Subgroups — C₃₁×#13407 #21470166

Order 7936.

566 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#357	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#357	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₃₁	—	2	1	no	—
7	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#127	—	2	1	no	—
9	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#493	—	2	4	no	—
11	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#389	—	2	1	no	—
15	C₃₁×#357	—	2	4	no	—
16	C₃₁×#357	—	2	4	no	—
17	C₃₁×#340	—	2	1	no	—
18	C₂²×D₄×C₃₁	—	2	2	no	—
19	C₂²×D₄×C₃₁	—	4	1	no	—
20	C₂²×D₄×C₃₁	—	4	1	no	—
21	C₂×D₈×C₃₁	—	2	4	no	—
22	C₂×D₈×C₃₁	—	2	2	no	—
23	C₂×D₈×C₃₁	—	2	2	no	—
24	C₂×D₈×C₃₁	—	2	2	no	—
25	C₂×D₈×C₃₁	—	2	2	no	—
26	C₂×D₈×C₃₁	—	2	2	no	—
27	C₂×D₈×C₃₁	—	4	2	no	—
28	C₂×D₈×C₃₁	—	4	2	no	—
29	C₂×D₈×C₃₁	D₄	1	4	yes	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×#127	—	2	4	no	—
33	C₃₁×#127	—	4	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
37	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₃₁	—	2	2	no	—
39	C₂×C₁₆×C₃₁	—	2	2	no	—
40	C₂×C₁₆×C₃₁	—	2	2	no	—
41	C₃₁×#107	—	2	1	no	—
42	C₃₁×#107	—	2	2	no	—
43	C₃₁×#107	D₄	1	2	yes	complemented
44	C₄×C₈×C₃₁	—	2	2	no	—
45	C₃₁×#95	—	4	1	no	—
46	C₂⁴×C₃₁	—	4	2	no	—
47	C₂×D₄×C₃₁	—	2	4	no	—
48	C₂×D₄×C₃₁	—	4	4	no	—
49	C₂×D₄×C₃₁	—	4	4	no	—
50	C₂×D₄×C₃₁	—	4	2	no	—
51	C₂×D₄×C₃₁	—	4	2	no	—
52	C₂×D₄×C₃₁	—	4	2	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	2	no	—
55	C₂×D₄×C₃₁	—	8	2	no	—
56	C₂×D₄×C₃₁	—	8	2	no	—
57	C₂×D₄×C₃₁	—	8	2	no	—
58	C₂×D₄×C₃₁	—	8	2	no	—
59	C₂²×C₄×C₃₁	—	4	1	no	—
60	C₂²×C₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	D₈×C₃₁	—	2	4	no	—
65	D₈×C₃₁	—	2	4	no	—
66	D₈×C₃₁	—	4	8	no	—
67	D₈×C₃₁	—	4	2	no	—
68	D₈×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	—	2	1	no	—
70	C₂×C₈×C₃₁	—	2	1	no	—
71	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
72	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₃₁	#41	1	2	yes	—
75	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	4	1	no	—
78	C₄⋊C₄×C₃₁	—	4	1	no	—
79	C₄⋊C₄×C₃₁	—	4	1	no	—
80	C₄⋊C₄×C₃₁	—	8	2	no	—
81	C₄²×C₃₁	—	2	1	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₁₆×C₃₁	—	4	2	no	—
84	#13407	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₃₁	—	4	4	no	—
86	C₂³×C₃₁	—	8	2	no	—
87	C₂³×C₃₁	—	8	2	no	—
88	C₂³×C₃₁	—	8	2	no	—
89	C₂³×C₃₁	—	8	2	no	—
90	C₂³×C₃₁	—	8	2	no	—
91	C₂³×C₃₁	—	8	2	no	—
92	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
93	D₄×C₃₁	—	4	4	no	—
94	D₄×C₃₁	—	4	4	no	—
95	D₄×C₃₁	—	8	8	no	—
96	D₄×C₃₁	—	8	8	no	—
97	D₄×C₃₁	—	8	2	no	—
98	D₄×C₃₁	—	8	2	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	8	1	no	—
105	C₂×C₄×C₃₁	—	8	1	no	—
106	C₂×C₄×C₃₁	—	8	1	no	—
107	C₂×C₄×C₃₁	—	8	1	no	—
108	C₂×C₄×C₃₁	—	8	1	no	—
109	C₂×C₄×C₃₁	D₁₆	1	2	yes	—
110	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
111	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₈×C₃₁	—	2	2	no	—
118	C₈×C₃₁	#121	1	2	yes	—
119	C₈×C₃₁	#123	1	2	yes	—
120	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
121	C₂×#357	C₂×C₃₁	1	2	yes	complemented
122	C₂×#357	C₂×C₃₁	1	2	yes	complemented
123	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#1374	C₂×C₃₁	1	1	yes	characteristic, complemented
125	C₂²×C₃₁	—	8	4	no	—
126	C₂²×C₃₁	—	8	4	no	—
127	C₂²×C₃₁	—	8	4	no	—
128	C₂²×C₃₁	—	16	2	no	—
129	C₂²×C₃₁	—	16	2	no	—
130	C₂²×C₃₁	—	16	2	no	—
131	C₂²×C₃₁	—	16	2	no	—
132	C₂²×C₃₁	#447	1	1	yes	characteristic
133	C₂²×C₃₁	#459	1	1	yes	characteristic
134	C₂²×C₃₁	#486	1	1	yes	characteristic
135	C₂²×C₃₁	C₂×D₁₆	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
137	C₂²×C₃₁	#509	1	1	yes	characteristic
138	C₂²×C₃₁	#509	1	1	yes	characteristic
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	8	2	no	—
142	C₄×C₃₁	—	16	1	no	—
143	C₄×C₃₁	#394	1	1	yes	characteristic
144	C₄×C₃₁	#450	1	1	yes	characteristic
145	C₄×C₃₁	#460	1	1	yes	characteristic
146	C₄×C₃₁	#490	1	1	yes	characteristic
147	C₂²×D₈	—	2	1	no	—
148	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
149	C₂×#127	—	2	1	no	—
150	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
151	#493	—	2	4	no	—
152	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
155	#389	—	2	1	no	—
156	#357	—	2	4	no	—
157	#357	—	2	4	no	—
158	#340	—	2	1	no	—
159	C₂×C₃₁	—	16	4	no	—
160	C₂×C₃₁	#2041	1	1	yes	characteristic
161	C₂×C₃₁	#2184	1	1	yes	characteristic
162	C₂×C₃₁	#2187	1	1	yes	characteristic
163	C₂×C₃₁	#2206	1	1	yes	characteristic
164	C₂×C₃₁	#2207	1	1	yes	characteristic
165	C₂×C₃₁	#2240	1	1	yes	characteristic
166	C₂×C₃₁	#2244	1	1	yes	characteristic
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	4	1	no	—
169	C₂²×D₄	—	4	1	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	—	4	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂×D₈	D₄×C₃₁	1	4	yes	—
179	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
181	#127	—	2	4	no	—
182	#127	—	4	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	2	2	no	—
190	#107	—	2	1	no	—
191	#107	—	2	2	no	—
192	#107	D₄×C₃₁	1	2	yes	complemented
193	C₄×C₈	—	2	2	no	—
194	#95	—	4	1	no	—
195	C₃₁	#13407	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	—	4	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	8	2	no	—
206	C₂×D₄	—	8	2	no	—
207	C₂×D₄	—	8	2	no	—
208	C₂×D₄	—	8	2	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
214	D₈	—	2	4	no	—
215	D₈	—	2	4	no	—
216	D₈	—	4	8	no	—
217	D₈	—	4	2	no	—
218	D₈	—	4	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
224	C₂×C₈	#41×C₃₁	1	2	yes	—
225	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	8	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
242	D₄	—	4	4	no	—
243	D₄	—	4	4	no	—
244	D₄	—	8	8	no	—
245	D₄	—	8	8	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	C₃₁×D₁₆	1	2	yes	—
259	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
260	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
261	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
262	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
263	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	C₃₁×#121	1	2	yes	—
268	C₈	C₃₁×#123	1	2	yes	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	16	2	no	—
273	C₂²	—	16	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₃₁×#447	1	1	yes	characteristic
277	C₂²	C₃₁×#459	1	1	yes	characteristic
278	C₂²	C₃₁×#486	1	1	yes	characteristic
279	C₂²	C₂×C₃₁×D₁₆	1	1	yes	characteristic
280	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
281	C₂²	C₃₁×#509	1	1	yes	characteristic
282	C₂²	C₃₁×#509	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	8	2	no	—
286	C₄	—	16	1	no	—
287	C₄	C₃₁×#394	1	1	yes	characteristic
288	C₄	C₃₁×#450	1	1	yes	characteristic
289	C₄	C₃₁×#460	1	1	yes	characteristic
290	C₄	C₃₁×#490	1	1	yes	characteristic
291	C₂	—	16	4	no	—
292	C₂	C₃₁×#2041	1	1	yes	characteristic
293	C₂	C₃₁×#2184	1	1	yes	characteristic
294	C₂	C₃₁×#2187	1	1	yes	characteristic
295	C₂	C₃₁×#2206	1	1	yes	characteristic
296	C₂	C₃₁×#2207	1	1	yes	characteristic
297	C₂	C₃₁×#2240	1	1	yes	characteristic
298	C₂	C₃₁×#2244	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#493	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#357	C₂	semidirect
3. C₂×C₃₁×#357	C₂	semidirect
4. C₃₁×#1942	C₂	semidirect
5. C₃₁×#1374	C₂	semidirect
43. C₃₁×#107	D₄	semidirect
84. #13407	C₃₁	direct (N × H)
121. C₂×#357	C₂×C₃₁	semidirect
122. C₂×#357	C₂×C₃₁	semidirect
123. #1942	C₂×C₃₁	semidirect
124. #1374	C₂×C₃₁	semidirect
192. #107	D₄×C₃₁	semidirect
195. C₃₁	#13407	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×#493	2
C₂×C₃₁×#357	2
C₃₁×#1942	2
C₃₁×#1374	2
#13407	31

Subgroups — C31×#13407 #21470166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13407 #21470166