Finite Group Databasebeta

← #38743130

Subgroups — #38743130

Order 25216.

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

normal subgroups only

This group has 342 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
1#27161586C211yesmaximal, characteristic
2#27161063C211yesmaximal, characteristic, complemented
3#27161042C212yesmaximal, complemented
4#27160850C212yesmaximal, complemented
5#568×C197C211yesmaximal, characteristic, complemented
6C22×C4×D197C2211yescharacteristic
7C2×#15691342C2212yes—
8#20263307C412yes—
9#20263302—21no—
10#20263297—21no—
11#20263294C412yes—
12C4×C22⋊D197—22no—
13#20263285C2211yescharacteristic
14#20263245—21no—
15#20263243—21no—
16#20263238—24no—
17C2×#41×C197C2211yescharacteristic
18#131×C197—24no—
19C2×M16×C197—22no—
20C2×M16×C197C2212yes—
21C23×D197C2×C411yescharacteristic
22C2×C22⋊D197—22no—
23C22×Dic197C2×C411yescharacteristic
24C2×C4×D197—22no—
25C2×C4×D197—21no—
26C2×C4×D197—21no—
27C2×C4×D197—22no—
28#15691342—22no—
29#15691342—22no—
30C2×#13096802—22no—
31C4⋊Dic197—21no—
32#15691337—22no—
33C22⋊Dic197—22no—
34C22⋊Dic197C2×C412yes—
35#15691335—22no—
36C4×Dic197—21no—
37#41×C197—22no—
38#41×C197—22no—
39C2×Q8×C197D411yescharacteristic
40C2×D4×C197C2×C412yes—
41C2×D4×C197D411yescharacteristic
42C22×C4×C197D412yes—
43C22×C4×C197C2311yescharacteristic
44M16×C197—24no—
45M16×C197—22no—
46M16×C197—22no—
47C2×C8×C197—24no—
48C2×C8×C197D414yescomplemented
49C22×D197—21no—
50C22×D197—22no—
51C22×D197—42no—
52C22⋊D197—44no—
53C2×Dic197—22no—
54C2×Dic197—21no—
55C2×Dic197—22no—
56C4×D197—41no—
57C4×D197—42no—
58C4×D197—41no—
59#13096802—42no—
60C23×C197C22⋊C412yes—
61C23×C197C22×C411yescharacteristic
62Q8×C197—22no—
63D4×C197—24no—
64D4×C197—22no—
65C2×C4×C197—22no—
66C2×C4×C197—22no—
67C2×C4×C197C22⋊C412yes—
68C2×C4×C197C2×D412yes—
69C2×C4×C197C2×D411yescharacteristic
70C2×C4×C197C2×D411yescharacteristic
71C2×C4×C197#4112yes—
72C8×C197—24no—
73C22×C197—22no—
74C22×C197—41no—
75C22×C197—42no—
76C22×C197C2×C22⋊C411yescharacteristic
77C22×C197C4×D412yes—
78C2×D197—22no—
79C2×D197—24no—
80C2×D197—41no—
81C2×D197—42no—
82C2×D197—41no—
83C4×C197—22no—
84C4×C197#12011yescharacteristic
85C4×C197#12112yes—
86C4×C197#12311yescharacteristic
87Dic197—42no—
88Dic197—41no—
89C2×C197—21no—
90C2×C197—22no—
91C2×C197—42no—
92C2×C197#38611yescharacteristic
93D197—42no—
94C197#189111yescharacteristic, complemented, Sylow
95#1891—1971nomaximal, Sylow
96#568D19711yescharacteristic, complemented
97#525—1971no—
98#413—1971no—
99#407—1972no—
100#344—1972no—
101C2×#41C2×D19711yescharacteristic
102C23×C4—1971no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1972no—
106C2×M16C2×D19712yes—
107#123—1972no—
108C22⋊Q8—3941no—
109#121—3941no—
110#120—1972no—
111C4×D4—3942no—
112#117—1971no—
113#101—3941no—
114#100—3941no—
115C22⋊C8—3944no—
116C24—1971no—
117#41—22no—
118#41—22no—
119C2×Q8D78811yescharacteristic
120C2×D4—3942no—
121C2×D4C4×D19712yes—
122C2×D4D78811yescharacteristic
123C22×C4—3942no—
124C22×C4—3941no—
125C22×C4—3941no—
126C22×C4—3942no—
127C22×C4—1971no—
128C22×C4C22⋊D19712yes—
129C22×C4C22×D19711yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—3942no—
134M16—3942no—
135C2×C8—24no—
136C2×C8—3942no—
137C2×C8C22⋊D19714yescomplemented
138C4⋊C4—3942no—
139C4⋊C4—3941no—
140C22⋊C4—3942no—
141C22⋊C4—3942no—
142C22⋊C4—1972no—
143C42—3941no—
144C23—7882no—
145C23—3941no—
146C23—3942no—
147C23#1569133512yes—
148C23C2×C4×D19711yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—7884no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—7881no—
156C2×C4—7882no—
157C2×C4—7881no—
158C2×C4—3942no—
159C2×C4—3941no—
160C2×C4—3942no—
161C2×C4#1569133512yes—
162C2×C4C2×C22⋊D19712yes—
163C2×C4C2×C22⋊D19711yescharacteristic
164C2×C4C2×D78811yescharacteristic
165C2×C4#1569135812yes—
166C8—24no—
167C8—7882no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—7881no—
172C22—7882no—
173C22—7881no—
174C22—3942no—
175C22—3944no—
176C22C2×#1569133511yescharacteristic
177C22C4×C22⋊D19712yes—
178C4—22no—
179C4—7882no—
180C4—7881no—
181C4#2026329511yescharacteristic
182C4#2026330012yes—
183C4#2026330911yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—7882no—
188C2#2716095811yescharacteristic
subgroupG/H
centerC4#20263295
commutatorC23×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#568×C197C2
socleC2×C197#386
prime radical#27161586C2
cosocleC22×C4×C197C23
hypercenter#568D197
hypocenterC197#1891
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27161063C2semidirect
3. #27161042C2semidirect
4. #27160850C2semidirect
5. #568×C197C2semidirect
48. C2×C8×C197D4semidirect
94. C197#1891semidirect
96. #568D197semidirect
137. C2×C8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271615862
#271610632
#271610422
#271608502
#568×C1972
#1891197