Finite Group Databasebeta

← #38769380

Subgroups — #38769380

Order 25472.

362 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 85 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#27164585C211yesmaximal, characteristic
2#27164493C211yesmaximal, characteristic, complemented
3#27164034C212yesmaximal, complemented
4#27163775C212yesmaximal, complemented
5C2×#100×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C2211yescharacteristic
7#20263951C412yescomplemented
8#20263942C412yescomplemented
9#20263926—22no—
10#20263924—22no—
11D4×Dic199—24no—
12#20263914C2211yescharacteristic
13#20263914C2212yes—
14C2×C22⋊Dic199—22no—
15#20263881—24no—
16C22×D4×C199C2211yescharacteristic
17C2×M16×C199C2212yes—
18#100×C199—22no—
19#15691886—22no—
20#15691886—22no—
21C2×Dic398C2×C411yescharacteristic, complemented
22C2×D796C2×C411yescharacteristic, complemented
23C2×C22⋊D199—24no—
24C2×C22⋊D199—21no—
25C22×Dic199—24no—
26C2×C4×D199—21no—
27C4⋊Dic199—22no—
28C4⋊Dic199—21no—
29C22⋊Dic199—24no—
30C22⋊Dic199—24no—
31C22⋊Dic199—22no—
32C22⋊Dic199—21no—
33C4×Dic199C2×C412yescomplemented
34C4×Dic199D414yescomplemented
35C24×C199D412yes—
36C2×D4×C199—22no—
37C2×D4×C199—22no—
38C2×D4×C199C2×C412yes—
39C2×D4×C199D412yes—
40C22×C4×C199C2311yescharacteristic
41M16×C199—22no—
42M16×C199—22no—
43C2×C8×C199—22no—
44C22×D199—21no—
45Dic398—22no—
46D796—22no—
47C22⋊D199—41no—
48C22⋊D199—41no—
49C22⋊D199—84no—
50C2×Dic199—22no—
51C2×Dic199—21no—
52C2×Dic199—24no—
53C2×Dic199—84no—
54C4×D199—42no—
55C23×C199—22no—
56C23×C199—22no—
57C23×C199—82no—
58C23×C199C22⋊C414yes—
59C23×C199C2×D411yescharacteristic
60D4×C199—24no—
61D4×C199—24no—
62C2×C4×C199C22×C411yescharacteristic
63C2×C4×C199C2×D412yes—
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199#4112yes—
66C8×C199—42no—
67C22×C199—24no—
68C22×C199—41no—
69C22×C199—44no—
70C22×C199—44no—
71C22×C199—82no—
72C22×C199C2×C22⋊C411yescharacteristic
73C22×C199#12011yescharacteristic
74C22×C199#12311yescharacteristic
75C2×D199—42no—
76C2×D199—41no—
77C4×C199C4×D412yes—
78C4×C199#12112yes—
79Dic199—44no—
80Dic199—42no—
81Dic199—41no—
82C2×C199—21no—
83C2×C199—21no—
84C2×C199—21no—
85C2×C199—44no—
86C2×C199#38611yescharacteristic
87D199—81no—
88C199#191111yescharacteristic, complemented, Sylow
89#1911—1991nomaximal, Sylow
90#535—1991no—
91#518—1991no—
92#421—1992no—
93C2×#100D19911yescharacteristic, complemented
94H(3,ℤ/4)—1992no—
95C2×#41—1991no—
96C22×D4C2×D19911yescharacteristic
97C2×M16C2×D19912yes—
98#127—1992no—
99#124—1992no—
100#121—3982no—
101#120—3982no—
102C4×D4—3984no—
103#117—1992no—
104#117—1991no—
105C2×C22⋊C4—3982no—
106#104—3984no—
107#100—22no—
108C24D79612yes—
109#41—3982no—
110#41—3982no—
111C2×Q8—1991no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—3981no—
115C2×D4—3984no—
116C2×D4—1991no—
117C2×D4C4×D19912yes—
118C2×D4C22⋊D19912yes—
119C22×C4—3981no—
120C22×C4—3984no—
121C22×C4C22×D19911yescharacteristic
122M16—22no—
123M16—22no—
124C2×C8—22no—
125C4⋊C4—3982no—
126C4⋊C4—3981no—
127C22⋊C4—3984no—
128C22⋊C4—3984no—
129C22⋊C4—3982no—
130C22⋊C4—3981no—
131C42—1992no—
132C42—1994no—
133C23—22no—
134C23—22no—
135C23—82no—
136C23—3981no—
137C23(C2×D199)⋊C414yes—
138C23C2×D79611yescharacteristic
139Q8—3982no—
140D4—24no—
141D4—24no—
142D4—7961no—
143D4—7961no—
144D4—15924no—
145D4—3982no—
146C2×C4—7962no—
147C2×C4—15924no—
148C2×C4—3982no—
149C2×C4—3981no—
150C2×C4—3984no—
151C2×C4C2×C4×D19911yescharacteristic
152C2×C4D4×D19912yes—
153C2×C4C2×C22⋊D19911yescharacteristic
154C2×C4#1569188512yes—
155C8—42no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—7962no—
162C22—7961no—
163C22C2×(C2×D199)⋊C411yescharacteristic
164C22#2026392511yescharacteristic
165C22#2026393911yescharacteristic
166C4—7964no—
167C4—7962no—
168C4—7961no—
169C4(C22⋊D199)⋊C412yes—
170C4#2026392712yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—15921no—
176C2#2716389511yescharacteristic
subgroupG/H
centerC2#27163895
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×#100×C199C2
socleC2×C199#386
prime radical#27164585C2
cosocleC22×C4×C199C23
hypercenterC2×#100D199
hypocenterC199#1911
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #27164493C2semidirect
3. #27164034C2semidirect
4. #27163775C2semidirect
5. C2×#100×C199C2semidirect
7. #20263951C4semidirect
8. #20263942C4semidirect
21. C2×Dic398C2×C4semidirect
22. C2×D796C2×C4semidirect
33. C4×Dic199C2×C4semidirect
34. C4×Dic199D4semidirect
88. C199#1911semidirect
93. C2×#100D199semidirect

Maximal subgroups

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

subgroupindex
#271645852
#271644932
#271640342
#271637752
C2×#100×C1992
#1911199