Finite Group Databasebeta

← C22×#15668037 #26034984

Subgroups — C22×#15668037 #26034984

Order 11392.

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
1C23×D356C211yesmaximal, characteristic, complemented
2C2×#15668037C2124yesmaximal, direct product factor
3C22×#13083561C212yesmaximal, complemented
4C22×#13083559C212yesmaximal, complemented
5C22×#13083558C211yesmaximal, characteristic, complemented
6C22×#41×C89C211yesmaximal, characteristic, complemented
7C24×D89—21no—
8C22×D356—21no—
9C22×D356—26no—
10C22×D356C2212yes—
11C22×D356C22112yescomplemented
12#15668037C22164yesdirect product factor
13C2×#13083561C22124yescomplemented
14C2×#13083559C22124yescomplemented
15C2×#13083558C22112yescomplemented
16C22×#10623494C2212yescomplemented
17C2×#41×C89—26no—
18C2×#41×C89C22112yescomplemented
19C22×Q8×C89C2211yescharacteristic, complemented
20C22×D4×C89—21no—
21C22×D4×C89C2211yescharacteristic, complemented
22C23×C4×C89—21no—
23C23×C4×C89C2211yescharacteristic
24C23×D89—22no—
25C23×D89—212no—
26C23×D89—48no—
27C2×D356—26no—
28C2×D356—224no—
29C2×D356—212no—
30C2×D356C23112yes—
31C2×D356C23116yescomplemented
32#13083561C23132yescomplemented
33#13083559C23132yescomplemented
34#13083558C23116yescomplemented
35C2×#10623494C23112yescomplemented
36C24×C89—21no—
37C24×C89D411yescharacteristic
38#41×C89—224no—
39#41×C89C23116yescomplemented
40C2×Q8×C89—23no—
41C2×Q8×C89C2316yescomplemented
42C2×D4×C89—212no—
43C2×D4×C89—23no—
44C2×D4×C89C2316yescomplemented
45C22×C4×C89—21no—
46C22×C4×C89—212no—
47C22×C4×C89D411yescharacteristic
48C22×C4×C89D416yes—
49C22×C4×C89C2311yescharacteristic
50C22×C4×C89C2316yes—
51C23×C89—24no—
52C23×C89—26no—
53C23×C89—44no—
54C23×C89C2×D416yes—
55C23×C89C2×D411yescharacteristic
56C22×D89—212no—
57C22×D89—216no—
58C22×D89—448no—
59C22×D89—48no—
60Q8×C89—26no—
61Q8×C89C2414yescomplemented
62D4×C89—216no—
63D4×C89—26no—
64D4×C89C2414yescomplemented
65C2×C4×C89—26no—
66C2×C4×C89—216no—
67C2×C4×C89C2×D416yes—
68C2×C4×C89C2×D413yes—
69C2×C4×C89C2×D4112yes—
70C2×C4×C89C2413yes—
71C2×C4×C89C2414yes—
72D356—212no—
73D356—216no—
74D356C2418yes—
75#10623494C2418yescomplemented
76C22×C89—212no—
77C22×C89—24no—
78C22×C89—412no—
79C22×C89AGL(1,ℤ/8)14yescomplemented
80C22×C89C22×D414yes—
81C22×C89C22×D413yes—
82C2×D89—28no—
83C2×D89—432no—
84C2×D89—424no—
85C4×C89—24no—
86C4×C89C22×D414yes—
87C4×C89C22×D413yes—
88C4×C89C2511yescharacteristic
89C2×C89—24no—
90C2×C89—44no—
91C2×C89C2×AGL(1,ℤ/8)16yescomplemented
92C2×C89C23×D411yescharacteristic
93D89—48no—
94C22×AGL(1,ℤ/8)—891nomaximal, Sylow
95C89C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
96C22×#41D8911yescharacteristic, complemented
97C23×D4—891no—
98C2×AGL(1,ℤ/8)—8924no—
99C22×SD16—892no—
100C22×D8—892no—
101C22×M16—891no—
102C25—1781no—
103C2×#41—26no—
104C2×#41C2×D89112yescomplemented
105C22×Q8C2×D8911yescharacteristic, complemented
106C22×D4—21no—
107C22×D4—892no—
108C22×D4—8912no—
109C22×D4—1786no—
110C22×D4—1781no—
111C22×D4C2×D8911yescharacteristic, complemented
112C23×C4—21no—
113C23×C4C2×D8911yescharacteristic
114AGL(1,ℤ/8)—8964no—
115C2×SD16—8924no—
116C2×D8—8924no—
117C2×M16—8912no—
118C22×C8—892no—
119C24—21no—
120C24—3568no—
121C24—17812no—
122C24—1782no—
123C24C22⋊D8911yescharacteristic
124#41—224no—
125#41C22×D89116yescomplemented
126C2×Q8—23no—
127C2×Q8C22×D8916yescomplemented
128C2×D4—212no—
129C2×D4—23no—
130C2×D4—8912no—
131C2×D4—8916no—
132C2×D4—17812no—
133C2×D4—1786no—
134C2×D4—17824no—
135C2×D4C22×D8916yescomplemented
136C22×C4—21no—
137C22×C4—212no—
138C22×C4C22⋊D8916yes—
139C22×C4C22⋊D8911yescharacteristic
140C22×C4C22×D8916yes—
141C22×C4C22×D8911yescharacteristic
142SD16—8932no—
143D8—8932no—
144M16—8916no—
145C2×C8—8912no—
146C23—24no—
147C23—26no—
148C23—44no—
149C23—35648no—
150C23—3568no—
151C23—17816no—
152C23—17812no—
153C23C2×C22⋊D8916yes—
154C23C2×C22⋊D8911yescharacteristic
155Q8—26no—
156Q8C23×D8914yescomplemented
157D4—216no—
158D4—26no—
159D4—898no—
160D4—17816no—
161D4—17812no—
162D4C23×D8914yescomplemented
163C2×C4—26no—
164C2×C4—216no—
165C2×C4C2×C22⋊D8916yes—
166C2×C4C2×C22⋊D8913yes—
167C2×C4C2×C22⋊D89112yes—
168C2×C4C23×D8913yes—
169C2×C4C23×D8914yes—
170C8—898no—
171C22—212no—
172C22—24no—
173C22—412no—
174C22—35632no—
175C22—35624no—
176C22—1788no—
177C22#1566803714yesdirect product factor
178C22C22×C22⋊D8914yes—
179C22C22×C22⋊D8913yes—
180C4—24no—
181C4C22×C22⋊D8914yes—
182C4C22×C22⋊D8913yes—
183C4C24×D8911yescharacteristic
184C2—24no—
185C2—44no—
186C2—3568no—
187C2C2×#1566803716yesdirect product factor
188C2C23×C22⋊D8911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D89
commutator, cosocleC4×C89C25
FrattiniC4C24×D89
FittingC22×#41×C89C2
socleC23×C89C2×D4
hypercenterC22×#41D89
hypocenterC89C22×AGL(1,ℤ/8)
LCSG ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C23×D356C2semidirect
2. C2×#15668037C2direct (N × H)
3. C22×#13083561C2semidirect
4. C22×#13083559C2semidirect
5. C22×#13083558C2semidirect
6. C22×#41×C89C2semidirect
11. C22×D356C22semidirect
12. #15668037C22direct (N × H)
13. C2×#13083561C22semidirect
14. C2×#13083559C22semidirect
15. C2×#13083558C22semidirect
16. C22×#10623494C22semidirect
18. C2×#41×C89C22semidirect
19. C22×Q8×C89C22semidirect
21. C22×D4×C89C22semidirect
31. C2×D356C23semidirect
32. #13083561C23semidirect
33. #13083559C23semidirect
34. #13083558C23semidirect
35. C2×#10623494C23semidirect
39. #41×C89C23semidirect
41. C2×Q8×C89C23semidirect
44. C2×D4×C89C23semidirect
61. Q8×C89C24semidirect
64. D4×C89C24semidirect
75. #10623494C24semidirect
79. C22×C89AGL(1,ℤ/8)semidirect
91. C2×C89C2×AGL(1,ℤ/8)semidirect
95. C89C22×AGL(1,ℤ/8)semidirect
96. C22×#41D89semidirect
104. C2×#41C2×D89semidirect
105. C22×Q8C2×D89semidirect
111. C22×D4C2×D89semidirect
125. #41C22×D89semidirect
127. C2×Q8C22×D89semidirect
135. C2×D4C22×D89semidirect
156. Q8C23×D89semidirect
162. D4C23×D89semidirect
177. C22#15668037direct (N × H)
187. C2C2×#15668037direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3562
C2×#156680372
C22×#130835612
C22×#130835592
C22×#130835582
C22×#41×C892
C22×AGL(1,ℤ/8)89