Finite Group Databasebeta

← #38741692

Subgroups — #38741692

Order 25216.

260 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161234C211yesmaximal, characteristic, complemented
2#27161217C211yesmaximal, characteristic, complemented
3#27160991C211yesmaximal, characteristic, complemented
4#27160816C211yesmaximal, characteristic, complemented
5#27160813C211yesmaximal, characteristic, complemented
6#27160803C211yesmaximal, characteristic, complemented
7#538×C197C211yesmaximal, characteristic
8C22×Dic394C2211yescharacteristic
9C2×#15691348—21no—
10C2×#15691346—21no—
11C2×C22⋊Dic197—21no—
12#20263249—21no—
13#20263249—21no—
14C22⋊#13096802C2211yescharacteristic
15C22⋊#13096802C2211yescharacteristic
16#20263233—21no—
17#20263233C2211yescharacteristic
18#126×C197—21no—
19#124×C197—21no—
20#123×C197—21no—
21C22⋊Q8×C197C2211yescharacteristic
22#121×C197—21no—
23#121×C197C2211yescharacteristic
24#120×C197—21no—
25C4×D4×C197—21no—
26C2×C22⋊C4×C197C2211yescharacteristic
27C2×Dic394—21no—
28C2×Dic394—21no—
29C2×Dic394—42no—
30C22×Dic197—21no—
31C22×Dic197—21no—
32#15691348—42no—
33#15691346—42no—
34C2×#13096802—21no—
35C2×#13096802—21no—
36C22⋊Dic197—42no—
37C24×C197D411yescharacteristic
38C2×Q8×C197D411yescharacteristic
39C2×D4×C197—21no—
40C2×D4×C197—21no—
41C2×D4×C197—21no—
42C2×D4×C197D411yescharacteristic
43C22×C4×C197—21no—
44C22×C4×C197D411yescharacteristic
45C22×C4×C197C2311yescharacteristic
46C4⋊C4×C197—21no—
47C4⋊C4×C197—21no—
48C4⋊C4×C197D411yescharacteristic
49C4⋊C4×C197D411yescharacteristic
50C22⋊C4×C197—21no—
51C22⋊C4×C197—21no—
52C22⋊C4×C197—21no—
53C22⋊C4×C197—21no—
54C22⋊C4×C197—21no—
55C22⋊C4×C197—21no—
56C22⋊C4×C197—21no—
57C42×C197—21no—
58Dic394—22no—
59Dic394—22no—
60Dic394—41no—
61Dic394—41no—
62C2×Dic197—22no—
63C2×Dic197—41no—
64C2×Dic197—41no—
65C2×Dic197—41no—
66C2×Dic197—41no—
67C2×Dic197—41no—
68#13096802—41no—
69#13096802—41no—
70C23×C197—21no—
71C23×C197—21no—
72C23×C197—21no—
73C23×C197—21no—
74C23×C197—41no—
75C23×C197—41no—
76C23×C197C2×D411yescharacteristic
77Q8×C197—41no—
78D4×C197—41no—
79D4×C197—41no—
80D4×C197—41no—
81D4×C197—81no—
82D4×C197—81no—
83C2×C4×C197—21no—
84C2×C4×C197—21no—
85C2×C4×C197—21no—
86C2×C4×C197—21no—
87C2×C4×C197—41no—
88C2×C4×C197—41no—
89C2×C4×C197—81no—
90C2×C4×C197C2×D411yescharacteristic
91C2×C4×C197C2×D411yescharacteristic
92C22×C197—21no—
93C22×C197—21no—
94C22×C197—42no—
95C22×C197—41no—
96C22×C197—41no—
97C22×C197—41no—
98C22×C197—41no—
99C22×C197—41no—
100C22×C197—41no—
101C22×C197—81no—
102C22×C197#12011yescharacteristic
103C22×C197#13511yescharacteristic
104C22×C197#13711yescharacteristic
105C4×C197—21no—
106C4×C197—21no—
107C4×C197—41no—
108C4×C197—41no—
109C4×C197—41no—
110C4×C197—41no—
111Dic197—42no—
112Dic197—81no—
113C2×C197—21no—
114C2×C197—21no—
115C2×C197—81no—
116C2×C197—81no—
117C2×C197#45211yescharacteristic
118C2×C197#45511yescharacteristic
119C2×C197B(4,𝔽2)11yescharacteristic
120C197#160611yescharacteristic, complemented, Sylow
121#1606—1971nomaximal, Sylow
122#538D19711yescharacteristic
123#451—1971no—
124#448—1971no—
125#393—1971no—
126#328—1971no—
127#327—1971no—
128#323—1971no—
129C22×Q8—1971no—
130C2×Q16—3941no—
131C2×SD16—3941no—
132#126—21no—
133#124—21no—
134#123—21no—
135C22⋊Q8C2×D19711yescharacteristic
136#121—21no—
137#121C2×D19711yescharacteristic
138#120—21no—
139C4×D4—21no—
140C2×C22⋊C4—3941no—
141C2×C22⋊C4C2×D19711yescharacteristic
142#103—3941no—
143#103—3941no—
144C22⋊C8—1971no—
145C22⋊C8—1971no—
146#95—3941no—
147#95—1971no—
148C24C22⋊D19711yescharacteristic
149C2×Q8—7882no—
150C2×Q8—3941no—
151C2×Q8—3941no—
152C2×Q8C22⋊D19711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4C22⋊D19711yescharacteristic
157C22×C4—21no—
158C22×C4—3941no—
159C22×C4—3941no—
160C22×C4C22⋊D19711yescharacteristic
161C22×C4C22×D19711yescharacteristic
162Q16—7882no—
163SD16—7882no—
164C2×C8—3941no—
165C2×C8—3941no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4C22⋊D19711yescharacteristic
169C4⋊C4C22⋊D19711yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—7882no—
178C42—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23C2×C22⋊D19711yescharacteristic
186Q8—41no—
187Q8—7881no—
188Q8—7881no—
189Q8—3942no—
190Q8—3942no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—81no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—7881no—
204C2×C4—7881no—
205C2×C4—7881no—
206C2×C4—7881no—
207C2×C4—7881no—
208C2×C4—3942no—
209C2×C4C2×C22⋊D19711yescharacteristic
210C2×C4C2×C22⋊D19711yescharacteristic
211C8—7881no—
212C8—7881no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22#2026329511yescharacteristic
224C22#2026334611yescharacteristic
225C22#2026335611yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—7882no—
233C4—15761no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—81no—
238C2#2716124111yescharacteristic
239C2#2716125711yescharacteristic
240C2#2716126611yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting, prime radical#538×C197C2
socleC22×C197#120
hypercenter#538D197
hypocenterC197#1606
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #27161234C2semidirect
2. #27161217C2semidirect
3. #27160991C2semidirect
4. #27160816C2semidirect
5. #27160813C2semidirect
6. #27160803C2semidirect
120. C197#1606semidirect

Maximal subgroups

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

subgroupindex
#271612342
#271612172
#271609912
#271608162
#271608132
#271608032
#538×C1972
#1606197