Finite Group Databasebeta

← #38741635

Subgroups — #38741635

Order 25216.

316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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#27161229C212yesmaximal, complemented
2#27160988C211yesmaximal, characteristic
3#27160816C212yesmaximal, complemented
4#27160803C211yesmaximal, characteristic, complemented
5#543×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7C22×C22⋊D197—21no—
8C2×#15691345—22no—
9C2×#15691335—21no—
10#20263246—22no—
11C22⋊#13096802C2212yes—
12#20263233C2211yescharacteristic
13C22⋊Q8×C197—22no—
14C22⋊Q8×C197—21no—
15C22⋊Q8×C197—22no—
16C22⋊Q8×C197C2212yes—
17C2×C22⋊C4×C197—21no—
18C2×C22⋊C4×C197C2211yescharacteristic
19C23×D197—21no—
20C2×D788—22no—
21C2×D788—42no—
22C2×C22⋊D197—42no—
23C2×C22⋊D197—42no—
24C2×C22⋊D197—42no—
25C22×Dic197—21no—
26#15691345—44no—
27C2×#13096802—22no—
28#15691335—42no—
29C24×C197D411yescharacteristic
30C2×Q8×C197—21no—
31C2×Q8×C197D412yescomplemented
32C22×C4×C197—21no—
33C22×C4×C197—21no—
34C22×C4×C197D411yescharacteristic
35C22×C4×C197C2311yescharacteristic
36C4⋊C4×C197—21no—
37C4⋊C4×C197—22no—
38C4⋊C4×C197—22no—
39C4⋊C4×C197D412yescomplemented
40C22⋊C4×C197—22no—
41C22⋊C4×C197—22no—
42C22⋊C4×C197—22no—
43C22×D197—22no—
44C22×D197—41no—
45C22×D197—41no—
46C22×D197—41no—
47C22×D197—41no—
48C22×D197—41no—
49C22×D197—41no—
50D788—24no—
51D788—42no—
52C22⋊D197—48no—
53C2×Dic197—22no—
54C2×Dic197—41no—
55C2×Dic197—41no—
56#13096802—42no—
57C23×C197—21no—
58C23×C197—21no—
59C23×C197—41no—
60C23×C197—41no—
61C23×C197SD1612yescomplemented
62C23×C197C2×D411yescharacteristic
63Q8×C197—42no—
64Q8×C197—81no—
65C2×C4×C197—22no—
66C2×C4×C197—21no—
67C2×C4×C197—22no—
68C2×C4×C197—41no—
69C2×C4×C197—41no—
70C2×C4×C197—81no—
71C2×C4×C197—81no—
72C2×C4×C197C2×D412yes—
73C22×C197—21no—
74C22×C197—21no—
75C22×C197—22no—
76C22×C197—42no—
77C22×C197—41no—
78C22×C197—41no—
79C22×C197—81no—
80C22×C197#12011yescharacteristic
81C22×C197C2×SD1611yescharacteristic
82C22×C197#13711yescharacteristic
83C2×D197—42no—
84C2×D197—42no—
85C2×D197—42no—
86C2×D197—81no—
87C2×D197—81no—
88C2×D197—81no—
89C2×D197—81no—
90C4×C197—22no—
91C4×C197—41no—
92C4×C197—42no—
93C4×C197—42no—
94Dic197—42no—
95C2×C197—21no—
96C2×C197—21no—
97C2×C197—42no—
98C2×C197#44811yescharacteristic
99C2×C197#45311yescharacteristic
100C2×C197B(4,𝔽2)11yescharacteristic
101D197—82no—
102C197#159611yescharacteristic, complemented, Sylow
103#1596—1971nomaximal, Sylow
104#543D19711yescharacteristic, complemented
105#450—1972no—
106#392—1971no—
107#328—1972no—
108#323—1971no—
109C22×D4—3941no—
110C22×D4—1971no—
111C2×SD16—3942no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D19712yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—3941no—
118C2×C22⋊C4C2×D19711yescharacteristic
119#102—3942no—
120C22⋊C8—1972no—
121#95—1971no—
122C24—3941no—
123C24C22⋊D19711yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D19712yescomplemented
126C2×D4—7882no—
127C2×D4—7882no—
128C2×D4—7882no—
129C2×D4—7882no—
130C2×D4—3942no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—3941no—
134C22×C4C22⋊D19711yescharacteristic
135C22×C4C22×D19711yescharacteristic
136SD16—7884no—
137C2×C8—3942no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D19712yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—7882no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—7881no—
151C23—7881no—
152C23—7881no—
153C23—7881no—
154C23—7881no—
155C23—7881no—
156C23—3942no—
157C23#1569134512yescomplemented
158C23C2×C22⋊D19711yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—7882no—
162D4—7888no—
163D4—3944no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—7881no—
172C2×C4—7881no—
173C2×C4—3942no—
174C2×C4C2×C22⋊D19712yes—
175C8—7882no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—7882no—
184C22—7882no—
185C22—7882no—
186C22—15761no—
187C22—15761no—
188C22—15761no—
189C22—15761no—
190C22#2026329511yescharacteristic
191C22C2×#1569134511yescharacteristic
192C22#2026335711yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—7882no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—15762no—
202C2#2716121811yescharacteristic
203C2#2716124711yescharacteristic
204C2#2716126611yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#543×C197C2
socleC22×C197#120
prime radical#27160988C2
hypercenter#543D197
hypocenterC197#1596
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #27161229C2semidirect
3. #27160816C2semidirect
4. #27160803C2semidirect
5. #543×C197C2semidirect
31. C2×Q8×C197D4semidirect
39. C4⋊C4×C197D4semidirect
61. C23×C197SD16semidirect
102. C197#1596semidirect
104. #543D197semidirect
125. C2×Q8C22⋊D197semidirect
141. C4⋊C4C22⋊D197semidirect
157. C23#15691345semidirect

Maximal subgroups

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

subgroupindex
#271612292
#271609882
#271608162
#271608032
#543×C1972
#1596197