Finite Group Databasebeta

← #38745991

Subgroups — #38745991

Order 25216.

840 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C2×C4×D788C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D197C214yesmaximal, complemented
3#27161015C212yesmaximal, complemented
4#27161014C212yesmaximal, complemented
5#27160988C212yesmaximal, complemented
6#27160987C212yesmaximal, complemented
7#27160925C211yesmaximal, characteristic, complemented
8C2×C4×D4×C197C211yesmaximal, characteristic, complemented
9C22×D788C2211yescharacteristic
10C22×C22⋊D197C2214yes—
11C22×C4×D197C2212yes—
12C4×D788—24no—
13C4×C22⋊D197—216no—
14C2×C4⋊Dic197C2211yescharacteristic
15C2×#15691337C2214yes—
16C2×C22⋊Dic197C2214yes—
17C2×#15691335C2212yes—
18C2×#15691335C2214yes—
19C2×C4×Dic197C2212yes—
20#20263233C2212yes—
21#20263233C2212yes—
22C22×D4×C197C2211yescharacteristic
23C23×C4×C197C2212yes—
24C4×D4×C197—24no—
25C2×C4⋊C4×C197C2211yescharacteristic
26C2×C22⋊C4×C197C2212yes—
27C2×C42×C197C2211yescharacteristic
28C23×D197C2312yes—
29C2×D788—24no—
30C2×D788—22no—
31C2×C22⋊D197—28no—
32C2×C22⋊D197—28no—
33C2×C22⋊D197—28no—
34C22×Dic197C2312yes—
35C22×Dic197C2314yes—
36C2×C4×D197—28no—
37C2×C4×D197—24no—
38C4⋊Dic197—22no—
39#15691337—28no—
40C22⋊Dic197—28no—
41#15691335—28no—
42#15691335—24no—
43C4×Dic197D418yescomplemented
44C24×C197C2312yes—
45C2×D4×C197—24no—
46C2×D4×C197—22no—
47C22×C4×C197—24no—
48C22×C4×C197—28no—
49C22×C4×C197C2312yes—
50C22×C4×C197C2311yescharacteristic
51C22×C4×C197C2311yescharacteristic
52C22×C4×C197C2311yescharacteristic
53C4⋊C4×C197—22no—
54C22⋊C4×C197—24no—
55C42×C197D414yescomplemented
56C22×D197—24no—
57C22×D197—22no—
58C22×D197—24no—
59C22×D197—22no—
60C22×D197#4114yes—
61D788—28no—
62C22⋊D197—232no—
63C2×Dic197—24no—
64C2×Dic197—24no—
65C2×Dic197—22no—
66C2×Dic197—24no—
67C2×Dic197C2×D418yes—
68C4×D197—48no—
69C23×C197—22no—
70C23×C197—24no—
71C23×C197—22no—
72C23×C197—24no—
73C23×C197#4114yes—
74C23×C197C2411yescharacteristic
75D4×C197—28no—
76C2×C4×C197—22no—
77C2×C4×C197—22no—
78C2×C4×C197—22no—
79C2×C4×C197—24no—
80C2×C4×C197—48no—
81C2×C4×C197C2×D414yes—
82C2×C4×C197C2×D412yes—
83C2×C4×C197C2×D412yes—
84C2×C4×C197C2×D412yes—
85C22×C197—24no—
86C22×C197—28no—
87C22×C197—42no—
88C22×C197—44no—
89C22×C197—42no—
90C22×C197C22×D412yes—
91C22×C197C22×D411yescharacteristic
92C22×C197C2×#4112yes—
93C22×C197C2×#4112yes—
94C2×D197—28no—
95C2×D197—24no—
96C2×D197—44no—
97C2×D197—42no—
98C2×D197—42no—
99C4×C197—22no—
100C4×C197—22no—
101C4×C197—42no—
102C4×C197#12714yes—
103Dic197—28no—
104Dic197—42no—
105C2×C197—44no—
106C2×C197#52511yescharacteristic
107C2×C197#52514yes—
108C2×C197#52511yescharacteristic
109C2×C197C2×#12711yescharacteristic
110D197—44no—
111C197#239711yescharacteristic, complemented, Sylow
112#2397—1971nomaximal, Sylow
113C2×C4×D4—1974no—
114C2×C4×D4—1971no—
115C2×C4×D4D19711yescharacteristic, complemented
116#399—1972no—
117#399—1972no—
118#392—1972no—
119#392—1972no—
120#376—1971no—
121C22×D4—1974no—
122C22×D4—1971no—
123C22×D4C2×D19711yescharacteristic
124C23×C4—1972no—
125C23×C4C2×D19712yes—
126C4×D4—24no—
127C4×D4—39416no—
128C4×D4—3944no—
129C2×C4⋊C4—1974no—
130C2×C4⋊C4—1971no—
131C2×C4⋊C4C2×D19711yescharacteristic
132C2×C22⋊C4—1974no—
133C2×C22⋊C4—1974no—
134C2×C22⋊C4—1972no—
135C2×C22⋊C4C2×D19712yes—
136C2×C42—1972no—
137C2×C42C2×D19711yescharacteristic
138#95—1972no—
139#95—1972no—
140C24—1972no—
141C24C22×D19712yes—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—3944no—
145C2×D4—3948no—
146C2×D4—3948no—
147C2×D4—3948no—
148C2×D4—3942no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—3948no—
152C22×C4—3944no—
153C22×C4—1972no—
154C22×C4—1974no—
155C22×C4C22×D19712yes—
156C22×C4C22×D19711yescharacteristic
157C22×C4C22×D19711yescharacteristic
158C22×C4C22×D19711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—3948no—
161C4⋊C4—3942no—
162C22⋊C4—24no—
163C22⋊C4—3948no—
164C22⋊C4—3948no—
165C22⋊C4—3944no—
166C42—1978no—
167C42C22⋊D19714yescomplemented
168C23—24no—
169C23—22no—
170C23—24no—
171C23—22no—
172C23—3944no—
173C23—3942no—
174C23—3942no—
175C23—3944no—
176C23—1974no—
177C23#1569135814yes—
178C23C23×D19711yescharacteristic
179D4—28no—
180D4—3948no—
181D4—39432no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—7888no—
188C2×C4—3944no—
189C2×C4—3942no—
190C2×C4—3944no—
191C2×C4—3944no—
192C2×C4—1978no—
193C2×C4D4×D19714yes—
194C2×C4C2×C22⋊D19712yes—
195C2×C4C2×C22⋊D19712yes—
196C2×C4C2×C22⋊D19712yes—
197C22—28no—
198C22—24no—
199C22—42no—
200C22—42no—
201C22—44no—
202C22—7882no—
203C22—7884no—
204C22—7882no—
205C22—3944no—
206C22—3948no—
207C22C2×D4×D19712yes—
208C22C22×C22⋊D19711yescharacteristic
209C22#41×D19712yes—
210C22C2×#1569135812yes—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—7882no—
215C4—3948no—
216C4#2026332214yes—
217C2—44no—
218C2—7884no—
219C2#2716158611yescharacteristic
220C2#2716158814yes—
221C2#2716159011yescharacteristic
222C2C2×#2026332211yescharacteristic
subgroupG/H
centerC22×C4C22×D197
commutator, socle, cosocleC23×C197C24
FrattiniC23C23×D197
FittingC2×C4×D4×C197C2
hypercenterC2×C4×D4D197
hypocenterC197#2397
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D788C2semidirect
2. C2×C4×C22⋊D197C2semidirect
3. #27161015C2semidirect
4. #27161014C2semidirect
5. #27160988C2semidirect
6. #27160987C2semidirect
7. #27160925C2semidirect
8. C2×C4×D4×C197C2semidirect
43. C4×Dic197D4semidirect
55. C42×C197D4semidirect
111. C197#2397semidirect
115. C2×C4×D4D197semidirect
167. C42C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D7882
C2×C4×C22⋊D1972
#271610152
#271610142
#271609882
#271609872
#271609252
C2×C4×D4×C1972
#2397197