Finite Group Databasebeta

← #38763087

Subgroups — #38763087

Order 25408.

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

normal subgroups only

This group has 334 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×#20263730C211yesmaximal, characteristic, complemented
2C22×C22⋊D397C211yesmaximal, characteristic, complemented
3#27163092C212yesmaximal, complemented
4#27163082C212yesmaximal, complemented
5#27163069C212yesmaximal, complemented
6#27163068C212yesmaximal, complemented
7#27163066C211yesmaximal, characteristic, complemented
8#27163065C211yesmaximal, characteristic, complemented
9#27163064C211yesmaximal, characteristic, complemented
10#27163052C211yesmaximal, characteristic, complemented
11#120×C397C211yesmaximal, characteristic, complemented
12C23×D397C2211yescharacteristic
13#20263730—24no—
14C2×Dic794C2211yescharacteristic, complemented
15C2×D1588C2212yescomplemented
16C2×C22⋊D397—24no—
17C2×C22⋊D397C2211yescharacteristic
18C2×C22⋊D397C2212yes—
19C2×C22⋊D397C2212yes—
20C2×C22⋊D397C2211yescharacteristic, complemented
21C2×C22⋊D397C2212yescomplemented
22C2×C22⋊D397C2212yescomplemented
23C22×Dic397C2211yescharacteristic
24C22×Dic397C2211yescharacteristic, complemented
25C2×C4×D397C2211yescharacteristic, complemented
26#20263710C2212yescomplemented
27C22⋊Dic397C2211yescharacteristic
28C22⋊Dic397C2212yescomplemented
29#20263708C2212yescomplemented
30#20263708C2212yescomplemented
31C4×Dic397C2212yescomplemented
32C24×C397C2211yescharacteristic
33C2×D4×C397C2211yescharacteristic
34C2×D4×C397C2212yescomplemented
35C22⋊C4×C397C2211yescharacteristic
36C22⋊C4×C397C2212yescomplemented
37C22×D397—22no—
38C22×D397—42no—
39C22×D397C2312yes—
40C22×D397C2311yescharacteristic
41Dic794—22no—
42D1588—42no—
43C22⋊D397—22no—
44C22⋊D397—24no—
45C22⋊D397—24no—
46C22⋊D397—22no—
47C22⋊D397—42no—
48C22⋊D397—42no—
49C2×Dic397—22no—
50C2×Dic397D414yescomplemented
51C2×Dic397C2312yes—
52C2×Dic397C2311yescharacteristic
53C2×Dic397C2311yescharacteristic
54C2×Dic397C2311yescharacteristic
55C4×D397—22no—
56C23×C397—22no—
57C23×C397—42no—
58C23×C397C2312yes—
59C23×C397C2311yescharacteristic
60C23×C397C2311yescharacteristic
61D4×C397—22no—
62D4×C397—42no—
63C2×C4×C397C2312yes—
64C2×C4×C397C2311yescharacteristic
65C22×C397—22no—
66C22×C397—21no—
67C22×C397—22no—
68C22×C397—22no—
69C22×C397—22no—
70C22×C397—22no—
71C22×C397—44no—
72C22×C397C2×D412yes—
73C22×C397C2411yescharacteristic
74C2×D397—21no—
75C2×D397—22no—
76C2×D397—22no—
77C2×D397—22no—
78C2×D397—22no—
79C2×D397—44no—
80C4×C397—21no—
81C4×C397—22no—
82Dic397—21no—
83Dic397—22no—
84Dic397C2×D412yes—
85Dic397C2×D412yes—
86C2×C397—22no—
87C2×C397—41no—
88C2×C397—42no—
89C2×C397C22×D411yescharacteristic
90C2×C397H(5,𝔽2)12yes—
91D397—41no—
92D397—42no—
93C397#53511yescharacteristic, complemented, Sylow
94#535—3971nomaximal, Sylow
95C2×#41—3971no—
96C22×D4—3971no—
97#127—3972no—
98#124—3972no—
99#121—3972no—
100#121—3972no—
101#120—3971no—
102#120—3971no—
103#120—3971no—
104#120D39711yescharacteristic, complemented
105#117—3971no—
106C24—3971no—
107C24C2×D39711yescharacteristic
108#41—7944no—
109C2×Q8—3971no—
110C2×D4—7944no—
111C2×D4—3971no—
112C2×D4—3972no—
113C2×D4—3972no—
114C2×D4—3971no—
115C2×D4—3972no—
116C2×D4—3972no—
117C2×D4—3972no—
118C2×D4C2×D39711yescharacteristic
119C2×D4C2×D39712yescomplemented
120C22×C4—3971no—
121C22×C4—3971no—
122C22×C4—3971no—
123C4⋊C4—3972no—
124C22⋊C4—3971no—
125C22⋊C4—3972no—
126C22⋊C4—3972no—
127C22⋊C4—3972no—
128C22⋊C4C2×D39711yescharacteristic
129C22⋊C4C2×D39712yescomplemented
130C42—3972no—
131C23—22no—
132C23—42no—
133C23—7942no—
134C23—15882no—
135C23—3972no—
136C23—3971no—
137C23C22×D39712yes—
138C23C22×D39711yescharacteristic
139C23C22×D39711yescharacteristic
140Q8—7942no—
141D4—22no—
142D4—42no—
143D4—7942no—
144D4—7944no—
145D4—7944no—
146D4—7942no—
147D4—15882no—
148D4—15882no—
149D4—15882no—
150C2×C4—7942no—
151C2×C4—7942no—
152C2×C4—3972no—
153C2×C4—3971no—
154C2×C4—3974no—
155C2×C4—3971no—
156C2×C4—3971no—
157C2×C4C22×D39712yes—
158C2×C4C22×D39711yescharacteristic
159C22—22no—
160C22—22no—
161C22—22no—
162C22—22no—
163C22—21no—
164C22—22no—
165C22—44no—
166C22—7942no—
167C22—7942no—
168C22—7942no—
169C22—7941no—
170C22—7942no—
171C22—15884no—
172C22D4×D39712yes—
173C22C23×D39711yescharacteristic
174C4—22no—
175C4—21no—
176C4—7942no—
177C4—7941no—
178C4—3972no—
179C4—3972no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—15882no—
184C2—15881no—
185C2C2×D4×D39711yescharacteristic
186C2#2716314112yes—
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
Fitting#120×C397C2
hypercenter#120D397
hypocenterC397#535
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20263730C2semidirect
2. C22×C22⋊D397C2semidirect
3. #27163092C2semidirect
4. #27163082C2semidirect
5. #27163069C2semidirect
6. #27163068C2semidirect
7. #27163066C2semidirect
8. #27163065C2semidirect
9. #27163064C2semidirect
10. #27163052C2semidirect
11. #120×C397C2semidirect
14. C2×Dic794C22semidirect
15. C2×D1588C22semidirect
20. C2×C22⋊D397C22semidirect
21. C2×C22⋊D397C22semidirect
22. C2×C22⋊D397C22semidirect
24. C22×Dic397C22semidirect
25. C2×C4×D397C22semidirect
26. #20263710C22semidirect
28. C22⋊Dic397C22semidirect
29. #20263708C22semidirect
30. #20263708C22semidirect
31. C4×Dic397C22semidirect
34. C2×D4×C397C22semidirect
36. C22⋊C4×C397C22semidirect
50. C2×Dic397D4semidirect
93. C397#535semidirect
104. #120D397semidirect
119. C2×D4C2×D397semidirect
129. C22⋊C4C2×D397semidirect

Maximal subgroups

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

subgroupindex
C2×#202637302
C22×C22⋊D3972
#271630922
#271630822
#271630692
#271630682
#271630662
#271630652
#271630642
#271630522
#120×C3972
#535397