Finite Group Databasebeta

← #36378345

Subgroups — #36378345

Order 22592.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×D4×D353C212yesmaximal, complemented
2#26014977C212yesmaximal, complemented
3#26014974C212yesmaximal, complemented
4#26014966C211yesmaximal, characteristic, complemented
5#26014964C212yesmaximal, complemented
6#26014962C212yesmaximal, complemented
7#26014960C212yesmaximal, complemented
8C2×#19150714C211yesmaximal, characteristic, complemented
9#121×C353C211yesmaximal, characteristic, complemented
10C23×D353C2211yescharacteristic
11C23×D353C2212yescomplemented
12C2×D1412C2212yes—
13C2×D1412C2211yescharacteristic, complemented
14C2×C22⋊D353C2212yes—
15C2×C22⋊D353C2212yescomplemented
16C2×C22⋊D353C2212yescomplemented
17D4×D353—28no—
18C22×Dic353C2211yescharacteristic, complemented
19C2×C4×D353C2212yescomplemented
20C4⋊Dic353C2211yescharacteristic, complemented
21#19150716C2212yescomplemented
22C22⋊Dic353C2212yescomplemented
23#19150714C2212yescomplemented
24#19150714C2212yescomplemented
25#19150714C2212yescomplemented
26#19150714C2212yescomplemented
27C2×D4×C353C2212yescomplemented
28C2×D4×C353C2211yescharacteristic, complemented
29C22×C4×C353C2211yescharacteristic, complemented
30C4⋊C4×C353C2211yescharacteristic, complemented
31C22⋊C4×C353C2212yescomplemented
32C22×D353—24no—
33C22×D353—41no—
34C22×D353—41no—
35C22×D353—42no—
36C22×D353—42no—
37C22×D353D414yescomplemented
38C22×D353C2312yes—
39C22×D353C2312yes—
40D1412—24no—
41D1412—41no—
42C22⋊D353—24no—
43C22⋊D353—24no—
44C22⋊D353—42no—
45C2×Dic353—41no—
46C2×Dic353C2312yes—
47C2×Dic353C2312yes—
48C4×D353—24no—
49C23×C353C2312yes—
50C23×C353C2311yescharacteristic
51D4×C353—24no—
52D4×C353—41no—
53C2×C4×C353—41no—
54C2×C4×C353C2311yescharacteristic
55C2×C4×C353C2312yes—
56C2×C4×C353C2311yescharacteristic
57C22×C353—21no—
58C22×C353—21no—
59C22×C353—22no—
60C22×C353—22no—
61C22×C353—22no—
62C22×C353C2×D412yes—
63C22×C353C2411yescharacteristic
64C2×D353—22no—
65C2×D353—22no—
66C2×D353—22no—
67C2×D353—22no—
68C2×D353—22no—
69C2×D353—44no—
70C2×D353—48no—
71C2×D353C2×D414yes—
72C4×C353—22no—
73C4×C353—21no—
74C4×C353—21no—
75Dic353—22no—
76Dic353—22no—
77C2×C353—22no—
78C2×C353—42no—
79C2×C353C22×D411yescharacteristic
80C2×C353H(5,𝔽2)11yescharacteristic
81C2×C353H(5,𝔽2)11yescharacteristic
82D353—24no—
83D353—42no—
84C353#53411yescharacteristic, complemented, Sylow
85#534—3531nomaximal, Sylow
86C22×D4—3532no—
87#123—3532no—
88#123—3532no—
89#121—3532no—
90#121—3531no—
91#121D35311yescharacteristic, complemented
92#120—3532no—
93#120—3532no—
94C2×C22⋊C4—3531no—
95C24—3531no—
96C24—3532no—
97C2×D4—3532no—
98C2×D4—3532no—
99C2×D4—3532no—
100C2×D4—3531no—
101C2×D4—3532no—
102C2×D4—7068no—
103C2×D4C2×D35312yescomplemented
104C2×D4C2×D35311yescharacteristic, complemented
105C22×C4—3531no—
106C22×C4—3532no—
107C22×C4C2×D35311yescharacteristic, complemented
108C4⋊C4—3532no—
109C4⋊C4—3531no—
110C4⋊C4C2×D35311yescharacteristic, complemented
111C22⋊C4—3532no—
112C22⋊C4—3532no—
113C22⋊C4—3532no—
114C22⋊C4—3532no—
115C22⋊C4—3532no—
116C22⋊C4C2×D35312yescomplemented
117C23—3534no—
118C23—3532no—
119C23—3532no—
120C23—14121no—
121C23—14121no—
122C23—14122no—
123C23—14122no—
124C23—7064no—
125C23C22×D35312yes—
126C23C22×D35311yescharacteristic
127D4—24no—
128D4—41no—
129D4—14121no—
130D4—14122no—
131D4—7064no—
132D4—7064no—
133D4—7064no—
134C2×C4—41no—
135C2×C4—3532no—
136C2×C4—3532no—
137C2×C4—14121no—
138C2×C4—7064no—
139C2×C4C22×D35311yescharacteristic
140C2×C4C22×D35311yescharacteristic
141C2×C4C22×D35312yes—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—3534no—
148C22—14124no—
149C22—14128no—
150C22—7062no—
151C22—7062no—
152C22—7062no—
153C22—7062no—
154C22—7062no—
155C22D4×D35312yes—
156C22C23×D35311yescharacteristic
157C4—22no—
158C4—21no—
159C4—21no—
160C4—7062no—
161C4—7062no—
162C2—22no—
163C2—42no—
164C2—14122no—
165C2—7064no—
166C2C2×D4×D35311yescharacteristic
167C2#2601503711yescharacteristic
168C2#2601503811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D353
commutator, socle, cosocleC22×C353C24
Fitting#121×C353C2
hypercenter#121D353
hypocenterC353#534
LCSG ▷ C22×C353 ▷ C353
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D353C2semidirect
2. #26014977C2semidirect
3. #26014974C2semidirect
4. #26014966C2semidirect
5. #26014964C2semidirect
6. #26014962C2semidirect
7. #26014960C2semidirect
8. C2×#19150714C2semidirect
9. #121×C353C2semidirect
11. C23×D353C22semidirect
13. C2×D1412C22semidirect
15. C2×C22⋊D353C22semidirect
16. C2×C22⋊D353C22semidirect
18. C22×Dic353C22semidirect
19. C2×C4×D353C22semidirect
20. C4⋊Dic353C22semidirect
21. #19150716C22semidirect
22. C22⋊Dic353C22semidirect
23. #19150714C22semidirect
24. #19150714C22semidirect
25. #19150714C22semidirect
26. #19150714C22semidirect
27. C2×D4×C353C22semidirect
28. C2×D4×C353C22semidirect
29. C22×C4×C353C22semidirect
30. C4⋊C4×C353C22semidirect
31. C22⋊C4×C353C22semidirect
37. C22×D353D4semidirect
84. C353#534semidirect
91. #121D353semidirect
103. C2×D4C2×D353semidirect
104. C2×D4C2×D353semidirect
107. C22×C4C2×D353semidirect
110. C4⋊C4C2×D353semidirect
116. C22⋊C4C2×D353semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3532
#260149772
#260149742
#260149662
#260149642
#260149622
#260149602
C2×#191507142
#121×C3532
#534353