Finite Group Databasebeta

← #36341813

Subgroups — #36341813

Order 21568.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 107 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×#19129184C211yesmaximal, characteristic, complemented
2C22×C4×D337C211yesmaximal, characteristic, complemented
3#24924942C211yesmaximal, characteristic, complemented
4#24924937C212yesmaximal, complemented
5#24924932C212yesmaximal, complemented
6#24924929C211yesmaximal, characteristic, complemented
7#24924927C211yesmaximal, characteristic, complemented
8#24924921C212yesmaximal, complemented
9#24924919C212yesmaximal, complemented
10#24924915C211yesmaximal, characteristic, complemented
11#123×C337C211yesmaximal, characteristic, complemented
12C23×D337C2211yescharacteristic
13#19129184—24no—
14C2×Dic674C2211yescharacteristic, complemented
15C2×D1348C2212yes—
16C2×C22⋊D337C2211yescharacteristic
17C2×C22⋊D337C2212yes—
18C2×C22⋊D337C2211yescharacteristic, complemented
19C22×Dic337C2211yescharacteristic
20C22×Dic337C2211yescharacteristic, complemented
21C2×C4×D337—24no—
22C2×C4×D337C2212yes—
23C2×C4×D337C2212yes—
24C2×C4×D337C2211yescharacteristic, complemented
25#19129164C2212yescomplemented
26#19129164C2212yescomplemented
27C22⋊Dic337C2211yescharacteristic
28#19129162C2212yes—
29#19129162C2212yes—
30#19129162C2212yescomplemented
31C4×Dic337C2212yescomplemented
32C2×D4×C337C2211yescharacteristic
33C22×C4×C337C2211yescharacteristic
34C4⋊C4×C337C2212yescomplemented
35C22⋊C4×C337C2211yescharacteristic
36C22⋊C4×C337C2212yescomplemented
37C22×D337—22no—
38C22×D337—24no—
39C22×D337C2311yescharacteristic
40C22×D337C2312yes—
41Dic674—22no—
42D1348—24no—
43C22⋊D337—22no—
44C22⋊D337—24no—
45C22⋊D337—22no—
46C2×Dic337—22no—
47C2×Dic337D414yescomplemented
48C2×Dic337C2312yes—
49C2×Dic337C2311yescharacteristic
50C2×Dic337C2311yescharacteristic
51C2×Dic337C2311yescharacteristic
52C4×D337—24no—
53C4×D337—24no—
54C4×D337—22no—
55C23×C337C2311yescharacteristic
56C23×C337C2311yescharacteristic
57D4×C337—22no—
58C2×C4×C337—22no—
59C2×C4×C337C2312yes—
60C2×C4×C337C2312yes—
61C2×C4×C337C2311yescharacteristic
62C22×C337—22no—
63C22×C337—21no—
64C22×C337—22no—
65C22×C337C2×D412yes—
66C22×C337C2411yescharacteristic
67C2×D337—28no—
68C2×D337—22no—
69C2×D337—22no—
70C2×D337—21no—
71C2×D337—22no—
72C2×D337#4114yescomplemented
73C4×C337—22no—
74C4×C337—21no—
75C4×C337—22no—
76Dic337—21no—
77Dic337—22no—
78Dic337C2×D412yes—
79Dic337C2×D412yes—
80C2×C337—22no—
81C2×C337—41no—
82C2×C337C22×D411yescharacteristic
83C2×C337C2×#4112yes—
84D337—24no—
85D337—41no—
86C337#52511yescharacteristic, complemented, Sylow
87#525—3371nomaximal, Sylow
88C2×#41—3371no—
89C23×C4—3371no—
90#123—3371no—
91#123D33711yescharacteristic, complemented
92C22⋊Q8—3372no—
93#121—3372no—
94#120—3371no—
95#120—3371no—
96C4×D4—3372no—
97C4×D4—3372no—
98#117—3371no—
99C24—3371no—
100#41—6744no—
101C2×Q8—3371no—
102C2×D4—3372no—
103C2×D4—3371no—
104C2×D4—3371no—
105C2×D4—3372no—
106C2×D4C2×D33711yescharacteristic
107C22×C4—3371no—
108C22×C4—3371no—
109C22×C4—3372no—
110C22×C4—3372no—
111C22×C4—3371no—
112C22×C4—6744no—
113C22×C4C2×D33711yescharacteristic
114C4⋊C4—3372no—
115C4⋊C4—3372no—
116C4⋊C4C2×D33712yescomplemented
117C22⋊C4—3371no—
118C22⋊C4—3372no—
119C22⋊C4—3372no—
120C22⋊C4—3372no—
121C22⋊C4C2×D33711yescharacteristic
122C22⋊C4C2×D33712yescomplemented
123C42—3372no—
124C23—3372no—
125C23—3371no—
126C23—6744no—
127C23—6742no—
128C23C22×D33711yescharacteristic
129C23C22×D33711yescharacteristic
130Q8—6742no—
131D4—22no—
132D4—6744no—
133D4—6742no—
134D4—6744no—
135D4—6742no—
136C2×C4—22no—
137C2×C4—3374no—
138C2×C4—3371no—
139C2×C4—3371no—
140C2×C4—3372no—
141C2×C4—3371no—
142C2×C4—6744no—
143C2×C4—6744no—
144C2×C4—6742no—
145C2×C4—6742no—
146C2×C4C22×D33712yes—
147C2×C4C22×D33711yescharacteristic
148C2×C4C22×D33712yes—
149C22—22no—
150C22—21no—
151C22—22no—
152C22—3374no—
153C22—6748no—
154C22—6741no—
155C22—6742no—
156C22—6742no—
157C22—6742no—
158C22D4×D33712yes—
159C22C23×D33711yescharacteristic
160C4—22no—
161C4—21no—
162C4—22no—
163C4—3372no—
164C4—3372no—
165C4—6741no—
166C4—6742no—
167C2—22no—
168C2—41no—
169C2—13481no—
170C2—6744no—
171C2C2×D4×D33711yescharacteristic
172C2#41×D33712yes—
subgroupG/H
center, FrattiniC22C23×D337
commutator, socle, cosocleC22×C337C24
Fitting#123×C337C2
hypercenter#123D337
hypocenterC337#525
LCSG ▷ C22×C337 ▷ C337
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#19129184C2semidirect
2. C22×C4×D337C2semidirect
3. #24924942C2semidirect
4. #24924937C2semidirect
5. #24924932C2semidirect
6. #24924929C2semidirect
7. #24924927C2semidirect
8. #24924921C2semidirect
9. #24924919C2semidirect
10. #24924915C2semidirect
11. #123×C337C2semidirect
14. C2×Dic674C22semidirect
18. C2×C22⋊D337C22semidirect
20. C22×Dic337C22semidirect
24. C2×C4×D337C22semidirect
25. #19129164C22semidirect
26. #19129164C22semidirect
30. #19129162C22semidirect
31. C4×Dic337C22semidirect
34. C4⋊C4×C337C22semidirect
36. C22⋊C4×C337C22semidirect
47. C2×Dic337D4semidirect
72. C2×D337#41semidirect
86. C337#525semidirect
91. #123D337semidirect
116. C4⋊C4C2×D337semidirect
122. C22⋊C4C2×D337semidirect

Maximal subgroups

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

subgroupindex
C2×#191291842
C22×C4×D3372
#249249422
#249249372
#249249322
#249249292
#249249272
#249249212
#249249192
#249249152
#123×C3372
#525337