Finite Group Databasebeta

← #38767355

Subgroups — #38767355

Order 25472.

334 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 77 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×C199⋊#95C211yesmaximal, characteristic, complemented
3#27163756C212yesmaximal, complemented
4#27163753C212yesmaximal, complemented
5#407×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C411yescharacteristic, complemented
7C23×Dic199—21no—
8#20263938—21no—
9#20263934C411yescharacteristic, complemented
10#20263934C2211yescharacteristic
11#20263929C411yescharacteristic, complemented
12#20263929C2211yescharacteristic
13#20263924—21no—
14C4×C22⋊D199—21no—
15C4×C22⋊D199—21no—
16#20263914C411yescharacteristic, complemented
17C199⋊#95—21no—
18C199⋊#95C2212yes—
19C23×C4×C199C2211yescharacteristic
20C2×M16×C199—21no—
21C22⋊C8×C199—21no—
22C22⋊C8×C199C2212yes—
23#15691886—42no—
24C2×Dic398C2×C411yescharacteristic, complemented
25C2×D796C2×C411yescharacteristic, complemented
26C2×C22⋊D199—21no—
27C2×C22⋊D199—21no—
28C2×C22⋊D199—21no—
29C22×Dic199—24no—
30C22×Dic199—22no—
31C22×Dic199—42no—
32C22×Dic199—42no—
33C2×C4×D199—21no—
34C4⋊Dic199C2×C411yescharacteristic, complemented
35C4⋊Dic199C2×C411yescharacteristic, complemented
36Dic199⋊C4—21no—
37Dic199⋊C4—21no—
38C22⋊Dic199—21no—
39C22⋊Dic199—21no—
40C22⋊Dic199—21no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43C4×Dic199—21no—
44C24×C199D411yescharacteristic
45C22×C4×C199—42no—
46C22×C4×C199C2×C411yescharacteristic
47C22×C4×C199C2×C411yescharacteristic
48C22×C4×C199D411yescharacteristic
49C22×C4×C199D411yescharacteristic
50C22×C4×C199D411yescharacteristic
51C22×C4×C199C2311yescharacteristic
52M16×C199—42no—
53C2×C8×C199—22no—
54C22×D199—21no—
55Dic398—41no—
56D796—41no—
57C22⋊D199—42no—
58C22⋊D199—42no—
59C22⋊D199—41no—
60C22⋊D199—41no—
61C2×Dic199—24no—
62C2×Dic199—21no—
63C2×Dic199—21no—
64C2×Dic199—21no—
65C2×Dic199—48no—
66C2×Dic199—42no—
67C2×Dic199—42no—
68C4×D199—42no—
69C23×C199—22no—
70C23×C199—22no—
71C23×C199—21no—
72C23×C199—21no—
73C23×C199C22⋊C411yescharacteristic
74C23×C199C22⋊C411yescharacteristic
75C23×C199C2×D411yescharacteristic
76C2×C4×C199—22no—
77C2×C4×C199—22no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199—41no—
81C2×C4×C199—41no—
82C2×C4×C199C22⋊C411yescharacteristic
83C2×C4×C199C22⋊C411yescharacteristic
84C2×C4×C199C22×C411yescharacteristic
85C2×C4×C199C2×D411yescharacteristic
86C8×C199—42no—
87C22×C199—24no—
88C22×C199—24no—
89C22×C199—21no—
90C22×C199—21no—
91C22×C199—21no—
92C22×C199—21no—
93C22×C199—21no—
94C22×C199—21no—
95C22×C199#9912yes—
96C22×C199#10412yes—
97C22×C199C2×C22⋊C411yescharacteristic
98C22×C199AGL(1,ℤ/8)11yescharacteristic
99C22×C199#13711yescharacteristic
100C2×D199—41no—
101C2×D199—41no—
102C2×D199—41no—
103C4×C199—21no—
104C4×C199—21no—
105C4×C199—21no—
106C4×C199—21no—
107Dic199—44no—
108Dic199—41no—
109Dic199—41no—
110Dic199—41no—
111C2×C199—21no—
112C2×C199—22no—
113C2×C199—22no—
114C2×C199—21no—
115C2×C199C2×#9911yescharacteristic
116C2×C199#41711yescharacteristic
117C2×C199C2×#10411yescharacteristic
118D199—81no—
119C199#150411yescharacteristic, complemented, Sylow
120#1504—1991nomaximal, Sylow
121#525—1991no—
122#407D19911yescharacteristic, complemented
123C2×#95—1991no—
124#328—1992no—
125#327—1992no—
126C2×#41—1991no—
127C23×C4—3981no—
128C23×C4C2×D19911yescharacteristic
129C2×M16—21no—
130#123—3981no—
131C22⋊Q8—1991no—
132C22⋊Q8—1991no—
133#121—1991no—
134#121—1991no—
135#120—3981no—
136C4×D4—3981no—
137C4×D4—3981no—
138#117—1991no—
139C22⋊C8—21no—
140C22⋊C8C2×D19912yes—
141#95—3981no—
142#95—1992no—
143C24D79611yescharacteristic
144#41—7962no—
145C2×Q8—1991no—
146C2×D4—3981no—
147C2×D4—3981no—
148C2×D4—3981no—
149C2×D4—1991no—
150C22×C4—42no—
151C22×C4—7962no—
152C22×C4—7962no—
153C22×C4—3984no—
154C22×C4—3982no—
155C22×C4—3981no—
156C22×C4C4×D19911yescharacteristic
157C22×C4C4×D19911yescharacteristic
158C22×C4C22⋊D19911yescharacteristic
159C22×C4C22⋊D19911yescharacteristic
160C22×C4D79611yescharacteristic
161C22×C4C22×D19911yescharacteristic
162M16—42no—
163C2×C8—22no—
164C4⋊C4—3981no—
165C4⋊C4—3981no—
166C4⋊C4—1991no—
167C4⋊C4—1991no—
168C22⋊C4—3981no—
169C22⋊C4—3981no—
170C22⋊C4—3981no—
171C22⋊C4—3981no—
172C22⋊C4—3981no—
173C42—3981no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—3981no—
179C23(C2×D199)⋊C411yescharacteristic
180C23(C2×D199)⋊C411yescharacteristic
181C23C2×D79611yescharacteristic
182Q8—7961no—
183D4—7961no—
184D4—7961no—
185D4—7961no—
186D4—7962no—
187D4—7962no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—7962no—
195C2×C4—7968no—
196C2×C4—7962no—
197C2×C4—7962no—
198C2×C4—3984no—
199C2×C4—3981no—
200C2×C4—3981no—
201C2×C4—3981no—
202C2×C4(C2×D199)⋊C411yescharacteristic
203C2×C4(C2×D199)⋊C411yescharacteristic
204C2×C4C2×C4×D19911yescharacteristic
205C2×C4C2×C22⋊D19911yescharacteristic
206C8—42no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—7961no—
216C22—7961no—
217C22—7961no—
218C22#2026386912yes—
219C22#2026388112yes—
220C22C2×(C2×D199)⋊C411yescharacteristic
221C22#2026398311yescharacteristic
222C22#2026398811yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—7964no—
228C4—7961no—
229C4—7961no—
230C4—7961no—
231C2—21no—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—15921no—
236C2C2×#2026386911yescharacteristic
237C2#2716402011yescharacteristic
238C2C2×#2026388111yescharacteristic
subgroupG/H
centerC22C2×(C2×D199)⋊C4
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#407×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164529C2
cosocleC22×C4×C199C23
hypercenter#407D199
hypocenterC199#1504
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×C199⋊#95C2semidirect
3. #27163756C2semidirect
4. #27163753C2semidirect
5. #407×C199C2semidirect
6. C2×#15691886C4semidirect
9. #20263934C4semidirect
11. #20263929C4semidirect
16. #20263914C4semidirect
24. C2×Dic398C2×C4semidirect
25. C2×D796C2×C4semidirect
34. C4⋊Dic199C2×C4semidirect
35. C4⋊Dic199C2×C4semidirect
119. C199#1504semidirect
122. #407D199semidirect

Maximal subgroups

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

subgroupindex
#271645292
C2×C199⋊#952
#271637562
#271637532
#407×C1992
#1504199