Finite Group Databasebeta

← #38741983

Subgroups — #38741983

Order 25216.

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

normal subgroups only

This group has 200 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#27161803C211yesmaximal, characteristic
2#27161475C211yesmaximal, characteristic, complemented
3#27161360C211yesmaximal, characteristic
4#27161290C211yesmaximal, characteristic, complemented
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic
7#332×C197C211yesmaximal, characteristic, complemented
8C2×Dic788—21no—
9#20263326C2211yescharacteristic
10#20263321—22no—
11#20263315—21no—
12#20263306—21no—
13Q8×Dic197—21no—
14C4×Dic394C2211yescharacteristic
15C4×D788C2211yescharacteristic
16#20263284—21no—
17#20263260—21no—
18#20263257—22no—
19#20263251—21no—
20#20263248—21no—
21C4×#13096802C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25C2×Dic394—21no—
26C2×Dic394D411yescharacteristic
27C2×D788D411yescharacteristic
28C2×C4×D197—21no—
29Dic788—42no—
30C2×#13096802—21no—
31C4⋊Dic197—22no—
32C4⋊Dic197—21no—
33C4⋊Dic197—21no—
34C4⋊Dic197D411yescharacteristic
35C4⋊Dic197D411yescharacteristic
36#15691337—21no—
37#15691335—22no—
38#15691335—22no—
39#15691335—21no—
40C4×Dic197—22no—
41C4×Dic197—21no—
42C2×Q8×C197—21no—
43C2×C8×C197—21no—
44C2×C8×C197—21no—
45C4⋊C4×C197—21no—
46C4⋊C4×C197D412yes—
47C42×C197C2311yescharacteristic
48C22×D197—21no—
49Dic394—21no—
50Dic394—21no—
51Dic394—41no—
52Dic394—41no—
53D788—21no—
54D788—21no—
55C2×Dic197—21no—
56C2×Dic197—21no—
57C2×Dic197—21no—
58C2×Dic197—41no—
59C4×D197—41no—
60C4×D197—41no—
61#13096802—22no—
62Q8×C197—41no—
63Q8×C197—41no—
64C2×C4×C197—22no—
65C2×C4×C197C2×D411yescharacteristic
66C2×C4×C197C2×D411yescharacteristic
67C2×C4×C197C2×D411yescharacteristic
68C8×C197—41no—
69C8×C197—41no—
70C22×C197#12011yescharacteristic
71C2×D197—41no—
72C2×D197—41no—
73C2×D197—41no—
74C4×C197—21no—
75C4×C197—42no—
76C4×C197#13511yescharacteristic
77C4×C197#13511yescharacteristic
78C4×C197AGL(1,ℤ/8)11yescharacteristic
79C4×C197#13711yescharacteristic
80Dic197—41no—
81Dic197—41no—
82Dic197—41no—
83Dic197—81no—
84C2×C197#44911yescharacteristic
85C2×C197#45211yescharacteristic
86C2×C197#45611yescharacteristic
87D197—81no—
88C197#165211yescharacteristic, complemented, Sylow
89#1652—1971nomaximal, Sylow
90#556—1971no—
91#498—1971no—
92#474—1971no—
93#462—1971no—
94#332D19711yescharacteristic, complemented
95Q8⋊C8—1971no—
96D4⋊C8—1971no—
97C2×Q16—3941no—
98C4⋊Q8—1971no—
99C4⋊Q8C2×D19711yescharacteristic
100#126—3942no—
101#124—3941no—
102C22⋊Q8—3941no—
103C4×Q8—3941no—
104C4×Q8—1971no—
105C4×D4—1971no—
106#117—3941no—
107#107—3941no—
108#106—3942no—
109C4⋊C8C2×D19711yescharacteristic
110C4⋊C8C2×D19711yescharacteristic
111#103—3941no—
112#102—3941no—
113C4×C8—1971no—
114C2×Q8—21no—
115C2×Q8—3941no—
116C2×Q8—1971no—
117C2×D4—1971no—
118C22×C4—3941no—
119Q16—7882no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3941no—
123C4⋊C4—21no—
124C4⋊C4—3942no—
125C4⋊C4—3941no—
126C4⋊C4—3941no—
127C4⋊C4—3941no—
128C4⋊C4—1971no—
129C4⋊C4—1971no—
130C4⋊C4D78812yes—
131C22⋊C4—3942no—
132C22⋊C4—3942no—
133C22⋊C4—3941no—
134C42—3942no—
135C42—3941no—
136C42C22×D19711yescharacteristic
137C23—3941no—
138Q8—41no—
139Q8—41no—
140Q8—7881no—
141Q8—7881no—
142Q8—3941no—
143Q8—3941no—
144D4—3941no—
145D4—3941no—
146C2×C4—22no—
147C2×C4—7881no—
148C2×C4—7881no—
149C2×C4—7881no—
150C2×C4—3941no—
151C2×C4—3941no—
152C2×C4—3941no—
153C2×C4D4×D19711yescharacteristic
154C2×C4D4×D19711yescharacteristic
155C2×C4C2×D78811yescharacteristic
156C8—41no—
157C8—41no—
158C8—3942no—
159C22—7881no—
160C22—7881no—
161C22—7881no—
162C22#2026329611yescharacteristic
163C4—21no—
164C4—42no—
165C4—7881no—
166C4—7881no—
167C4—7881no—
168C4—15761no—
169C4#2026334511yescharacteristic
170C4#2026334911yescharacteristic
171C4#2026335311yescharacteristic
172C4#2026335511yescharacteristic
173C2—15761no—
174C2#2716122711yescharacteristic
175C2#2716124211yescharacteristic
176C2#2716126411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#332×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#332D197
hypocenterC197#1652
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27161475C2semidirect
4. #27161290C2semidirect
5. #27160809C2semidirect
7. #332×C197C2semidirect
88. C197#1652semidirect
94. #332D197semidirect

Maximal subgroups

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

subgroupindex
#271618032
#271614752
#271613602
#271612902
#271608092
#271608062
#332×C1972
#1652197