Finite Group Databasebeta

← C22×#19149704 #36368295

Subgroups — C22×#19149704 #36368295

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
1C23×D692C211yesmaximal, characteristic, complemented
2C2×#19149704C2124yesmaximal, direct product factor
3C22×#14583813C212yesmaximal, complemented
4C22×#14583811C212yesmaximal, complemented
5C22×#14583810C211yesmaximal, characteristic, complemented
6C22×#41×C173C211yesmaximal, characteristic, complemented
7C24×D173—21no—
8C22×D692—26no—
9C22×D692—21no—
10C22×D692C2212yes—
11C22×D692C22112yescomplemented
12#19149704C22164yesdirect product factor
13C2×#14583813C22124yescomplemented
14C2×#14583811C22124yescomplemented
15C2×#14583810C22112yescomplemented
16C22×#13063436C2212yescomplemented
17C2×#41×C173—26no—
18C2×#41×C173C22112yescomplemented
19C22×Q8×C173C2211yescharacteristic, complemented
20C22×D4×C173—21no—
21C22×D4×C173C2211yescharacteristic, complemented
22C23×C4×C173—21no—
23C23×C4×C173C2211yescharacteristic
24C23×D173—22no—
25C23×D173—212no—
26C23×D173—48no—
27C2×D692—26no—
28C2×D692—212no—
29C2×D692—224no—
30C2×D692C23112yes—
31C2×D692C23116yescomplemented
32#14583813C23132yescomplemented
33#14583811C23132yescomplemented
34#14583810C23116yescomplemented
35C2×#13063436C23112yescomplemented
36C24×C173—21no—
37C24×C173D411yescharacteristic
38#41×C173—224no—
39#41×C173C23116yescomplemented
40C2×Q8×C173—23no—
41C2×Q8×C173C2316yescomplemented
42C2×D4×C173—212no—
43C2×D4×C173—23no—
44C2×D4×C173C2316yescomplemented
45C22×C4×C173—21no—
46C22×C4×C173—212no—
47C22×C4×C173D411yescharacteristic
48C22×C4×C173D416yes—
49C22×C4×C173C2311yescharacteristic
50C22×C4×C173C2316yes—
51C22×D173—212no—
52C22×D173—216no—
53C22×D173—448no—
54C22×D173—48no—
55D692—212no—
56D692—216no—
57D692C2418yes—
58#13063436C2418yescomplemented
59C23×C173—24no—
60C23×C173—26no—
61C23×C173—44no—
62C23×C173C2×D416yes—
63C23×C173C2×D411yescharacteristic
64Q8×C173—26no—
65Q8×C173C2414yescomplemented
66D4×C173—216no—
67D4×C173—26no—
68D4×C173C2414yescomplemented
69C2×C4×C173—26no—
70C2×C4×C173—216no—
71C2×C4×C173C2×D416yes—
72C2×C4×C173C2×D413yes—
73C2×C4×C173C2×D4112yes—
74C2×C4×C173C2413yes—
75C2×C4×C173C2414yes—
76C22×C173—212no—
77C22×C173—24no—
78C22×C173—412no—
79C22×C173AGL(1,ℤ/8)14yescomplemented
80C22×C173C22×D414yes—
81C22×C173C22×D413yes—
82C2×D173—28no—
83C2×D173—432no—
84C2×D173—424no—
85C4×C173—24no—
86C4×C173C22×D414yes—
87C4×C173C22×D413yes—
88C4×C173C2511yescharacteristic
89C2×C173—24no—
90C2×C173—44no—
91C2×C173C2×AGL(1,ℤ/8)16yescomplemented
92C2×C173C23×D411yescharacteristic
93D173—48no—
94C173C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1731nomaximal, Sylow
96C22×#41D17311yescharacteristic, complemented
97C23×D4—1731no—
98C2×AGL(1,ℤ/8)—17324no—
99C22×SD16—1732no—
100C22×D8—1732no—
101C22×M16—1731no—
102C25—3461no—
103C2×#41—26no—
104C2×#41C2×D173112yescomplemented
105C22×Q8C2×D17311yescharacteristic, complemented
106C22×D4—21no—
107C22×D4—3466no—
108C22×D4—3461no—
109C22×D4—1732no—
110C22×D4—17312no—
111C22×D4C2×D17311yescharacteristic, complemented
112C23×C4—21no—
113C23×C4C2×D17311yescharacteristic
114AGL(1,ℤ/8)—17364no—
115C2×SD16—17324no—
116C2×D8—17324no—
117C2×M16—17312no—
118C22×C8—1732no—
119C24—21no—
120C24—34612no—
121C24—3462no—
122C24—6928no—
123C24C22⋊D17311yescharacteristic
124#41—224no—
125#41C22×D173116yescomplemented
126C2×Q8—23no—
127C2×Q8C22×D17316yescomplemented
128C2×D4—212no—
129C2×D4—23no—
130C2×D4—34612no—
131C2×D4—3466no—
132C2×D4—34624no—
133C2×D4—17312no—
134C2×D4—17316no—
135C2×D4C22×D17316yescomplemented
136C22×C4—21no—
137C22×C4—212no—
138C22×C4C22⋊D17316yes—
139C22×C4C22⋊D17311yescharacteristic
140C22×C4C22×D17316yes—
141C22×C4C22×D17311yescharacteristic
142SD16—17332no—
143D8—17332no—
144M16—17316no—
145C2×C8—17312no—
146C23—24no—
147C23—26no—
148C23—44no—
149C23—34616no—
150C23—34612no—
151C23—69248no—
152C23—6928no—
153C23C2×C22⋊D17316yes—
154C23C2×C22⋊D17311yescharacteristic
155Q8—26no—
156Q8C23×D17314yescomplemented
157D4—216no—
158D4—26no—
159D4—34616no—
160D4—34612no—
161D4—1738no—
162D4C23×D17314yescomplemented
163C2×C4—26no—
164C2×C4—216no—
165C2×C4C2×C22⋊D17316yes—
166C2×C4C2×C22⋊D17313yes—
167C2×C4C2×C22⋊D173112yes—
168C2×C4C23×D17313yes—
169C2×C4C23×D17314yes—
170C8—1738no—
171C22—212no—
172C22—24no—
173C22—412no—
174C22—3468no—
175C22—69232no—
176C22—69224no—
177C22#1914970414yesdirect product factor
178C22C22×C22⋊D17314yes—
179C22C22×C22⋊D17313yes—
180C4—24no—
181C4C22×C22⋊D17314yes—
182C4C22×C22⋊D17313yes—
183C4C24×D17311yescharacteristic
184C2—24no—
185C2—44no—
186C2—6928no—
187C2C2×#1914970416yesdirect product factor
188C2C23×C22⋊D17311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC22×#41×C173C2
socleC23×C173C2×D4
hypercenterC22×#41D173
hypocenterC173C22×AGL(1,ℤ/8)
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C23×D692C2semidirect
2. C2×#19149704C2direct (N × H)
3. C22×#14583813C2semidirect
4. C22×#14583811C2semidirect
5. C22×#14583810C2semidirect
6. C22×#41×C173C2semidirect
11. C22×D692C22semidirect
12. #19149704C22direct (N × H)
13. C2×#14583813C22semidirect
14. C2×#14583811C22semidirect
15. C2×#14583810C22semidirect
16. C22×#13063436C22semidirect
18. C2×#41×C173C22semidirect
19. C22×Q8×C173C22semidirect
21. C22×D4×C173C22semidirect
31. C2×D692C23semidirect
32. #14583813C23semidirect
33. #14583811C23semidirect
34. #14583810C23semidirect
35. C2×#13063436C23semidirect
39. #41×C173C23semidirect
41. C2×Q8×C173C23semidirect
44. C2×D4×C173C23semidirect
58. #13063436C24semidirect
65. Q8×C173C24semidirect
68. D4×C173C24semidirect
79. C22×C173AGL(1,ℤ/8)semidirect
91. C2×C173C2×AGL(1,ℤ/8)semidirect
94. C173C22×AGL(1,ℤ/8)semidirect
96. C22×#41D173semidirect
104. C2×#41C2×D173semidirect
105. C22×Q8C2×D173semidirect
111. C22×D4C2×D173semidirect
125. #41C22×D173semidirect
127. C2×Q8C22×D173semidirect
135. C2×D4C22×D173semidirect
156. Q8C23×D173semidirect
162. D4C23×D173semidirect
177. C22#19149704direct (N × H)
187. C2C2×#19149704direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D6922
C2×#191497042
C22×#145838132
C22×#145838112
C22×#145838102
C22×#41×C1732
C22×AGL(1,ℤ/8)173