Finite Group Databasebeta

← #38741424

Subgroups — #38741424

Order 25216.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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#127×D197C211yesmaximal, characteristic
2#27161135C212yesmaximal, complemented
3#27160823C211yesmaximal, characteristic, complemented
4#27160822C212yesmaximal, complemented
5#331×C197C211yesmaximal, characteristic, complemented
6C2×D4×D197—22no—
7C2×D4×D197C412yes—
8#20263323C2211yescharacteristic
9#20263322—22no—
10#20263322C412yes—
11C42×D197C2211yescharacteristic
12C4⋊#13096802C2212yescomplemented
13#20263238—22no—
14#20263236—22no—
15#127×C197C2211yescharacteristic
16C4⋊C8×C197C2212yescomplemented
17C23×D197—22no—
18C2×D788—22no—
19C2×D788C2×C412yes—
20C2×C22⋊D197—22no—
21C2×C22⋊D197—24no—
22D4×D197—44no—
23D4×D197—44no—
24D4×D197—44no—
25C2×C4×D197C2×C411yescharacteristic
26C2×C4×D197D412yes—
27C2×#13096802—22no—
28C4×Dic197C2×C411yescharacteristic
29C4×Dic197D412yes—
30C2×D4×C197—22no—
31C2×D4×C197C2×C412yes—
32C2×C8×C197—22no—
33C42×C197C2311yescharacteristic
34C22×D197—22no—
35C22×D197—48no—
36C22×D197—44no—
37C22×D197C22⋊C411yescharacteristic
38D788—42no—
39D788—42no—
40D788—42no—
41C22⋊D197—44no—
42C22⋊D197—48no—
43C2×Dic197C22⋊C412yes—
44C2×Dic197C22⋊C411yescharacteristic
45C4×D197—22no—
46C4×D197—22no—
47C4×D197—22no—
48#13096802—42no—
49C23×C197—22no—
50D4×C197—42no—
51D4×C197—42no—
52D4×C197—42no—
53C2×C4×C197C22×C411yescharacteristic
54C2×C4×C197C2×D412yes—
55C8×C197—42no—
56C22×C197—42no—
57C22×C197—44no—
58C22×C197C2×C22⋊C411yescharacteristic
59C2×D197—22no—
60C2×D197—42no—
61C2×D197—44no—
62C2×D197—88no—
63C2×D197#10012yes—
64C4×C197—21no—
65C4×C197AGL(1,ℤ/8)12yes—
66C4×C197AGL(1,ℤ/8)12yes—
67Dic197—21no—
68Dic197—22no—
69C2×C197—82no—
70C2×C197C2×#10011yescharacteristic
71C2×C197#41812yes—
72D197—22no—
73D197—82no—
74C197#155511yescharacteristic, complemented, Sylow
75#1555—1971nomaximal, Sylow
76C2×#127—1971no—
77#432—1972no—
78#331—1971no—
79#331—1972no—
80#331D19711yescharacteristic, complemented
81C22×D4—3942no—
82C22×D4—1972no—
83#127—3942no—
84#127—1972no—
85#127—1971no—
86#127C2×D19711yescharacteristic
87C2×C42—1971no—
88C4⋊C8—1972no—
89C4⋊C8C2×D19712yescomplemented
90C22⋊C8—3942no—
91#97—3942no—
92C24—3942no—
93C2×D4—22no—
94C2×D4—7884no—
95C2×D4—7884no—
96C2×D4—7884no—
97C2×D4—3942no—
98C2×D4—3944no—
99C2×D4—3942no—
100C2×D4—1972no—
101C2×D4C4×D19712yes—
102C22×C4—1971no—
103C22×C4—1972no—
104C2×C8—22no—
105C2×C8—3942no—
106C42—1971no—
107C42—1972no—
108C42C22×D19711yescharacteristic
109C23—22no—
110C23—7888no—
111C23—7884no—
112C23—3942no—
113C23—1971no—
114D4—42no—
115D4—42no—
116D4—42no—
117D4—7882no—
118D4—7884no—
119D4—7888no—
120D4—7882no—
121D4—7882no—
122C2×C4—3942no—
123C2×C4—3942no—
124C2×C4—3942no—
125C2×C4—1971no—
126C2×C4—1972no—
127C2×C4C2×C4×D19711yescharacteristic
128C2×C4D4×D19712yes—
129C8—42no—
130C8—7882no—
131C22—42no—
132C22—44no—
133C22—7882no—
134C22—7884no—
135C22—15768no—
136C22—3942no—
137C22—1972no—
138C22C22⋊C4×D19711yescharacteristic
139C4—21no—
140C4—3941no—
141C4—3942no—
142C4#2026335012yes—
143C4#2026335312yes—
144C2—82no—
145C2—15762no—
146C2—3942no—
147C2#100×D19711yescharacteristic
148C2#2716108212yes—
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#331×C197C2
socleC22×C197C2×C22⋊C4
prime radical#127×D197C2
cosocleC42×C197C23
hypercenter#331D197
hypocenterC197#1555
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27161135C2semidirect
3. #27160823C2semidirect
4. #27160822C2semidirect
5. #331×C197C2semidirect
12. C4⋊#13096802C22semidirect
16. C4⋊C8×C197C22semidirect
74. C197#1555semidirect
80. #331D197semidirect
89. C4⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#127×D1972
#271611352
#271608232
#271608222
#331×C1972
#1555197