Subgroups — C₃₁×#13155 #21469914

Order 7936.

350 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2140	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
12	C₃₁×#427	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
14	C₃₁×#424	—	2	1	no	—
15	C₂×C₃₁×#102	—	2	1	no	—
16	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
17	C₃₁×#390	—	2	1	no	—
18	C₃₁×#361	—	2	2	no	—
19	C₃₁×#357	C₄	1	4	yes	complemented
20	C₃₁×#347	—	2	2	no	—
21	C₃₁×#339	—	2	1	no	—
22	C₂²×D₄×C₃₁	—	2	1	no	—
23	C₂×D₈×C₃₁	—	2	1	no	—
24	C₂×D₈×C₃₁	—	2	1	no	—
25	C₂×D₈×C₃₁	—	2	1	no	—
26	C₂×D₈×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₃₁	—	2	1	no	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×#117	—	2	1	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
35	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
36	C₃₁×M₃₂	—	2	1	no	—
37	C₃₁×M₃₂	—	2	1	no	—
38	C₃₁×M₃₂	—	2	1	no	—
39	C₃₁×M₃₂	D₄	1	2	yes	—
40	C₂×C₁₆×C₃₁	—	2	1	no	—
41	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
42	C₃₁×#108	—	2	1	no	—
43	C₃₁×#108	—	2	1	no	—
44	C₃₁×#107	C₂×C₄	1	2	yes	—
45	C₃₁×#106	—	2	1	no	—
46	C₃₁×#106	—	2	1	no	—
47	C₃₁×#106	—	2	1	no	—
48	C₃₁×C₄⋊C₈	—	4	2	no	—
49	C₃₁×#102	—	4	2	no	—
50	C₃₁×#97	—	2	1	no	—
51	C₃₁×#97	—	2	1	no	—
52	C₂⁴×C₃₁	—	4	1	no	—
53	C₂×D₄×C₃₁	—	2	2	no	—
54	C₂×D₄×C₃₁	—	4	1	no	—
55	C₂×D₄×C₃₁	—	4	1	no	—
56	C₂×D₄×C₃₁	—	4	1	no	—
57	C₂×D₄×C₃₁	—	4	1	no	—
58	C₂×D₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	—	2	1	no	—
60	C₂²×C₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	D₈×C₃₁	—	2	2	no	—
63	D₈×C₃₁	—	2	2	no	—
64	D₈×C₃₁	—	4	1	no	—
65	D₈×C₃₁	—	4	1	no	—
66	M₁₆×C₃₁	—	4	1	no	—
67	M₁₆×C₃₁	—	4	1	no	—
68	C₂×C₈×C₃₁	—	2	1	no	—
69	C₂×C₈×C₃₁	—	2	1	no	—
70	C₂×C₈×C₃₁	—	2	1	no	—
71	C₂×C₈×C₃₁	—	4	1	no	—
72	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
75	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
76	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
77	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	4	1	no	—
80	C₄⋊C₄×C₃₁	—	4	1	no	—
81	C₂²⋊C₄×C₃₁	—	4	1	no	—
82	C₂²⋊C₄×C₃₁	—	8	4	no	—
83	C₄²×C₃₁	—	2	1	no	—
84	C₄²×C₃₁	—	2	1	no	—
85	C₄²×C₃₁	—	4	1	no	—
86	C₁₆×C₃₁	—	2	2	no	—
87	C₁₆×C₃₁	—	2	2	no	—
88	#13155	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	—	4	2	no	—
90	C₂³×C₃₁	—	8	2	no	—
91	C₂³×C₃₁	—	8	2	no	—
92	C₂³×C₃₁	—	8	1	no	—
93	C₂³×C₃₁	—	8	1	no	—
94	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
95	D₄×C₃₁	—	4	2	no	—
96	D₄×C₃₁	—	4	2	no	—
97	D₄×C₃₁	—	8	1	no	—
98	D₄×C₃₁	—	8	1	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	4	1	no	—
102	C₂×C₄×C₃₁	—	8	2	no	—
103	C₂×C₄×C₃₁	—	8	1	no	—
104	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
107	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
110	C₈×C₃₁	—	4	1	no	—
111	C₈×C₃₁	—	8	1	no	—
112	C₈×C₃₁	#117	1	1	yes	characteristic
113	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
114	C₈×C₃₁	#123	1	1	yes	characteristic
115	C₈×C₃₁	#126	1	1	yes	characteristic
116	C₂×#361	C₂×C₃₁	1	1	yes	characteristic
117	#2140	C₂×C₃₁	1	1	yes	characteristic
118	C₂×#347	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#1972	C₂×C₃₁	1	1	yes	characteristic
120	#1946	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#1386	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#1383	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₂²×C₃₁	—	2	2	no	—
124	C₂²×C₃₁	—	8	4	no	—
125	C₂²×C₃₁	—	8	2	no	—
126	C₂²×C₃₁	—	16	2	no	—
127	C₂²×C₃₁	—	16	1	no	—
128	C₂²×C₃₁	—	16	1	no	—
129	C₂²×C₃₁	#442	1	1	yes	characteristic
130	C₂²×C₃₁	#468	1	1	yes	characteristic
131	C₂²×C₃₁	#490	1	1	yes	characteristic
132	C₄×C₃₁	—	8	1	no	—
133	C₄×C₃₁	—	8	2	no	—
134	C₄×C₃₁	—	16	1	no	—
135	C₄×C₃₁	#388	1	1	yes	characteristic
136	C₄×C₃₁	#440	1	1	yes	characteristic
137	C₄×C₃₁	#469	1	1	yes	characteristic
138	C₄×C₃₁	#489	1	1	yes	characteristic
139	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
141	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
143	#427	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
145	#424	—	2	1	no	—
146	C₂×#102	—	2	1	no	—
147	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
148	#390	—	2	1	no	—
149	#361	—	2	2	no	—
150	#357	C₄×C₃₁	1	4	yes	complemented
151	#347	—	2	2	no	—
152	#339	—	2	1	no	—
153	C₂×C₃₁	—	2	1	no	—
154	C₂×C₃₁	—	2	1	no	—
155	C₂×C₃₁	—	16	2	no	—
156	C₂×C₃₁	#1936	1	1	yes	characteristic
157	C₂×C₃₁	AGL(1,ℤ/16)	1	2	yes	—
158	C₂²×D₄	—	2	1	no	—
159	C₂×D₈	—	2	1	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	#117	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₂²⋊C₄	—	4	2	no	—
171	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
172	M₃₂	—	2	1	no	—
173	M₃₂	—	2	1	no	—
174	M₃₂	—	2	1	no	—
175	M₃₂	D₄×C₃₁	1	2	yes	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
178	#108	—	2	1	no	—
179	#108	—	2	1	no	—
180	#107	C₂×C₄×C₃₁	1	2	yes	—
181	#106	—	2	1	no	—
182	#106	—	2	1	no	—
183	#106	—	2	1	no	—
184	C₄⋊C₈	—	4	2	no	—
185	#102	—	4	2	no	—
186	#97	—	2	1	no	—
187	#97	—	2	1	no	—
188	C₃₁	#13155	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	—	4	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
199	D₈	—	2	2	no	—
200	D₈	—	2	2	no	—
201	D₈	—	4	1	no	—
202	D₈	—	4	1	no	—
203	M₁₆	—	4	1	no	—
204	M₁₆	—	4	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
212	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
213	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
214	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	1	no	—
219	C₂²⋊C₄	—	8	4	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	4	1	no	—
223	C₁₆	—	2	2	no	—
224	C₁₆	—	2	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	1	no	—
229	C₂³	—	8	1	no	—
230	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
242	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
243	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
246	C₈	—	4	1	no	—
247	C₈	—	8	1	no	—
248	C₈	C₃₁×#117	1	1	yes	characteristic
249	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
250	C₈	C₃₁×#123	1	1	yes	characteristic
251	C₈	C₃₁×#126	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	—	16	1	no	—
257	C₂²	—	16	1	no	—
258	C₂²	C₃₁×#442	1	1	yes	characteristic
259	C₂²	C₃₁×#468	1	1	yes	characteristic
260	C₂²	C₃₁×#490	1	1	yes	characteristic
261	C₄	—	8	1	no	—
262	C₄	—	8	2	no	—
263	C₄	—	16	1	no	—
264	C₄	C₃₁×#388	1	1	yes	characteristic
265	C₄	C₃₁×#440	1	1	yes	characteristic
266	C₄	C₃₁×#469	1	1	yes	characteristic
267	C₄	C₃₁×#489	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	16	2	no	—
271	C₂	C₃₁×#1936	1	1	yes	characteristic
272	C₂	C₃₁×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#442
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#347	C₂	semidirect
5. C₃₁×#1946	C₂	semidirect
6. C₃₁×#1386	C₂	semidirect
7. C₃₁×#1383	C₂	semidirect
19. C₃₁×#357	C₄	semidirect
88. #13155	C₃₁	direct (N × H)
118. C₂×#347	C₂×C₃₁	semidirect
120. #1946	C₂×C₃₁	semidirect
121. #1386	C₂×C₃₁	semidirect
122. #1383	C₂×C₃₁	semidirect
150. #357	C₄×C₃₁	semidirect
188. C₃₁	#13155	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×#361	2
C₃₁×#2140	2
C₂×C₃₁×#347	2
C₃₁×#1972	2
C₃₁×#1946	2
C₃₁×#1386	2
C₃₁×#1383	2
#13155	31

Subgroups — C31×#13155 #21469914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13155 #21469914