Finite Group Databasebeta

← #38741748

Subgroups — #38741748

Order 25216.

246 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161660C211yesmaximal, characteristic
2#27161397C212yesmaximal, complemented
3#27160812C212yesmaximal, complemented
4#27160805C211yesmaximal, characteristic, complemented
5#400×C197C211yesmaximal, characteristic, complemented
6C2×#15691358—21no—
7#20263306—22no—
8#20263301—22no—
9#20263300—21no—
10#20263300C2212yes—
11#20263257—22no—
12#20263246—22no—
13C22⋊#13096802C2212yes—
14C2×C4⋊C4×C197C2211yescharacteristic
15C2×C4⋊C4×C197C2211yescharacteristic
16#95×C197—21no—
17#95×C197—21no—
18#95×C197C2211yescharacteristic
19#15691358—42no—
20#15691358—42no—
21C2×Dic394—21no—
22C2×D788—21no—
23C2×D788—21no—
24C2×D788D412yes—
25C2×C22⋊D197—22no—
26C2×C4×D197—22no—
27C2×#13096802—22no—
28C4⋊Dic197—21no—
29C4⋊Dic197D412yes—
30#15691335—22no—
31#15691335—22no—
32C22×C4×C197—21no—
33C22×C4×C197—21no—
34C22×C4×C197D411yescharacteristic
35C22×C4×C197D411yescharacteristic
36C22×C4×C197C2311yescharacteristic
37C4⋊C4×C197—22no—
38C4⋊C4×C197—22no—
39C22×D197—22no—
40Dic394—42no—
41D788—42no—
42D788—42no—
43D788—81no—
44C22⋊D197—42no—
45C22⋊D197—42no—
46C2×Dic197—22no—
47C4×D197—44no—
48#13096802—42no—
49C23×C197C2×D411yescharacteristic
50C2×C4×C197—21no—
51C2×C4×C197—21no—
52C2×C4×C197—21no—
53C2×C4×C197—41no—
54C2×C4×C197—41no—
55C2×C4×C197—41no—
56C2×C4×C197—41no—
57C2×C4×C197—42no—
58C2×C4×C197—41no—
59C2×C4×C197—41no—
60C2×C4×C197SD1612yes—
61C2×C4×C197C2×D412yes—
62C22×C197—21no—
63C22×C197—21no—
64C22×C197#12011yescharacteristic
65C22×C197C2×SD1611yescharacteristic
66C22×C197AGL(1,ℤ/8)11yescharacteristic
67C2×D197—42no—
68C2×D197—42no—
69C2×D197—42no—
70C4×C197—22no—
71C4×C197—22no—
72C4×C197—41no—
73C4×C197—81no—
74C4×C197—81no—
75Dic197—42no—
76C2×C197—21no—
77C2×C197—21no—
78C2×C197#45011yescharacteristic
79C2×C197#45411yescharacteristic
80C2×C197#45811yescharacteristic
81D197—82no—
82C197#161611yescharacteristic, complemented, Sylow
83#1616—1971nomaximal, Sylow
84#537—1971no—
85#481—1972no—
86#400D19711yescharacteristic, complemented
87#327—1972no—
88#324—1971no—
89C2×#41—3941no—
90C22⋊Q8—3942no—
91#121—3942no—
92#121—3941no—
93#121—1972no—
94C2×C4⋊C4C2×D19711yescharacteristic
95C2×C4⋊C4C2×D19711yescharacteristic
96#106—3942no—
97#102—3942no—
98C22⋊C8—1972no—
99#95—21no—
100#95—21no—
101#95C2×D19711yescharacteristic
102#41—7882no—
103#41—7882no—
104C2×Q8—3941no—
105C2×D4—3941no—
106C2×D4—3941no—
107C2×D4—3942no—
108C2×D4—1972no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—3942no—
112C22×C4C22⋊D19711yescharacteristic
113C22×C4C22⋊D19711yescharacteristic
114C22×C4C22×D19711yescharacteristic
115C2×C8—3942no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—3941no—
119C4⋊C4—1972no—
120C22⋊C4—3942no—
121C22⋊C4—3942no—
122C23—3942no—
123C23C2×C22⋊D19711yescharacteristic
124Q8—7882no—
125D4—7882no—
126D4—7882no—
127D4—7882no—
128D4—7882no—
129D4—15761no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—7884no—
141C2×C4—3942no—
142C2×C4#1569134512yes—
143C2×C4D4×D19712yes—
144C8—7882no—
145C22—21no—
146C22—21no—
147C22—7882no—
148C22—7882no—
149C22—7882no—
150C22#2026329411yescharacteristic
151C22C2×#1569134511yescharacteristic
152C22#2026335111yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—81no—
158C4—7882no—
159C2—21no—
160C2—21no—
161C2—15762no—
162C2#2716122911yescharacteristic
163C2#2716125111yescharacteristic
164C2#2716126811yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#400×C197C2
socleC22×C197#120
prime radical#27161660C2
hypercenter#400D197
hypocenterC197#1616
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #27161397C2semidirect
3. #27160812C2semidirect
4. #27160805C2semidirect
5. #400×C197C2semidirect
82. C197#1616semidirect
86. #400D197semidirect

Maximal subgroups

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

subgroupindex
#271616602
#271613972
#271608122
#271608052
#400×C1972
#1616197