C₂²	semidirect
26. C₃₁×#322	C₂²	semidirect
27. C₈²×C₃₁	C₂²	semidirect
30. C₂×D₈×C₃₁	C₈	semidirect
64. C₂×D₄×C₃₁	C₂×C₈	semidirect
73. D₈×C₃₁	C₂×C₈	semidirect
74. D₈×C₃₁	C₂×C₈	semidirect
79. C₂×C₈×C₃₁	C₂×C₈	semidirect
93. #17854	C₃₁	direct (N × H)
123. #2972	C₂×C₃₁	semidirect
125. #1580	C₂×C₃₁	semidirect
126. C₈×D₈	C₂×C₃₁	semidirect
127. #1564	C₂×C₃₁	semidirect
128. #1478	C₂×C₃₁	semidirect
129. #1450	C₂×C₃₁	semidirect
153. C₈×D₄	C₂²×C₃₁	semidirect
161. #335	C₂²×C₃₁	semidirect
162. #334	C₂²×C₃₁	semidirect
163. D₄⋊C₈	C₂²×C₃₁	semidirect
164. #322	C₂²×C₃₁	semidirect
165. C₈²	C₂²×C₃₁	semidirect
173. C₂×D₈	C₈×C₃₁	semidirect
204. C₃₁	#17854	direct (N × H)
208. C₂×D₄	C₂×C₈×C₃₁	semidirect
217. D₈	C₂×C₈×C₃₁	semidirect
218. D₈	C₂×C₈×C₃₁	semidirect
223. C₂×C₈	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2972	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#1580	2
C₈×D₈×C₃₁	2
C₃₁×#1564	2
C₃₁×#1478	2
C₃₁×#1450	2
#17854	31

Subgroups — C31×#17854 #21474613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17854 #21474613