Finite Group Databasebeta

← #38769290

Subgroups — #38769290

Order 25472.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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#27164525C211yesmaximal, characteristic
2C2×C42×D199C211yesmaximal, characteristic, complemented
3C2×#20263882C211yesmaximal, characteristic, complemented
4#27163981C211yesmaximal, characteristic, complemented
5#27163779C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7C2×#104×C199C211yesmaximal, characteristic, complemented
8C22×C4×D199—21no—
9C22×C4×D199C2211yescharacteristic
10C2×#15691870C2211yescharacteristic
11#20263936—21no—
12#20263931C412yescomplemented
13#20263926C412yescomplemented
14#20263923—21no—
15C4×C22⋊D199—22no—
16#20263914C2211yescharacteristic
17C2×C4×Dic199—21no—
18C2×C4×Dic199C2211yescharacteristic
19C42×D199—22no—
20C42×D199—22no—
21#20263882—22no—
22(C2×D199)⋊C8—22no—
23C2×#41×C199C2211yescharacteristic
24C2×M16×C199C2211yescharacteristic
25C2×C42×C199C2211yescharacteristic
26#104×C199—22no—
27C23×D199D411yescharacteristic
28C2×C22⋊D199—22no—
29C22×Dic199—21no—
30C22×Dic199D411yescharacteristic
31C2×C4×D199—28no—
32C2×C4×D199—24no—
33C2×C4×D199—21no—
34C2×C4×D199—21no—
35C2×C4×D199—21no—
36C2×C4×D199—21no—
37C2×C4×D199C2×C412yes—
38C2×C4×D199D412yes—
39#15691870—21no—
40#15691870—21no—
41C2×C199⋊C8—21no—
42C4⋊Dic199C2×C412yescomplemented
43Dic199⋊C4—22no—
44C22⋊Dic199—22no—
45C22⋊Dic199—21no—
46(C2×D199)⋊C4—22no—
47C4×Dic199—24no—
48C4×Dic199—21no—
49C4×Dic199—21no—
50C4×Dic199D412yescomplemented
51#41×C199—21no—
52#41×C199—21no—
53#41×C199—21no—
54#41×C199—21no—
55C2×Q8×C199C2×C411yescharacteristic, complemented
56C2×D4×C199—21no—
57C2×D4×C199C2×C411yescharacteristic, complemented
58C22×C4×C199—21no—
59C22×C4×C199—21no—
60C22×C4×C199C2311yescharacteristic
61M16×C199—21no—
62M16×C199—21no—
63C2×C8×C199—21no—
64C42×C199—21no—
65C42×C199D412yescomplemented
66C22×D199—21no—
67C22×D199—21no—
68C22×D199—24no—
69C22×D199C22⋊C412yes—
70C22⋊D199—44no—
71C2×Dic199—24no—
72C2×Dic199—21no—
73C2×Dic199—22no—
74C2×Dic199—21no—
75C2×Dic199—22no—
76C2×Dic199C22⋊C412yes—
77C4×D199—216no—
78C4×D199—22no—
79C4×D199—22no—
80C4×D199—22no—
81C4×D199—22no—
82C199⋊C8—41no—
83C23×C199—21no—
84C23×C199C2×D411yescharacteristic
85Q8×C199—21no—
86Q8×C199—21no—
87D4×C199—21no—
88D4×C199—21no—
89D4×C199—41no—
90D4×C199—41no—
91C2×C4×C199—24no—
92C2×C4×C199—22no—
93C2×C4×C199—21no—
94C2×C4×C199—41no—
95C2×C4×C199—41no—
96C2×C4×C199C22×C411yescharacteristic
97C2×C4×C199C2×D411yescharacteristic
98C2×C4×C199C2×D411yescharacteristic
99C2×C4×C199C2×D411yescharacteristic
100C2×C4×C199#4111yescharacteristic
101C2×C4×C199#4111yescharacteristic
102C8×C199—41no—
103C22×C199—22no—
104C22×C199—41no—
105C22×C199—42no—
106C22×C199C2×C22⋊C411yescharacteristic
107C22×C199#12011yescharacteristic
108C22×C199#12311yescharacteristic
109C2×D199—24no—
110C2×D199—24no—
111C2×D199—22no—
112C2×D199—22no—
113C2×D199#10414yescomplemented
114C4×C199—24no—
115C4×C199—41no—
116C4×C199C4×D411yescharacteristic
117C4×C199C4×D411yescharacteristic
118C4×C199#12111yescharacteristic
119C4×C199#12111yescharacteristic
120Dic199—24no—
121Dic199—22no—
122Dic199—42no—
123C2×C199—21no—
124C2×C199—21no—
125C2×C199—81no—
126C2×C199#38611yescharacteristic
127C2×C199C2×#10412yes—
128D199—24no—
129C199#189611yescharacteristic, complemented, Sylow
130#1896—1991nomaximal, Sylow
131#525—1991no—
132C22×C42—1991no—
133C2×#104—1991no—
134C2×#104D19911yescharacteristic, complemented
135#407—1991no—
136#339—1991no—
137#339—1991no—
138C2×#41C2×D19911yescharacteristic
139C23×C4—3981no—
140C23×C4—1991no—
141C2×M16—1991no—
142C2×M16C2×D19911yescharacteristic
143#123—3981no—
144C22⋊Q8—1992no—
145#121—1992no—
146#120—3981no—
147C4×D4—3982no—
148#117—1991no—
149C2×C42—3982no—
150C2×C42—3981no—
151C2×C42—3982no—
152C2×C42—1991no—
153C2×C42C2×D19911yescharacteristic
154#104—22no—
155#104—3982no—
156C22⋊C8—3982no—
157C24—1991no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—21no—
162C2×Q8C4×D19911yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—3982no—
165C2×D4C4×D19911yescharacteristic, complemented
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—3984no—
169C22×C4—3988no—
170C22×C4—3981no—
171C22×C4—3981no—
172C22×C4—3981no—
173C22×C4—3981no—
174C22×C4—3981no—
175C22×C4—1991no—
176C22×C4—1992no—
177C22×C4—1992no—
178C22×C4C22×D19911yescharacteristic
179M16—21no—
180M16—21no—
181M16—3981no—
182M16—3981no—
183C2×C8—21no—
184C2×C8—3981no—
185C4⋊C4—3982no—
186C4⋊C4—1992no—
187C22⋊C4—3982no—
188C22⋊C4—3981no—
189C22⋊C4—3982no—
190C42—21no—
191C42—3984no—
192C42—3981no—
193C42—3981no—
194C42—1992no—
195C42C22⋊D19912yescomplemented
196C23—21no—
197C23—3984no—
198C23—3981no—
199C23—3981no—
200C23—1992no—
201C23D4×D19911yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—7964no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—39816no—
215C2×C4—3982no—
216C2×C4—3984no—
217C2×C4—3981no—
218C2×C4—3981no—
219C2×C4—3982no—
220C2×C4—3982no—
221C2×C4—3982no—
222C2×C4—3982no—
223C2×C4—3982no—
224C2×C4—1992no—
225C2×C4C2×C4×D19911yescharacteristic
226C2×C4D4×D19911yescharacteristic
227C2×C4D4×D19911yescharacteristic
228C2×C4C2×C22⋊D19911yescharacteristic
229C2×C4#1569188511yescharacteristic
230C2×C4#1569188611yescharacteristic
231C8—41no—
232C8—7961no—
233C22—22no—
234C22—42no—
235C22—41no—
236C22—3984no—
237C22—3984no—
238C22—3982no—
239C22—3982no—
240C22—1994no—
241C22C22⋊C4×D19911yescharacteristic
242C22#2026392311yescharacteristic
243C22#2026393711yescharacteristic
244C4—24no—
245C4—41no—
246C4—7962no—
247C4—3984no—
248C4—3982no—
249C4(C22⋊D199)⋊C411yescharacteristic
250C4C4×C22⋊D19911yescharacteristic
251C4#2026392611yescharacteristic
252C4#2026392811yescharacteristic
253C2—21no—
254C2—21no—
255C2—81no—
256C2—3984no—
257C2#2716389411yescharacteristic
258C2#104×D19912yes—
subgroupG/H
centerC2×C4D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×#104×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164525C2
cosocleC22×C4×C199C23
hypercenterC2×#104D199
hypocenterC199#1896
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×C42×D199C2semidirect
3. C2×#20263882C2semidirect
4. #27163981C2semidirect
5. #27163779C2semidirect
6. #27163778C2semidirect
7. C2×#104×C199C2semidirect
12. #20263931C4semidirect
13. #20263926C4semidirect
42. C4⋊Dic199C2×C4semidirect
50. C4×Dic199D4semidirect
55. C2×Q8×C199C2×C4semidirect
57. C2×D4×C199C2×C4semidirect
65. C42×C199D4semidirect
113. C2×D199#104semidirect
129. C199#1896semidirect
134. C2×#104D199semidirect
162. C2×Q8C4×D199semidirect
165. C2×D4C4×D199semidirect
195. C42C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271645252
C2×C42×D1992
C2×#202638822
#271639812
#271637792
#271637782
C2×#104×C1992
#1896199