Finite Group Databasebeta

← #38743123

Subgroups — #38743123

Order 25216.

386 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20263307C211yesmaximal, characteristic
2#27161554C211yesmaximal, characteristic
3#27160940C212yesmaximal, complemented
4#27160845C212yesmaximal, complemented
5C2×#99×C197C211yesmaximal, characteristic
6C23×Dic197C2211yescharacteristic
7#20263307—22no—
8#20263307C414yescomplemented
9D4×Dic197—24no—
10#20263285—21no—
11C2×#15691337—21no—
12C2×C22⋊Dic197—21no—
13C2×C22⋊Dic197—21no—
14C2×C22⋊Dic197C2212yes—
15C2×C22⋊Dic197C2211yescharacteristic
16#20263233—24no—
17C22×D4×C197C2211yescharacteristic
18C2×C22⋊C4×C197C2212yes—
19#99×C197—22no—
20C22×Dic197—22no—
21C22×Dic197—22no—
22C22×Dic197—21no—
23C22×Dic197—21no—
24C22×Dic197—21no—
25C22×Dic197—22no—
26C22×Dic197C2×C412yescomplemented
27C22×Dic197D414yes—
28C4⋊Dic197—22no—
29#15691337—24no—
30C22⋊Dic197—22no—
31C22⋊Dic197—24no—
32C22⋊Dic197—22no—
33C22⋊Dic197—22no—
34C22⋊Dic197—21no—
35C22⋊Dic197C2×C412yescomplemented
36C4×Dic197—22no—
37C24×C197D411yescharacteristic
38C24×C197C2311yescharacteristic
39C2×D4×C197—22no—
40C2×D4×C197—22no—
41C2×D4×C197C2×C412yes—
42C2×D4×C197D412yes—
43C22×C4×C197—22no—
44C22×C4×C197D411yescharacteristic
45C22⋊C4×C197—22no—
46C22⋊C4×C197—22no—
47C2×Dic197—22no—
48C2×Dic197—22no—
49C2×Dic197—24no—
50C2×Dic197—24no—
51C2×Dic197—42no—
52C2×Dic197—41no—
53C2×Dic197—42no—
54C2×Dic197—41no—
55C2×Dic197—44no—
56C2×Dic197—82no—
57C23×C197—21no—
58C23×C197—21no—
59C23×C197—42no—
60C23×C197—42no—
61C23×C197C22⋊C412yes—
62C23×C197C22×C411yescharacteristic
63C23×C197C2×D412yes—
64C23×C197C2×D411yescharacteristic
65C23×C197C2×D411yescharacteristic
66C23×C197#4112yes—
67D4×C197—24no—
68D4×C197—24no—
69C2×C4×C197—21no—
70C2×C4×C197—21no—
71C2×C4×C197—44no—
72C2×C4×C197—42no—
73C2×C4×C197C22⋊C412yes—
74C22×C197—24no—
75C22×C197—22no—
76C22×C197—22no—
77C22×C197—41no—
78C22×C197—41no—
79C22×C197—42no—
80C22×C197—42no—
81C22×C197—42no—
82C22×C197—81no—
83C22×C197C2×C22⋊C411yescharacteristic
84C22×C197C4×D412yes—
85C22×C197#12011yescharacteristic
86C22×C197#12112yes—
87C22×C197#12311yescharacteristic
88C4×C197—22no—
89C4×C197—82no—
90Dic197—42no—
91Dic197—42no—
92Dic197—44no—
93C2×C197—21no—
94C2×C197—22no—
95C2×C197—21no—
96C2×C197—22no—
97C2×C197—42no—
98C2×C197#38611yescharacteristic
99C2×C197#45812yes—
100C197#189011yescharacteristic, complemented, Sylow
101#1890—1971nomaximal, Sylow
102C2×#123—1971no—
103#518—1971no—
104C2×#99D19711yescharacteristic
105#381—1972no—
106#342—1972no—
107C22×D4C2×D19711yescharacteristic
108C23×C4—1971no—
109#123—3942no—
110#123—1974no—
111C4×D4—3944no—
112#117—3941no—
113C2×C4⋊C4—3941no—
114C2×C22⋊C4—3941no—
115C2×C22⋊C4—3941no—
116C2×C22⋊C4—1972no—
117C2×C22⋊C4—1971no—
118C2×C22⋊C4C2×D19712yes—
119#99—22no—
120#95—3944no—
121C24D78811yescharacteristic
122C24C22×D19711yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C4×D19712yes—
126C2×D4C22⋊D19712yes—
127C22×C4—22no—
128C22×C4—3942no—
129C22×C4—3941no—
130C22×C4—3941no—
131C22×C4—3941no—
132C22×C4—3942no—
133C22×C4—3942no—
134C22×C4—1974no—
135C22×C4—1972no—
136C22×C4D78811yescharacteristic
137C4⋊C4—3942no—
138C4⋊C4—3944no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—3942no—
142C22⋊C4—3944no—
143C22⋊C4—3942no—
144C22⋊C4—3941no—
145C22⋊C4—3942no—
146C22⋊C4—1972no—
147C42—3942no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23#1569133512yes—
153C23C2×C4×D19711yescharacteristic
154C23D4×D19712yes—
155C23C2×C22⋊D19711yescharacteristic
156C23C2×D78811yescharacteristic
157C23#1569135712yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4—7882no—
165C2×C4—7881no—
166C2×C4—7882no—
167C2×C4—7881no—
168C2×C4—7884no—
169C2×C4—15762no—
170C2×C4—3942no—
171C2×C4—3942no—
172C2×C4—3944no—
173C2×C4—3944no—
174C2×C4#1569133512yes—
175C22—24no—
176C22—22no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×#1569133511yescharacteristic
185C22#2026328612yes—
186C22#2026329611yescharacteristic
187C22#2026329812yes—
188C22#2026331011yescharacteristic
189C4—22no—
190C4—82no—
191C4—7882no—
192C4—7882no—
193C4—7884no—
194C2—22no—
195C2—21no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#2716095611yescharacteristic
200C2#2716127012yes—
subgroupG/H
centerC22#20263296
commutatorC23×C197C22×C4
FrattiniC24C22×D197
FittingC2×#99×C197C2
socleC22×C197#120
prime radicalC22×D4×C197C22
cosocleC24×C197C23
hypercenterC2×#99D197
hypocenterC197#1890
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #27160940C2semidirect
4. #27160845C2semidirect
8. #20263307C4semidirect
26. C22×Dic197C2×C4semidirect
35. C22⋊Dic197C2×C4semidirect
100. C197#1890semidirect

Maximal subgroups

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

subgroupindex
C2×#202633072
#271615542
#271609402
#271608452
C2×#99×C1972
#1890197