Finite Group Databasebeta

← C22×#16862274 #30680696

Subgroups — C22×#16862274 #30680696

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×#14523513C211yesmaximal, characteristic, complemented
2C23×D524C211yesmaximal, characteristic, complemented
3C2×#16862274C2124yesmaximal, direct product factor
4C22×#14523502C212yesmaximal, complemented
5C22×D1048C212yesmaximal, complemented
6C22×M16×C131C211yesmaximal, characteristic, complemented
7C24×D131—21no—
8C2×#14523513—26no—
9C2×#14523513C22112yescomplemented
10C22×Dic262C2211yescharacteristic, complemented
11C22×D524—21no—
12C22×D524—26no—
13C22×D524C2212yes—
14C22×D524C22112yescomplemented
15C22×D524C2211yescharacteristic, complemented
16C22×C22⋊D131—21no—
17C22×C4×D131—21no—
18#16862274C22164yesdirect product factor
19C2×#14523502C22124yescomplemented
20C2×D1048C22124yescomplemented
21C23×C4×C131C2211yescharacteristic
22C2×M16×C131C22112yescomplemented
23C22×C8×C131C2212yescomplemented
24C23×D131—21no—
25C23×D131—22no—
26C23×D131—212no—
27C23×D131—48no—
28#14523513—224no—
29#14523513C23116yescomplemented
30C2×Dic262—23no—
31C2×Dic262C2316yescomplemented
32C2×D524—26no—
33C2×D524—212no—
34C2×D524—224no—
35C2×D524—23no—
36C2×D524C23112yes—
37C2×D524C23116yescomplemented
38C2×D524C2316yescomplemented
39C2×C22⋊D131—212no—
40C22×Dic131—21no—
41C2×C4×D131—212no—
42#14523502C23132yescomplemented
43D1048C23132yescomplemented
44C24×C131D411yescharacteristic
45C22×C4×C131D411yescharacteristic
46C22×C4×C131D416yes—
47C22×C4×C131C2311yescharacteristic
48C22×C4×C131C2316yes—
49M16×C131C23116yescomplemented
50C2×C8×C131C23112yescomplemented
51C22×D131—26no—
52C22×D131—212no—
53C22×D131—216no—
54C22×D131—448no—
55C22×D131—44no—
56C22×D131—48no—
57Dic262—26no—
58Dic262C2414yescomplemented
59D524—212no—
60D524—216no—
61D524—26no—
62D524C2418yes—
63D524C2414yescomplemented
64C22⋊D131—216no—
65C2×Dic131—26no—
66C4×D131—216no—
67C23×C131—24no—
68C23×C131C2×D416yes—
69C23×C131C2×D411yescharacteristic
70C2×C4×C131C2×D416yes—
71C2×C4×C131C2×D413yes—
72C2×C4×C131C2×D4112yes—
73C2×C4×C131C2413yes—
74C2×C4×C131C2414yes—
75C8×C131C2418yescomplemented
76C22×C131—212no—
77C22×C131AGL(1,ℤ/8)14yescomplemented
78C22×C131C22×D414yes—
79C22×C131C22×D413yes—
80C2×D131—24no—
81C2×D131—28no—
82C2×D131—432no—
83C2×D131—412no—
84C2×D131—424no—
85C4×C131C22×D414yes—
86C4×C131C22×D413yes—
87C4×C131C2511yescharacteristic
88Dic131—24no—
89C2×C131—24no—
90C2×C131C2×AGL(1,ℤ/8)16yescomplemented
91C2×C131C23×D411yescharacteristic
92D131—44no—
93D131—48no—
94C131C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1311nomaximal, Sylow
96C22×#41—1311no—
97C23×D4—1311no—
98C2×AGL(1,ℤ/8)—13124no—
99C22×SD16—1312no—
100C22×D8—1312no—
101C22×M16D13111yescharacteristic, complemented
102C25—2621no—
103C2×#41—2626no—
104C2×#41—13112no—
105C22×Q8—1311no—
106C22×D4—2621no—
107C22×D4—2626no—
108C22×D4—2621no—
109C22×D4—1311no—
110C22×D4—1312no—
111C22×D4—13112no—
112C23×C4—2621no—
113C23×C4C2×D13111yescharacteristic
114AGL(1,ℤ/8)—13164no—
115C2×SD16—13124no—
116C2×D8—13124no—
117C2×M16C2×D131112yescomplemented
118C22×C8C2×D13112yescomplemented
119C24—26212no—
120C24—2621no—
121C24—2622no—
122C24—5248no—
123C24D52411yescharacteristic
124#41—26224no—
125#41—13116no—
126C2×Q8—2623no—
127C2×Q8—1316no—
128C2×D4—26212no—
129C2×D4—26212no—
130C2×D4—2626no—
131C2×D4—26224no—
132C2×D4—2623no—
133C2×D4—13112no—
134C2×D4—1316no—
135C2×D4—13116no—
136C22×C4—2621no—
137C22×C4—26212no—
138C22×C4D52416yes—
139C22×C4D52411yescharacteristic
140C22×C4C22×D13116yes—
141C22×C4C22×D13111yescharacteristic
142SD16—13132no—
143D8—13132no—
144M16C22×D131116yescomplemented
145C2×C8C22×D131112yescomplemented
146C23—24no—
147C23—2626no—
148C23—26216no—
149C23—26212no—
150C23—52448no—
151C23—5248no—
152C23—5244no—
153C23C2×D52416yes—
154C23C2×D52411yescharacteristic
155Q8—2626no—
156Q8—1314no—
157D4—26216no—
158D4—26216no—
159D4—26212no—
160D4—2626no—
161D4—1314no—
162D4—1318no—
163C2×C4—2626no—
164C2×C4—26216no—
165C2×C4C2×D52416yes—
166C2×C4C2×D52413yes—
167C2×C4C2×D524112yes—
168C2×C4C23×D13113yes—
169C2×C4C23×D13114yes—
170C8C23×D13118yescomplemented
171C22—212no—
172C22—2624no—
173C22—2628no—
174C22—52432no—
175C22—52424no—
176C22—52412no—
177C22#1686227414yesdirect product factor
178C22C22×D52414yes—
179C22C22×D52413yes—
180C4—2624no—
181C4C22×D52414yes—
182C4C22×D52413yes—
183C4C24×D13111yescharacteristic
184C2—24no—
185C2—5248no—
186C2—5244no—
187C2C2×#1686227416yesdirect product factor
188C2C23×D52411yescharacteristic
subgroupG/H
centerC23C2×D524
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
FittingC22×M16×C131C2
socleC23×C131C2×D4
hypercenterC22×M16D131
hypocenterC131C22×AGL(1,ℤ/8)
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#14523513C2semidirect
2. C23×D524C2semidirect
3. C2×#16862274C2direct (N × H)
4. C22×#14523502C2semidirect
5. C22×D1048C2semidirect
6. C22×M16×C131C2semidirect
9. C2×#14523513C22semidirect
10. C22×Dic262C22semidirect
14. C22×D524C22semidirect
15. C22×D524C22semidirect
18. #16862274C22direct (N × H)
19. C2×#14523502C22semidirect
20. C2×D1048C22semidirect
22. C2×M16×C131C22semidirect
23. C22×C8×C131C22semidirect
29. #14523513C23semidirect
31. C2×Dic262C23semidirect
37. C2×D524C23semidirect
38. C2×D524C23semidirect
42. #14523502C23semidirect
43. D1048C23semidirect
49. M16×C131C23semidirect
50. C2×C8×C131C23semidirect
58. Dic262C24semidirect
63. D524C24semidirect
75. C8×C131C24semidirect
77. C22×C131AGL(1,ℤ/8)semidirect
90. C2×C131C2×AGL(1,ℤ/8)semidirect
94. C131C22×AGL(1,ℤ/8)semidirect
101. C22×M16D131semidirect
117. C2×M16C2×D131semidirect
118. C22×C8C2×D131semidirect
144. M16C22×D131semidirect
145. C2×C8C22×D131semidirect
170. C8C23×D131semidirect
177. C22#16862274direct (N × H)
187. C2C2×#16862274direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145235132
C23×D5242
C2×#168622742
C22×#145235022
C22×D10482
C22×M16×C1312
C22×AGL(1,ℤ/8)131