Finite Group Databasebeta

← #38742661

Subgroups — #38742661

Order 25216.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#27161915C211yesmaximal, characteristic
2#27161061C212yesmaximal, complemented
3#27161047C212yesmaximal, complemented
4#27161046C211yesmaximal, characteristic, complemented
5#413×C197C211yesmaximal, characteristic, complemented
6#20263374—21no—
7#20263373—22no—
8#20263372—22no—
9#20263371—21no—
10#20263371C414yes—
11C2×#15691358—22no—
12C2×#15691358C2212yes—
13#41×D197—24no—
14C2×#15691342C2211yescharacteristic, complemented
15C22×#13096802C2211yescharacteristic, complemented
16#20263244—22no—
17#20263242—22no—
18C22⋊#13096802—22no—
19#20263238—24no—
20C2×#41×C197C2211yescharacteristic
21C2×M16×C197C2212yescomplemented
22#101×C197—21no—
23#100×C197—21no—
24#15691359—24no—
25#15691358—24no—
26#15691358—24no—
27#15691358—24no—
28#15691358—22no—
29#15691358—22no—
30#15691358—22no—
31#15691358—22no—
32#15691357—24no—
33#15691357—28no—
34C2×Dic394—22no—
35C2×Dic394D412yes—
36Q8×D197—24no—
37C2×D788—22no—
38C2×D788D412yes—
39C2×C22⋊D197—24no—
40C2×C22⋊D197C2×C414yes—
41D4×D197—24no—
42D4×D197—28no—
43C2×C4×D197—24no—
44C2×C4×D197D414yes—
45#15691342—22no—
46C2×#13096802—22no—
47C2×#13096802—21no—
48C2×#13096802—21no—
49#41×C197—22no—
50#41×C197—22no—
51C2×Q8×C197D411yescharacteristic
52C2×D4×C197C2×C412yes—
53C2×D4×C197D411yescharacteristic
54C22×C4×C197D412yes—
55C22×C4×C197C2311yescharacteristic
56M16×C197—22no—
57M16×C197—22no—
58C2×C8×C197—22no—
59C22×D197—24no—
60C22×D197C22⋊C414yes—
61Dic394—24no—
62Dic394—22no—
63Dic394—22no—
64D788—24no—
65D788—22no—
66D788—22no—
67C22⋊D197—28no—
68C22⋊D197—24no—
69C22⋊D197—24no—
70C2×Dic197—24no—
71C2×Dic197C22⋊C414yes—
72C4×D197—28no—
73C4×D197—24no—
74C4×D197—24no—
75#13096802—41no—
76#13096802—41no—
77C23×C197C22⋊C412yes—
78C23×C197C22×C411yescharacteristic
79Q8×C197—22no—
80D4×C197—24no—
81D4×C197—22no—
82C2×C4×C197—22no—
83C2×C4×C197—22no—
84C2×C4×C197C22⋊C412yes—
85C2×C4×C197C2×D412yes—
86C2×C4×C197C2×D412yes—
87C2×C4×C197C2×D411yescharacteristic
88C2×C4×C197C2×D411yescharacteristic
89C8×C197—42no—
90C22×C197—22no—
91C22×C197—41no—
92C22×C197—42no—
93C22×C197C2×C22⋊C412yes—
94C22×C197C2×C22⋊C411yescharacteristic
95C2×D197—24no—
96C2×D197—44no—
97C2×D197—84no—
98C4×C197—22no—
99C4×C197#12012yes—
100C4×C197#12011yescharacteristic
101C4×C197#12011yescharacteristic
102Dic197—24no—
103C2×C197—22no—
104C2×C197—21no—
105C2×C197—42no—
106C2×C197#37911yescharacteristic
107D197—44no—
108C197#179111yescharacteristic, complemented, Sylow
109#1791—1971nomaximal, Sylow
110#585—1971no—
111#413—1972no—
112#413D19711yescharacteristic, complemented
113#408—1971no—
114#408—1972no—
1152−1+4—3942no—
1162−1+4—3941no—
117H(5,𝔽2)—3942no—
118H(5,𝔽2)—3941no—
119H(5,𝔽2)—1974no—
120C2×#41—3944no—
121C2×#41—3942no—
122C2×#41—1972no—
123C2×#41C2×D19711yescharacteristic
124C2×M16—1971no—
125C2×M16C2×D19712yescomplemented
126C22×C8—1971no—
127#101—21no—
128#101—3942no—
129#100—21no—
130#100—3942no—
131C22⋊C8—3944no—
132C22⋊C8—3942no—
133#41—22no—
134#41—22no—
135#41—3944no—
136#41—3944no—
137#41—3944no—
138#41—3944no—
139#41—3948no—
140#41—3944no—
141#41—3942no—
142#41—3942no—
143#41—3942no—
144#41—3942no—
145C2×Q8—3942no—
146C2×Q8—3944no—
147C2×Q8—1972no—
148C2×Q8C22⋊D19711yescharacteristic
149C2×D4—3942no—
150C2×D4—3944no—
151C2×D4—3948no—
152C2×D4—3944no—
153C2×D4—1974no—
154C2×D4—1972no—
155C2×D4C4×D19712yes—
156C2×D4C22⋊D19711yescharacteristic
157C22×C4—3944no—
158C22×C4—1974no—
159C22×C4D78812yes—
160C22×C4C22×D19711yescharacteristic
161M16—22no—
162M16—22no—
163M16—3942no—
164C2×C8—22no—
165C2×C8—3942no—
166C2×C8—3941no—
167C2×C8—3941no—
168C23—3944no—
169C23—1974no—
170C23#1569133512yes—
171C23C2×C4×D19711yescharacteristic
172Q8—22no—
173Q8—3944no—
174Q8—3942no—
175Q8—3942no—
176D4—24no—
177D4—22no—
178D4—3944no—
179D4—3948no—
180D4—3944no—
181D4—3944no—
182D4—3942no—
183D4—3942no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3948no—
187C2×C4—3944no—
188C2×C4—3944no—
189C2×C4—3944no—
190C2×C4—1974no—
191C2×C4#1569133512yes—
192C2×C4D4×D19712yes—
193C2×C4D4×D19712yes—
194C2×C4C2×C22⋊D19711yescharacteristic
195C2×C4C2×D78811yescharacteristic
196C8—42no—
197C8—7881no—
198C8—7881no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—7884no—
203C22—15764no—
204C22—3944no—
205C22C22⋊C4×D19712yes—
206C22C2×#1569133511yescharacteristic
207C4—22no—
208C4—3944no—
209C4#2026329412yes—
210C4#2026329611yescharacteristic
211C4#2026329611yescharacteristic
212C2—21no—
213C2—22no—
214C2—42no—
215C2—7884no—
216C2#2716093411yescharacteristic
subgroupG/H
centerC4#20263296
commutatorC23×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#413×C197C2
socleC2×C197#379
prime radical#27161915C2
cosocleC22×C4×C197C23
hypercenter#413D197
hypocenterC197#1791
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #27161061C2semidirect
3. #27161047C2semidirect
4. #27161046C2semidirect
5. #413×C197C2semidirect
14. C2×#15691342C22semidirect
15. C22×#13096802C22semidirect
21. C2×M16×C197C22semidirect
108. C197#1791semidirect
112. #413D197semidirect
125. C2×M16C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
#271610612
#271610472
#271610462
#413×C1972
#1791197