Finite Group Databasebeta

← #38741733

Subgroups — #38741733

Order 25216.

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

normal subgroups only

This group has 288 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#27161398C211yesmaximal, characteristic, complemented
3#27161395C211yesmaximal, characteristic, complemented
4#27161016C211yesmaximal, 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⋊Dic197—21no—
16C2×C4⋊Dic197C2211yescharacteristic, complemented
17C2×#15691335—21no—
18C2×#15691335C2211yescharacteristic
19#20263258—21no—
20#20263257—21no—
21#20263248—21no—
22#20263247—21no—
23C22⋊#13096802C2211yescharacteristic, complemented
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⋊Dic197—42no—
43C4⋊Dic197D411yescharacteristic
44#15691337—21no—
45C22⋊Dic197—21no—
46#15691335—21no—
47#15691335—21no—
48#15691335—21no—
49#15691335—21no—
50#15691335—42no—
51C2×D4×C197—21no—
52C2×D4×C197D411yescharacteristic
53C22×C4×C197—21no—
54C22×C4×C197C2311yescharacteristic
55C2×C8×C197—21no—
56C4⋊C4×C197D411yescharacteristic
57C22⋊C4×C197—21no—
58C22×D197—21no—
59C22×D197—21no—
60C22×D197—21no—
61C22×D197—42no—
62C22×D197—41no—
63C22×D197—41no—
64C22×D197—81no—
65C22×D197—81no—
66C22×D197SD1612yes—
67D788—41no—
68D788—42no—
69C22⋊D197—42no—
70C22⋊D197—41no—
71C22⋊D197—41no—
72C22⋊D197—81no—
73C2×Dic197—21no—
74C2×Dic197—21no—
75C2×Dic197—41no—
76C2×Dic197—41no—
77C2×Dic197—41no—
78C2×Dic197—41no—
79C4×D197—42no—
80#13096802—41no—
81C23×C197—21no—
82C23×C197C2×D411yescharacteristic
83D4×C197—41no—
84D4×C197—41no—
85D4×C197—41no—
86C2×C4×C197—22no—
87C2×C4×C197—21no—
88C2×C4×C197—41no—
89C2×C4×C197—41no—
90C2×C4×C197—41no—
91C2×C4×C197C2×D411yescharacteristic
92C2×C4×C197C2×D411yescharacteristic
93C8×C197—41no—
94C22×C197—21no—
95C22×C197—21no—
96C22×C197—41no—
97C22×C197—41no—
98C22×C197—41no—
99C22×C197#12011yescharacteristic
100C22×C197C2×SD1611yescharacteristic
101C22×C197AGL(1,ℤ/8)11yescharacteristic
102C2×D197—22no—
103C2×D197—41no—
104C2×D197—41no—
105C2×D197—42no—
106C2×D197—41no—
107C2×D197—41no—
108C2×D197—41no—
109C2×D197—81no—
110C2×D197—84no—
111C4×C197—21no—
112C4×C197—21no—
113C4×C197—42no—
114C4×C197—41no—
115Dic197—41no—
116Dic197—41no—
117Dic197—81no—
118C2×C197—21no—
119C2×C197—21no—
120C2×C197—81no—
121C2×C197#45011yescharacteristic
122C2×C197#45311yescharacteristic
123C2×C197#45811yescharacteristic
124D197—42no—
125D197—81no—
126C197#161311yescharacteristic, complemented, Sylow
127#1613—1971nomaximal, Sylow
128#534—1971no—
129#481—1971no—
130#481—1971no—
131#399—1971no—
132#327—1971no—
133#327D19711yescharacteristic, complemented
134#323—1971no—
135C22×D4—3941no—
136#123—3941no—
137#123—3941no—
138#121—3941no—
139#121—1971no—
140#121C2×D19711yescharacteristic
141#120—3941no—
142#120—3941no—
143C2×C4⋊C4—3941no—
144C2×C4⋊C4—1971no—
145C2×C22⋊C4—3941no—
146C2×C22⋊C4—1971no—
147#106—3941no—
148#106—3941no—
149#102—3941no—
150#102—3941no—
151C22⋊C8—1971no—
152C22⋊C8C2×D19711yescharacteristic, complemented
153#95C2×D19711yescharacteristic, complemented
154C24—3941no—
155C24—1971no—
156C2×D4—21no—
157C2×D4—7882no—
158C2×D4—7882no—
159C2×D4—3941no—
160C2×D4—3941no—
161C2×D4—3941no—
162C2×D4—3941no—
163C2×D4—1971no—
164C2×D4D78811yescharacteristic
165C22×C4—21no—
166C22×C4—3941no—
167C22×C4—3941no—
168C22×C4—1971no—
169C22×C4C22×D19711yescharacteristic
170C2×C8—21no—
171C2×C8—3941no—
172C4⋊C4—7882no—
173C4⋊C4—3941no—
174C4⋊C4—3941no—
175C4⋊C4—3941no—
176C4⋊C4—1971no—
177C4⋊C4D78811yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—7882no—
180C22⋊C4—3941no—
181C22⋊C4—3941no—
182C22⋊C4—3941no—
183C22⋊C4—3941no—
184C22⋊C4—3941no—
185C23—21no—
186C23—7881no—
187C23—7881no—
188C23—7882no—
189C23—15761no—
190C23—15761no—
191C23—3941no—
192C23—3941no—
193C23—3941no—
194C23—1972no—
195C23D4×D19711yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—7881no—
200D4—7882no—
201D4—7882no—
202D4—7881no—
203D4—7881no—
204D4—15761no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—7882no—
211C2×C4—7881no—
212C2×C4—7881no—
213C2×C4—7881no—
214C2×C4—7881no—
215C2×C4—3941no—
216C2×C4—3941no—
217C2×C4D4×D19711yescharacteristic
218C2×C4C2×D78811yescharacteristic
219C8—41no—
220C8—7881no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—7882no—
227C22—7881no—
228C22—7881no—
229C22—7881no—
230C22—7881no—
231C22—7881no—
232C22—15761no—
233C22—15764no—
234C22—3942no—
235C22#2026329611yescharacteristic
236C22SD16×D19711yescharacteristic
237C22#2026335011yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—7881no—
243C4—7881no—
244C4—15761no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—7882no—
249C2—15761no—
250C2#2716123111yescharacteristic
251C2#2716124811yescharacteristic
252C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197#120
prime radical#27161637C2
hypercenter#327D197
hypocenterC197#1613
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. #27161398C2semidirect
3. #27161395C2semidirect
4. #27161016C2semidirect
5. #27160812C2semidirect
6. #27160802C2semidirect
7. #327×C197C2semidirect
16. C2×C4⋊Dic197C22semidirect
23. C22⋊#13096802C22semidirect
25. C22⋊C8×C197C22semidirect
26. #95×C197C22semidirect
126. C197#1613semidirect
133. #327D197semidirect
152. C22⋊C8C2×D197semidirect
153. #95C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616372
#271613982
#271613952
#271610162
#271608122
#271608022
#327×C1972
#1613197