Finite Group Databasebeta

← #38741699

Subgroups — #38741699

Order 25216.

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

normal subgroups only

This group has 280 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#27161637C211yesmaximal, characteristic
2#27161393C211yesmaximal, characteristic, complemented
3#27161390C211yesmaximal, characteristic, complemented
4#27161006C211yesmaximal, characteristic, complemented
5#27160812C211yesmaximal, characteristic, complemented
6#27160802C211yesmaximal, characteristic, complemented
7#327×C197C211yesmaximal, characteristic, complemented
8C2×D4×D197—21no—
9#20263311—21no—
10#20263308—21no—
11#20263300C2211yescharacteristic
12#20263298—21no—
13#20263296—21no—
14#20263294—21no—
15C2×C4⋊Dic197C2211yescharacteristic, complemented
16C2×#15691335—21no—
17C2×#15691335C2211yescharacteristic
18#20263260—21no—
19#20263259—21no—
20#20263248—21no—
21#20263247—21no—
22C22⋊#13096802C2211yescharacteristic, complemented
23#20263233—21no—
24#121×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic, complemented
26#95×C197C2211yescharacteristic, complemented
27C23×D197—21no—
28C23×D197D411yescharacteristic
29C2×D788—21no—
30C2×D788D411yescharacteristic
31C2×C22⋊D197—21no—
32C2×C22⋊D197—21no—
33C2×C22⋊D197—21no—
34D4×D197—42no—
35D4×D197—42no—
36C22×Dic197—21no—
37C22×Dic197D411yescharacteristic
38C2×C4×D197—21no—
39C2×#13096802—21no—
40C4⋊Dic197—21no—
41C4⋊Dic197—21no—
42C4⋊Dic197D411yescharacteristic
43#15691337—21no—
44C22⋊Dic197—21no—
45#15691335—21no—
46#15691335—21no—
47#15691335—21no—
48#15691335—21no—
49#15691335—42no—
50C2×D4×C197—21no—
51C2×D4×C197D411yescharacteristic
52C22×C4×C197—21no—
53C22×C4×C197C2311yescharacteristic
54C2×C8×C197—21no—
55C4⋊C4×C197D411yescharacteristic
56C22⋊C4×C197—21no—
57C22×D197—21no—
58C22×D197—21no—
59C22×D197—21no—
60C22×D197—42no—
61C22×D197—41no—
62C22×D197—41no—
63C22×D197—81no—
64C22×D197—81no—
65C22×D197D812yes—
66D788—41no—
67D788—42no—
68C22⋊D197—42no—
69C22⋊D197—41no—
70C22⋊D197—41no—
71C22⋊D197—81no—
72C2×Dic197—21no—
73C2×Dic197—21no—
74C2×Dic197—41no—
75C2×Dic197—41no—
76C2×Dic197—41no—
77C2×Dic197—41no—
78C4×D197—42no—
79#13096802—41no—
80C23×C197—21no—
81C23×C197C2×D411yescharacteristic
82D4×C197—41no—
83D4×C197—41no—
84D4×C197—41no—
85C2×C4×C197—21no—
86C2×C4×C197—41no—
87C2×C4×C197—41no—
88C2×C4×C197—41no—
89C2×C4×C197—41no—
90C2×C4×C197C2×D411yescharacteristic
91C2×C4×C197C2×D411yescharacteristic
92C8×C197—41no—
93C22×C197—21no—
94C22×C197—21no—
95C22×C197—41no—
96C22×C197—41no—
97C22×C197—41no—
98C22×C197#12011yescharacteristic
99C22×C197C2×D811yescharacteristic
100C22×C197AGL(1,ℤ/8)11yescharacteristic
101C2×D197—22no—
102C2×D197—41no—
103C2×D197—41no—
104C2×D197—42no—
105C2×D197—41no—
106C2×D197—41no—
107C2×D197—41no—
108C2×D197—81no—
109C2×D197—84no—
110C4×C197—21no—
111C4×C197—21no—
112C4×C197—41no—
113C4×C197—81no—
114Dic197—41no—
115Dic197—41no—
116Dic197—81no—
117C2×C197—21no—
118C2×C197—21no—
119C2×C197—81no—
120C2×C197#44711yescharacteristic
121C2×C197#45511yescharacteristic
122C2×C197#45811yescharacteristic
123D197—42no—
124D197—81no—
125C197#160711yescharacteristic, complemented, Sylow
126#1607—1971nomaximal, Sylow
127#534—1971no—
128#480—1971no—
129#480—1971no—
130#397—1971no—
131#327—1971no—
132#327D19711yescharacteristic, complemented
133#323—1971no—
134C22×D4—3941no—
135#123—3941no—
136#123—3941no—
137#121—3941no—
138#121—1971no—
139#121C2×D19711yescharacteristic
140#120—3941no—
141#120—3941no—
142C2×C4⋊C4—1971no—
143C2×C22⋊C4—3941no—
144C2×C22⋊C4—1971no—
145#107—3941no—
146#107—3941no—
147#102—3941no—
148#102—3941no—
149C22⋊C8—1971no—
150C22⋊C8C2×D19711yescharacteristic, complemented
151#95—3941no—
152#95C2×D19711yescharacteristic, complemented
153C24—3941no—
154C24—1971no—
155C2×D4—21no—
156C2×D4—7882no—
157C2×D4—7882no—
158C2×D4—3941no—
159C2×D4—3941no—
160C2×D4—3941no—
161C2×D4—3941no—
162C2×D4—1971no—
163C2×D4D78811yescharacteristic
164C22×C4—21no—
165C22×C4—3941no—
166C22×C4—3941no—
167C22×C4—1971no—
168C22×C4C22×D19711yescharacteristic
169C2×C8—21no—
170C2×C8—3941no—
171C4⋊C4—3941no—
172C4⋊C4—3941no—
173C4⋊C4—3941no—
174C4⋊C4—1971no—
175C4⋊C4D78811yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—7882no—
178C22⋊C4—3941no—
179C22⋊C4—3941no—
180C22⋊C4—3941no—
181C22⋊C4—3941no—
182C22⋊C4—3941no—
183C23—21no—
184C23—7881no—
185C23—7881no—
186C23—7882no—
187C23—15761no—
188C23—15761no—
189C23—3941no—
190C23—3941no—
191C23—3941no—
192C23—1972no—
193C23D4×D19711yescharacteristic
194D4—41no—
195D4—41no—
196D4—41no—
197D4—7881no—
198D4—7882no—
199D4—7882no—
200D4—7881no—
201D4—7881no—
202D4—15761no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—7882no—
209C2×C4—7881no—
210C2×C4—7881no—
211C2×C4—7881no—
212C2×C4—7881no—
213C2×C4—3941no—
214C2×C4—3941no—
215C2×C4D4×D19711yescharacteristic
216C2×C4C2×D78811yescharacteristic
217C8—41no—
218C8—7881no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—7882no—
225C22—7881no—
226C22—7881no—
227C22—7881no—
228C22—7881no—
229C22—7881no—
230C22—15761no—
231C22—15764no—
232C22—3942no—
233C22#2026329611yescharacteristic
234C22D8×D19711yescharacteristic
235C22#2026335311yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—81no—
240C4—7881no—
241C4—7881no—
242C4—15761no—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—7882no—
247C2—15761no—
248C2#2716121411yescharacteristic
249C2#2716125811yescharacteristic
250C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197#120
prime radical#27161637C2
hypercenter#327D197
hypocenterC197#1607
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27161393C2semidirect
3. #27161390C2semidirect
4. #27161006C2semidirect
5. #27160812C2semidirect
6. #27160802C2semidirect
7. #327×C197C2semidirect
15. C2×C4⋊Dic197C22semidirect
22. C22⋊#13096802C22semidirect
25. C22⋊C8×C197C22semidirect
26. #95×C197C22semidirect
125. C197#1607semidirect
132. #327D197semidirect
150. C22⋊C8C2×D197semidirect
152. #95C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616372
#271613932
#271613902
#271610062
#271608122
#271608022
#327×C1972
#1607197