Subgroups — C₃₁×#13282 #21470041

Order 7936.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1380	C₂	1	4	yes	maximal
5	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
7	C₃₁×#500	—	2	2	no	—
8	C₃₁×#500	C₄	1	4	yes	complemented
9	C₂×C₃₁×#107	—	2	1	no	—
10	C₂×C₃₁×#107	—	2	1	no	—
11	C₂×C₃₁×#107	C₂²	1	2	yes	—
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#384	—	2	1	no	—
15	C₄×C₁₆×C₃₁	—	2	2	no	—
16	C₂²×Q₈×C₃₁	—	4	1	no	—
17	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
19	C₃₁×C₄⋊Q₈	—	2	4	no	—
20	C₃₁×C₄⋊Q₈	—	4	2	no	—
21	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
24	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
25	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
26	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₃₁	—	2	2	no	—
28	C₂×C₁₆×C₃₁	—	2	2	no	—
29	C₂×C₁₆×C₃₁	Q₈	1	4	yes	complemented
30	C₃₁×#107	—	2	4	no	—
31	C₃₁×#107	—	2	2	no	—
32	C₃₁×#107	—	2	4	no	—
33	C₃₁×#107	—	4	2	no	—
34	C₃₁×#107	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₃₁	D₄	1	2	yes	—
37	C₂×Q₈×C₃₁	—	4	4	no	—
38	C₂×Q₈×C₃₁	—	8	2	no	—
39	C₂×Q₈×C₃₁	—	8	2	no	—
40	C₂²×C₄×C₃₁	—	4	2	no	—
41	C₂²×C₄×C₃₁	—	4	2	no	—
42	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
47	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
48	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₃₁	#41	1	2	yes	—
52	C₄⋊C₄×C₃₁	—	2	4	no	—
53	C₄⋊C₄×C₃₁	—	4	4	no	—
54	C₄⋊C₄×C₃₁	—	4	2	no	—
55	C₄⋊C₄×C₃₁	—	4	2	no	—
56	C₄⋊C₄×C₃₁	—	4	2	no	—
57	C₄⋊C₄×C₃₁	—	8	2	no	—
58	C₄⋊C₄×C₃₁	—	8	2	no	—
59	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₃₁	SD₁₆	1	2	yes	—
61	C₁₆×C₃₁	—	2	4	no	—
62	#13282	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
64	Q₈×C₃₁	—	8	8	no	—
65	C₂×C₄×C₃₁	—	4	4	no	—
66	C₂×C₄×C₃₁	—	8	2	no	—
67	C₂×C₄×C₃₁	—	8	2	no	—
68	C₂×C₄×C₃₁	—	8	2	no	—
69	C₂×C₄×C₃₁	—	8	2	no	—
70	C₂×C₄×C₃₁	—	8	2	no	—
71	C₂×C₄×C₃₁	#102	1	2	yes	—
72	C₂×C₄×C₃₁	#102	1	2	yes	—
73	C₂×C₄×C₃₁	D₁₆	1	2	yes	—
74	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
75	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
76	C₂×C₄×C₃₁	Q₃₂	1	2	yes	—
77	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
83	C₈×C₃₁	#103	1	4	yes	—
84	C₈×C₃₁	C₄×Q₈	1	2	yes	—
85	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
86	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
87	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
88	#1845	C₂×C₃₁	1	1	yes	characteristic
89	#1380	C₂×C₃₁	1	4	yes	—
90	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
91	C₂²×C₃₁	#486	1	1	yes	characteristic
92	C₂²×C₃₁	#500	1	1	yes	characteristic
93	C₂²×C₃₁	C₂×D₁₆	1	1	yes	characteristic
94	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
95	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
96	C₂²×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
97	C₄×C₃₁	—	8	4	no	—
98	C₄×C₃₁	—	16	2	no	—
99	C₄×C₃₁	#357	1	4	yes	—
100	C₄×C₃₁	#358	1	4	yes	—
101	C₄×C₃₁	#391	1	1	yes	characteristic
102	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
103	C₄×C₃₁	#487	1	1	yes	characteristic
104	C₄×C₃₁	#498	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	—	2	1	no	—
106	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
107	#500	—	2	2	no	—
108	#500	C₄×C₃₁	1	4	yes	complemented
109	C₂×#107	—	2	1	no	—
110	C₂×#107	—	2	1	no	—
111	C₂×#107	C₂²×C₃₁	1	2	yes	—
112	C₂×#107	C₂²×C₃₁	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
114	#384	—	2	1	no	—
115	C₄×C₁₆	—	2	2	no	—
116	C₂×C₃₁	#1990	1	1	yes	characteristic
117	C₂×C₃₁	C₂×#357	1	1	yes	characteristic
118	C₂×C₃₁	C₂×#358	1	1	yes	characteristic
119	C₂×C₃₁	#2240	1	1	yes	characteristic
120	C₂×C₃₁	#2241	1	1	yes	characteristic
121	C₂×C₃₁	#2252	1	1	yes	characteristic
122	C₂×C₃₁	#2254	1	1	yes	characteristic
123	C₂²×Q₈	—	4	1	no	—
124	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	4	no	—
127	C₄⋊Q₈	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	Q₈×C₃₁	1	4	yes	complemented
137	#107	—	2	4	no	—
138	#107	—	2	2	no	—
139	#107	—	2	4	no	—
140	#107	—	4	2	no	—
141	#107	C₂×C₄×C₃₁	1	4	yes	—
142	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
143	C₄×C₈	D₄×C₃₁	1	2	yes	—
144	C₃₁	#13282	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	8	2	no	—
147	C₂×Q₈	—	8	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
151	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
154	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
155	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
156	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
158	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
159	C₂×C₈	#41×C₃₁	1	2	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	8	2	no	—
166	C₄⋊C₄	—	8	2	no	—
167	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
168	C₄²	SD₁₆×C₃₁	1	2	yes	—
169	C₁₆	—	2	4	no	—
170	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₃₁×#102	1	2	yes	—
179	C₂×C₄	C₃₁×#102	1	2	yes	—
180	C₂×C₄	C₃₁×D₁₆	1	2	yes	—
181	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
182	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
183	C₂×C₄	C₃₁×Q₃₂	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
186	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
190	C₈	C₃₁×#103	1	4	yes	—
191	C₈	C₄×Q₈×C₃₁	1	2	yes	—
192	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
193	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
194	C₂²	C₃₁×#486	1	1	yes	characteristic
195	C₂²	C₃₁×#500	1	1	yes	characteristic
196	C₂²	C₂×C₃₁×D₁₆	1	1	yes	characteristic
197	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
198	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
199	C₂²	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	16	2	no	—
202	C₄	C₃₁×#357	1	4	yes	—
203	C₄	C₃₁×#358	1	4	yes	—
204	C₄	C₃₁×#391	1	1	yes	characteristic
205	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
206	C₄	C₃₁×#487	1	1	yes	characteristic
207	C₄	C₃₁×#498	1	1	yes	characteristic
208	C₂	C₃₁×#1990	1	1	yes	characteristic
209	C₂	C₂×C₃₁×#357	1	1	yes	characteristic
210	C₂	C₂×C₃₁×#358	1	1	yes	characteristic
211	C₂	C₃₁×#2240	1	1	yes	characteristic
212	C₂	C₃₁×#2241	1	1	yes	characteristic
213	C₂	C₃₁×#2252	1	1	yes	characteristic
214	C₂	C₃₁×#2254	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
LCS	G ▷ C₂×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
8. C₃₁×#500	C₄	semidirect
29. C₂×C₁₆×C₃₁	Q₈	semidirect
62. #13282	C₃₁	direct (N × H)
108. #500	C₄×C₃₁	semidirect
136. C₂×C₁₆	Q₈×C₃₁	semidirect
144. C₃₁	#13282	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×#500	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1845	2
C₃₁×#1380	2
#13282	31

Subgroups — C31×#13282 #21470041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13282 #21470041