Finite Group Databasebeta

← #38741641

Subgroups — #38741641

Order 25216.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161229C211yesmaximal, characteristic, complemented
2#27161212C211yesmaximal, characteristic
3#27160996C211yesmaximal, characteristic, complemented
4#27160816C211yesmaximal, characteristic, complemented
5#27160813C211yesmaximal, characteristic, complemented
6#27160805C211yesmaximal, characteristic, complemented
7#539×C197C211yesmaximal, characteristic, complemented
8C22×D788C2211yescharacteristic
9C2×#15691345—21no—
10C2×#15691343—21no—
11C2×#15691335—21no—
12C2×#15691335—21no—
13#20263246—21no—
14#20263246—21no—
15C22⋊#13096802C2211yescharacteristic
16C22⋊#13096802C2211yescharacteristic
17#20263233C2211yescharacteristic
18#126×C197—21no—
19#125×C197—21no—
20#123×C197—21no—
21#123×C197—21no—
22C22⋊Q8×C197—21no—
23C22⋊Q8×C197C2211yescharacteristic
24#121×C197C2211yescharacteristic
25C4×D4×C197—21no—
26C2×C4⋊C4×C197C2211yescharacteristic
27C23×D197—21no—
28C2×D788—21no—
29C2×D788—21no—
30C2×D788—42no—
31C22×Dic197—21no—
32#15691345—42no—
33#15691343—42no—
34C2×#13096802—21no—
35C2×#13096802—21no—
36#15691335—42no—
37#15691335—42no—
38C2×Q8×C197D411yescharacteristic
39C2×D4×C197—21no—
40C2×D4×C197D411yescharacteristic
41C22×C4×C197—21no—
42C22×C4×C197D411yescharacteristic
43C22×C4×C197D411yescharacteristic
44C22×C4×C197C2311yescharacteristic
45C4⋊C4×C197—21no—
46C4⋊C4×C197—21no—
47C4⋊C4×C197—21no—
48C4⋊C4×C197—21no—
49C4⋊C4×C197—21no—
50C4⋊C4×C197—21no—
51C4⋊C4×C197D411yescharacteristic
52C4⋊C4×C197D411yescharacteristic
53C22⋊C4×C197—21no—
54C22⋊C4×C197—21no—
55C22⋊C4×C197—21no—
56C22⋊C4×C197—21no—
57C22⋊C4×C197—21no—
58C42×C197—21no—
59C22×D197—22no—
60C22×D197—41no—
61C22×D197—41no—
62C22×D197—41no—
63C22×D197—41no—
64C22×D197—41no—
65C22×D197—41no—
66D788—22no—
67D788—22no—
68D788—41no—
69D788—41no—
70C2×Dic197—41no—
71C2×Dic197—41no—
72C2×Dic197—41no—
73#13096802—41no—
74#13096802—41no—
75C23×C197—21no—
76C23×C197C2×D411yescharacteristic
77Q8×C197—41no—
78D4×C197—41no—
79D4×C197—41no—
80D4×C197—41no—
81C2×C4×C197—21no—
82C2×C4×C197—21no—
83C2×C4×C197—21no—
84C2×C4×C197—21no—
85C2×C4×C197—21no—
86C2×C4×C197—21no—
87C2×C4×C197—21no—
88C2×C4×C197—41no—
89C2×C4×C197—41no—
90C2×C4×C197—81no—
91C2×C4×C197C2×D411yescharacteristic
92C2×C4×C197C2×D411yescharacteristic
93C22×C197—21no—
94C22×C197—21no—
95C22×C197—41no—
96C22×C197—41no—
97C22×C197—41no—
98C22×C197#12011yescharacteristic
99C22×C197#13511yescharacteristic
100C22×C197AGL(1,ℤ/8)11yescharacteristic
101C2×D197—42no—
102C2×D197—42no—
103C2×D197—42no—
104C2×D197—81no—
105C2×D197—81no—
106C2×D197—81no—
107C2×D197—81no—
108C4×C197—21no—
109C4×C197—21no—
110C4×C197—41no—
111C4×C197—41no—
112C4×C197—41no—
113C4×C197—41no—
114C4×C197—41no—
115Dic197—81no—
116C2×C197—21no—
117C2×C197—21no—
118C2×C197—81no—
119C2×C197#44911yescharacteristic
120C2×C197#45411yescharacteristic
121C2×C197B(4,𝔽2)11yescharacteristic
122D197—82no—
123C197#159711yescharacteristic, complemented, Sylow
124#1597—1971nomaximal, Sylow
125#539D19711yescharacteristic, complemented
126#450—1971no—
127#447—1971no—
128#394—1971no—
129#328—1971no—
130#327—1971no—
131#324—1971no—
132C22×D4—1971no—
133C2×SD16—3941no—
134C2×D8—3941no—
135#126—21no—
136#125—21no—
137#123—21no—
138#123—21no—
139C22⋊Q8—21no—
140C22⋊Q8C2×D19711yescharacteristic
141#121C2×D19711yescharacteristic
142C4×D4—21no—
143C2×C4⋊C4C2×D19711yescharacteristic
144C2×C22⋊C4—3941no—
145C2×C22⋊C4—3941no—
146#102—3941no—
147#102—3941no—
148C22⋊C8—1971no—
149C22⋊C8—1971no—
150#95—1971no—
151C24—3941no—
152C2×Q8C22⋊D19711yescharacteristic
153C2×D4—21no—
154C2×D4—7882no—
155C2×D4—3941no—
156C2×D4—3941no—
157C2×D4C22⋊D19711yescharacteristic
158C22×C4—21no—
159C22×C4—3941no—
160C22×C4C22⋊D19711yescharacteristic
161C22×C4C22⋊D19711yescharacteristic
162C22×C4C22×D19711yescharacteristic
163SD16—7882no—
164D8—7882no—
165C2×C8—3941no—
166C2×C8—3941no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22⋊D19711yescharacteristic
174C4⋊C4C22⋊D19711yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—7882no—
181C22⋊C4—7882no—
182C42—21no—
183C23—21no—
184C23—7881no—
185C23—7881no—
186C23—7881no—
187C23—7881no—
188C23—7881no—
189C23—7881no—
190C23—3942no—
191C23C2×C22⋊D19711yescharacteristic
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—7881no—
197D4—7881no—
198D4—3942no—
199D4—3942no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—7881no—
211C2×C4—7881no—
212C2×C4—7881no—
213C2×C4C2×C22⋊D19711yescharacteristic
214C2×C4C2×C22⋊D19711yescharacteristic
215C8—7881no—
216C8—7881no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—7882no—
223C22—7882no—
224C22—7882no—
225C22—15761no—
226C22—15761no—
227C22—15761no—
228C22—15761no—
229C22#2026329511yescharacteristic
230C22#2026334611yescharacteristic
231C22#2026335111yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—15761no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—15762no—
244C2#2716122411yescharacteristic
245C2#2716125211yescharacteristic
246C2#2716126611yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#539×C197C2
socleC22×C197#120
prime radical#27161212C2
hypercenter#539D197
hypocenterC197#1597
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #27161229C2semidirect
3. #27160996C2semidirect
4. #27160816C2semidirect
5. #27160813C2semidirect
6. #27160805C2semidirect
7. #539×C197C2semidirect
123. C197#1597semidirect
125. #539D197semidirect

Maximal subgroups

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

subgroupindex
#271612292
#271612122
#271609962
#271608162
#271608132
#271608052
#539×C1972
#1597197