Finite Group Databasebeta

← #38742785

Subgroups — #38742785

Order 25216.

504 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 81 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×C22⋊C4×D197C211yesmaximal, characteristic
2#27160934C212yesmaximal
3#27160846C211yesmaximal, characteristic, complemented
4#27160845C212yesmaximal, complemented
5#342×C197C211yesmaximal, characteristic, complemented
6C24×D197C2211yescharacteristic
7C22×C4×D197C412yescomplemented
8C2×C22⋊Dic197C2212yes—
9C2×C22⋊Dic197C2211yescharacteristic
10C2×#15691335—24no—
11C2×#15691335C412yescomplemented
12C22⋊C4×D197—24no—
13C2×C22⋊C4×C197C2212yes—
14C2×C22⋊C4×C197C2211yescharacteristic
15C23×D197—21no—
16C23×D197—22no—
17C23×D197—21no—
18C23×D197—44no—
19C23×D197C2×C411yescharacteristic
20C23×D197C2×C411yescharacteristic
21C23×D197D414yes—
22C22×Dic197—22no—
23C22×Dic197C2×C412yes—
24C2×C4×D197—22no—
25C2×C4×D197—22no—
26C2×C4×D197—44no—
27C22⋊Dic197—22no—
28C22⋊Dic197—22no—
29C22⋊Dic197—22no—
30#15691335—24no—
31#15691335—48no—
32C24×C197C2311yescharacteristic
33C22×C4×C197—22no—
34C22×C4×C197C2×C412yes—
35C22⋊C4×C197—22no—
36C22⋊C4×C197—22no—
37C22⋊C4×C197—22no—
38C22×D197—21no—
39C22×D197—21no—
40C22×D197—24no—
41C22×D197—21no—
42C22×D197—21no—
43C22×D197—24no—
44C22×D197—24no—
45C22×D197—28no—
46C22×D197—42no—
47C22×D197—44no—
48C22×D197—44no—
49C22×D197—48no—
50C22×D197—44no—
51C22×D197C22⋊C412yes—
52C22×D197C22⋊C412yes—
53C2×Dic197—22no—
54C2×Dic197—42no—
55C2×Dic197—44no—
56C2×Dic197—42no—
57C4×D197—44no—
58C4×D197—44no—
59C23×C197—42no—
60C23×C197C22×C411yescharacteristic
61C23×C197C2×D412yes—
62C23×C197#4111yescharacteristic
63C23×C197#4112yes—
64C23×C197#4111yescharacteristic
65C2×C4×C197—22no—
66C2×C4×C197—42no—
67C2×C4×C197—44no—
68C2×C4×C197—42no—
69C22×C197—22no—
70C22×C197—24no—
71C22×C197—41no—
72C22×C197—42no—
73C22×C197—41no—
74C22×C197C2×C22⋊C411yescharacteristic
75C22×C197#11711yescharacteristic
76C22×C197#11711yescharacteristic
77C22×C197#12312yes—
78C22×C197#12312yes—
79C2×D197—24no—
80C2×D197—24no—
81C2×D197—22no—
82C2×D197—22no—
83C2×D197—22no—
84C2×D197—24no—
85C2×D197—48no—
86C2×D197—48no—
87C2×D197—42no—
88C2×D197—42no—
89C2×D197#9914yes—
90C4×C197—42no—
91C4×C197—82no—
92Dic197—42no—
93Dic197—82no—
94C2×C197—22no—
95C2×C197—22no—
96C2×C197—21no—
97C2×C197—21no—
98C2×C197#38111yescharacteristic
99C2×C197C2×#9912yes—
100D197—24no—
101D197—42no—
102C197#181911yescharacteristic, complemented, Sylow
103#1819—1971nomaximal, Sylow
104C22×C22⋊C4—1971no—
105#379—1972no—
106#342—1972no—
107#342—1971no—
108#342D19711yescharacteristic, complemented
109C25—1971no—
110C23×C4—1972no—
111C2×C22⋊C4—3944no—
112C2×C22⋊C4—3944no—
113C2×C22⋊C4—1972no—
114C2×C22⋊C4—1971no—
115C2×C22⋊C4—1972no—
116C2×C22⋊C4C2×D19712yes—
117C2×C22⋊C4C2×D19711yescharacteristic
118C24—7884no—
119C24—3941no—
120C24—3942no—
121C24—3941no—
122C24—1971no—
123C24—1974no—
124C24—1971no—
125C24C22×D19711yescharacteristic
126C22×C4—22no—
127C22×C4—7884no—
128C22×C4—3942no—
129C22×C4—3942no—
130C22×C4—3942no—
131C22×C4—1972no—
132C22×C4C4×D19712yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—7888no—
137C22⋊C4—3944no—
138C22⋊C4—3942no—
139C22⋊C4—3942no—
140C22⋊C4—3942no—
141C23—42no—
142C23—7882no—
143C23—7884no—
144C23—7884no—
145C23—7888no—
146C23—7884no—
147C23—3948no—
148C23—3944no—
149C23—3941no—
150C23—3941no—
151C23—3944no—
152C23—3944no—
153C23—3941no—
154C23—3941no—
155C23—1972no—
156C23—1972no—
157C23C2×C4×D19711yescharacteristic
158C23D4×D19712yes—
159C23#1569135711yescharacteristic
160C23#1569135812yes—
161C23#1569135911yescharacteristic
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—7884no—
167C2×C4—7884no—
168C2×C4—7882no—
169C2×C4—7884no—
170C2×C4—7882no—
171C2×C4—3942no—
172C22—22no—
173C22—24no—
174C22—41no—
175C22—42no—
176C22—41no—
177C22—7882no—
178C22—7882no—
179C22—7888no—
180C22—7888no—
181C22—3944no—
182C22—3944no—
183C22—3942no—
184C22—3944no—
185C22—3942no—
186C22—3942no—
187C22—1974no—
188C22C22⋊C4×D19711yescharacteristic
189C22#2026328311yescharacteristic
190C22#2026328411yescharacteristic
191C22#2026330812yes—
192C22#2026331112yes—
193C4—42no—
194C4—82no—
195C4—7882no—
196C4—15762no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—21no—
201C2—7882no—
202C2—3944no—
203C2#2716094111yescharacteristic
204C2#99×D19712yes—
subgroupG/H
centerC22C22⋊C4×D197
commutatorC23×C197C22×C4
FrattiniC24C22×D197
Fitting#342×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC24×D197C22
cosocleC24×C197C23
hypercenter#342D197
hypocenterC197#1819
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #27160846C2semidirect
4. #27160845C2semidirect
5. #342×C197C2semidirect
7. C22×C4×D197C4semidirect
11. C2×#15691335C4semidirect
102. C197#1819semidirect
108. #342D197semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1972
#271609342
#271608462
#271608452
#342×C1972
#1819197