Subgroups — C₃₁×#13059 #21469818

Order 7936.

578 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#357	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#121	—	2	1	no	—
9	C₂²×C₁₆×C₃₁	—	2	2	no	—
10	C₂²×C₁₆×C₃₁	—	2	4	no	—
11	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#466	—	2	2	no	—
13	C₃₁×#466	C₄	1	4	yes	complemented
14	C₂×C₃₁×#107	C₂²	1	2	yes	—
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#102	—	2	1	no	—
17	C₃₁×#380	—	2	1	no	—
18	C₃₁×#357	—	2	4	no	—
19	C₃₁×#340	—	2	1	no	—
20	C₂²×D₄×C₃₁	—	2	1	no	—
21	C₂²×D₄×C₃₁	—	4	1	no	—
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂×D₈×C₃₁	—	2	2	no	—
24	C₂×D₈×C₃₁	—	2	1	no	—
25	C₂×D₈×C₃₁	—	2	2	no	—
26	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×#121	—	2	4	no	—
34	C₃₁×#121	—	4	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
39	C₂×C₁₆×C₃₁	—	2	8	no	—
40	C₂×C₁₆×C₃₁	—	2	2	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
43	C₃₁×#107	—	2	1	no	—
44	C₃₁×#107	—	2	2	no	—
45	C₃₁×#107	—	2	2	no	—
46	C₃₁×#107	C₂×C₄	1	2	yes	—
47	C₃₁×#102	—	2	4	no	—
48	C₃₁×#95	—	4	1	no	—
49	C₂⁴×C₃₁	—	4	1	no	—
50	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	4	4	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	1	no	—
55	C₂×D₄×C₃₁	—	4	2	no	—
56	C₂×D₄×C₃₁	—	8	2	no	—
57	C₂×D₄×C₃₁	—	8	2	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	D₈×C₃₁	—	2	4	no	—
67	D₈×C₃₁	—	4	1	no	—
68	D₈×C₃₁	—	4	1	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	#41	1	2	yes	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	4	1	no	—
80	C₄⋊C₄×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	4	4	no	—
83	C₂²⋊C₄×C₃₁	—	8	2	no	—
84	C₁₆×C₃₁	—	2	4	no	—
85	#13059	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	—	2	2	no	—
87	C₂³×C₃₁	—	2	2	no	—
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	—	8	2	no	—
90	C₂³×C₃₁	—	8	1	no	—
91	C₂³×C₃₁	—	8	1	no	—
92	C₂³×C₃₁	—	8	2	no	—
93	C₂³×C₃₁	#102	1	2	yes	—
94	C₂³×C₃₁	D₁₆	1	2	yes	—
95	C₂³×C₃₁	SD₃₂	1	2	yes	—
96	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
97	D₄×C₃₁	—	4	2	no	—
98	D₄×C₃₁	—	4	2	no	—
99	D₄×C₃₁	—	8	8	no	—
100	D₄×C₃₁	—	8	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	—	2	4	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	8	1	no	—
107	C₂×C₄×C₃₁	—	8	1	no	—
108	C₂×C₄×C₃₁	—	8	2	no	—
109	C₂×C₄×C₃₁	—	8	2	no	—
110	C₂×C₄×C₃₁	—	8	2	no	—
111	C₂×C₄×C₃₁	#102	1	2	yes	—
112	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#121	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#135	1	2	yes	—
116	C₈×C₃₁	—	2	2	no	—
117	C₈×C₃₁	C₄×D₄	1	2	yes	—
118	C₈×C₃₁	#120	1	1	yes	characteristic
119	C₈×C₃₁	#123	1	1	yes	characteristic
120	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#357	C₂×C₃₁	1	2	yes	complemented
123	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#1380	C₂×C₃₁	1	2	yes	complemented
125	C₂²×C₃₁	—	2	2	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	4	no	—
128	C₂²×C₃₁	—	2	4	no	—
129	C₂²×C₃₁	—	8	2	no	—
130	C₂²×C₃₁	—	8	2	no	—
131	C₂²×C₃₁	—	8	2	no	—
132	C₂²×C₃₁	—	16	2	no	—
133	C₂²×C₃₁	—	16	2	no	—
134	C₂²×C₃₁	#357	1	4	yes	complemented
135	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
136	C₂²×C₃₁	#466	1	2	yes	—
137	C₂²×C₃₁	C₂×D₁₆	1	1	yes	characteristic
138	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
139	C₂²×C₃₁	#508	1	2	yes	—
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	8	2	no	—
142	C₄×C₃₁	—	16	2	no	—
143	C₄×C₃₁	#386	1	1	yes	characteristic
144	C₄×C₃₁	#416	1	1	yes	characteristic
145	C₄×C₃₁	#465	1	2	yes	—
146	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×#121	—	2	1	no	—
149	C₂²×C₁₆	—	2	2	no	—
150	C₂²×C₁₆	—	2	4	no	—
151	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
152	#466	—	2	2	no	—
153	#466	C₄×C₃₁	1	4	yes	complemented
154	C₂×#107	C₂²×C₃₁	1	2	yes	—
155	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×#102	—	2	1	no	—
157	#380	—	2	1	no	—
158	#357	—	2	4	no	—
159	#340	—	2	1	no	—
160	C₂×C₃₁	—	2	4	no	—
161	C₂×C₃₁	—	16	2	no	—
162	C₂×C₃₁	#1893	1	1	yes	characteristic
163	C₂×C₃₁	C₂×#357	1	1	yes	characteristic
164	C₂×C₃₁	#2138	1	1	yes	characteristic
165	C₂×C₃₁	#2215	1	2	yes	—
166	C₂×C₃₁	#2217	1	2	yes	—
167	C₂²×D₄	—	2	1	no	—
168	C₂²×D₄	—	4	1	no	—
169	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
170	C₂×D₈	—	2	2	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
177	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	D₄×C₃₁	1	2	yes	—
179	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
180	#121	—	2	4	no	—
181	#121	—	4	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₂²⋊C₄	—	4	1	no	—
185	C₂×C₂²⋊C₄	—	4	1	no	—
186	C₂×C₁₆	—	2	8	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
190	#107	—	2	1	no	—
191	#107	—	2	2	no	—
192	#107	—	2	2	no	—
193	#107	C₂×C₄×C₃₁	1	2	yes	—
194	#102	—	2	4	no	—
195	#95	—	4	1	no	—
196	C₃₁	#13059	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	—	4	1	no	—
198	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	8	2	no	—
205	C₂×D₄	—	8	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
211	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
214	D₈	—	2	4	no	—
215	D₈	—	4	1	no	—
216	D₈	—	4	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂×C₈	#41×C₃₁	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	8	2	no	—
232	C₁₆	—	2	4	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	1	no	—
239	C₂³	—	8	2	no	—
240	C₂³	C₃₁×#102	1	2	yes	—
241	C₂³	C₃₁×D₁₆	1	2	yes	—
242	C₂³	C₃₁×SD₃₂	1	2	yes	—
243	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	8	8	no	—
247	D₄	—	8	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	C₃₁×#102	1	2	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#121	1	2	yes	—
261	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
262	C₂×C₄	C₃₁×#135	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	C₄×D₄×C₃₁	1	2	yes	—
265	C₈	C₃₁×#120	1	1	yes	characteristic
266	C₈	C₃₁×#123	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₃₁×#357	1	4	yes	complemented
277	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
278	C₂²	C₃₁×#466	1	2	yes	—
279	C₂²	C₂×C₃₁×D₁₆	1	1	yes	characteristic
280	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
281	C₂²	C₃₁×#508	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	16	2	no	—
285	C₄	C₃₁×#386	1	1	yes	characteristic
286	C₄	C₃₁×#416	1	1	yes	characteristic
287	C₄	C₃₁×#465	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	—	16	2	no	—
290	C₂	C₃₁×#1893	1	1	yes	characteristic
291	C₂	C₂×C₃₁×#357	1	1	yes	characteristic
292	C₂	C₃₁×#2138	1	1	yes	characteristic
293	C₂	C₃₁×#2215	1	2	yes	—
294	C₂	C₃₁×#2217	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₆×C₃₁	C₂	semidirect
3. C₂×C₃₁×#357	C₂	semidirect
4. C₃₁×#1808	C₂	semidirect
5. C₃₁×#1380	C₂	semidirect
13. C₃₁×#466	C₄	semidirect
42. C₂×C₁₆×C₃₁	D₄	semidirect
85. #13059	C₃₁	direct (N × H)
120. C₂³×C₁₆	C₂×C₃₁	semidirect
122. C₂×#357	C₂×C₃₁	semidirect
123. #1808	C₂×C₃₁	semidirect
124. #1380	C₂×C₃₁	semidirect
134. C₂²×C₃₁	#357	semidirect
153. #466	C₄×C₃₁	semidirect
189. C₂×C₁₆	D₄×C₃₁	semidirect
196. C₃₁	#13059	direct (N × H)
276. C₂²	C₃₁×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₃₁	2
C₂×C₃₁×#466	2
C₂×C₃₁×#357	2
C₃₁×#1808	2
C₃₁×#1380	2
#13059	31

Subgroups — C31×#13059 #21469818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13059 #21469818