Finite Group Databasebeta

← C31×#3019 #16842952

Subgroups — C31×#3019 #16842952

Order 3968.

762 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C31×#575C212yesmaximal, complemented
3C31×#535C211yesmaximal, characteristic, complemented
4C31×#455C214yesmaximal, complemented
5C31×#453C214yesmaximal, complemented
6C31×#421C212yesmaximal, complemented
7C2×C31×#100C211yesmaximal, characteristic, complemented
8C31×H(5,𝔽2)—24no—
9C31×H(5,𝔽2)—22no—
10C31×H(5,𝔽2)C2214yes—
11C2×#41×C31—22no—
12C2×#41×C31C2212yes—
13C2×#41×C31C2211yescharacteristic, complemented
14C22×D4×C31—22no—
15C22×D4×C31—22no—
16C22×D4×C31C2211yescharacteristic
17C31×#137—22no—
18C31×#137C2214yescomplemented
19C31×AGL(1,ℤ/8)—22no—
20C31×AGL(1,ℤ/8)C2214yescomplemented
21C31×#135—24no—
22C2×M16×C31C2212yescomplemented
23C31×#127C2212yescomplemented
24C31×#124C2212yescomplemented
25C31×#121—22no—
26C31×#120—22no—
27C31×#117C2211yescharacteristic, complemented
28C31×#104C2218yescomplemented
29C31×#100C2214yescomplemented
30C24×C31—22no—
31C24×C31D412yes—
32#41×C31—28no—
33#41×C31—28no—
34#41×C31—22no—
35#41×C31—22no—
36#41×C31—22no—
37#41×C31—22no—
38#41×C31D414yescomplemented
39#41×C31C2314yes—
40C2×Q8×C31D412yescomplemented
41C2×Q8×C31C2311yescharacteristic, complemented
42C2×D4×C31—24no—
43C2×D4×C31—28no—
44C2×D4×C31—28no—
45C2×D4×C31—22no—
46C2×D4×C31—28no—
47C2×D4×C31—28no—
48C2×D4×C31—21no—
49C2×D4×C31—24no—
50C2×D4×C31—22no—
51C2×D4×C31—22no—
52C2×D4×C31D412yes—
53C2×D4×C31D412yescomplemented
54C2×D4×C31C2312yes—
55C2×D4×C31C2311yescharacteristic, complemented
56C22×C4×C31—22no—
57C22×C4×C31—22no—
58C22×C4×C31—21no—
59C22×C4×C31C2311yescharacteristic
60Q16×C31—24no—
61SD16×C31—24no—
62SD16×C31—24no—
63D8×C31—24no—
64M16×C31—22no—
65M16×C31C2314yescomplemented
66C2×C8×C31—22no—
67C4⋊C4×C31—21no—
68C22⋊C4×C31—21no—
69C22⋊C4×C31—24no—
70C42×C31C2312yescomplemented
71C23×C31—28no—
72C23×C31—22no—
73C23×C31—22no—
74C23×C31—22no—
75C23×C31—21no—
76C23×C31—82no—
77C23×C31—84no—
78C23×C31C2×D414yes—
79C23×C31C2×D411yescharacteristic
80Q8×C31—22no—
81Q8×C31—22no—
82Q8×C31C2×D414yes—
83D4×C31—216no—
84D4×C31—24no—
85D4×C31—24no—
86D4×C31—24no—
87D4×C31—22no—
88D4×C31—24no—
89D4×C31—22no—
90D4×C31—41no—
91D4×C31—41no—
92D4×C31—84no—
93D4×C31C2×D414yes—
94C2×C4×C31—28no—
95C2×C4×C31—22no—
96C2×C4×C31—24no—
97C2×C4×C31—24no—
98C2×C4×C31—22no—
99C2×C4×C31—21no—
100C2×C4×C31—42no—
101C2×C4×C31C2×D411yescharacteristic
102C2×C4×C31C2×D412yes—
103C2×C4×C31C2×D412yes—
104C2×C4×C31C2411yescharacteristic
105C8×C31—24no—
106#3019C3111yesmaximal, characteristic, direct product factor, Sylow
107C22×C31—24no—
108C22×C31—24no—
109C22×C31—41no—
110C22×C31—44no—
111C22×C31—44no—
112C22×C31—44no—
113C22×C31—42no—
114C22×C31—41no—
115C22×C31—82no—
116C22×C31—88no—
117C22×C31#12011yescharacteristic
118C22×C31#12011yescharacteristic
119C22×C31C22×D411yescharacteristic
120C4×C31—24no—
121C4×C31—42no—
122C4×C31—41no—
123C4×C31#12012yes—
124C4×C31C22×D412yes—
125C2×H(5,𝔽2)C2×C3111yescharacteristic, complemented
126#575C2×C3112yescomplemented
127#535C2×C3111yescharacteristic, complemented
128#455C2×C3114yescomplemented
129#453C2×C3114yescomplemented
130#421C2×C3112yescomplemented
131C2×#100C2×C3111yescharacteristic, complemented
132C2×C31—21no—
133C2×C31—21no—
134C2×C31—21no—
135C2×C31—44no—
136C2×C31—44no—
137C2×C31—81no—
138C2×C31C2×#12011yescharacteristic
139H(5,𝔽2)—24no—
140H(5,𝔽2)—22no—
141H(5,𝔽2)C22×C3114yes—
142C2×#41—22no—
143C2×#41C22×C3112yes—
144C2×#41C22×C3111yescharacteristic, complemented
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C22×C3111yescharacteristic
148#137—22no—
149#137C22×C3114yescomplemented
150AGL(1,ℤ/8)—22no—
151AGL(1,ℤ/8)C22×C3114yescomplemented
152#135—24no—
153C2×M16C22×C3112yescomplemented
154#127C22×C3112yescomplemented
155#124C22×C3112yescomplemented
156#121—22no—
157#120—22no—
158#117C22×C3111yescharacteristic, complemented
159#104C22×C3118yescomplemented
160#100C22×C3114yescomplemented
161C31#301911yescharacteristic, direct product factor, Sylow
162C24—22no—
163C24D4×C3112yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—22no—
169#41—22no—
170#41D4×C3114yescomplemented
171#41C23×C3114yes—
172C2×Q8D4×C3112yescomplemented
173C2×Q8C23×C3111yescharacteristic, complemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—21no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4D4×C3112yes—
185C2×D4D4×C3112yescomplemented
186C2×D4C23×C3112yes—
187C2×D4C23×C3111yescharacteristic, complemented
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4C23×C3111yescharacteristic
192Q16—24no—
193SD16—24no—
194SD16—24no—
195D8—24no—
196M16—22no—
197M16C23×C3114yescomplemented
198C2×C8—22no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C23×C3112yescomplemented
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—82no—
209C23—84no—
210C23C2×D4×C3114yes—
211C23C2×D4×C3111yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8C2×D4×C3114yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—22no—
222D4—41no—
223D4—41no—
224D4—84no—
225D4C2×D4×C3114yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—42no—
233C2×C4C2×D4×C3111yescharacteristic
234C2×C4C2×D4×C3112yes—
235C2×C4C2×D4×C3112yes—
236C2×C4C24×C3111yescharacteristic
237C8—24no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—42no—
245C22—41no—
246C22—82no—
247C22—88no—
248C22C31×#12011yescharacteristic
249C22C31×#12011yescharacteristic
250C22C22×D4×C3111yescharacteristic
251C4—24no—
252C4—42no—
253C4—41no—
254C4C31×#12012yes—
255C4C22×D4×C3112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—81no—
262C2C2×C31×#12011yescharacteristic
subgroupG/H
center, socleC2×C31C2×#120
commutator, Frattini, cosocleC2×C4C24×C31
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×H(5,𝔽2)C2semidirect
2. C31×#575C2semidirect
3. C31×#535C2semidirect
4. C31×#455C2semidirect
5. C31×#453C2semidirect
6. C31×#421C2semidirect
7. C2×C31×#100C2semidirect
13. C2×#41×C31C22semidirect
18. C31×#137C22semidirect
20. C31×AGL(1,ℤ/8)C22semidirect
22. C2×M16×C31C22semidirect
23. C31×#127C22semidirect
24. C31×#124C22semidirect
27. C31×#117C22semidirect
28. C31×#104C22semidirect
29. C31×#100C22semidirect
38. #41×C31D4semidirect
40. C2×Q8×C31D4semidirect
41. C2×Q8×C31C23semidirect
53. C2×D4×C31D4semidirect
55. C2×D4×C31C23semidirect
65. M16×C31C23semidirect
70. C42×C31C23semidirect
106. #3019C31direct (N × H)
125. C2×H(5,𝔽2)C2×C31semidirect
126. #575C2×C31semidirect
127. #535C2×C31semidirect
128. #455C2×C31semidirect
129. #453C2×C31semidirect
130. #421C2×C31semidirect
131. C2×#100C2×C31semidirect
144. C2×#41C22×C31semidirect
149. #137C22×C31semidirect
151. AGL(1,ℤ/8)C22×C31semidirect
153. C2×M16C22×C31semidirect
154. #127C22×C31semidirect
155. #124C22×C31semidirect
158. #117C22×C31semidirect
159. #104C22×C31semidirect
160. #100C22×C31semidirect
161. C31#3019direct (N × H)
170. #41D4×C31semidirect
172. C2×Q8D4×C31semidirect
173. C2×Q8C23×C31semidirect
185. C2×D4D4×C31semidirect
187. C2×D4C23×C31semidirect
197. M16C23×C31semidirect
202. C42C23×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×H(5,𝔽2)2
C31×#5752
C31×#5352
C31×#4552
C31×#4532
C31×#4212
C2×C31×#1002
#301931