Finite Group Databasebeta

← #38763032

Subgroups — #38763032

Order 25408.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×C4×D397C211yesmaximal, characteristic, complemented
3#27163078C212yesmaximal, complemented
4#27163076C211yesmaximal, characteristic, complemented
5#27163075C211yesmaximal, characteristic, complemented
6#27163067C211yesmaximal, characteristic, complemented
7#27163064C212yesmaximal, complemented
8D4×Dic397C212yesmaximal, complemented
9#27163056C212yesmaximal, complemented
10#27163054C211yesmaximal, characteristic, complemented
11#121×C397C211yesmaximal, characteristic, complemented
12C23×D397C2211yescharacteristic
13#20263730—24no—
14C2×Dic794C2211yescharacteristic, complemented
15C2×C22⋊D397C2212yes—
16C2×C22⋊D397C2212yes—
17C22×Dic397C2211yescharacteristic
18C22×Dic397C2212yescomplemented
19C2×C4×D397—24no—
20C2×C4×D397C2211yescharacteristic
21C2×C4×D397C2211yescharacteristic
22C2×C4×D397C2211yescharacteristic
23C2×C4×D397C2211yescharacteristic
24C4⋊Dic397C2211yescharacteristic, complemented
25C4⋊Dic397C2212yescomplemented
26#20263710C2212yescomplemented
27C22⋊Dic397C2212yes—
28C22⋊Dic397C2212yescomplemented
29#20263708C2212yes—
30#20263708C2212yes—
31C4×Dic397C2212yescomplemented
32C2×D4×C397C2212yes—
33C2×D4×C397C2211yescharacteristic, complemented
34C22×C4×C397C2211yescharacteristic
35C4⋊C4×C397C2211yescharacteristic, complemented
36C22⋊C4×C397C2212yescomplemented
37C22×D397—22no—
38C22×D397—22no—
39C22×D397—22no—
40C22×D397C2311yescharacteristic
41C22×D397C2311yescharacteristic
42Dic794—22no—
43C22⋊D397—24no—
44C22⋊D397—24no—
45C2×Dic397—22no—
46C2×Dic397—24no—
47C2×Dic397C2312yes—
48C2×Dic397C2312yes—
49C2×Dic397C2311yescharacteristic
50C2×Dic397C2311yescharacteristic
51C4×D397—22no—
52C4×D397—22no—
53C4×D397—22no—
54C4×D397D414yescomplemented
55C23×C397C2312yes—
56C23×C397C2311yescharacteristic
57D4×C397—24no—
58D4×C397—22no—
59C2×C4×C397—22no—
60C2×C4×C397C2311yescharacteristic
61C2×C4×C397C2312yes—
62C2×C4×C397C2311yescharacteristic
63C22×C397—21no—
64C22×C397—21no—
65C22×C397—22no—
66C22×C397—22no—
67C22×C397—22no—
68C22×C397#4112yescomplemented
69C22×C397C2411yescharacteristic
70C2×D397—28no—
71C2×D397—21no—
72C2×D397—21no—
73C2×D397—21no—
74C2×D397—21no—
75C2×D397C2×D412yes—
76C2×D397#4112yescomplemented
77C4×C397—21no—
78C4×C397—22no—
79C4×C397C2×D412yes—
80Dic397—21no—
81Dic397—22no—
82Dic397—22no—
83Dic397C2×D412yes—
84C2×C397—22no—
85C2×C397—42no—
86C2×C397C22×D411yescharacteristic
87C2×C397C2×#4111yescharacteristic
88C2×C397C2×#4111yescharacteristic
89D397—22no—
90D397—22no—
91C397#52511yescharacteristic, complemented, Sylow
92#525—3971nomaximal, Sylow
93C2×#41—3971no—
94C23×C4—3971no—
95#123—3972no—
96C22⋊Q8—3971no—
97C22⋊Q8—3971no—
98#121—3971no—
99#121D39711yescharacteristic, complemented
100#120—3972no—
101C4×D4—3972no—
102C4×D4—3972no—
103#117—3971no—
104C24—3971no—
105#41—7944no—
106C2×Q8—3971no—
107C2×D4—3972no—
108C2×D4—3972no—
109C2×D4C2×D39712yes—
110C2×D4C2×D39711yescharacteristic, complemented
111C22×C4—7944no—
112C22×C4—3971no—
113C22×C4—3972no—
114C22×C4—3971no—
115C22×C4—3971no—
116C22×C4—3971no—
117C22×C4—3971no—
118C22×C4C2×D39711yescharacteristic
119C4⋊C4—3972no—
120C4⋊C4—3971no—
121C4⋊C4—3972no—
122C4⋊C4C2×D39711yescharacteristic, complemented
123C22⋊C4—3972no—
124C22⋊C4—3972no—
125C22⋊C4—3972no—
126C22⋊C4—3972no—
127C22⋊C4C2×D39712yescomplemented
128C42—3972no—
129C23—7942no—
130C23—7942no—
131C23—7942no—
132C23—3971no—
133C23—3971no—
134C23C22×D39712yes—
135C23C22×D39711yescharacteristic
136Q8—7942no—
137D4—24no—
138D4—22no—
139D4—7944no—
140D4—7944no—
141C2×C4—22no—
142C2×C4—7942no—
143C2×C4—7942no—
144C2×C4—7942no—
145C2×C4—7944no—
146C2×C4—7942no—
147C2×C4—3972no—
148C2×C4—3971no—
149C2×C4—3972no—
150C2×C4—3971no—
151C2×C4—3974no—
152C2×C4C22×D39712yes—
153C2×C4C22×D39711yescharacteristic
154C2×C4C22×D39711yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—7948no—
161C22—7941no—
162C22—7941no—
163C22—7941no—
164C22—7941no—
165C22—3972no—
166C22—3972no—
167C22#2026373012yescomplemented
168C22C23×D39711yescharacteristic
169C4—22no—
170C4—21no—
171C4—7942no—
172C4—7941no—
173C4—7942no—
174C4—3972no—
175C4D4×D39712yes—
176C2—22no—
177C2—42no—
178C2—7942no—
179C2—7942no—
180C2C2×D4×D39711yescharacteristic
181C2#41×D39711yescharacteristic
182C2C2×#2026373011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
Fitting#121×C397C2
hypercenter#121D397
hypocenterC397#525
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20263730C2semidirect
2. C22×C4×D397C2semidirect
3. #27163078C2semidirect
4. #27163076C2semidirect
5. #27163075C2semidirect
6. #27163067C2semidirect
7. #27163064C2semidirect
8. D4×Dic397C2semidirect
9. #27163056C2semidirect
10. #27163054C2semidirect
11. #121×C397C2semidirect
14. C2×Dic794C22semidirect
18. C22×Dic397C22semidirect
24. C4⋊Dic397C22semidirect
25. C4⋊Dic397C22semidirect
26. #20263710C22semidirect
28. C22⋊Dic397C22semidirect
31. C4×Dic397C22semidirect
33. C2×D4×C397C22semidirect
35. C4⋊C4×C397C22semidirect
36. C22⋊C4×C397C22semidirect
54. C4×D397D4semidirect
68. C22×C397#41semidirect
76. C2×D397#41semidirect
91. C397#525semidirect
99. #121D397semidirect
110. C2×D4C2×D397semidirect
122. C4⋊C4C2×D397semidirect
127. C22⋊C4C2×D397semidirect
167. C22#20263730semidirect

Maximal subgroups

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

subgroupindex
C2×#202637302
C22×C4×D3972
#271630782
#271630762
#271630752
#271630672
#271630642
D4×Dic3972
#271630562
#271630542
#121×C3972
#525397