Finite Group Databasebeta

← #38769879

Subgroups — #38769879

Order 25472.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20263986C211yesmaximal, characteristic, complemented
2#27164585C211yesmaximal, characteristic
3C2×#20263883C211yesmaximal, characteristic, complemented
4#27164027C211yesmaximal, characteristic, complemented
5C2×#20263872C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7C2×#124×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9#20263986—22no—
10#20263986—22no—
11C2×#15691876—21no—
12C2×#15691873—21no—
13C2×#15691870C2211yescharacteristic
14C2×#15691870C2211yescharacteristic
15#20263951—21no—
16#20263942—21no—
17#20263926—22no—
18#20263924—22no—
19#20263914C2211yescharacteristic
20#20263883—22no—
21#20263879—22no—
22#20263872—22no—
23C22×Q8×C199C2211yescharacteristic
24C22×D4×C199C2211yescharacteristic
25#124×C199—22no—
26#124×C199—22no—
27C2×C22⋊C4×C199—22no—
28C2×C42×C199C2211yescharacteristic
29#15691886—21no—
30#15691886—21no—
31#15691886—21no—
32#15691886—21no—
33C2×Dic398D411yescharacteristic
34C2×D796D411yescharacteristic
35C2×C22⋊D199—24no—
36C2×C22⋊D199—21no—
37C2×C4×D199—21no—
38#15691876—24no—
39#15691873—24no—
40#15691870—21no—
41#15691870—21no—
42#15691870—21no—
43#15691870D412yescomplemented
44C2×C199⋊C8—21no—
45C2×C199⋊C8—21no—
46C4⋊Dic199D412yes—
47C22⋊Dic199—24no—
48C22⋊Dic199—21no—
49C4×Dic199—21no—
50C24×C199D412yes—
51C2×Q8×C199—21no—
52C2×Q8×C199—21no—
53C2×Q8×C199—21no—
54C2×Q8×C199—21no—
55C2×Q8×C199—21no—
56C2×Q8×C199D412yescomplemented
57C2×D4×C199—21no—
58C2×D4×C199—21no—
59C2×D4×C199—21no—
60C2×D4×C199—21no—
61C2×D4×C199—21no—
62C2×D4×C199—21no—
63C22×C4×C199—21no—
64C22×C4×C199—21no—
65C22×C4×C199C2311yescharacteristic
66C22⋊C4×C199—28no—
67C42×C199—21no—
68C42×C199D412yescomplemented
69C22×D199—21no—
70Dic398—21no—
71Dic398—21no—
72D796—21no—
73D796—21no—
74C22⋊D199—41no—
75C22⋊D199—41no—
76C22⋊D199—84no—
77C2×Dic199—22no—
78C2×Dic199—21no—
79C4×D199—41no—
80C4×D199—41no—
81C199⋊C8—22no—
82C199⋊C8—41no—
83C23×C199—22no—
84C23×C199—22no—
85C23×C199—22no—
86C23×C199—44no—
87C23×C199C2×D411yescharacteristic
88Q8×C199—22no—
89Q8×C199—22no—
90Q8×C199—41no—
91Q8×C199—41no—
92D4×C199—41no—
93D4×C199—41no—
94D4×C199—41no—
95D4×C199—41no—
96C2×C4×C199—22no—
97C2×C4×C199—22no—
98C2×C4×C199—42no—
99C2×C4×C199—41no—
100C2×C4×C199—41no—
101C2×C4×C199C2×D411yescharacteristic
102C2×C4×C199C2×D411yescharacteristic
103C2×C4×C199C2×D411yescharacteristic
104C2×C4×C199C2×D411yescharacteristic
105C2×C4×C199C2×D411yescharacteristic
106C2×C4×C199#4111yescharacteristic
107C22×C199—22no—
108C22×C199—44no—
109C22×C199—42no—
110C22×C199—44no—
111C22×C199—82no—
112C22×C199#12011yescharacteristic
113C22×C199#12011yescharacteristic
114C22×C199#12111yescharacteristic
115C2×D199—41no—
116C2×D199—42no—
117C4×C199—42no—
118C4×C199—42no—
119C4×C199#12011yescharacteristic
120C4×C199#12111yescharacteristic
121C4×C199#12111yescharacteristic
122C4×C199#12711yescharacteristic
123Dic199—41no—
124Dic199—42no—
125C2×C199—21no—
126C2×C199—21no—
127C2×C199—82no—
128C2×C199#39211yescharacteristic
129C2×C199#45512yes—
130D199—81no—
131C199#200411yescharacteristic, complemented, Sylow
132#2004—1991nomaximal, Sylow
133C2×#137—1991no—
134#535—1991no—
135C2×#124D19911yescharacteristic, complemented
136C2×#104—1991no—
137#419—1991no—
138C2×#100—1991no—
139#339—1991no—
140C2×#41—1991no—
141C22×Q8C2×D19911yescharacteristic
142C22×D4C2×D19911yescharacteristic
143#137—3982no—
144#137—3982no—
145C2×Q16—3981no—
146C2×SD16—3981no—
147C2×M16—1991no—
148C2×M16—1991no—
149#127—3981no—
150#124—22no—
151#124—22no—
152#124—3981no—
153#121—3982no—
154#120—3982no—
155#117—1991no—
156C2×C22⋊C4—22no—
157C2×C42C2×D19911yescharacteristic
158#104—3982no—
159#103—3982no—
160#100—3982no—
161C24C22⋊D19912yes—
162#41—3981no—
163#41—3981no—
164#41—3981no—
165#41—3981no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1991no—
172C2×Q8C22⋊D19912yescomplemented
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—3981no—
180C2×D4—3984no—
181C2×D4—1991no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—3981no—
185C22×C4C22×D19911yescharacteristic
186Q16—3984no—
187SD16—3984no—
188M16—3981no—
189M16—3981no—
190M16—3981no—
191M16—1992no—
192C2×C8—3981no—
193C2×C8—3981no—
194C4⋊C4—1992no—
195C22⋊C4—28no—
196C22⋊C4—3984no—
197C22⋊C4—3981no—
198C42—21no—
199C42—3981no—
200C42C22⋊D19912yescomplemented
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—3981no—
206C23C2×C22⋊D19911yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—3981no—
212Q8—3981no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—7961no—
218D4—7961no—
219D4—15924no—
220D4—3981no—
221D4—3981no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—7961no—
228C2×C4—7961no—
229C2×C4—3982no—
230C2×C4—3981no—
231C2×C4D4×D19911yescharacteristic
232C2×C4D4×D19911yescharacteristic
233C2×C4D4×D19911yescharacteristic
234C2×C4C2×C22⋊D19911yescharacteristic
235C2×C4C2×C22⋊D19911yescharacteristic
236C2×C4#1569188511yescharacteristic
237C8—7961no—
238C8—3982no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—7961no—
245C22—7962no—
246C22#2026392311yescharacteristic
247C22#2026392411yescharacteristic
248C22#2026392711yescharacteristic
249C4—42no—
250C4—42no—
251C4—7961no—
252C4—7962no—
253C4#2026392311yescharacteristic
254C4#2026392611yescharacteristic
255C4#2026392711yescharacteristic
256C4#2026395111yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—15921no—
261C2#2716392611yescharacteristic
262C2#2716419512yes—
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
FittingC2×#124×C199C2
socleC22×C199#120
prime radical#27164585C2
hypercenterC2×#124D199
hypocenterC199#2004
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#20263986C2semidirect
3. C2×#20263883C2semidirect
4. #27164027C2semidirect
5. C2×#20263872C2semidirect
6. #27163778C2semidirect
7. C2×#124×C199C2semidirect
43. #15691870D4semidirect
56. C2×Q8×C199D4semidirect
68. C42×C199D4semidirect
131. C199#2004semidirect
135. C2×#124D199semidirect
172. C2×Q8C22⋊D199semidirect
200. C42C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639862
#271645852
C2×#202638832
#271640272
C2×#202638722
#271637782
C2×#124×C1992
#2004199