Finite Group Databasebeta

← #38741058

Subgroups — #38741058

Order 25216.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20263309C211yesmaximal, characteristic
2#27161043C211yesmaximal, characteristic, complemented
3#27160804C212yesmaximal, complemented
4#27160802C212yesmaximal, complemented
5C2×C22⋊C8×C197C211yesmaximal, characteristic, complemented
6C22×C22⋊D197C411yescharacteristic
7C2×#15691342—21no—
8#20263309—21no—
9#20263309—21no—
10#20263309—21no—
11#20263309—21no—
12C2×#15691337C411yescharacteristic
13C2×#15691337C2211yescharacteristic
14C2×C22⋊Dic197C411yescharacteristic
15C2×#15691335C411yescharacteristic
16C2×#15691335C2211yescharacteristic
17C22⋊#13096802—21no—
18C22⋊#13096802C2212yes—
19C23×C4×C197C2211yescharacteristic
20C22×C8×C197—21no—
21C22⋊C8×C197—21no—
22C22⋊C8×C197C2212yes—
23C23×D197C2×C411yescharacteristic
24C2×C22⋊D197—21no—
25C2×C22⋊D197—21no—
26C2×C22⋊D197—42no—
27C22×Dic197C2×C411yescharacteristic
28C22×Dic197C2×C411yescharacteristic
29C22×Dic197C2×C411yescharacteristic
30#15691342—42no—
31C2×#13096802—22no—
32#15691337—21no—
33#15691337—21no—
34#15691337—21no—
35#15691337—21no—
36C22⋊Dic197—21no—
37C22⋊Dic197—21no—
38#15691335—21no—
39#15691335—21no—
40#15691335—21no—
41#15691335—21no—
42C24×C197C2×C411yescharacteristic
43C22×C4×C197—42no—
44C22×C4×C197C2×C411yescharacteristic
45C22×C4×C197D411yescharacteristic
46C22×C4×C197D411yescharacteristic
47C22×C4×C197D411yescharacteristic
48C22×C4×C197D411yescharacteristic
49C22×C4×C197C2311yescharacteristic
50C2×C8×C197—22no—
51C2×C8×C197—42no—
52C22×D197—21no—
53C22×D197—22no—
54C22×D197—41no—
55C22×D197—41no—
56C22⋊D197—42no—
57C22⋊D197—42no—
58C2×Dic197—21no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—22no—
62C2×Dic197—41no—
63C2×Dic197—41no—
64#13096802—42no—
65C23×C197—22no—
66C23×C197—22no—
67C23×C197—41no—
68C23×C197C22⋊C411yescharacteristic
69C23×C197C22⋊C411yescharacteristic
70C23×C197C22×C411yescharacteristic
71C2×C4×C197—22no—
72C2×C4×C197—22no—
73C2×C4×C197—41no—
74C2×C4×C197—41no—
75C2×C4×C197—41no—
76C2×C4×C197—41no—
77C2×C4×C197C22⋊C411yescharacteristic
78C2×C4×C197C22⋊C411yescharacteristic
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C8×C197—42no—
82C22×C197—21no—
83C22×C197—21no—
84C22×C197—21no—
85C22×C197—21no—
86C22×C197—21no—
87C22×C197—21no—
88C22×C197—21no—
89C22×C197—44no—
90C22×C197#9912yes—
91C22×C197C2×C22⋊C411yescharacteristic
92C22×C197#13112yes—
93C2×D197—41no—
94C2×D197—41no—
95C2×D197—41no—
96C2×D197—82no—
97C4×C197—21no—
98C4×C197—21no—
99C4×C197—21no—
100C4×C197—21no—
101Dic197—41no—
102Dic197—41no—
103Dic197—41no—
104C2×C197—22no—
105C2×C197—22no—
106C2×C197—41no—
107C2×C197#40811yescharacteristic
108C2×C197C2×#9911yescharacteristic
109C2×C197#41311yescharacteristic
110D197—81no—
111C197#146811yescharacteristic, complemented, Sylow
112#1468—1971nomaximal, Sylow
113C2×#123—1971no—
114#407—1971no—
115C2×C22⋊C8D19711yescharacteristic, complemented
116#324—1972no—
117#323—1972no—
118C22×D4—1971no—
119C23×C4C2×D19711yescharacteristic
120C2×M16—3941no—
121C22×C8—21no—
122#123—3941no—
123#123—3941no—
124#123—3941no—
125#123—3941no—
126C2×C4⋊C4—1971no—
127C2×C4⋊C4—1971no—
128C2×C22⋊C4—1971no—
129C2×C22⋊C4—1971no—
130C2×C22⋊C4—1971no—
131C22⋊C8—21no—
132C22⋊C8—3941no—
133C22⋊C8—1972no—
134C22⋊C8C2×D19712yes—
135C24—1971no—
136C24C4×D19711yescharacteristic
137C2×D4—7882no—
138C2×D4—3941no—
139C2×D4—3941no—
140C22×C4—42no—
141C22×C4—1971no—
142C22×C4—1971no—
143C22×C4—1971no—
144C22×C4C4×D19711yescharacteristic
145C22×C4C22⋊D19711yescharacteristic
146C22×C4C22⋊D19711yescharacteristic
147C22×C4D78811yescharacteristic
148C22×C4D78811yescharacteristic
149C22×C4C22×D19711yescharacteristic
150M16—7882no—
151C2×C8—22no—
152C2×C8—42no—
153C2×C8—3942no—
154C4⋊C4—3941no—
155C4⋊C4—3941no—
156C4⋊C4—3941no—
157C4⋊C4—3941no—
158C22⋊C4—3941no—
159C22⋊C4—3941no—
160C22⋊C4—3941no—
161C22⋊C4—3941no—
162C22⋊C4—3941no—
163C22⋊C4—3941no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—7881no—
168C23—7881no—
169C23—3941no—
170C23—3942no—
171C23#1569133511yescharacteristic
172C23#1569133511yescharacteristic
173C23C2×C4×D19711yescharacteristic
174D4—7882no—
175D4—7882no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—7881no—
183C2×C4—7881no—
184C2×C4—3941no—
185C2×C4—3941no—
186C2×C4—3941no—
187C2×C4—3942no—
188C2×C4#1569133511yescharacteristic
189C2×C4#1569133511yescharacteristic
190C2×C4C2×C22⋊D19711yescharacteristic
191C2×C4C2×D78811yescharacteristic
192C8—42no—
193C8—7882no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—7881no—
203C22—7881no—
204C22—7881no—
205C22—15762no—
206C22#2026324012yes—
207C22C2×#1569133511yescharacteristic
208C22#2026333212yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—7881no—
214C4—7881no—
215C4—7881no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—15761no—
220C2#2716104511yescharacteristic
221C2C2×#2026324011yescharacteristic
222C2#2716106111yescharacteristic
subgroupG/H
centerC22C2×#15691335
commutatorC23×C197C22×C4
FrattiniC22×C4C22×D197
FittingC2×C22⋊C8×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC22×C22⋊D197C4
cosocleC22×C4×C197C23
hypercenterC2×C22⋊C8D197
hypocenterC197#1468
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #27161043C2semidirect
3. #27160804C2semidirect
4. #27160802C2semidirect
5. C2×C22⋊C8×C197C2semidirect
111. C197#1468semidirect
115. C2×C22⋊C8D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633092
#271610432
#271608042
#271608022
C2×C22⋊C8×C1972
#1468197