Finite Group Databasebeta

← #38741689

Subgroups — #38741689

Order 25216.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161666C211yesmaximal, characteristic
2#27161235C211yesmaximal, characteristic, complemented
3#27160990C211yesmaximal, characteristic, complemented
4#27160817C211yesmaximal, characteristic, complemented
5#27160813C211yesmaximal, characteristic, complemented
6#27160802C211yesmaximal, characteristic, complemented
7#448×C197C211yesmaximal, characteristic, complemented
8C2×#15691348—21no—
9#20263321—21no—
10#20263312—21no—
11#20263310—21no—
12#20263305C2211yescharacteristic
13#20263298—21no—
14#20263294—21no—
15D4×Dic197—21no—
16C2×#15691335—21no—
17C2×#15691335C2211yescharacteristic
18#20263250—21no—
19C22⋊#13096802C2211yescharacteristic
20#20263233—21no—
21#20263233C2211yescharacteristic
22C22×Q8×C197C2211yescharacteristic
23C2×SD16×C197—21no—
24#121×C197C2211yescharacteristic
25#103×C197—21no—
26C22⋊C8×C197C2211yescharacteristic
27C23×D197D411yescharacteristic
28C2×Dic394D411yescharacteristic
29C2×C22⋊D197—21no—
30C2×C22⋊D197—21no—
31C22×Dic197—21no—
32C22×Dic197—21no—
33C22×Dic197D411yescharacteristic
34#15691348—42no—
35C2×#13096802—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197D411yescharacteristic
38#15691337—21no—
39C22⋊Dic197—21no—
40C22⋊Dic197—21no—
41#15691335—21no—
42#15691335—21no—
43#15691335—21no—
44#15691335—21no—
45#15691335—42no—
46C4×Dic197—21no—
47C2×Q8×C197—21no—
48C2×Q8×C197—21no—
49C2×Q8×C197—42no—
50C2×D4×C197—21no—
51C2×D4×C197D411yescharacteristic
52C22×C4×C197—21no—
53C22×C4×C197C2311yescharacteristic
54SD16×C197—42no—
55C2×C8×C197—21no—
56C4⋊C4×C197D411yescharacteristic
57C22⋊C4×C197—21no—
58C22×D197—21no—
59C22×D197—21no—
60C22×D197—21no—
61C22×D197—41no—
62C22×D197—41no—
63Dic394—41no—
64C22⋊D197—81no—
65C22⋊D197—81no—
66C2×Dic197—21no—
67C2×Dic197—21no—
68C2×Dic197—21no—
69C2×Dic197—41no—
70C2×Dic197—41no—
71C2×Dic197—41no—
72C2×Dic197—41no—
73C2×Dic197—81no—
74#13096802—41no—
75C23×C197—21no—
76C23×C197C2×D411yescharacteristic
77Q8×C197—22no—
78Q8×C197—22no—
79Q8×C197—41no—
80Q8×C197—41no—
81D4×C197—41no—
82D4×C197—41no—
83D4×C197—41no—
84C2×C4×C197—22no—
85C2×C4×C197—21no—
86C2×C4×C197—41no—
87C2×C4×C197—41no—
88C2×C4×C197—41no—
89C2×C4×C197C2×D411yescharacteristic
90C2×C4×C197C2×D411yescharacteristic
91C8×C197—41no—
92C22×C197—21no—
93C22×C197—21no—
94C22×C197—41no—
95C22×C197—41no—
96C22×C197—41no—
97C22×C197#12011yescharacteristic
98C22×C197#13511yescharacteristic
99C22×C197#13711yescharacteristic
100C2×D197—42no—
101C2×D197—41no—
102C2×D197—41no—
103C2×D197—41no—
104C2×D197—81no—
105C4×C197—21no—
106C4×C197—21no—
107C4×C197—41no—
108C4×C197—42no—
109Dic197—41no—
110Dic197—41no—
111Dic197—41no—
112Dic197—81no—
113C2×C197—21no—
114C2×C197—21no—
115C2×C197—81no—
116C2×C197#45211yescharacteristic
117C2×C197#45511yescharacteristic
118C2×C197B(4,𝔽2)11yescharacteristic
119D197—81no—
120C197#160611yescharacteristic, complemented, Sylow
121#1606—1971nomaximal, Sylow
122#538—1971no—
123#451—1971no—
124#448D19711yescharacteristic, complemented
125#393—1971no—
126#328—1971no—
127#327—1971no—
128#323—1971no—
129C22×Q8C2×D19711yescharacteristic
130C2×Q16—3941no—
131C2×SD16—21no—
132#126—3941no—
133#124—3941no—
134#123—3941no—
135C22⋊Q8—1971no—
136#121—3941no—
137#121C2×D19711yescharacteristic
138#120—3941no—
139C4×D4—3941no—
140C2×C22⋊C4—3941no—
141C2×C22⋊C4—1971no—
142#103—21no—
143#103—3941no—
144C22⋊C8—1971no—
145C22⋊C8C2×D19711yescharacteristic
146#95—3941no—
147#95—1971no—
148C24—1971no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—1971no—
153C2×D4—21no—
154C2×D4—3941no—
155C2×D4—3941no—
156C2×D4C22⋊D19711yescharacteristic
157C22×C4—21no—
158C22×C4—3941no—
159C22×C4—3941no—
160C22×C4—1971no—
161C22×C4C22×D19711yescharacteristic
162Q16—7882no—
163SD16—42no—
164C2×C8—21no—
165C2×C8—3941no—
166C4⋊C4—3941no—
167C4⋊C4—3941no—
168C4⋊C4—1971no—
169C4⋊C4C22⋊D19711yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—7882no—
172C22⋊C4—3941no—
173C22⋊C4—3941no—
174C22⋊C4—3941no—
175C22⋊C4—3941no—
176C22⋊C4—3941no—
177C22⋊C4—3941no—
178C42—3941no—
179C23—21no—
180C23—7881no—
181C23—7881no—
182C23—3941no—
183C23—3941no—
184C23—3941no—
185C23D4×D19711yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—7881no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—15761no—
195D4—15761no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—7881no—
202C2×C4—7881no—
203C2×C4—7881no—
204C2×C4—7881no—
205C2×C4—15761no—
206C2×C4—3941no—
207C2×C4—3941no—
208C2×C4—3941no—
209C2×C4D4×D19711yescharacteristic
210C2×C4C2×C22⋊D19711yescharacteristic
211C8—41no—
212C8—7881no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—7882no—
219C22—7881no—
220C22—7881no—
221C22—7881no—
222C22—15761no—
223C22#2026329411yescharacteristic
224C22#2026334711yescharacteristic
225C22#2026335511yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—7881no—
231C4—7881no—
232C4—7881no—
233C4—15761no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—15761no—
238C2#2716123811yescharacteristic
239C2#2716125511yescharacteristic
240C2#2716126511yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#448×C197C2
socleC22×C197#120
prime radical#27161666C2
hypercenter#448D197
hypocenterC197#1606
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #27161235C2semidirect
3. #27160990C2semidirect
4. #27160817C2semidirect
5. #27160813C2semidirect
6. #27160802C2semidirect
7. #448×C197C2semidirect
120. C197#1606semidirect
124. #448D197semidirect

Maximal subgroups

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

subgroupindex
#271616662
#271612352
#271609902
#271608172
#271608132
#271608022
#448×C1972
#1606197