Finite Group Databasebeta

← #38769878

Subgroups — #38769878

Order 25472.

394 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20263985C211yesmaximal, characteristic, complemented
2#27164582C211yesmaximal, characteristic
3C2×#20263943C211yesmaximal, characteristic, complemented
4#27164028C211yesmaximal, characteristic, complemented
5C2×#20263871C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7C2×#104×C199C211yesmaximal, characteristic, complemented
8C22×Dic398C2211yescharacteristic
9C22×D796C2211yescharacteristic
10#20263985—22no—
11#20263985—22no—
12C2×#15691876—21no—
13C2×#15691874—21no—
14C2×#15691870C2211yescharacteristic
15#20263951—21no—
16#20263944—21no—
17#20263943—22no—
18#20263943—22no—
19#20263929—22no—
20#20263923—22no—
21#20263914C2211yescharacteristic
22C2×(C2×D199)⋊C4—22no—
23#20263880—22no—
24#20263871—22no—
25C2×#41×C199C2211yescharacteristic
26C2×M16×C199C2211yescharacteristic
27C2×C42×C199C2211yescharacteristic
28#104×C199—22no—
29C23×D199D412yes—
30C2×Dic398—21no—
31C2×Dic398—21no—
32C2×Dic398—21no—
33C2×Dic398—21no—
34C2×Dic398—21no—
35C2×Dic398D412yescomplemented
36C2×D796—21no—
37C2×D796—21no—
38C2×D796—21no—
39C2×D796—21no—
40C2×D796—21no—
41C2×D796—21no—
42C2×C22⋊D199—24no—
43C22×Dic199—21no—
44#15691876—24no—
45#15691874—24no—
46#15691870—21no—
47#15691870D412yescomplemented
48C2×C199⋊C8—21no—
49C4⋊Dic199D412yes—
50C22⋊Dic199—21no—
51(C2×D199)⋊C4—28no—
52(C2×D199)⋊C4—24no—
53C4×Dic199—21no—
54#41×C199—21no—
55#41×C199—21no—
56#41×C199—21no—
57#41×C199—21no—
58C2×Q8×C199D411yescharacteristic
59C2×D4×C199—21no—
60C2×D4×C199D411yescharacteristic
61C22×C4×C199—21no—
62C22×C4×C199—21no—
63C22×C4×C199C2311yescharacteristic
64M16×C199—21no—
65M16×C199—21no—
66C2×C8×C199—21no—
67C42×C199—21no—
68C42×C199D412yescomplemented
69C22×D199—22no—
70C22×D199—22no—
71C22×D199—22no—
72C22×D199—44no—
73Dic398—22no—
74Dic398—22no—
75Dic398—41no—
76Dic398—41no—
77D796—41no—
78D796—41no—
79D796—41no—
80D796—41no—
81C22⋊D199—84no—
82C2×Dic199—22no—
83C2×Dic199—22no—
84C2×Dic199—41no—
85C2×Dic199—41no—
86C199⋊C8—22no—
87C23×C199—21no—
88C23×C199C2×D411yescharacteristic
89Q8×C199—21no—
90Q8×C199—21no—
91D4×C199—21no—
92D4×C199—21no—
93D4×C199—41no—
94D4×C199—41no—
95C2×C4×C199—22no—
96C2×C4×C199—21no—
97C2×C4×C199—42no—
98C2×C4×C199—41no—
99C2×C4×C199—41no—
100C2×C4×C199C2×D411yescharacteristic
101C2×C4×C199C2×D411yescharacteristic
102C2×C4×C199C2×D411yescharacteristic
103C2×C4×C199C2×D411yescharacteristic
104C2×C4×C199C2×D411yescharacteristic
105C2×C4×C199#4111yescharacteristic
106C8×C199—41no—
107C22×C199—22no—
108C22×C199—41no—
109C22×C199—42no—
110C22×C199#12011yescharacteristic
111C22×C199#12011yescharacteristic
112C22×C199#12111yescharacteristic
113C2×D199—44no—
114C2×D199—42no—
115C2×D199—44no—
116C2×D199—82no—
117C4×C199—42no—
118C4×C199—41no—
119C4×C199#12011yescharacteristic
120C4×C199#12111yescharacteristic
121C4×C199#12111yescharacteristic
122C4×C199#12711yescharacteristic
123Dic199—42no—
124Dic199—42no—
125C2×C199—21no—
126C2×C199—21no—
127C2×C199—81no—
128C2×C199#39211yescharacteristic
129C2×C199#45512yes—
130D199—82no—
131C199#200411yescharacteristic, complemented, Sylow
132#2004—1991nomaximal, Sylow
133C2×#137—1991no—
134#535—1991no—
135C2×#124—1991no—
136C2×#104D19911yescharacteristic, complemented
137#419—1991no—
138C2×#100—1991no—
139#339—1991no—
140C2×#41C2×D19911yescharacteristic
141C22×Q8—1991no—
142C22×D4—1991no—
143#137—3982no—
144#137—3982no—
145C2×Q16—3981no—
146C2×SD16—3981no—
147C2×M16—1991no—
148C2×M16C2×D19911yescharacteristic
149#127—3981no—
150#124—3982no—
151#124—3981no—
152#124—3982no—
153#121—3982no—
154#120—3982no—
155#117—1991no—
156C2×C22⋊C4—3982no—
157C2×C42C2×D19911yescharacteristic
158#104—22no—
159#103—3982no—
160#100—3982no—
161C24—1992no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—3981no—
167C2×Q8—3981no—
168C2×Q8—3981no—
169C2×Q8—3981no—
170C2×Q8—3981no—
171C2×Q8—1992no—
172C2×Q8D79611yescharacteristic
173C2×D4—21no—
174C2×D4—3984no—
175C2×D4—3981no—
176C2×D4—3981no—
177C2×D4—3981no—
178C2×D4—3981no—
179C2×D4—3981no—
180C2×D4—3981no—
181C2×D4D79611yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—3981no—
185C22×C4C22×D19911yescharacteristic
186Q16—3984no—
187SD16—3984no—
188M16—21no—
189M16—21no—
190M16—3981no—
191M16—1992no—
192C2×C8—21no—
193C2×C8—3981no—
194C4⋊C4—1992no—
195C22⋊C4—3988no—
196C22⋊C4—3984no—
197C22⋊C4—3981no—
198C42—21no—
199C42—3981no—
200C42C22⋊D19912yescomplemented
201C23—21no—
202C23—7964no—
203C23—3982no—
204C23—3982no—
205C23—3982no—
206C23D4×D19911yescharacteristic
207Q8—21no—
208Q8—21no—
209Q8—7961no—
210Q8—7961no—
211Q8—3982no—
212Q8—3982no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—7961no—
218D4—7961no—
219D4—7961no—
220D4—7961no—
221D4—15924no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—7961no—
228C2×C4—7961no—
229C2×C4—3982no—
230C2×C4—3982no—
231C2×C4D4×D19911yescharacteristic
232C2×C4D4×D19911yescharacteristic
233C2×C4D4×D19911yescharacteristic
234C2×C4C2×C22⋊D19911yescharacteristic
235C2×C4C2×D79611yescharacteristic
236C2×C4#1569188611yescharacteristic
237C8—41no—
238C8—3982no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—7964no—
243C22—7962no—
244C22—7964no—
245C22—15922no—
246C22#2026392311yescharacteristic
247C22#2026392511yescharacteristic
248C22#2026392811yescharacteristic
249C4—42no—
250C4—41no—
251C4—7962no—
252C4—7962no—
253C4#2026392511yescharacteristic
254C4#2026392811yescharacteristic
255C4#2026392911yescharacteristic
256C4#2026395111yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—15922no—
261C2#2716392711yescharacteristic
262C2#2716419712yes—
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
FittingC2×#104×C199C2
socleC22×C199#120
prime radical#27164582C2
hypercenterC2×#104D199
hypocenterC199#2004
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#20263985C2semidirect
3. C2×#20263943C2semidirect
4. #27164028C2semidirect
5. C2×#20263871C2semidirect
6. #27163778C2semidirect
7. C2×#104×C199C2semidirect
35. C2×Dic398D4semidirect
47. #15691870D4semidirect
68. C42×C199D4semidirect
131. C199#2004semidirect
136. C2×#104D199semidirect
200. C42C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639852
#271645822
C2×#202639432
#271640282
C2×#202638712
#271637782
C2×#104×C1992
#2004199