Finite Group Databasebeta

← #38743114

Subgroups — #38743114

Order 25216.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#20263314C211yesmaximal, characteristic
2#27161556C211yesmaximal, characteristic, complemented
3#27161085C211yesmaximal, characteristic
4#27161083C211yesmaximal, characteristic
5C2×#20263244C211yesmaximal, characteristic, complemented
6#27160839C211yesmaximal, characteristic, complemented
7#339×C197C211yesmaximal, characteristic, complemented
8C22×Dic394C2211yescharacteristic
9C22×D788C2211yescharacteristic
10C2×#15691342C2211yescharacteristic
11#20263314—22no—
12#20263314C414yescomplemented
13C4×Dic394—22no—
14#20263291—22no—
15#20263285C2211yescharacteristic
16#20263282—22no—
17C2×#15691335—22no—
18#20263248—22no—
19#20263246—22no—
20#20263244—22no—
21C2×M16×C197C2211yescharacteristic
22#117×C197C2211yescharacteristic
23C2×C42×C197C2211yescharacteristic
24C23×D197—21no—
25C2×Dic394—21no—
26C2×Dic394—21no—
27C2×Dic394—21no—
28C2×Dic394—21no—
29C2×Dic394C2×C412yes—
30C2×Dic394D412yes—
31C2×D788—21no—
32C2×D788—21no—
33C2×D788—21no—
34C2×D788—21no—
35C2×D788—21no—
36C2×D788C2×C412yes—
37C22×Dic197D412yes—
38#15691342—21no—
39#15691342—21no—
40C2×#13096802—21no—
41C4⋊Dic197D412yes—
42#15691337—24no—
43C22⋊Dic197—21no—
44#15691335—28no—
45C4×Dic197—24no—
46C4×Dic197—21no—
47C22×C4×C197—21no—
48C22×C4×C197C2311yescharacteristic
49M16×C197—21no—
50M16×C197—21no—
51C2×C8×C197—21no—
52C4⋊C4×C197D412yes—
53C22⋊C4×C197—21no—
54C42×C197—21no—
55C42×C197—21no—
56C42×C197C2×C412yes—
57C22×D197—22no—
58C22×D197—44no—
59C22×D197—41no—
60C22×D197—41no—
61Dic394—22no—
62Dic394—22no—
63Dic394—22no—
64Dic394—22no—
65D788—22no—
66D788—22no—
67D788—41no—
68D788—41no—
69C2×Dic197—22no—
70C2×Dic197—22no—
71C2×Dic197—22no—
72C2×Dic197C22⋊C414yes—
73#13096802—41no—
74C23×C197C2×D411yescharacteristic
75C2×C4×C197—22no—
76C2×C4×C197—22no—
77C2×C4×C197—42no—
78C2×C4×C197C22×C411yescharacteristic
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D411yescharacteristic
82C2×C4×C197#4111yescharacteristic
83C2×C4×C197#4111yescharacteristic
84C8×C197—41no—
85C22×C197—22no—
86C22×C197C2×C22⋊C411yescharacteristic
87C22×C197#12011yescharacteristic
88C22×C197#12311yescharacteristic
89C2×D197—44no—
90C2×D197—42no—
91C2×D197—82no—
92C2×D197—82no—
93C4×C197—42no—
94C4×C197—42no—
95C4×C197C4×D411yescharacteristic
96C4×C197C4×D411yescharacteristic
97C4×C197#12111yescharacteristic
98C4×C197#12111yescharacteristic
99Dic197—24no—
100Dic197—42no—
101C2×C197—21no—
102C2×C197—21no—
103C2×C197#38611yescharacteristic
104C2×C197#45412yes—
105D197—82no—
106C197#188811yescharacteristic, complemented, Sylow
107#1888—1971nomaximal, Sylow
108C2×#124—1971no—
109#519—1971no—
110#418—1971no—
111#418—1971no—
112C2×#101—1971no—
113#339—1971no—
114#339D19711yescharacteristic, complemented
115C22×Q8—1971no—
116C22×D4—1971no—
117C2×M16—1971no—
118C2×M16C2×D19711yescharacteristic
119#124—3942no—
120#124—1974no—
121C4×Q8—3942no—
122C4×Q8—3942no—
123#117—3942no—
124#117—1971no—
125#117C2×D19711yescharacteristic
126C2×C22⋊C4—3942no—
127C2×C42C2×D19711yescharacteristic
128#102—3942no—
129#102—3942no—
130#101—3942no—
131C24—3941no—
132C2×Q8—3941no—
133C2×Q8—3941no—
134C2×Q8—3941no—
135C2×Q8—3941no—
136C2×Q8—1972no—
137C2×Q8—1972no—
138C2×D4—3941no—
139C2×D4—3941no—
140C2×D4—3941no—
141C2×D4—3941no—
142C2×D4—3941no—
143C2×D4—1972no—
144C22×C4—21no—
145C22×C4—1972no—
146C22×C4C22×D19711yescharacteristic
147M16—21no—
148M16—21no—
149M16—3941no—
150M16—3941no—
151C2×C8—21no—
152C2×C8—3941no—
153C4⋊C4—3944no—
154C4⋊C4—1972no—
155C4⋊C4D78812yes—
156C22⋊C4—21no—
157C22⋊C4—3948no—
158C22⋊C4—3941no—
159C42—21no—
160C42—21no—
161C42—3944no—
162C42—3941no—
163C42C4×D19712yes—
164C23—7884no—
165C23—7881no—
166C23—7881no—
167C23—3942no—
168C23D4×D19711yescharacteristic
169Q8—3942no—
170Q8—3942no—
171Q8—3942no—
172Q8—3942no—
173D4—7881no—
174D4—7881no—
175D4—3942no—
176D4—3942no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—3942no—
181C2×C4—3942no—
182C2×C4—3942no—
183C2×C4—1974no—
184C2×C4C2×C4×D19711yescharacteristic
185C2×C4D4×D19711yescharacteristic
186C2×C4D4×D19711yescharacteristic
187C2×C4C2×D78811yescharacteristic
188C2×C4#1569135811yescharacteristic
189C2×C4#1569135911yescharacteristic
190C8—41no—
191C8—7881no—
192C22—22no—
193C22—7884no—
194C22—7882no—
195C22—15762no—
196C22—15762no—
197C22C22⋊C4×D19711yescharacteristic
198C22#2026329611yescharacteristic
199C22#2026331111yescharacteristic
200C4—42no—
201C4—42no—
202C4—7882no—
203C4—3944no—
204C4#2026328811yescharacteristic
205C4C4×D78811yescharacteristic
206C4#2026329911yescharacteristic
207C4#2026330111yescharacteristic
208C2—21no—
209C2—21no—
210C2—15762no—
211C2#2716095711yescharacteristic
212C2#2716125312yes—
subgroupG/H
centerC22#20263296
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#339×C197C2
socleC22×C197#120
prime radicalC22×D788C22
cosocleC22×C4×C197C23
hypercenter#339D197
hypocenterC197#1888
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #27161556C2semidirect
5. C2×#20263244C2semidirect
6. #27160839C2semidirect
7. #339×C197C2semidirect
12. #20263314C4semidirect
106. C197#1888semidirect
114. #339D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633142
#271615562
#271610852
#271610832
C2×#202632442
#271608392
#339×C1972
#1888197