Finite Group Databasebeta

← #38769876

Subgroups — #38769876

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
1#27164582C211yesmaximal, characteristic
2C2×#20263944C211yesmaximal, characteristic, complemented
3C2×#20263882C211yesmaximal, characteristic, complemented
4#27164027C211yesmaximal, characteristic, complemented
5C2×#20263871C211yesmaximal, characteristic, complemented
6#27163779C211yesmaximal, characteristic, complemented
7C2×#137×C199C211yesmaximal, characteristic, complemented
8C22×D796C2211yescharacteristic
9C2×#15691870C2211yescharacteristic
10#20263951—21no—
11#20263944—22no—
12#20263944—21no—
13#20263944—22no—
14#20263929—22no—
15#20263923—22no—
16#20263914C2211yescharacteristic
17C2×(C2×D199)⋊C4—22no—
18C2×C4×Dic199C2211yescharacteristic
19#20263882—22no—
20#20263879—22no—
21#20263871—22no—
22C2×#41×C199C2211yescharacteristic
23C22×Q8×C199C2211yescharacteristic
24#137×C199—22no—
25#137×C199—22no—
26C2×Q16×C199—21no—
27C2×SD16×C199—21no—
28C2×M16×C199C2211yescharacteristic
29C23×D199D412yes—
30C2×D796—21no—
31C2×D796—21no—
32C2×D796—21no—
33C2×D796—21no—
34C2×D796—21no—
35C2×D796—21no—
36C2×C22⋊D199—24no—
37C22×Dic199—21no—
38#15691870—21no—
39#15691870—21no—
40C2×C199⋊C8—21no—
41C4⋊Dic199D412yes—
42C22⋊Dic199—21no—
43(C2×D199)⋊C4—28no—
44(C2×D199)⋊C4—24no—
45C4×Dic199—21no—
46C4×Dic199—21no—
47C4×Dic199D412yescomplemented
48#41×C199—21no—
49#41×C199—21no—
50#41×C199—21no—
51#41×C199—21no—
52C2×Q8×C199—21no—
53C2×Q8×C199—21no—
54C2×Q8×C199—21no—
55C2×Q8×C199—21no—
56C2×Q8×C199—21no—
57C2×Q8×C199D411yescharacteristic
58C2×Q8×C199D412yescomplemented
59C2×D4×C199—21no—
60C2×D4×C199D411yescharacteristic
61C22×C4×C199—21no—
62C22×C4×C199—21no—
63C22×C4×C199C2311yescharacteristic
64Q16×C199—24no—
65SD16×C199—24no—
66M16×C199—21no—
67M16×C199D412yescomplemented
68C2×C8×C199—21no—
69C22×D199—22no—
70C22×D199—22no—
71C22×D199—22no—
72C22×D199—44no—
73D796—41no—
74D796—41no—
75D796—41no—
76D796—41no—
77C22⋊D199—84no—
78C2×Dic199—22no—
79C2×Dic199—22no—
80C2×Dic199—42no—
81C199⋊C8—41no—
82C23×C199—21no—
83C23×C199C2×D411yescharacteristic
84Q8×C199—22no—
85Q8×C199—21no—
86Q8×C199—22no—
87Q8×C199—21no—
88Q8×C199—41no—
89Q8×C199—41no—
90D4×C199—21no—
91D4×C199—21no—
92D4×C199—41no—
93D4×C199—41no—
94C2×C4×C199—21no—
95C2×C4×C199—22no—
96C2×C4×C199—41no—
97C2×C4×C199—41no—
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—22no—
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—41no—
118C4×C199—42no—
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×#137D19911yescharacteristic, complemented
134#535—1991no—
135C2×#124—1991no—
136C2×#104—1991no—
137#419—1991no—
138C2×#100—1991no—
139#339—1991no—
140C2×#41C2×D19911yescharacteristic
141C22×Q8C2×D19911yescharacteristic
142C22×D4—1991no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—1991no—
148C2×M16C2×D19911yescharacteristic
149#127—3981no—
150#124—3981no—
151#124—3982no—
152#124—3982no—
153#121—3982no—
154#120—3982no—
155#117—1991no—
156C2×C22⋊C4—3982no—
157C2×C42—1991no—
158#104—3982no—
159#103—3982no—
160#100—3982no—
161C24—1992no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D19911yescharacteristic
172C2×Q8C22⋊D19912yescomplemented
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×D4C22⋊D19911yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—3981no—
185C22×C4C22×D19911yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—3981no—
190M16—3981no—
191M16C22⋊D19912yescomplemented
192C2×C8—21no—
193C2×C8—3981no—
194C4⋊C4—1992no—
195C22⋊C4—3988no—
196C22⋊C4—3984no—
197C22⋊C4—3981no—
198C42—3981no—
199C42—3981no—
200C42—1992no—
201C23—21no—
202C23—7964no—
203C23—3982no—
204C23—3982no—
205C23—3982no—
206C23D4×D19911yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
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—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—7962no—
229C2×C4—3982no—
230C2×C4—3982no—
231C2×C4D4×D19911yescharacteristic
232C2×C4D4×D19911yescharacteristic
233C2×C4C2×C22⋊D19911yescharacteristic
234C2×C4C2×C22⋊D19911yescharacteristic
235C2×C4C2×C22⋊D19911yescharacteristic
236C2×C4#1569188511yescharacteristic
237C8—22no—
238C8—7961no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—7964no—
243C22—7962no—
244C22—7964no—
245C22—15922no—
246C22#2026392311yescharacteristic
247C22#2026392311yescharacteristic
248C22#2026392611yescharacteristic
249C4—41no—
250C4—42no—
251C4—7962no—
252C4—7962no—
253C4#2026392411yescharacteristic
254C4#2026392711yescharacteristic
255C4#2026392711yescharacteristic
256C4#2026395111yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—15922no—
261C2#2716392611yescharacteristic
262C2#2716419412yes—
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
FittingC2×#137×C199C2
socleC22×C199#120
prime radical#27164582C2
hypercenterC2×#137D199
hypocenterC199#2004
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#20263944C2semidirect
3. C2×#20263882C2semidirect
4. #27164027C2semidirect
5. C2×#20263871C2semidirect
6. #27163779C2semidirect
7. C2×#137×C199C2semidirect
47. C4×Dic199D4semidirect
58. C2×Q8×C199D4semidirect
67. M16×C199D4semidirect
131. C199#2004semidirect
133. C2×#137D199semidirect
172. C2×Q8C22⋊D199semidirect
191. M16C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271645822
C2×#202639442
C2×#202638822
#271640272
C2×#202638712
#271637792
C2×#137×C1992
#2004199