Subgroups — C₃₁×#1926 #16841859

Order 3968.

290 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#390	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
15	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
16	C₂×C₄²×C₃₁	—	2	1	no	—
17	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#106	—	2	2	no	—
19	C₃₁×#102	—	2	2	no	—
20	C₃₁×#102	C₄	1	4	yes	complemented
21	C₃₁×#95	—	2	1	no	—
22	C₂⁴×C₃₁	—	2	1	no	—
23	C₂×D₄×C₃₁	—	2	1	no	—
24	C₂×D₄×C₃₁	—	2	1	no	—
25	C₂×D₄×C₃₁	—	2	1	no	—
26	C₂×D₄×C₃₁	—	2	1	no	—
27	C₂×D₄×C₃₁	—	2	1	no	—
28	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₃₁	—	2	1	no	—
30	C₂²×C₄×C₃₁	—	2	1	no	—
31	C₂²×C₄×C₃₁	—	2	1	no	—
32	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₃₁	—	2	1	no	—
36	C₂×C₈×C₃₁	—	2	1	no	—
37	C₂×C₈×C₃₁	—	2	1	no	—
38	C₂×C₈×C₃₁	—	2	1	no	—
39	C₂×C₈×C₃₁	—	2	1	no	—
40	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
41	C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
47	C₂²⋊C₄×C₃₁	—	4	2	no	—
48	C₂²⋊C₄×C₃₁	—	4	2	no	—
49	C₄²×C₃₁	—	2	1	no	—
50	C₄²×C₃₁	—	2	1	no	—
51	C₄²×C₃₁	—	4	2	no	—
52	C₂³×C₃₁	—	2	2	no	—
53	C₂³×C₃₁	—	4	1	no	—
54	C₂³×C₃₁	—	4	1	no	—
55	C₂³×C₃₁	—	4	1	no	—
56	C₂³×C₃₁	—	4	1	no	—
57	C₂³×C₃₁	—	4	1	no	—
58	C₂³×C₃₁	—	4	1	no	—
59	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
60	D₄×C₃₁	—	2	2	no	—
61	D₄×C₃₁	—	2	2	no	—
62	D₄×C₃₁	—	4	1	no	—
63	D₄×C₃₁	—	4	1	no	—
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	2	1	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	—	2	1	no	—
69	C₂×C₄×C₃₁	—	2	1	no	—
70	C₂×C₄×C₃₁	—	2	1	no	—
71	C₂×C₄×C₃₁	—	4	1	no	—
72	C₂×C₄×C₃₁	—	4	1	no	—
73	C₂×C₄×C₃₁	—	4	1	no	—
74	C₂×C₄×C₃₁	—	4	1	no	—
75	C₂×C₄×C₃₁	—	4	1	no	—
76	C₂×C₄×C₃₁	—	4	1	no	—
77	C₂×C₄×C₃₁	—	4	1	no	—
78	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
79	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
82	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
83	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
84	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
85	C₈×C₃₁	—	2	2	no	—
86	C₈×C₃₁	—	4	1	no	—
87	#1926	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₃₁	—	4	2	no	—
89	C₂²×C₃₁	—	4	2	no	—
90	C₂²×C₃₁	—	4	2	no	—
91	C₂²×C₃₁	—	8	1	no	—
92	C₂²×C₃₁	—	8	1	no	—
93	C₂²×C₃₁	—	8	1	no	—
94	C₂²×C₃₁	—	8	1	no	—
95	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
96	C₂²×C₃₁	#121	1	1	yes	characteristic
97	C₂²×C₃₁	#123	1	1	yes	characteristic
98	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
99	C₂²×C₃₁	#135	1	1	yes	characteristic
100	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	4	2	no	—
104	C₄×C₃₁	—	4	2	no	—
105	C₄×C₃₁	—	4	1	no	—
106	C₄×C₃₁	—	8	1	no	—
107	C₄×C₃₁	#117	1	1	yes	characteristic
108	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
109	C₄×C₃₁	#124	1	1	yes	characteristic
110	C₄×C₃₁	#126	1	1	yes	characteristic
111	C₂×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
112	C₂×#102	C₂×C₃₁	1	1	yes	characteristic
113	C₂×#102	C₂×C₃₁	1	1	yes	characteristic
114	#390	C₂×C₃₁	1	1	yes	characteristic
115	C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
116	#340	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#336	C₂×C₃₁	1	1	yes	characteristic, complemented
118	C₂×C₃₁	—	8	2	no	—
119	C₂×C₃₁	#388	1	1	yes	characteristic
120	C₂×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
121	C₂×C₃₁	#442	1	1	yes	characteristic
122	C₂×C₃₁	#460	1	1	yes	characteristic
123	C₂×C₃₁	#463	1	1	yes	characteristic
124	C₂×C₃₁	#481	1	1	yes	characteristic
125	C₂×C₃₁	#482	1	1	yes	characteristic
126	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
127	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	—	2	1	no	—
133	C₂×C₂²⋊C₄	—	2	1	no	—
134	C₂×C₄²	—	2	1	no	—
135	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
136	#106	—	2	2	no	—
137	#102	—	2	2	no	—
138	#102	C₄×C₃₁	1	4	yes	complemented
139	#95	—	2	1	no	—
140	C₃₁	#1926	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴	—	2	1	no	—
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	C₂×C₄×C₃₁	1	2	yes	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
152	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	C₂×C₄×C₃₁	1	2	yes	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
165	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
166	C₂²⋊C₄	—	4	2	no	—
167	C₂²⋊C₄	—	4	2	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	4	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	1	no	—
173	C₂³	—	4	1	no	—
174	C₂³	—	4	1	no	—
175	C₂³	—	4	1	no	—
176	C₂³	—	4	1	no	—
177	C₂³	—	4	1	no	—
178	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	4	1	no	—
182	D₄	—	4	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	SD₁₆×C₃₁	1	2	yes	—
198	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄	#41×C₃₁	1	1	yes	characteristic
201	C₂×C₄	#41×C₃₁	1	1	yes	characteristic
202	C₂×C₄	#41×C₃₁	1	1	yes	characteristic
203	C₂×C₄	#41×C₃₁	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	1	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	8	1	no	—
210	C₂²	—	8	1	no	—
211	C₂²	—	8	1	no	—
212	C₂²	—	8	1	no	—
213	C₂²	C₄×D₄×C₃₁	1	1	yes	characteristic
214	C₂²	C₃₁×#121	1	1	yes	characteristic
215	C₂²	C₃₁×#123	1	1	yes	characteristic
216	C₂²	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
217	C₂²	C₃₁×#135	1	1	yes	characteristic
218	C₂²	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂²	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	8	1	no	—
225	C₄	C₃₁×#117	1	1	yes	characteristic
226	C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
227	C₄	C₃₁×#124	1	1	yes	characteristic
228	C₄	C₃₁×#126	1	1	yes	characteristic
229	C₂	—	8	2	no	—
230	C₂	C₃₁×#388	1	1	yes	characteristic
231	C₂	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
232	C₂	C₃₁×#442	1	1	yes	characteristic
233	C₂	C₃₁×#460	1	1	yes	characteristic
234	C₂	C₃₁×#463	1	1	yes	characteristic
235	C₂	C₃₁×#481	1	1	yes	characteristic
236	C₂	C₃₁×#482	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#106	C₂	semidirect
5. C₄×C₄⋊C₄×C₃₁	C₂	semidirect
6. C₃₁×#340	C₂	semidirect
7. C₃₁×#336	C₂	semidirect
20. C₃₁×#102	C₄	semidirect
87. #1926	C₃₁	direct (N × H)
111. C₂×#106	C₂×C₃₁	semidirect
115. C₄×C₄⋊C₄	C₂×C₃₁	semidirect
116. #340	C₂×C₃₁	semidirect
117. #336	C₂×C₃₁	semidirect
138. #102	C₄×C₃₁	semidirect
140. C₃₁	#1926	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×#106	2
C₂×C₃₁×#102	2
C₃₁×#390	2
C₄×C₄⋊C₄×C₃₁	2
C₃₁×#340	2
C₃₁×#336	2
#1926	31

Subgroups — C31×#1926 #16841859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#1926 #16841859