Finite Group Databasebeta

← #38741920

Subgroups — #38741920

Order 25216.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161790C211yesmaximal, characteristic
2#27161429C211yesmaximal, characteristic, complemented
3#27161365C211yesmaximal, characteristic, complemented
4#27161285C211yesmaximal, characteristic
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic
7#333×C197C211yesmaximal, characteristic, complemented
8C2×#15691347—21no—
9#20263326C2211yescharacteristic
10#20263317—21no—
11#20263306—21no—
12#20263302—22no—
13C4×Dic394C2211yescharacteristic
14C4×D788C2211yescharacteristic
15#20263288—21no—
16C4⋊C4×D197—21no—
17#20263258—21no—
18#20263251—21no—
19#20263251—21no—
20#20263250—22no—
21C4×#13096802C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25C2×Dic394—21no—
26C2×Dic394D411yescharacteristic
27C2×D788D411yescharacteristic
28C2×C4×D197—22no—
29C2×C4×D197—21no—
30#15691347—42no—
31C2×#13096802—21no—
32C4⋊Dic197—22no—
33C4⋊Dic197—21no—
34C4⋊Dic197—21no—
35C4⋊Dic197D411yescharacteristic
36C4⋊Dic197D411yescharacteristic
37#15691337—22no—
38#15691337—21no—
39#15691335—22no—
40#15691335—21no—
41C4×Dic197—21no—
42C2×Q8×C197D412yes—
43C2×C8×C197—21no—
44C2×C8×C197—21no—
45C4⋊C4×C197—21no—
46C4⋊C4×C197—21no—
47C42×C197C2311yescharacteristic
48C22×D197—21no—
49Dic394—21no—
50Dic394—21no—
51Dic394—41no—
52Dic394—41no—
53D788—21no—
54D788SD1612yes—
55C2×Dic197—21no—
56C2×Dic197—21no—
57C2×Dic197—21no—
58C2×Dic197—41no—
59C4×D197—41no—
60C4×D197—41no—
61C4×D197—82no—
62#13096802—41no—
63Q8×C197—42no—
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—22no—
72C2×D197—41no—
73C2×D197—41no—
74C4×C197—21no—
75C4×C197—42no—
76C4×C197C2×SD1611yescharacteristic
77C4×C197#13511yescharacteristic
78C4×C197#13711yescharacteristic
79C4×C197#13711yescharacteristic
80Dic197—41no—
81Dic197—41no—
82Dic197—41no—
83Dic197—81no—
84C2×C197#44811yescharacteristic
85C2×C197#45211yescharacteristic
86C2×C197#45511yescharacteristic
87D197—42no—
88C197#164311yescharacteristic, complemented, Sylow
89#1643—1971nomaximal, Sylow
90#554—1971no—
91#487—1971no—
92#475—1971no—
93#461—1971no—
94#333D19711yescharacteristic, complemented
95Q8⋊C8—1971no—
96D4⋊C8—1971no—
97C2×SD16—3941no—
98C4⋊Q8—1971no—
99C4⋊Q8C2×D19711yescharacteristic
100#125—3941no—
101C22⋊Q8—3941no—
102C22⋊Q8—3942no—
103C4×Q8—1971no—
104C4×D4—3941no—
105C4×D4—1971no—
106C2×C4⋊C4—3941no—
107#106—3941no—
108C4⋊C8C2×D19711yescharacteristic
109C4⋊C8C2×D19711yescharacteristic
110#103—3941no—
111#103—3942no—
112#103—3941no—
113C4×C8—1971no—
114C2×Q8—3941no—
115C2×Q8—1971no—
116C2×Q8D78812yes—
117C2×D4—1971no—
118C22×C4—3942no—
119C22×C4—3941no—
120SD16—7882no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—3941no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3942no—
127C4⋊C4—3942no—
128C4⋊C4—3941no—
129C4⋊C4—3941no—
130C4⋊C4—3941no—
131C4⋊C4—1971no—
132C4⋊C4—1971no—
133C22⋊C4—3942no—
134C22⋊C4—3941no—
135C42—3941no—
136C42C22×D19711yescharacteristic
137C23—3941no—
138Q8—42no—
139Q8—7881no—
140Q8—7881no—
141Q8—3941no—
142Q8—3941no—
143D4—3941no—
144D4—1972no—
145C2×C4—22no—
146C2×C4—7881no—
147C2×C4—7881no—
148C2×C4—7881no—
149C2×C4—15762no—
150C2×C4—3941no—
151C2×C4—3941no—
152C2×C4—3941no—
153C2×C4D4×D19711yescharacteristic
154C2×C4D4×D19711yescharacteristic
155C2×C4C2×D78811yescharacteristic
156C8—41no—
157C8—41no—
158C8—7881no—
159C22—7881no—
160C22—7881no—
161C22—3942no—
162C22#2026329611yescharacteristic
163C4—21no—
164C4—42no—
165C4—7881no—
166C4—7881no—
167C4—7881no—
168C4—15761no—
169C4SD16×D19711yescharacteristic
170C4#2026334711yescharacteristic
171C4#2026335811yescharacteristic
172C4#2026335811yescharacteristic
173C2—7882no—
174C2#2716121911yescharacteristic
175C2#2716124411yescharacteristic
176C2#2716125911yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#333×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#333D197
hypocenterC197#1643
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #27161429C2semidirect
3. #27161365C2semidirect
5. #27160809C2semidirect
7. #333×C197C2semidirect
88. C197#1643semidirect
94. #333D197semidirect

Maximal subgroups

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

subgroupindex
#271617902
#271614292
#271613652
#271612852
#271608092
#271608062
#333×C1972
#1643197