Finite Group Databasebeta

← #38741679

Subgroups — #38741679

Order 25216.

282 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 51 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#27161656C211yesmaximal, characteristic
2#27161218C211yesmaximal, characteristic, complemented
3#27160998C211yesmaximal, characteristic, complemented
4#27160814C211yesmaximal, characteristic, complemented
5#27160813C211yesmaximal, characteristic, complemented
6#27160804C211yesmaximal, characteristic, complemented
7#448×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9C2×#15691345—21no—
10#20263304—21no—
11#20263303—21no—
12#20263300C2211yescharacteristic
13#20263299—21no—
14#20263298—21no—
15#20263298—21no—
16C2×#15691337—21no—
17C2×#15691337C2211yescharacteristic
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
27#15691357—42no—
28#15691357—42no—
29C2×Dic394—21no—
30C2×D788D411yescharacteristic
31C2×C22⋊D197—21no—
32C2×C22⋊D197—21no—
33C2×C22⋊D197—21no—
34C22×Dic197—21no—
35C22×Dic197—21no—
36C22×Dic197D411yescharacteristic
37C22×Dic197D411yescharacteristic
38C2×C4×D197—21no—
39#15691345—42no—
40C2×#13096802—21no—
41C4⋊Dic197D411yescharacteristic
42#15691337—21no—
43#15691337—21no—
44#15691337—21no—
45#15691337—42no—
46C22⋊Dic197—21no—
47#15691335—21no—
48#15691335—21no—
49C2×Q8×C197—21no—
50C2×Q8×C197—21no—
51C2×Q8×C197—42no—
52C2×D4×C197—21no—
53C2×D4×C197D411yescharacteristic
54C22×C4×C197—21no—
55C22×C4×C197C2311yescharacteristic
56SD16×C197—42no—
57C2×C8×C197—21no—
58C4⋊C4×C197D411yescharacteristic
59C22⋊C4×C197—21no—
60C22×D197—21no—
61Dic394—42no—
62D788—41no—
63C22⋊D197—42no—
64C22⋊D197—41no—
65C22⋊D197—41no—
66C22⋊D197—81no—
67C2×Dic197—21no—
68C2×Dic197—21no—
69C2×Dic197—22no—
70C2×Dic197—21no—
71C2×Dic197—21no—
72C2×Dic197—42no—
73C2×Dic197—41no—
74C2×Dic197—41no—
75C2×Dic197—41no—
76C2×Dic197SD1612yescomplemented
77C4×D197—42no—
78#13096802—41no—
79C23×C197—21no—
80C23×C197C2×D411yescharacteristic
81Q8×C197—22no—
82Q8×C197—22no—
83Q8×C197—41no—
84Q8×C197—41no—
85D4×C197—41no—
86D4×C197—41no—
87D4×C197—41no—
88C2×C4×C197—22no—
89C2×C4×C197—21no—
90C2×C4×C197—41no—
91C2×C4×C197—41no—
92C2×C4×C197—41no—
93C2×C4×C197C2×D411yescharacteristic
94C2×C4×C197C2×D411yescharacteristic
95C8×C197—41no—
96C22×C197—21no—
97C22×C197—21no—
98C22×C197—41no—
99C22×C197—41no—
100C22×C197—41no—
101C22×C197#12011yescharacteristic
102C22×C197C2×SD1611yescharacteristic
103C22×C197AGL(1,ℤ/8)11yescharacteristic
104C2×D197—41no—
105C2×D197—41no—
106C2×D197—41no—
107C4×C197—21no—
108C4×C197—21no—
109C4×C197—41no—
110C4×C197—42no—
111Dic197—22no—
112Dic197—42no—
113Dic197—41no—
114Dic197—41no—
115C2×C197—21no—
116C2×C197—21no—
117C2×C197—81no—
118C2×C197#45011yescharacteristic
119C2×C197#45611yescharacteristic
120C2×C197B(4,𝔽2)11yescharacteristic
121D197—81no—
122C197#160411yescharacteristic, complemented, Sylow
123#1604—1971nomaximal, Sylow
124#537—1971no—
125#448—1971no—
126#448D19711yescharacteristic, complemented
127#395—1971no—
128#327—1971no—
129#327—1971no—
130#324—1971no—
131C2×#41—3941no—
132C22×Q8C2×D19711yescharacteristic
133C2×SD16—21no—
134C2×SD16—3941no—
135C22⋊Q8—3941no—
136C22⋊Q8—3941no—
137#121—3941no—
138#121—3941no—
139#121—3941no—
140#121—1971no—
141#121C2×D19711yescharacteristic
142C2×C4⋊C4—3941no—
143C2×C4⋊C4—1971no—
144#103—21no—
145#103—3941no—
146C22⋊C8—1971no—
147C22⋊C8C2×D19711yescharacteristic
148#95—3941no—
149#95—1971no—
150#41—7882no—
151#41—7882no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—42no—
155C2×Q8—3941no—
156C2×D4—21no—
157C2×D4—3941no—
158C2×D4—3941no—
159C2×D4—3941no—
160C2×D4—1971no—
161C2×D4C22⋊D19711yescharacteristic
162C22×C4—21no—
163C22×C4—3941no—
164C22×C4—3941no—
165C22×C4—3941no—
166C22×C4—1971no—
167C22×C4—1971no—
168C22×C4C22×D19711yescharacteristic
169SD16—42no—
170SD16—7882no—
171C2×C8—21no—
172C2×C8—3941no—
173C4⋊C4—7882no—
174C4⋊C4—3941no—
175C4⋊C4—3941no—
176C4⋊C4—3941no—
177C4⋊C4—1971no—
178C4⋊C4C22⋊D19711yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—3941no—
181C22⋊C4—3941no—
182C22⋊C4—3941no—
183C23—21no—
184C23—3941no—
185C23D4×D19711yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—7882no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—7881no—
195D4—7881no—
196D4—7881no—
197D4—7882no—
198D4—15761no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—7882no—
205C2×C4—7881no—
206C2×C4—7881no—
207C2×C4—7882no—
208C2×C4—7881no—
209C2×C4—3942no—
210C2×C4—3941no—
211C2×C4—3941no—
212C2×C4—3941no—
213C2×C4—3941no—
214C2×C4—1972no—
215C2×C4D4×D19711yescharacteristic
216C2×C4C2×C22⋊D19711yescharacteristic
217C8—41no—
218C8—7881no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—7881no—
225C22—7881no—
226C22—7881no—
227C22#2026329411yescharacteristic
228C22SD16×D19711yescharacteristic
229C22#2026335311yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—42no—
234C4—7882no—
235C4—7881no—
236C4—7881no—
237C4—3942no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—15761no—
242C2#2716122811yescharacteristic
243C2#2716126011yescharacteristic
244C2#2716126511yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#448×C197C2
socleC22×C197#120
prime radical#27161656C2
hypercenter#448D197
hypocenterC197#1604
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. #27161218C2semidirect
3. #27160998C2semidirect
4. #27160814C2semidirect
5. #27160813C2semidirect
6. #27160804C2semidirect
7. #448×C197C2semidirect
76. C2×Dic197SD16semidirect
122. C197#1604semidirect
126. #448D197semidirect

Maximal subgroups

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

subgroupindex
#271616562
#271612182
#271609982
#271608142
#271608132
#271608042
#448×C1972
#1604197