Finite Group Databasebeta

← #38767802

Subgroups — #38767802

Order 25472.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#27164156C212yesmaximal, complemented
2#27163930C211yesmaximal, characteristic, complemented
3#27163752C212yesmaximal, complemented
4#27163742C211yesmaximal, characteristic, complemented
5#534×C199C211yesmaximal, characteristic
6C22×Dic398C2211yescharacteristic, complemented
7C2×#15691874—22no—
8C2×C22⋊Dic199—21no—
9#20263878—22no—
10C22⋊(C199⋊C8)C2212yescomplemented
11C199⋊#95—21no—
12C199⋊#95C2211yescharacteristic, complemented
13C22×D4×C199—21no—
14#123×C199—22no—
15#121×C199—21no—
16#121×C199C2212yes—
17#120×C199—22no—
18C2×C22⋊C4×C199C2211yescharacteristic
19C2×Dic398—22no—
20C2×Dic398—42no—
21C22×Dic199—21no—
22C22×Dic199—21no—
23#15691874—44no—
24C2×C199⋊C8—22no—
25C22⋊Dic199—42no—
26C24×C199—21no—
27C24×C199D411yescharacteristic
28C2×D4×C199—22no—
29C2×D4×C199—22no—
30C2×D4×C199—21no—
31C2×D4×C199—42no—
32C2×D4×C199—42no—
33C2×D4×C199D412yes—
34C22×C4×C199—21no—
35C22×C4×C199D411yescharacteristic
36C22×C4×C199C2311yescharacteristic
37C4⋊C4×C199—21no—
38C4⋊C4×C199D412yes—
39C22⋊C4×C199—22no—
40C22⋊C4×C199—22no—
41C22⋊C4×C199—22no—
42Dic398—24no—
43Dic398—42no—
44C2×Dic199—22no—
45C2×Dic199—41no—
46C2×Dic199—41no—
47C2×Dic199—41no—
48C2×Dic199—41no—
49C2×Dic199—41no—
50C199⋊C8—42no—
51C23×C199—21no—
52C23×C199—21no—
53C23×C199—22no—
54C23×C199—41no—
55C23×C199—41no—
56C23×C199—42no—
57C23×C199—81no—
58C23×C199—81no—
59C23×C199SD1612yescomplemented
60C23×C199C2×D411yescharacteristic
61D4×C199—42no—
62D4×C199—44no—
63D4×C199—42no—
64D4×C199—42no—
65D4×C199—81no—
66C2×C4×C199—21no—
67C2×C4×C199—22no—
68C2×C4×C199—41no—
69C2×C4×C199—42no—
70C2×C4×C199—41no—
71C2×C4×C199C2×D412yes—
72C22×C199—21no—
73C22×C199—21no—
74C22×C199—22no—
75C22×C199—42no—
76C22×C199—41no—
77C22×C199—41no—
78C22×C199—42no—
79C22×C199—42no—
80C22×C199—42no—
81C22×C199—81no—
82C22×C199—84no—
83C22×C199#12011yescharacteristic
84C22×C199C2×SD1611yescharacteristic
85C22×C199AGL(1,ℤ/8)11yescharacteristic
86C4×C199—22no—
87C4×C199—42no—
88C4×C199—41no—
89Dic199—42no—
90Dic199—81no—
91C2×C199—21no—
92C2×C199—21no—
93C2×C199—42no—
94C2×C199—82no—
95C2×C199#45011yescharacteristic
96C2×C199#45511yescharacteristic
97C2×C199B(4,𝔽2)11yescharacteristic
98C199#160111yescharacteristic, complemented, Sylow
99#1601—1991nomaximal, Sylow
100#534D19911yescharacteristic
101#448—1992no—
102#393—1991no—
103#327—1992no—
104#323—1991no—
105C22×Q8—1991no—
106C22×D4—21no—
107C2×SD16—3982no—
108#123—22no—
109#121—21no—
110#121C2×D19912yes—
111#120—22no—
112C2×C22⋊C4—3981no—
113C2×C22⋊C4C2×D19911yescharacteristic
114#103—3982no—
115C22⋊C8—1992no—
116#95—3981no—
117#95—1991no—
118C24—21no—
119C24C22⋊D19911yescharacteristic
120C2×Q8—7962no—
121C2×Q8—3982no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4C22⋊D19912yes—
128C22×C4—21no—
129C22×C4—3981no—
130C22×C4—3981no—
131C22×C4C22⋊D19911yescharacteristic
132C22×C4C22×D19911yescharacteristic
133SD16—7964no—
134C2×C8—3982no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D19912yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—7962no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23#1569187412yescomplemented
150C23C2×C22⋊D19911yescharacteristic
151Q8—7962no—
152Q8—3984no—
153D4—42no—
154D4—44no—
155D4—42no—
156D4—42no—
157D4—81no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—7961no—
164C2×C4—7961no—
165C2×C4—7961no—
166C2×C4—7961no—
167C2×C4—7961no—
168C2×C4—3982no—
169C2×C4C2×C22⋊D19912yes—
170C8—7962no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—81no—
181C22—84no—
182C22#2026392411yescharacteristic
183C22C2×#1569187411yescharacteristic
184C22#2026398111yescharacteristic
185C4—22no—
186C4—42no—
187C4—41no—
188C4—7962no—
189C4—15921no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2#2716416911yescharacteristic
195C2#2716419611yescharacteristic
196C2#2716420511yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting, prime radical#534×C199C2
socleC22×C199#120
hypercenter#534D199
hypocenterC199#1601
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #27164156C2semidirect
2. #27163930C2semidirect
3. #27163752C2semidirect
4. #27163742C2semidirect
6. C22×Dic398C22semidirect
10. C22⋊(C199⋊C8)C22semidirect
12. C199⋊#95C22semidirect
59. C23×C199SD16semidirect
98. C199#1601semidirect
149. C23#15691874semidirect

Maximal subgroups

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

subgroupindex
#271641562
#271639302
#271637522
#271637422
#534×C1992
#1601199