Finite Group Databasebeta

← C22×SD16×D131 #30680678

Subgroups — C22×SD16×D131 #30680678

Order 16768.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C22×Q8×D131C211yesmaximal, characteristic, complemented
2C22×D4×D131C211yesmaximal, characteristic
3C22×#14523502C211yesmaximal, characteristic, complemented
4C22×#14523501C211yesmaximal, characteristic, complemented
5C22×#14523500C211yesmaximal, characteristic, complemented
6C2×SD16×D131C2124yesmaximal, direct product factor
7C22×C8×D131C211yesmaximal, characteristic, complemented
8C22×SD16×C131C211yesmaximal, characteristic, complemented
9C24×D131—21no—
10C22×Dic262—21no—
11C22×Dic262C2211yescharacteristic, complemented
12C2×Q8×D131—26no—
13C2×Q8×D131C22112yescomplemented
14C22×D524C2211yescharacteristic
15C22×C22⋊D131—21no—
16C2×D4×D131—26no—
17C2×D4×D131C22112yes—
18C22×C4×D131—21no—
19C22×C4×D131C2211yescharacteristic
20C2×#14523502C22112yescomplemented
21C2×#14523501C22112yescomplemented
22C2×#14523500C22112yescomplemented
23SD16×D131C22164yesdirect product factor
24C22×C131⋊C8C2211yescharacteristic, complemented
25C2×C8×D131C22112yescomplemented
26C22×Q8×C131C2211yescharacteristic, complemented
27C22×D4×C131C2211yescharacteristic
28C2×SD16×C131C22112yescomplemented
29C22×C8×C131C2211yescharacteristic, complemented
30C23×D131—212no—
31C23×D131—21no—
32C23×D131—48no—
33C23×D131D411yescharacteristic
34C2×Dic262—212no—
35C2×Dic262—23no—
36C2×Dic262C2316yescomplemented
37Q8×D131—224no—
38Q8×D131C23116yescomplemented
39C2×D524—23no—
40C2×D524C2316yes—
41C2×C22⋊D131—212no—
42D4×D131—224no—
43D4×D131C23116yes—
44C22×Dic131—21no—
45C22×Dic131D411yescharacteristic
46C2×C4×D131—212no—
47C2×C4×D131D416yes—
48C2×C4×D131C2316yes—
49#14523502C23116yescomplemented
50#14523501C23116yescomplemented
51#14523500C23116yescomplemented
52C2×C131⋊C8C2316yescomplemented
53C8×D131C23116yescomplemented
54C24×C131—21no—
55C2×Q8×C131—23no—
56C2×Q8×C131C2316yescomplemented
57C2×D4×C131—23no—
58C2×D4×C131C2316yes—
59C22×C4×C131—21no—
60C22×C4×C131C2311yescharacteristic
61SD16×C131C23116yescomplemented
62C2×C8×C131C2316yescomplemented
63C22×D131—26no—
64C22×D131—216no—
65C22×D131—44no—
66C22×D131—448no—
67C22×D131SD1618yesdirect product factor
68C22×D131C2×D416yes—
69Dic262—26no—
70Dic262—216no—
71Dic262C2414yescomplemented
72D524—26no—
73D524C2414yes—
74C22⋊D131—216no—
75C2×Dic131—26no—
76C2×Dic131C2×D416yes—
77C4×D131—216no—
78C4×D131C2×D4112yes—
79C4×D131C2414yes—
80C131⋊C8C2414yescomplemented
81C23×C131—26no—
82C23×C131—44no—
83C23×C131C2×D411yescharacteristic
84Q8×C131—26no—
85Q8×C131C2414yescomplemented
86D4×C131—26no—
87D4×C131C2414yes—
88C2×C4×C131—26no—
89C2×C4×C131C2×D413yes—
90C2×C4×C131C2413yes—
91C8×C131C2414yescomplemented
92C22×C131—24no—
93C22×C131—412no—
94C22×C131C2×SD1614yescomplemented
95C22×C131C22×D413yes—
96C2×D131—24no—
97C2×D131—412no—
98C2×D131—432no—
99C2×D131C2×SD16124yesdirect product factor
100C2×D131C22×D414yes—
101C4×C131—24no—
102C4×C131C22×D413yes—
103C4×C131C2511yescharacteristic
104Dic131—24no—
105Dic131C22×D414yes—
106C2×C131—44no—
107C2×C131C22×SD1616yescomplemented
108C2×C131C23×D411yescharacteristic
109D131—44no—
110D131C22×SD1618yesdirect product factor
111C131C23×SD1611yescharacteristic, complemented, Sylow
112C23×SD16—1311nomaximal, Sylow
113C23×Q8—1311no—
114C23×D4—1311no—
115C22×SD16—1311no—
116C22×SD16—1311no—
117C22×SD16—13124no—
118C22×SD16—1311no—
119C22×SD16D13111yescharacteristic, direct product factor
120C23×C8—1311no—
121C25—2621no—
122C22×Q8—2621no—
123C22×Q8—2626no—
124C22×Q8—13112no—
125C22×Q8—1311no—
126C22×Q8C2×D13111yescharacteristic, complemented
127C22×D4—2626no—
128C22×D4—2621no—
129C22×D4—13112no—
130C22×D4—1311no—
131C22×D4C2×D13111yescharacteristic
132C23×C4—2621no—
133C23×C4—1311no—
134C2×SD16—13112no—
135C2×SD16—13112no—
136C2×SD16—13164no—
137C2×SD16—13112no—
138C2×SD16C2×D131112yesdirect product factor
139C22×C8—1311no—
140C22×C8—13112no—
141C22×C8C2×D13111yescharacteristic, complemented
142C24—21no—
143C24—26212no—
144C24—2621no—
145C24—5248no—
146C24—1311no—
147C2×Q8—23no—
148C2×Q8—26212no—
149C2×Q8—26224no—
150C2×Q8—2623no—
151C2×Q8—13116no—
152C2×Q8—1316no—
153C2×Q8C22×D13116yescomplemented
154C2×D4—23no—
155C2×D4—26224no—
156C2×D4—26212no—
157C2×D4—2623no—
158C2×D4—13116no—
159C2×D4—1316no—
160C2×D4C22×D13116yes—
161C22×C4—21no—
162C22×C4—26212no—
163C22×C4—2621no—
164C22×C4—1311no—
165C22×C4—1316no—
166C22×C4—1316no—
167C22×C4C22×D13111yescharacteristic
168SD16—13116no—
169SD16—13116no—
170SD16—13116no—
171SD16C22×D131116yesdirect product factor
172C2×C8—1316no—
173C2×C8—13116no—
174C2×C8C22×D13116yescomplemented
175C23—26no—
176C23—44no—
177C23—26216no—
178C23—2626no—
179C23—52448no—
180C23—5244no—
181C23—1318no—
182C23—1316no—
183C23D4×D13111yescharacteristic
184Q8—26no—
185Q8—26216no—
186Q8—2626no—
187Q8—1314no—
188Q8C23×D13114yescomplemented
189D4—26no—
190D4—26216no—
191D4—2626no—
192D4—1314no—
193D4C23×D13114yes—
194C2×C4—26no—
195C2×C4—26216no—
196C2×C4—2626no—
197C2×C4—1316no—
198C2×C4—13112no—
199C2×C4—1314no—
200C2×C4D4×D13113yes—
201C2×C4C23×D13113yes—
202C8—1314no—
203C8C23×D13114yescomplemented
204C22—24no—
205C22—412no—
206C22—2624no—
207C22—52432no—
208C22—52412no—
209C22—1314no—
210C22—13124no—
211C22SD16×D13114yesdirect product factor
212C22C2×D4×D13113yes—
213C4—24no—
214C4—2624no—
215C4—1314no—
216C4C2×D4×D13113yes—
217C4C24×D13111yescharacteristic
218C2—44no—
219C2—5244no—
220C2—1318no—
221C2C2×SD16×D13116yesdirect product factor
222C2C22×D4×D13111yescharacteristic
subgroupG/H
centerC23D4×D131
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
FittingC22×SD16×C131C2
socleC23×C131C2×D4
prime radicalC22×D4×D131C2
hypercenterC22×SD16D131
hypocenterC131C23×SD16
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D131C2semidirect
3. C22×#14523502C2semidirect
4. C22×#14523501C2semidirect
5. C22×#14523500C2semidirect
6. C2×SD16×D131C2direct (N × H)
7. C22×C8×D131C2semidirect
8. C22×SD16×C131C2semidirect
11. C22×Dic262C22semidirect
13. C2×Q8×D131C22semidirect
20. C2×#14523502C22semidirect
21. C2×#14523501C22semidirect
22. C2×#14523500C22semidirect
23. SD16×D131C22direct (N × H)
24. C22×C131⋊C8C22semidirect
25. C2×C8×D131C22semidirect
26. C22×Q8×C131C22semidirect
28. C2×SD16×C131C22semidirect
29. C22×C8×C131C22semidirect
36. C2×Dic262C23semidirect
38. Q8×D131C23semidirect
49. #14523502C23semidirect
50. #14523501C23semidirect
51. #14523500C23semidirect
52. C2×C131⋊C8C23semidirect
53. C8×D131C23semidirect
56. C2×Q8×C131C23semidirect
61. SD16×C131C23semidirect
62. C2×C8×C131C23semidirect
67. C22×D131SD16direct (N × H)
71. Dic262C24semidirect
80. C131⋊C8C24semidirect
85. Q8×C131C24semidirect
91. C8×C131C24semidirect
94. C22×C131C2×SD16semidirect
99. C2×D131C2×SD16direct (N × H)
107. C2×C131C22×SD16semidirect
110. D131C22×SD16direct (N × H)
111. C131C23×SD16semidirect
119. C22×SD16D131direct (N × H)
126. C22×Q8C2×D131semidirect
138. C2×SD16C2×D131direct (N × H)
141. C22×C8C2×D131semidirect
153. C2×Q8C22×D131semidirect
171. SD16C22×D131direct (N × H)
174. C2×C8C22×D131semidirect
188. Q8C23×D131semidirect
203. C8C23×D131semidirect
211. C22SD16×D131direct (N × H)
221. C2C2×SD16×D131direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1312
C22×D4×D1312
C22×#145235022
C22×#145235012
C22×#145235002
C2×SD16×D1312
C22×C8×D1312
C22×SD16×C1312
C23×SD16131