Finite Group Databasebeta

← #398×D197 #38745941

Subgroups — #398×D197 #38745941

Order 25216.

692 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D197C211yesmaximal, characteristic
2C2×C4⋊C4×D197C211yesmaximal, characteristic
3C2×C4⋊C4×D197C212yesmaximal
4#27161013C212yesmaximal, complemented
5#27161012C212yesmaximal, complemented
6#27161011C211yesmaximal, characteristic, complemented
7#27161010C211yesmaximal, characteristic, complemented
8#27161009C211yesmaximal, characteristic, complemented
9#95×D197C211yesmaximal, characteristic
10#95×D197C212yesmaximal
11#398×C197C211yesmaximal, characteristic, complemented
12C22×C4×D197C2211yescharacteristic
13C22×C4×D197C2212yes—
14C22×C4×D197C2211yescharacteristic
15C22×C4×D197C2211yescharacteristic
16C22×C4×D197C2212yes—
17C2×C4⋊Dic197C2211yescharacteristic
18C2×C4⋊Dic197C2212yes—
19C2×C4⋊Dic197C2211yescharacteristic
20C2×#15691337C2212yes—
21C2×#15691337C2211yescharacteristic
22C2×#15691337C2212yes—
23C2×#15691337C2212yes—
24C2×#15691337C2211yescharacteristic
25C4⋊C4×D197—24no—
26C4⋊C4×D197—28no—
27C4⋊C4×D197—24no—
28#20263233C2211yescharacteristic
29#20263233C2212yes—
30#20263233C2212yes—
31#20263233C2212yes—
32#20263233C2212yes—
33C2×C4⋊C4×C197C2211yescharacteristic
34C2×C4⋊C4×C197C2211yescharacteristic
35C2×C4⋊C4×C197C2212yes—
36#95×C197C2211yescharacteristic
37#95×C197C2212yes—
38C23×D197C2311yescharacteristic
39C22×Dic197C2311yescharacteristic
40C22×Dic197C2312yes—
41C22×Dic197C2311yescharacteristic
42C22×Dic197C2311yescharacteristic
43C22×Dic197C2312yes—
44C2×C4×D197—22no—
45C2×C4×D197—24no—
46C2×C4×D197—22no—
47C2×C4×D197—24no—
48C2×C4×D197—24no—
49C2×C4×D197—24no—
50C2×C4×D197—24no—
51C2×C4×D197—22no—
52C2×C4×D197—24no—
53C2×C4×D197—24no—
54C2×C4×D197D414yes—
55C2×C4×D197D414yes—
56C2×C4×D197Q818yes—
57C4⋊Dic197—22no—
58C4⋊Dic197—24no—
59C4⋊Dic197—22no—
60#15691337—22no—
61#15691337—24no—
62#15691337—24no—
63#15691337—24no—
64#15691337—22no—
65C22×C4×C197C2311yescharacteristic
66C22×C4×C197C2312yes—
67C22×C4×C197C2311yescharacteristic
68C22×C4×C197C2311yescharacteristic
69C22×C4×C197C2312yes—
70C4⋊C4×C197—24no—
71C4⋊C4×C197—22no—
72C4⋊C4×C197—22no—
73C22×D197C2×D412yes—
74C22×D197C2×D412yes—
75C22×D197C2×Q814yes—
76C22×D197#4112yes—
77C22×D197#4114yes—
78C2×Dic197—21no—
79C2×Dic197—22no—
80C2×Dic197—21no—
81C2×Dic197—22no—
82C2×Dic197—22no—
83C2×Dic197—22no—
84C2×Dic197—22no—
85C2×Dic197—21no—
86C2×Dic197—22no—
87C2×Dic197—22no—
88C2×Dic197C2×D412yes—
89C2×Dic197C2×D412yes—
90C2×Dic197C2×Q814yes—
91C4×D197—28no—
92C4×D197—216no—
93C4×D197—28no—
94C4×D197—48no—
95C4×D197—44no—
96C23×C197C2411yescharacteristic
97C2×C4×C197—21no—
98C2×C4×C197—22no—
99C2×C4×C197—21no—
100C2×C4×C197—22no—
101C2×C4×C197—22no—
102C2×C4×C197—22no—
103C2×C4×C197—22no—
104C2×C4×C197—21no—
105C2×C4×C197—22no—
106C2×C4×C197—22no—
107C2×C4×C197C2×D412yes—
108C2×C4×C197C2×D412yes—
109C2×C4×C197C2×Q814yes—
110C22×C197C22×D411yescharacteristic
111C22×C197C22×D411yescharacteristic
112C22×C197C22×Q812yes—
113C22×C197C2×#4111yescharacteristic
114C22×C197C2×#4112yes—
115C2×D197#12114yes—
116C2×D197C22⋊Q818yes—
117C2×D197#12314yes—
118C2×D197#12518yes—
119C2×D197C4⋊Q814yes—
120C4×C197—22no—
121C4×C197—24no—
122C4×C197—22no—
123C4×C197—42no—
124C4×C197—41no—
125Dic197—22no—
126Dic197—24no—
127Dic197—22no—
128Dic197—42no—
129Dic197—41no—
130C2×C197C2×#12111yescharacteristic
131C2×C197C2×C22⋊Q812yes—
132C2×C197C2×#12311yescharacteristic
133C2×C197C2×#12512yes—
134C2×C197C2×C4⋊Q811yescharacteristic
135D197#39818yesdirect product factor
136C197C2×#39811yescharacteristic, complemented, Sylow
137C2×#398—1971nomaximal, Sylow
138C22×C4⋊C4—1971no—
139C22×C4⋊C4—1972no—
140C22×C4⋊C4—1971no—
141#398—1971no—
142#398—1971no—
143#398—1972no—
144#398—1971no—
145#398—1972no—
146#398D19711yescharacteristic, direct product factor
147C2×#95—1972no—
148C2×#95—1971no—
149C23×C4—1972no—
150C23×C4—1971no—
151C23×C4—1971no—
152C23×C4—1971no—
153C23×C4—1972no—
154C2×C4⋊C4—3944no—
155C2×C4⋊C4—3948no—
156C2×C4⋊C4—3944no—
157C2×C4⋊C4—1972no—
158C2×C4⋊C4—1972no—
159C2×C4⋊C4—1971no—
160C2×C4⋊C4—1971no—
161C2×C4⋊C4—1972no—
162C2×C4⋊C4—1971no—
163C2×C4⋊C4—1972no—
164C2×C4⋊C4—1971no—
165C2×C4⋊C4C2×D19711yescharacteristic
166C2×C4⋊C4C2×D19711yescharacteristic
167C2×C4⋊C4C2×D19712yes—
168#95—1972no—
169#95—1972no—
170#95—1972no—
171#95—1971no—
172#95—1972no—
173#95C2×D19712yes—
174#95C2×D19711yescharacteristic
175C24—1971no—
176C22×C4—3944no—
177C22×C4—3944no—
178C22×C4—3942no—
179C22×C4—3944no—
180C22×C4—3944no—
181C22×C4—3942no—
182C22×C4—3944no—
183C22×C4—3942no—
184C22×C4—3944no—
185C22×C4—3944no—
186C22×C4—1971no—
187C22×C4—1972no—
188C22×C4—1972no—
189C22×C4—1971no—
190C22×C4—1971no—
191C22×C4—1974no—
192C22×C4—1974no—
193C22×C4—1978no—
194C22×C4C22×D19711yescharacteristic
195C22×C4C22×D19712yes—
196C22×C4C22×D19711yescharacteristic
197C22×C4C22×D19711yescharacteristic
198C22×C4C22×D19712yes—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—3944no—
203C4⋊C4—3944no—
204C4⋊C4—3942no—
205C4⋊C4—3942no—
206C4⋊C4—3944no—
207C4⋊C4—3944no—
208C4⋊C4—3942no—
209C4⋊C4—3942no—
210C23—1972no—
211C23—1972no—
212C23—1974no—
213C23—1972no—
214C23—1974no—
215C23C23×D19711yescharacteristic
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—7888no—
227C2×C4—7884no—
228C2×C4—3948no—
229C2×C4—39416no—
230C2×C4—3948no—
231C2×C4—3942no—
232C2×C4—3942no—
233C2×C4—3941no—
234C2×C4—3942no—
235C2×C4—3942no—
236C2×C4—3941no—
237C2×C4—3942no—
238C2×C4—3941no—
239C2×C4—3942no—
240C2×C4—3942no—
241C2×C4—1972no—
242C2×C4—1972no—
243C2×C4—1974no—
244C2×C4D4×D19712yes—
245C2×C4D4×D19712yes—
246C2×C4Q8×D19714yes—
247C22—1974no—
248C22—1974no—
249C22—1978no—
250C22—1974no—
251C22—1978no—
252C22C2×D4×D19711yescharacteristic
253C22C2×D4×D19711yescharacteristic
254C22C2×Q8×D19712yes—
255C22#41×D19711yescharacteristic
256C22#41×D19712yes—
257C4—22no—
258C4—24no—
259C4—22no—
260C4—42no—
261C4—41no—
262C4—7882no—
263C4—7881no—
264C4—3942no—
265C4—3944no—
266C4—3942no—
267C2—1978no—
268C2#121×D19711yescharacteristic
269C2C22⋊Q8×D19712yes—
270C2#123×D19711yescharacteristic
271C2#125×D19712yes—
272C2C4⋊Q8×D19711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#398×C197C2
prime radicalC23×D197C23
hypercenter#398D197
hypocenterC197C2×#398
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
4. #27161013C2semidirect
5. #27161012C2semidirect
6. #27161011C2semidirect
7. #27161010C2semidirect
8. #27161009C2semidirect
11. #398×C197C2semidirect
135. D197#398direct (N × H)
136. C197C2×#398semidirect
146. #398D197direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1972
#271610132
#271610122
#271610112
#271610102
#271610092
#95×D1972
#398×C1972
C2×#398197