Finite Group Databasebeta

← C22×SD16×D173 #36368280

Subgroups — C22×SD16×D173 #36368280

Order 22144.

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×D173C211yesmaximal, characteristic, complemented
2C22×D4×D173C211yesmaximal, characteristic
3C22×#14583815C211yesmaximal, characteristic, complemented
4C22×#14583814C211yesmaximal, characteristic, complemented
5C22×#14583813C211yesmaximal, characteristic, complemented
6C2×SD16×D173C2124yesmaximal, direct product factor
7C22×C8×D173C211yesmaximal, characteristic, complemented
8C22×SD16×C173C211yesmaximal, characteristic, complemented
9C24×D173—21no—
10C22×Dic346—21no—
11C22×Dic346C2211yescharacteristic, complemented
12C2×Q8×D173—26no—
13C2×Q8×D173C22112yescomplemented
14C22×D692C2211yescharacteristic
15C22×C22⋊D173—21no—
16C2×D4×D173—26no—
17C2×D4×D173C22112yes—
18C22×C4×D173—21no—
19C22×C4×D173C2211yescharacteristic
20C2×#14583815C22112yescomplemented
21C2×#14583814C22112yescomplemented
22C2×#14583813C22112yescomplemented
23SD16×D173C22164yesdirect product factor
24C22×#13063436C2211yescharacteristic, complemented
25C2×C8×D173C22112yescomplemented
26C22×Q8×C173C2211yescharacteristic, complemented
27C22×D4×C173C2211yescharacteristic
28C2×SD16×C173C22112yescomplemented
29C22×C8×C173C2211yescharacteristic, complemented
30C23×D173—212no—
31C23×D173—21no—
32C23×D173—48no—
33C23×D173D411yescharacteristic
34C2×Dic346—212no—
35C2×Dic346—23no—
36C2×Dic346C2316yescomplemented
37Q8×D173—224no—
38Q8×D173C23116yescomplemented
39C2×D692—23no—
40C2×D692C2316yes—
41C2×C22⋊D173—212no—
42D4×D173—224no—
43D4×D173C23116yes—
44C22×Dic173—21no—
45C22×Dic173D411yescharacteristic
46C2×C4×D173—212no—
47C2×C4×D173D416yes—
48C2×C4×D173C2316yes—
49#14583815C23116yescomplemented
50#14583814C23116yescomplemented
51#14583813C23116yescomplemented
52C2×#13063436C2316yescomplemented
53C8×D173C23116yescomplemented
54C24×C173—21no—
55C2×Q8×C173—23no—
56C2×Q8×C173C2316yescomplemented
57C2×D4×C173—23no—
58C2×D4×C173C2316yes—
59C22×C4×C173—21no—
60C22×C4×C173C2311yescharacteristic
61SD16×C173C23116yescomplemented
62C2×C8×C173C2316yescomplemented
63C22×D173—26no—
64C22×D173—216no—
65C22×D173—44no—
66C22×D173—448no—
67C22×D173SD1618yesdirect product factor
68C22×D173C2×D416yes—
69Dic346—26no—
70Dic346—216no—
71Dic346C2414yescomplemented
72D692—26no—
73D692C2414yes—
74C22⋊D173—216no—
75C2×Dic173—26no—
76C2×Dic173C2×D416yes—
77C4×D173—216no—
78C4×D173C2×D4112yes—
79C4×D173C2414yes—
80#13063436C2414yescomplemented
81C23×C173—26no—
82C23×C173—44no—
83C23×C173C2×D411yescharacteristic
84Q8×C173—26no—
85Q8×C173C2414yescomplemented
86D4×C173—26no—
87D4×C173C2414yes—
88C2×C4×C173—26no—
89C2×C4×C173C2×D413yes—
90C2×C4×C173C2413yes—
91C8×C173C2414yescomplemented
92C22×C173—24no—
93C22×C173—412no—
94C22×C173C2×SD1614yescomplemented
95C22×C173C22×D413yes—
96C2×D173—24no—
97C2×D173—412no—
98C2×D173—432no—
99C2×D173C2×SD16124yesdirect product factor
100C2×D173C22×D414yes—
101C4×C173—24no—
102C4×C173C22×D413yes—
103C4×C173C2511yescharacteristic
104Dic173—24no—
105Dic173C22×D414yes—
106C2×C173—44no—
107C2×C173C22×SD1616yescomplemented
108C2×C173C23×D411yescharacteristic
109D173—44no—
110D173C22×SD1618yesdirect product factor
111C173C23×SD1611yescharacteristic, complemented, Sylow
112C23×SD16—1731nomaximal, Sylow
113C23×Q8—1731no—
114C23×D4—1731no—
115C22×SD16—1731no—
116C22×SD16—1731no—
117C22×SD16—17324no—
118C22×SD16—1731no—
119C22×SD16D17311yescharacteristic, direct product factor
120C23×C8—1731no—
121C25—3461no—
122C22×Q8—3461no—
123C22×Q8—3466no—
124C22×Q8—17312no—
125C22×Q8—1731no—
126C22×Q8C2×D17311yescharacteristic, complemented
127C22×D4—3466no—
128C22×D4—3461no—
129C22×D4—17312no—
130C22×D4—1731no—
131C22×D4C2×D17311yescharacteristic
132C23×C4—3461no—
133C23×C4—1731no—
134C2×SD16—17312no—
135C2×SD16—17312no—
136C2×SD16—17364no—
137C2×SD16—17312no—
138C2×SD16C2×D173112yesdirect product factor
139C22×C8—1731no—
140C22×C8—17312no—
141C22×C8C2×D17311yescharacteristic, complemented
142C24—21no—
143C24—34612no—
144C24—3461no—
145C24—1731no—
146C24—6928no—
147C2×Q8—23no—
148C2×Q8—34612no—
149C2×Q8—34624no—
150C2×Q8—3463no—
151C2×Q8—17316no—
152C2×Q8—1736no—
153C2×Q8C22×D17316yescomplemented
154C2×D4—23no—
155C2×D4—34624no—
156C2×D4—34612no—
157C2×D4—3463no—
158C2×D4—17316no—
159C2×D4—1736no—
160C2×D4C22×D17316yes—
161C22×C4—21no—
162C22×C4—34612no—
163C22×C4—3461no—
164C22×C4—1731no—
165C22×C4—1736no—
166C22×C4—1736no—
167C22×C4C22×D17311yescharacteristic
168SD16—17316no—
169SD16—17316no—
170SD16—17316no—
171SD16C22×D173116yesdirect product factor
172C2×C8—1736no—
173C2×C8—17316no—
174C2×C8C22×D17316yescomplemented
175C23—26no—
176C23—44no—
177C23—34616no—
178C23—3466no—
179C23—1738no—
180C23—1736no—
181C23—69248no—
182C23—6924no—
183C23D4×D17311yescharacteristic
184Q8—26no—
185Q8—34616no—
186Q8—3466no—
187Q8—1734no—
188Q8C23×D17314yescomplemented
189D4—26no—
190D4—34616no—
191D4—3466no—
192D4—1734no—
193D4C23×D17314yes—
194C2×C4—26no—
195C2×C4—34616no—
196C2×C4—3466no—
197C2×C4—1736no—
198C2×C4—17312no—
199C2×C4—1734no—
200C2×C4D4×D17313yes—
201C2×C4C23×D17313yes—
202C8—1734no—
203C8C23×D17314yescomplemented
204C22—24no—
205C22—412no—
206C22—3464no—
207C22—17324no—
208C22—1734no—
209C22—69232no—
210C22—69212no—
211C22SD16×D17314yesdirect product factor
212C22C2×D4×D17313yes—
213C4—24no—
214C4—3464no—
215C4—1734no—
216C4C2×D4×D17313yes—
217C4C24×D17311yescharacteristic
218C2—44no—
219C2—1738no—
220C2—6924no—
221C2C2×SD16×D17316yesdirect product factor
222C2C22×D4×D17311yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC22×SD16×C173C2
socleC23×C173C2×D4
prime radicalC22×D4×D173C2
hypercenterC22×SD16D173
hypocenterC173C23×SD16
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D173C2semidirect
3. C22×#14583815C2semidirect
4. C22×#14583814C2semidirect
5. C22×#14583813C2semidirect
6. C2×SD16×D173C2direct (N × H)
7. C22×C8×D173C2semidirect
8. C22×SD16×C173C2semidirect
11. C22×Dic346C22semidirect
13. C2×Q8×D173C22semidirect
20. C2×#14583815C22semidirect
21. C2×#14583814C22semidirect
22. C2×#14583813C22semidirect
23. SD16×D173C22direct (N × H)
24. C22×#13063436C22semidirect
25. C2×C8×D173C22semidirect
26. C22×Q8×C173C22semidirect
28. C2×SD16×C173C22semidirect
29. C22×C8×C173C22semidirect
36. C2×Dic346C23semidirect
38. Q8×D173C23semidirect
49. #14583815C23semidirect
50. #14583814C23semidirect
51. #14583813C23semidirect
52. C2×#13063436C23semidirect
53. C8×D173C23semidirect
56. C2×Q8×C173C23semidirect
61. SD16×C173C23semidirect
62. C2×C8×C173C23semidirect
67. C22×D173SD16direct (N × H)
71. Dic346C24semidirect
80. #13063436C24semidirect
85. Q8×C173C24semidirect
91. C8×C173C24semidirect
94. C22×C173C2×SD16semidirect
99. C2×D173C2×SD16direct (N × H)
107. C2×C173C22×SD16semidirect
110. D173C22×SD16direct (N × H)
111. C173C23×SD16semidirect
119. C22×SD16D173direct (N × H)
126. C22×Q8C2×D173semidirect
138. C2×SD16C2×D173direct (N × H)
141. C22×C8C2×D173semidirect
153. C2×Q8C22×D173semidirect
171. SD16C22×D173direct (N × H)
174. C2×C8C22×D173semidirect
188. Q8C23×D173semidirect
203. C8C23×D173semidirect
211. C22SD16×D173direct (N × H)
221. C2C2×SD16×D173direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1732
C22×D4×D1732
C22×#145838152
C22×#145838142
C22×#145838132
C2×SD16×D1732
C22×C8×D1732
C22×SD16×C1732
C23×SD16173