Finite Group Databasebeta

← #38769293

Subgroups — #38769293

Order 25472.

442 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 101 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#27164529C211yesmaximal, characteristic
2C2×#20263883C212yesmaximal, complemented
3#27163982C211yesmaximal, characteristic, complemented
4#27163778C212yesmaximal, complemented
5C22×C42×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C2211yescharacteristic
7C2×#15691870C2212yes—
8#20263938—21no—
9#20263934C412yescomplemented
10#20263929C412yescomplemented
11#20263924—21no—
12C4×C22⋊D199—22no—
13#20263914C2211yescharacteristic
14#20263883—24no—
15C22⋊(C199⋊C8)—22no—
16C23×C4×C199—21no—
17C23×C4×C199C2211yescharacteristic
18C2×C42×C199—21no—
19C2×C42×C199—24no—
20C2×C42×C199C2212yes—
21#15691886—22no—
22#15691886—22no—
23C2×Dic398C2×C411yescharacteristic, complemented
24C2×D796C2×C411yescharacteristic, complemented
25C2×C22⋊D199—22no—
26C2×C22⋊D199—21no—
27C2×C4×D199—21no—
28#15691870—22no—
29#15691870—22no—
30C2×C199⋊C8—22no—
31C4⋊Dic199C2×C412yescomplemented
32Dic199⋊C4—22no—
33C22⋊Dic199—22no—
34C22⋊Dic199—21no—
35(C2×D199)⋊C4—22no—
36C4×Dic199—21no—
37C24×C199D411yescharacteristic
38C22×C4×C199—24no—
39C22×C4×C199—28no—
40C22×C4×C199—22no—
41C22×C4×C199—22no—
42C22×C4×C199—22no—
43C22×C4×C199C2×C412yes—
44C22×C4×C199D411yescharacteristic
45C22×C4×C199D412yes—
46C22×C4×C199C2311yescharacteristic
47C42×C199—24no—
48C42×C199—22no—
49C42×C199D414yescomplemented
50C22×D199—21no—
51Dic398—22no—
52D796—22no—
53C22⋊D199—44no—
54C22⋊D199—41no—
55C22⋊D199—41no—
56C2×Dic199—22no—
57C2×Dic199—21no—
58C4×D199—42no—
59C199⋊C8—42no—
60C23×C199—24no—
61C23×C199—21no—
62C23×C199—21no—
63C23×C199C22⋊C412yes—
64C23×C199C2×D411yescharacteristic
65C2×C4×C199—216no—
66C2×C4×C199—28no—
67C2×C4×C199—21no—
68C2×C4×C199—24no—
69C2×C4×C199—21no—
70C2×C4×C199—24no—
71C2×C4×C199—24no—
72C2×C4×C199C22⋊C412yes—
73C2×C4×C199C22×C411yescharacteristic
74C2×C4×C199C2×D412yes—
75C2×C4×C199C2×D411yescharacteristic
76C2×C4×C199#4112yes—
77C22×C199—24no—
78C22×C199—24no—
79C22×C199—22no—
80C22×C199—22no—
81C22×C199—22no—
82C22×C199#10414yescomplemented
83C22×C199C2×C22⋊C411yescharacteristic
84C22×C199#12011yescharacteristic
85C22×C199#12311yescharacteristic
86C2×D199—41no—
87C2×D199—42no—
88C4×C199—28no—
89C4×C199—22no—
90C4×C199C4×D412yes—
91C4×C199#12112yes—
92Dic199—41no—
93Dic199—42no—
94C2×C199—21no—
95C2×C199—24no—
96C2×C199—21no—
97C2×C199#38611yescharacteristic
98C2×C199C2×#10412yes—
99D199—81no—
100C199#189611yescharacteristic, complemented, Sylow
101#1896—1991nomaximal, Sylow
102#525—1991no—
103C22×C42D19911yescharacteristic, complemented
104C2×#104—1992no—
105#407—1991no—
106#339—1992no—
107C2×#41—1991no—
108C23×C4—21no—
109C23×C4C2×D19911yescharacteristic
110C2×M16—1992no—
111#123—3981no—
112C22⋊Q8—1992no—
113#121—1992no—
114#120—3981no—
115C4×D4—3982no—
116#117—1991no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C2×D19912yes—
120#104—3984no—
121C22⋊C8—3982no—
122C24C22⋊D19911yescharacteristic
123#41—3982no—
124#41—3982no—
125C2×Q8—1991no—
126C2×D4—3982no—
127C2×D4—3981no—
128C2×D4—1991no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—3981no—
135C22×C4C4×D19912yes—
136C22×C4C22⋊D19911yescharacteristic
137C22×C4D79612yes—
138C22×C4C22×D19911yescharacteristic
139M16—3982no—
140M16—3982no—
141C2×C8—3982no—
142C4⋊C4—3982no—
143C4⋊C4—1992no—
144C22⋊C4—3982no—
145C22⋊C4—3981no—
146C22⋊C4—3982no—
147C42—24no—
148C42—22no—
149C42—3981no—
150C42C22⋊D19914yescomplemented
151C23—21no—
152C23—24no—
153C23—21no—
154C23—3981no—
155C23(C2×D199)⋊C412yes—
156C23C2×C22⋊D19911yescharacteristic
157Q8—3982no—
158D4—7961no—
159D4—7964no—
160D4—7961no—
161D4—3982no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—7962no—
170C2×C4—3982no—
171C2×C4—3981no—
172C2×C4(C2×D199)⋊C412yes—
173C2×C4C2×C4×D19911yescharacteristic
174C2×C4C2×C22⋊D19912yes—
175C2×C4C2×D79611yescharacteristic
176C2×C4#1569188612yes—
177C8—7962no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—7962no—
184C22—7961no—
185C22#2026388314yescomplemented
186C22C2×(C2×D199)⋊C411yescharacteristic
187C22#2026392411yescharacteristic
188C22#2026393811yescharacteristic
189C4—28no—
190C4—22no—
191C4—7962no—
192C4—7961no—
193C4C4×C22⋊D19912yes—
194C4#2026392912yes—
195C2—21no—
196C2—21no—
197C2—24no—
198C2—15921no—
199C2#2716389711yescharacteristic
200C2C2×#2026388312yes—
subgroupG/H
centerC2×C4C2×D796
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC22×C42×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164529C2
cosocleC22×C4×C199C23
hypercenterC22×C42D199
hypocenterC199#1896
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#20263883C2semidirect
3. #27163982C2semidirect
4. #27163778C2semidirect
5. C22×C42×C199C2semidirect
9. #20263934C4semidirect
10. #20263929C4semidirect
23. C2×Dic398C2×C4semidirect
24. C2×D796C2×C4semidirect
31. C4⋊Dic199C2×C4semidirect
49. C42×C199D4semidirect
82. C22×C199#104semidirect
100. C199#1896semidirect
103. C22×C42D199semidirect
150. C42C22⋊D199semidirect
185. C22#20263883semidirect

Maximal subgroups

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

subgroupindex
#271645292
C2×#202638832
#271639822
#271637782
C22×C42×C1992
#1896199