Finite Group Databasebeta

← #38769397

Subgroups — #38769397

Order 25472.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#27164582C211yesmaximal, characteristic
2#27164002C211yesmaximal, characteristic, complemented
3C2×#20263871C212yesmaximal, complemented
4#27163790C212yesmaximal, complemented
5#568×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7C2×#15691870C2212yes—
8#20263951—21no—
9#20263944—21no—
10#20263929—22no—
11#20263923C414yes—
12#20263914C2211yescharacteristic
13#20263874—21no—
14#20263872—21no—
15#20263871—24no—
16C2×#41×C199C2211yescharacteristic
17#131×C199—24no—
18C2×M16×C199—22no—
19C2×M16×C199C2212yes—
20C23×D199C2×C412yes—
21C2×D796—21no—
22C2×D796—22no—
23C2×D796—21no—
24C2×D796—22no—
25C2×C22⋊D199—24no—
26#15691870—22no—
27#15691870—22no—
28C2×C199⋊C8—22no—
29C4⋊Dic199—21no—
30C22⋊Dic199C2×C412yes—
31(C2×D199)⋊C4—24no—
32C4×Dic199—21no—
33#41×C199—22no—
34#41×C199—22no—
35C2×Q8×C199D411yescharacteristic
36C2×D4×C199C2×C412yes—
37C2×D4×C199D411yescharacteristic
38C22×C4×C199D412yes—
39C22×C4×C199C2311yescharacteristic
40M16×C199—24no—
41M16×C199—22no—
42M16×C199D414yescomplemented
43C2×C8×C199—24no—
44C2×C8×C199—22no—
45C22×D199—22no—
46C22×D199—24no—
47C22×D199—82no—
48D796—41no—
49D796—42no—
50D796—41no—
51C22⋊D199—84no—
52C2×Dic199—22no—
53C199⋊C8—42no—
54C23×C199C22⋊C412yes—
55C23×C199C22×C411yescharacteristic
56Q8×C199—22no—
57D4×C199—24no—
58D4×C199—22no—
59C2×C4×C199—22no—
60C2×C4×C199—22no—
61C2×C4×C199C22⋊C412yes—
62C2×C4×C199C2×D412yes—
63C2×C4×C199C2×D411yescharacteristic
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199#4112yes—
66C8×C199—24no—
67C22×C199—22no—
68C22×C199—41no—
69C22×C199—42no—
70C22×C199C2×C22⋊C411yescharacteristic
71C22×C199C4×D412yes—
72C2×D199—44no—
73C2×D199—42no—
74C2×D199—84no—
75C4×C199—22no—
76C4×C199#12011yescharacteristic
77C4×C199#12112yes—
78C4×C199#12311yescharacteristic
79Dic199—42no—
80C2×C199—21no—
81C2×C199—22no—
82C2×C199—42no—
83C2×C199#38611yescharacteristic
84D199—82no—
85C199#191411yescharacteristic, complemented, Sylow
86#1914—1991nomaximal, Sylow
87#568D19911yescharacteristic, complemented
88#535—1991no—
89#413—1991no—
90C2×#100—1992no—
91#344—1992no—
92C2×#41C2×D19911yescharacteristic
93C22×D4—1991no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1992no—
97C2×M16C2×D19912yes—
98#127—3981no—
99#124—3981no—
100#121—3982no—
101#120—1994no—
102#117—1991no—
103#101—3981no—
104#100—3981no—
105#100—3984no—
106C24—1992no—
107#41—22no—
108#41—22no—
109C2×Q8D79611yescharacteristic
110C2×D4—3984no—
111C2×D4—3982no—
112C2×D4—3981no—
113C2×D4—3981no—
114C2×D4—3982no—
115C2×D4C4×D19912yes—
116C2×D4D79611yescharacteristic
117C22×C4C22⋊D19912yes—
118C22×C4C22×D19911yescharacteristic
119M16—24no—
120M16—22no—
121M16—3982no—
122M16—3982no—
123M16C22⋊D19914yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—3982no—
127C4⋊C4—3981no—
128C22⋊C4—3984no—
129C22⋊C4—1992no—
130C42—3981no—
131C23—15922no—
132C23—3982no—
133C23—3984no—
134C23(C2×D199)⋊C412yes—
135C23C2×C4×D19911yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—7961no—
140D4—7961no—
141D4—7962no—
142D4—15924no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—3982no—
146C2×C4(C2×D199)⋊C412yes—
147C2×C4C2×C22⋊D19911yescharacteristic
148C2×C4C2×C22⋊D19912yes—
149C2×C4C2×D79611yescharacteristic
150C2×C4#1569188612yes—
151C8—24no—
152C8—7962no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—7964no—
157C22—7962no—
158C22—15924no—
159C22C2×(C2×D199)⋊C411yescharacteristic
160C22C4×C22⋊D19912yes—
161C4—22no—
162C4—7962no—
163C4#2026392411yescharacteristic
164C4#2026392912yes—
165C4#2026393811yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—15922no—
170C2#2716389711yescharacteristic
subgroupG/H
centerC2#27163897
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#568×C199C2
socleC2×C199#386
prime radical#27164582C2
cosocleC22×C4×C199C23
hypercenter#568D199
hypocenterC199#1914
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27164002C2semidirect
3. C2×#20263871C2semidirect
4. #27163790C2semidirect
5. #568×C199C2semidirect
42. M16×C199D4semidirect
85. C199#1914semidirect
87. #568D199semidirect
123. M16C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271645822
#271640022
C2×#202638712
#271637902
#568×C1992
#1914199