Finite Group Databasebeta

← #38769377

Subgroups — #38769377

Order 25472.

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

normal subgroups only

This group has 344 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#27164024C212yesmaximal, complemented
3#27163992C211yesmaximal, characteristic, complemented
4#27163775C212yesmaximal, complemented
5#568×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7#20263951C412yescomplemented
8#20263944C412yescomplemented
9#20263929—22no—
10#20263923—22no—
11#20263914C2212yes—
12#20263914C2211yescharacteristic
13#20263877—24no—
14#20263870—22no—
15C2×#41×C199C2211yescharacteristic
16#131×C199—24no—
17C2×M16×C199—22no—
18C2×M16×C199C2212yes—
19C23×D199—21no—
20C2×D796—22no—
21C2×D796—21no—
22C2×D796—22no—
23C2×D796C2×C412yes—
24C2×C22⋊D199—24no—
25C4⋊Dic199—21no—
26C4⋊Dic199—22no—
27C22⋊Dic199—21no—
28C22⋊Dic199—22no—
29(C2×D199)⋊C4—24no—
30C4×Dic199—22no—
31C4×Dic199C2×C412yes—
32#41×C199—22no—
33#41×C199—22no—
34C2×Q8×C199C2×C411yescharacteristic
35C2×D4×C199C2×C411yescharacteristic
36C2×D4×C199D412yes—
37C22×C4×C199D412yes—
38C22×C4×C199C2311yescharacteristic
39M16×C199—24no—
40M16×C199—22no—
41M16×C199—22no—
42C2×C8×C199—24no—
43C2×C8×C199D414yescomplemented
44C22×D199—22no—
45C22×D199—42no—
46C22×D199—82no—
47D796—24no—
48D796—41no—
49D796—41no—
50C22⋊D199—84no—
51C2×Dic199—22no—
52C2×Dic199—42no—
53C23×C199C22⋊C412yes—
54C23×C199C2×D411yescharacteristic
55Q8×C199—22no—
56D4×C199—24no—
57D4×C199—22no—
58C2×C4×C199—22no—
59C2×C4×C199—22no—
60C2×C4×C199C22⋊C412yes—
61C2×C4×C199C22×C411yescharacteristic
62C2×C4×C199C2×D412yes—
63C2×C4×C199C2×D411yescharacteristic
64C2×C4×C199#4112yes—
65C8×C199—24no—
66C22×C199—22no—
67C22×C199—41no—
68C22×C199—42no—
69C22×C199C2×C22⋊C411yescharacteristic
70C22×C199#12112yes—
71C2×D199—44no—
72C2×D199—42no—
73C2×D199—84no—
74C4×C199—22no—
75C4×C199C4×D412yes—
76C4×C199#12011yescharacteristic
77C4×C199#12311yescharacteristic
78Dic199—42no—
79Dic199—82no—
80C2×C199—21no—
81C2×C199—22no—
82C2×C199—42no—
83C2×C199#38611yescharacteristic
84D199—82no—
85C199#191011yescharacteristic, complemented, Sylow
86#1910—1991nomaximal, Sylow
87#568D19911yescharacteristic, complemented
88#535—1991no—
89#418—1992no—
90#410—1991no—
91H(3,ℤ/4)—1992no—
92C2×#41C2×D19911yescharacteristic
93C22×D4—1991no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D19912yes—
97#127—1992no—
98#124—1992no—
99#121—3982no—
100#120—3982no—
101#117—1992no—
102#117—1991no—
103#102—3984no—
104#99—3982no—
105C24—3981no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D19911yescharacteristic
109C2×D4—3984no—
110C2×D4—3981no—
111C2×D4—3982no—
112C2×D4—3982no—
113C2×D4—1992no—
114C2×D4C4×D19911yescharacteristic
115C2×D4D79612yes—
116C22×C4C22⋊D19912yes—
117C22×C4C22×D19911yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D19914yescomplemented
123C4⋊C4—3982no—
124C4⋊C4—3981no—
125C22⋊C4—3982no—
126C22⋊C4—3981no—
127C22⋊C4—3984no—
128C42—3982no—
129C42—1992no—
130C23—7962no—
131C23—15922no—
132C23—3982no—
133C23(C2×D199)⋊C412yes—
134C23C2×D79611yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—7961no—
139D4—7961no—
140D4—15924no—
141D4—3984no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—7962no—
145C2×C4—3982no—
146C2×C4(C2×D199)⋊C412yes—
147C2×C4C2×C4×D19911yescharacteristic
148C2×C4C2×C22⋊D19911yescharacteristic
149C2×C4C2×C22⋊D19912yes—
150C2×C4#1569188612yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—7964no—
156C22—7962no—
157C22—15924no—
158C22C2×(C2×D199)⋊C411yescharacteristic
159C22#2026392912yes—
160C4—22no—
161C4—7962no—
162C4—15922no—
163C4C4×C22⋊D19912yes—
164C4#2026392411yescharacteristic
165C4#2026393811yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—15922no—
170C2#2716389711yescharacteristic
subgroupG/H
centerC2#27163897
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#568×C199C2
socleC2×C199#386
prime radical#27164582C2
cosocleC22×C4×C199C23
hypercenter#568D199
hypocenterC199#1910
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27164024C2semidirect
3. #27163992C2semidirect
4. #27163775C2semidirect
5. #568×C199C2semidirect
7. #20263951C4semidirect
8. #20263944C4semidirect
43. C2×C8×C199D4semidirect
85. C199#1910semidirect
87. #568D199semidirect
122. C2×C8C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271645822
#271640242
#271639922
#271637752
#568×C1992
#1910199