Finite Group Databasebeta

← #37554818

Subgroups — #37554818

Order 23488.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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×C22⋊D367C212yesmaximal, complemented
2C2×D4×D367C212yesmaximal, complemented
3#26044257C211yesmaximal, characteristic, complemented
4#26044234C212yesmaximal, complemented
5#26044233C212yesmaximal, complemented
6#26044231C212yesmaximal, complemented
7#26044229C211yesmaximal, characteristic, complemented
8(C22⋊D367)⋊C4C212yesmaximal, complemented
9#120×C367C211yesmaximal, characteristic, complemented
10C23×D367C2211yescharacteristic, complemented
11C23×D367C2212yescomplemented
12C2×D1468C2212yescomplemented
13C2×D1468C2211yescharacteristic, complemented
14C2×C22⋊D367—28no—
15C2×C22⋊D367C2212yes—
16C2×C22⋊D367C2212yescomplemented
17C2×C22⋊D367C2212yescomplemented
18C2×C22⋊D367C2212yescomplemented
19C2×C22⋊D367C2212yescomplemented
20D4×D367—28no—
21C22×Dic367C2212yescomplemented
22C2×C4×D367C2212yescomplemented
23Dic367⋊C4C2212yescomplemented
24C22⋊Dic367C2211yescharacteristic, complemented
25(C2×D367)⋊C4C2212yescomplemented
26(C2×D367)⋊C4C2212yescomplemented
27C4×Dic367C2211yescharacteristic, complemented
28C24×C367C2211yescharacteristic, complemented
29C2×D4×C367C2212yescomplemented
30C2×D4×C367C2211yescharacteristic, complemented
31C22⋊C4×C367C2212yescomplemented
32C22⋊C4×C367C2211yescharacteristic, complemented
33C22×D367—24no—
34C22×D367—24no—
35C22×D367—24no—
36C22×D367—41no—
37C22×D367—42no—
38C22×D367C2312yes—
39C22×D367C2312yes—
40D1468—24no—
41D1468—41no—
42C22⋊D367—24no—
43C22⋊D367—24no—
44C22⋊D367—24no—
45C22⋊D367—24no—
46C22⋊D367D418yescomplemented
47C2×Dic367—24no—
48C2×Dic367C2312yes—
49C2×Dic367C2312yes—
50C4×D367—24no—
51C23×C367—24no—
52C23×C367—41no—
53C23×C367C2311yescharacteristic
54C23×C367C2311yescharacteristic
55C23×C367C2312yes—
56D4×C367—24no—
57D4×C367—41no—
58C2×C4×C367C2311yescharacteristic
59C2×C4×C367C2312yes—
60C22×C367—22no—
61C22×C367—22no—
62C22×C367—22no—
63C22×C367—22no—
64C22×C367—21no—
65C22×C367—21no—
66C22×C367—44no—
67C22×C367C2×D414yescomplemented
68C22×C367C2411yescharacteristic
69C2×D367—22no—
70C2×D367—22no—
71C2×D367—22no—
72C2×D367—22no—
73C2×D367—22no—
74C2×D367—44no—
75C2×D367—48no—
76C2×D367C2×D414yescomplemented
77C4×C367—22no—
78C4×C367—21no—
79Dic367—22no—
80Dic367C2×D414yescomplemented
81C2×C367—24no—
82C2×C367—41no—
83C2×C367—41no—
84C2×C367C22×D412yes—
85C2×C367H(5,𝔽2)11yescharacteristic
86D367—24no—
87D367—42no—
88C367D4211yescharacteristic, complemented, Sylow
89D42—3671nomaximal, Sylow
90C22×D4—3672no—
91C22×D4—3672no—
92#127—3671no—
93#121—3672no—
94#121—3672no—
95#120—3671no—
96#120—3672no—
97#120D36711yescharacteristic, complemented
98C4×D4—3672no—
99C24—3671no—
100C24—3672no—
101C24C2×D36711yescharacteristic, complemented
102C2×D4—3672no—
103C2×D4—3672no—
104C2×D4—3672no—
105C2×D4—3672no—
106C2×D4—3672no—
107C2×D4—3671no—
108C2×D4—3672no—
109C2×D4—7348no—
110C2×D4—7348no—
111C2×D4C2×D36712yescomplemented
112C2×D4C2×D36711yescharacteristic, complemented
113C22×C4—3672no—
114C22×C4—3672no—
115C4⋊C4—3672no—
116C22⋊C4—3671no—
117C22⋊C4—3672no—
118C22⋊C4—3672no—
119C22⋊C4C2×D36712yescomplemented
120C22⋊C4C2×D36711yescharacteristic, complemented
121C42—3671no—
122C23—24no—
123C23—41no—
124C23—3672no—
125C23—3672no—
126C23—14682no—
127C23—14681no—
128C23—7344no—
129C23—7344no—
130C23—7344no—
131C23C22×D36711yescharacteristic
132C23C22×D36711yescharacteristic
133C23C22×D36712yes—
134D4—24no—
135D4—41no—
136D4—3678no—
137D4—14681no—
138D4—7344no—
139D4—7344no—
140D4—7344no—
141D4—7344no—
142D4—7344no—
143C2×C4—3672no—
144C2×C4—3672no—
145C2×C4—7344no—
146C2×C4—7344no—
147C2×C4C22×D36711yescharacteristic
148C2×C4C22×D36712yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—3674no—
157C22—14688no—
158C22—14684no—
159C22—7342no—
160C22—7342no—
161C22—7342no—
162C22—7342no—
163C22—7342no—
164C22D4×D36714yescomplemented
165C22C23×D36711yescharacteristic
166C4—22no—
167C4—21no—
168C4—3674no—
169C4—7342no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—14682no—
174C2—7344no—
175C2C2×D4×D36712yes—
176C2#2604430611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D367
commutator, socle, cosocleC22×C367C24
Fitting#120×C367C2
hypercenter#120D367
hypocenterC367D42
LCSG ▷ C22×C367 ▷ C367
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D367C2semidirect
2. C2×D4×D367C2semidirect
3. #26044257C2semidirect
4. #26044234C2semidirect
5. #26044233C2semidirect
6. #26044231C2semidirect
7. #26044229C2semidirect
8. (C22⋊D367)⋊C4C2semidirect
9. #120×C367C2semidirect
10. C23×D367C22semidirect
11. C23×D367C22semidirect
12. C2×D1468C22semidirect
13. C2×D1468C22semidirect
16. C2×C22⋊D367C22semidirect
17. C2×C22⋊D367C22semidirect
18. C2×C22⋊D367C22semidirect
19. C2×C22⋊D367C22semidirect
21. C22×Dic367C22semidirect
22. C2×C4×D367C22semidirect
23. Dic367⋊C4C22semidirect
24. C22⋊Dic367C22semidirect
25. (C2×D367)⋊C4C22semidirect
26. (C2×D367)⋊C4C22semidirect
27. C4×Dic367C22semidirect
28. C24×C367C22semidirect
29. C2×D4×C367C22semidirect
30. C2×D4×C367C22semidirect
31. C22⋊C4×C367C22semidirect
32. C22⋊C4×C367C22semidirect
46. C22⋊D367D4semidirect
67. C22×C367C2×D4semidirect
76. C2×D367C2×D4semidirect
80. Dic367C2×D4semidirect
88. C367D42semidirect
97. #120D367semidirect
101. C24C2×D367semidirect
111. C2×D4C2×D367semidirect
112. C2×D4C2×D367semidirect
119. C22⋊C4C2×D367semidirect
120. C22⋊C4C2×D367semidirect
164. C22D4×D367semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D3672
C2×D4×D3672
#260442572
#260442342
#260442332
#260442312
#260442292
(C22⋊D367)⋊C42
#120×C3672
D42367