Finite Group Databasebeta

← #38752003

Subgroups — #38752003

Order 25216.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20263374C211yesmaximal, characteristic
2C2×#20263357C211yesmaximal, characteristic, complemented
3#27161653C211yesmaximal, characteristic, complemented
4#27161240C214yesmaximal, complemented
5#27161233C214yesmaximal, complemented
6#27161088C211yesmaximal, characteristic, complemented
7#27161076C211yesmaximal, characteristic, complemented
8#27161045C211yesmaximal, characteristic, complemented
9C22×Q16×C197C211yesmaximal, characteristic, complemented
10#20263374—24no—
11#20263374—24no—
12C2×#15691359—21no—
13C2×#15691359C2212yes—
14C2×#15691358—22no—
15C2×#15691358C2211yescharacteristic
16C2×#15691358C2211yescharacteristic
17C2×Q8×D197—21no—
18C2×Q8×D197C2212yes—
19#20263357—24no—
20C2×#15691348C2212yes—
21C2×#15691345C2212yes—
22C2×#15691342C2211yescharacteristic
23#20263324—21no—
24#20263315—21no—
25#20263309—22no—
26#20263302C2214yes—
27#20263285C2211yescharacteristic
28#20263251C2212yes—
29#20263250C2214yes—
30#20263248C2212yes—
31#20263238C2214yes—
32C22×Q8×C197C2211yescharacteristic
33C22×Q8×C197C2211yescharacteristic
34C2×Q16×C197—24no—
35C2×Q16×C197C2214yes—
36C22×C8×C197C2211yescharacteristic
37#15691359—28no—
38#15691359—24no—
39#15691359—24no—
40#15691358—216no—
41#15691358—22no—
42#15691358—22no—
43#15691358—22no—
44#15691358D414yes—
45C2×Dic394—22no—
46C2×Dic394D411yescharacteristic
47C2×Dic394C2311yescharacteristic
48Q8×D197—28no—
49Q8×D197—24no—
50Q8×D197—24no—
51C2×D788—22no—
52C2×D788D411yescharacteristic
53C2×D788C2311yescharacteristic
54C2×C22⋊D197—21no—
55C2×C22⋊D197D412yes—
56C2×C4×D197—22no—
57C2×C4×D197—22no—
58C2×C4×D197—21no—
59C2×C4×D197C2312yes—
60#15691348—24no—
61#15691345—24no—
62#15691342—22no—
63C2×#13096802C2312yes—
64C4⋊Dic197C2312yes—
65#15691337—24no—
66C22⋊Dic197—21no—
67#15691335—24no—
68C4×Dic197—21no—
69C2×Q8×C197—21no—
70C2×Q8×C197—22no—
71C2×Q8×C197—21no—
72C2×Q8×C197—22no—
73C2×Q8×C197—22no—
74C2×Q8×C197D414yes—
75C2×Q8×C197C2312yes—
76C2×Q8×C197C2312yes—
77C22×C4×C197—21no—
78C22×C4×C197—21no—
79C22×C4×C197C2311yescharacteristic
80Q16×C197—28no—
81C2×C8×C197—22no—
82C2×C8×C197C2312yes—
83C22×D197—21no—
84C22×D197C2×D412yes—
85Dic394—28no—
86Dic394—22no—
87Dic394—21no—
88Dic394C2×D412yes—
89D788—28no—
90D788—22no—
91D788—21no—
92D788C2×D412yes—
93C22⋊D197—24no—
94C22⋊D197—24no—
95C2×Dic197—21no—
96C2×Dic197—22no—
97C2×Dic197C2×D412yes—
98C4×D197—28no—
99C4×D197—28no—
100C4×D197—24no—
101C4×D197—22no—
102C4×D197—22no—
103#13096802—22no—
104C23×C197C2×D411yescharacteristic
105Q8×C197—24no—
106Q8×C197—22no—
107Q8×C197—22no—
108Q8×C197—22no—
109Q8×C197—42no—
110Q8×C197C2×D414yes—
111C2×C4×C197—24no—
112C2×C4×C197—22no—
113C2×C4×C197—22no—
114C2×C4×C197—42no—
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2411yescharacteristic
119C8×C197—22no—
120C22×C197—22no—
121C22×C197#12012yes—
122C22×C197C22×D411yescharacteristic
123C2×D197—22no—
124C2×D197—22no—
125C2×D197—24no—
126C2×D197—42no—
127C4×C197—24no—
128C4×C197—42no—
129C4×C197#12012yes—
130C4×C197C22×D411yescharacteristic
131C4×C197C22×D411yescharacteristic
132Dic197—22no—
133Dic197—22no—
134Dic197—42no—
135C2×C197—22no—
136C2×C197C2×#12011yescharacteristic
137C2×C197#57812yes—
138D197—42no—
139D197—42no—
140C197#301111yescharacteristic, complemented, Sylow
141#3011—1971nomaximal, Sylow
142C2×2−1+4—1971no—
143C2×#137—1971no—
144C22×Q16D19711yescharacteristic, complemented
145#536—1971no—
146#452—1974no—
147#451—1974no—
148#419—1971no—
149#416—1971no—
150#408—1971no—
1512−1+4—3944no—
1522−1+4—3944no—
153C2×#41—3942no—
154C2×#41—3941no—
155C2×#41—1972no—
156C2×#41—1971no—
157C2×#41—1971no—
158C22×Q8—3941no—
159C22×Q8—1972no—
160C22×Q8C2×D19711yescharacteristic
161C22×Q8C2×D19711yescharacteristic
162#137—3944no—
163C2×Q16—24no—
164C2×Q16—1972no—
165C2×Q16C2×D19714yes—
166C2×SD16—1972no—
167C2×M16—1971no—
168C22×C8C2×D19711yescharacteristic
169C4⋊Q8—3941no—
170#124—3941no—
171#123—3942no—
172C22⋊Q8—1974no—
173#117—1971no—
174#103—1974no—
175#103—1972no—
176#102—1972no—
177C22⋊C8—1974no—
178#41—39416no—
179#41—3948no—
180#41—3944no—
181#41—3944no—
182#41—3942no—
183#41—3942no—
184#41—3942no—
185#41—1974no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—3942no—
192C2×Q8—3948no—
193C2×Q8—3944no—
194C2×Q8—3944no—
195C2×Q8—1971no—
196C2×Q8—1971no—
197C2×Q8C22⋊D19714yes—
198C2×Q8C22×D19712yes—
199C2×Q8C22×D19712yes—
200C2×D4—3942no—
201C2×D4—3941no—
202C2×D4—1972no—
203C2×D4—1971no—
204C2×D4—1971no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—3942no—
208C22×C4—3942no—
209C22×C4—3941no—
210C22×C4—1972no—
211C22×C4C22×D19711yescharacteristic
212Q16—28no—
213Q16—3944no—
214SD16—3944no—
215M16—3942no—
216C2×C8—22no—
217C2×C8—1972no—
218C2×C8C22×D19712yes—
219C4⋊C4—3944no—
220C4⋊C4—1972no—
221C22⋊C4—3944no—
222C22⋊C4—3941no—
223C42—3941no—
224C23—3941no—
225C23—1972no—
226C23D4×D19711yescharacteristic
227Q8—24no—
228Q8—22no—
229Q8—22no—
230Q8—22no—
231Q8—42no—
232Q8—3948no—
233Q8—3942no—
234Q8—3941no—
235Q8—1972no—
236Q8C2×C22⋊D19714yes—
237D4—3948no—
238D4—3944no—
239D4—3944no—
240D4—3942no—
241D4—3941no—
242D4—1972no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—3948no—
248C2×C4—3948no—
249C2×C4—3942no—
250C2×C4—3941no—
251C2×C4—3944no—
252C2×C4—3942no—
253C2×C4—3942no—
254C2×C4—1972no—
255C2×C4D4×D19711yescharacteristic
256C2×C4D4×D19712yes—
257C2×C4C2×C22⋊D19712yes—
258C2×C4C23×D19711yescharacteristic
259C8—22no—
260C8—3942no—
261C22—22no—
262C22—7882no—
263C22—3944no—
264C22—3942no—
265C22—3942no—
266C22#2026329412yes—
267C22C2×D4×D19711yescharacteristic
268C4—24no—
269C4—42no—
270C4—7882no—
271C4—3942no—
272C4—3942no—
273C4#2026329412yes—
274C4C2×D4×D19711yescharacteristic
275C4C22×C22⋊D19711yescharacteristic
276C2—22no—
277C2—7882no—
278C2—7882no—
279C2C2×#2026329411yescharacteristic
280C2#2716189212yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×Q16×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263374C2
hypercenterC22×Q16D197
hypocenterC197#3011
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#20263357C2semidirect
3. #27161653C2semidirect
4. #27161240C2semidirect
5. #27161233C2semidirect
6. #27161088C2semidirect
7. #27161076C2semidirect
8. #27161045C2semidirect
9. C22×Q16×C197C2semidirect
140. C197#3011semidirect
144. C22×Q16D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633742
C2×#202633572
#271616532
#271612402
#271612332
#271610882
#271610762
#271610452
C22×Q16×C1972
#3011197