Finite Group Databasebeta

← #38742017

Subgroups — #38742017

Order 25216.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#27161821C211yesmaximal, characteristic
2#27161471C211yesmaximal, characteristic, complemented
3#27161441C211yesmaximal, characteristic, complemented
4#27160819C211yesmaximal, characteristic, complemented
5#27160818C211yesmaximal, characteristic, complemented
6#27160818C211yesmaximal, characteristic, complemented
7#489×C197C211yesmaximal, characteristic, complemented
8C2×#15691348—21no—
9C2×#15691345—21no—
10#20263315—21no—
11#20263314C2211yescharacteristic
12#20263314C2211yescharacteristic
13#20263312—21no—
14#20263312—21no—
15#20263296—21no—
16#20263296—21no—
17#20263294—21no—
18#20263249—21no—
19#20263246—21no—
20#20263237C2211yescharacteristic, complemented
21#20263237C2211yescharacteristic, complemented
22C4⋊Q8×C197C2211yescharacteristic, complemented
23#124×C197C2211yescharacteristic
24#103×C197—21no—
25#102×C197—21no—
26#97×C197C2211yescharacteristic, complemented
27C23×D197—21no—
28C2×Dic394D411yescharacteristic
29C2×Dic394D411yescharacteristic
30C2×D788—21no—
31C2×D788D411yescharacteristic
32C2×D788D411yescharacteristic
33C2×C22⋊D197—21no—
34C2×C22⋊D197—21no—
35#15691348—42no—
36#15691345—42no—
37C2×#13096802—21no—
38C2×#13096802—21no—
39C22⋊Dic197—21no—
40#15691335—21no—
41#15691335—21no—
42#15691335—21no—
43#15691335—21no—
44#15691335—21no—
45#15691335—21no—
46C4×Dic197—21no—
47C2×Q8×C197—21no—
48C2×Q8×C197D411yescharacteristic
49C2×D4×C197D411yescharacteristic
50C2×C8×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197—21no—
53C22⋊C4×C197—21no—
54C22⋊C4×C197—21no—
55C42×C197C2311yescharacteristic
56C22×D197—21no—
57C22×D197—21no—
58C22×D197—81no—
59C22×D197—81no—
60C22×D197—81no—
61Dic394—41no—
62Dic394—41no—
63D788—41no—
64D788—41no—
65D788—81no—
66C22⋊D197—81no—
67C22⋊D197—81no—
68C2×Dic197—21no—
69C2×Dic197—21no—
70#13096802—22no—
71#13096802—41no—
72C23×C197—21no—
73Q8×C197—41no—
74Q8×C197—41no—
75Q8×C197—41no—
76D4×C197—41no—
77C2×C4×C197—21no—
78C2×C4×C197—41no—
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D411yescharacteristic
82C8×C197—41no—
83C22×C197—41no—
84C22×C197—41no—
85C22×C197—41no—
86C22×C197#12011yescharacteristic
87C2×D197—41no—
88C2×D197—41no—
89C2×D197—41no—
90C2×D197—41no—
91C2×D197—41no—
92C2×D197—41no—
93C2×D197—84no—
94C4×C197—21no—
95C4×C197—21no—
96C4×C197—21no—
97C4×C197—41no—
98C4×C197—81no—
99Dic197—41no—
100Dic197—41no—
101C2×C197—81no—
102C2×C197#45311yescharacteristic
103C2×C197#45511yescharacteristic
104C2×C197#45511yescharacteristic
105D197—81no—
106D197—81no—
107C197#165911yescharacteristic, complemented, Sylow
108#1659—1971nomaximal, Sylow
109H(3,𝔽4)—1971no—
110#497—1971no—
111#489—1971no—
112#489D19711yescharacteristic, complemented
113#329—1971no—
114#329—1971no—
115#329—1971no—
116C2×Q16—3941no—
117C2×SD16—3941no—
118C4⋊Q8C2×D19711yescharacteristic, complemented
119#124—3941no—
120#124—3941no—
121#124—3941no—
122#124—1971no—
123#124—1971no—
124#124C2×D19711yescharacteristic
125#120—3941no—
126#120—3941no—
127#120—3941no—
128#103—21no—
129#103—3941no—
130#102—21no—
131#102—3941no—
132#97—1971no—
133#97—1971no—
134#97C2×D19711yescharacteristic, complemented
135C24—3941no—
136C2×Q8—21no—
137C2×Q8—1971no—
138C2×Q8—1971no—
139C2×Q8C22⋊D19711yescharacteristic
140C2×D4—3941no—
141C2×D4—3941no—
142C2×D4—3941no—
143C2×D4—1971no—
144C2×D4—1971no—
145C2×D4C22⋊D19711yescharacteristic
146Q16—7882no—
147SD16—7882no—
148C2×C8—21no—
149C2×C8—3941no—
150C2×C8—3941no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—3941no—
156C22⋊C4—3941no—
157C22⋊C4—3941no—
158C22⋊C4—3941no—
159C22⋊C4—3941no—
160C22⋊C4—3941no—
161C22⋊C4—3941no—
162C42—3941no—
163C42C22×D19711yescharacteristic
164C23—21no—
165C23—15761no—
166C23—15761no—
167C23—15761no—
168C23—3941no—
169C23—3941no—
170Q8—41no—
171Q8—41no—
172Q8—41no—
173Q8—7881no—
174Q8—7881no—
175D4—41no—
176D4—7881no—
177D4—7881no—
178D4—15761no—
179D4—15761no—
180D4—15761no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—3941no—
184C2×C4—3941no—
185C2×C4D4×D19711yescharacteristic
186C2×C4D4×D19711yescharacteristic
187C2×C4C2×C22⋊D19711yescharacteristic
188C8—41no—
189C8—7881no—
190C8—3942no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—7881no—
195C22—7881no—
196C22—7881no—
197C22—7881no—
198C22—7881no—
199C22—7881no—
200C22—15764no—
201C22#2026329411yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—81no—
207C4—7881no—
208C4—7881no—
209C2—81no—
210C2—15761no—
211C2—15761no—
212C2#2716124511yescharacteristic
213C2#2716125511yescharacteristic
214C2#2716125611yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#489×C197C2
socleC22×C197#120
prime radical#27161821C2
hypercenter#489D197
hypocenterC197#1659
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #27161471C2semidirect
3. #27161441C2semidirect
4. #27160819C2semidirect
5. #27160818C2semidirect
6. #27160818C2semidirect
7. #489×C197C2semidirect
20. #20263237C22semidirect
21. #20263237C22semidirect
22. C4⋊Q8×C197C22semidirect
26. #97×C197C22semidirect
107. C197#1659semidirect
112. #489D197semidirect
118. C4⋊Q8C2×D197semidirect
134. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618212
#271614712
#271614412
#271608192
#271608182
#489×C1972
#1659197