Finite Group Databasebeta

← C31×#3020 #16842953

Subgroups — C31×#3020 #16842953

Order 3968.

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

normal subgroups only

This group has 698 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×C31×2−1+4C211yesmaximal, characteristic, complemented
2C31×#575C212yesmaximal, complemented
3C31×#536C211yesmaximal, characteristic, complemented
4C31×#456C214yesmaximal, complemented
5C31×#454C214yesmaximal, complemented
6C31×#421C212yesmaximal, complemented
7C2×C31×#101C211yesmaximal, characteristic, complemented
8C31×2−1+4—24no—
9C31×2−1+4—22no—
10C31×2−1+4C2214yes—
11C2×#41×C31—22no—
12C2×#41×C31—22no—
13C2×#41×C31C2212yes—
14C2×#41×C31C2211yescharacteristic
15C22×Q8×C31—22no—
16C22×Q8×C31C2211yescharacteristic
17C31×#137—22no—
18C31×#137C2214yescomplemented
19C31×AGL(1,ℤ/8)—22no—
20C31×AGL(1,ℤ/8)C2214yescomplemented
21C31×#135—24no—
22C2×M16×C31C2212yes—
23C31×C4⋊Q8C2212yescomplemented
24C31×#124C2212yescomplemented
25C31×#123—22no—
26C31×C22⋊Q8—22no—
27C31×#117C2211yescharacteristic
28C31×#104C2218yescomplemented
29C31×#101C2214yescomplemented
30#41×C31—28no—
31#41×C31—28no—
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×C31D414yes—
39#41×C31C2314yes—
40C2×Q8×C31—22no—
41C2×Q8×C31—28no—
42C2×Q8×C31—28no—
43C2×Q8×C31—22no—
44C2×Q8×C31—22no—
45C2×Q8×C31D412yes—
46C2×Q8×C31D412yes—
47C2×Q8×C31C2311yescharacteristic
48C2×Q8×C31C2312yes—
49C2×D4×C31—22no—
50C2×D4×C31—22no—
51C2×D4×C31—21no—
52C2×D4×C31D412yes—
53C2×D4×C31C2311yescharacteristic
54C22×C4×C31—24no—
55C22×C4×C31—22no—
56C22×C4×C31—21no—
57C22×C4×C31D412yes—
58C22×C4×C31C2311yescharacteristic
59Q16×C31—24no—
60SD16×C31—24no—
61SD16×C31—24no—
62D8×C31—24no—
63M16×C31—22no—
64M16×C31C2314yes—
65C2×C8×C31—22no—
66C4⋊C4×C31—21no—
67C4⋊C4×C31—24no—
68C22⋊C4×C31—21no—
69C22⋊C4×C31—24no—
70C42×C31C2312yes—
71C23×C31—22no—
72C23×C31—21no—
73C23×C31C2×D411yescharacteristic
74Q8×C31—28no—
75Q8×C31—22no—
76Q8×C31—24no—
77Q8×C31—24no—
78Q8×C31—22no—
79Q8×C31C2×D414yes—
80D4×C31—28no—
81D4×C31—24no—
82D4×C31—24no—
83D4×C31—22no—
84D4×C31—22no—
85D4×C31—41no—
86D4×C31—41no—
87D4×C31C2×D414yes—
88C2×C4×C31—216no—
89C2×C4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—24no—
93C2×C4×C31—24no—
94C2×C4×C31—22no—
95C2×C4×C31—21no—
96C2×C4×C31—42no—
97C2×C4×C31C2×D414yes—
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2×D412yes—
101C2×C4×C31C2411yescharacteristic
102C8×C31—24no—
103#3020C3111yesmaximal, characteristic, direct product factor, Sylow
104C22×C31—24no—
105C22×C31—41no—
106C22×C31—44no—
107C22×C31—42no—
108C22×C31—41no—
109C22×C31#12011yescharacteristic
110C22×C31#12011yescharacteristic
111C22×C31C22×D411yescharacteristic
112C4×C31—24no—
113C4×C31—24no—
114C4×C31—42no—
115C4×C31—41no—
116C4×C31#12012yes—
117C4×C31C22×D412yes—
118C2×2−1+4C2×C3111yescharacteristic, complemented
119#575C2×C3112yescomplemented
120#536C2×C3111yescharacteristic, complemented
121#456C2×C3114yescomplemented
122#454C2×C3114yescomplemented
123#421C2×C3112yescomplemented
124C2×#101C2×C3111yescharacteristic, complemented
125C2×C31—21no—
126C2×C31—21no—
127C2×C31—21no—
128C2×C31—44no—
129C2×C31—81no—
130C2×C31C2×#12011yescharacteristic
1312−1+4—24no—
1322−1+4—22no—
1332−1+4C22×C3114yes—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41C22×C3112yes—
137C2×#41C22×C3111yescharacteristic
138C22×Q8—22no—
139C22×Q8C22×C3111yescharacteristic
140#137—22no—
141#137C22×C3114yescomplemented
142AGL(1,ℤ/8)—22no—
143AGL(1,ℤ/8)C22×C3114yescomplemented
144#135—24no—
145C2×M16C22×C3112yes—
146C4⋊Q8C22×C3112yescomplemented
147#124C22×C3112yescomplemented
148#123—22no—
149C22⋊Q8—22no—
150#117C22×C3111yescharacteristic
151#104C22×C3118yescomplemented
152#101C22×C3114yescomplemented
153C31#302011yescharacteristic, direct product factor, Sylow
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—22no—
161#41—22no—
162#41D4×C3114yes—
163#41C23×C3114yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8D4×C3112yes—
170C2×Q8D4×C3112yes—
171C2×Q8C23×C3111yescharacteristic
172C2×Q8C23×C3112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4D4×C3112yes—
177C2×D4C23×C3111yescharacteristic
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4D4×C3112yes—
182C22×C4C23×C3111yescharacteristic
183Q16—24no—
184SD16—24no—
185SD16—24no—
186D8—24no—
187M16—22no—
188M16C23×C3114yes—
189C2×C8—22no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C23×C3112yes—
195C23—22no—
196C23—21no—
197C23C2×D4×C3111yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—22no—
203Q8C2×D4×C3114yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211D4C2×D4×C3114yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4C2×D4×C3114yes—
222C2×C4C2×D4×C3111yescharacteristic
223C2×C4C2×D4×C3112yes—
224C2×C4C2×D4×C3112yes—
225C2×C4C24×C3111yescharacteristic
226C8—24no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—42no—
231C22—41no—
232C22C31×#12011yescharacteristic
233C22C31×#12011yescharacteristic
234C22C22×D4×C3111yescharacteristic
235C4—24no—
236C4—24no—
237C4—42no—
238C4—41no—
239C4C31×#12012yes—
240C4C22×D4×C3112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—81no—
246C2C2×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×C31×2−1+4C2semidirect
2. C31×#575C2semidirect
3. C31×#536C2semidirect
4. C31×#456C2semidirect
5. C31×#454C2semidirect
6. C31×#421C2semidirect
7. C2×C31×#101C2semidirect
18. C31×#137C22semidirect
20. C31×AGL(1,ℤ/8)C22semidirect
23. C31×C4⋊Q8C22semidirect
24. C31×#124C22semidirect
28. C31×#104C22semidirect
29. C31×#101C22semidirect
103. #3020C31direct (N × H)
118. C2×2−1+4C2×C31semidirect
119. #575C2×C31semidirect
120. #536C2×C31semidirect
121. #456C2×C31semidirect
122. #454C2×C31semidirect
123. #421C2×C31semidirect
124. C2×#101C2×C31semidirect
141. #137C22×C31semidirect
143. AGL(1,ℤ/8)C22×C31semidirect
146. C4⋊Q8C22×C31semidirect
147. #124C22×C31semidirect
151. #104C22×C31semidirect
152. #101C22×C31semidirect
153. C31#3020direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×2−1+42
C31×#5752
C31×#5362
C31×#4562
C31×#4542
C31×#4212
C2×C31×#1012
#302031