Finite Group Databasebeta

← #38741690

Subgroups — #38741690

Order 25216.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161667C211yesmaximal, characteristic
2#27161235C211yesmaximal, characteristic, complemented
3#27161218C211yesmaximal, characteristic, complemented
4#27160815C211yesmaximal, characteristic, complemented
5#27160812C211yesmaximal, characteristic, complemented
6#27160803C211yesmaximal, characteristic, complemented
7#393×C197C211yesmaximal, characteristic, complemented
8C2×#15691348—21no—
9C2×#15691345—21no—
10#20263319—21no—
11#20263312—21no—
12#20263308—21no—
13#20263305C2211yescharacteristic
14#20263300C2211yescharacteristic
15#20263299—21no—
16#20263295—21no—
17C4×C22⋊D197—21no—
18#20263250—21no—
19#20263250—21no—
20C22⋊#13096802C2211yescharacteristic
21C22⋊#13096802C2211yescharacteristic
22C22×Q8×C197C2211yescharacteristic
23C2×C22⋊C4×C197—21no—
24C2×C22⋊C4×C197C2211yescharacteristic
25#95×C197—21no—
26#95×C197C2211yescharacteristic
27C2×Dic394D411yescharacteristic
28C2×D788D411yescharacteristic
29C2×C22⋊D197—21no—
30C2×C22⋊D197—21no—
31C2×C22⋊D197—21no—
32C2×C4×D197—21no—
33#15691348—42no—
34#15691345—42no—
35C2×#13096802—21no—
36C2×#13096802—21no—
37C4⋊Dic197D411yescharacteristic
38C4⋊Dic197D411yescharacteristic
39#15691337—21no—
40#15691337—21no—
41C22⋊Dic197—21no—
42C22⋊Dic197—21no—
43C22⋊Dic197—21no—
44#15691335—21no—
45#15691335—21no—
46C4×Dic197—21no—
47C24×C197D411yescharacteristic
48C2×Q8×C197—21no—
49C2×Q8×C197—21no—
50C2×Q8×C197—42no—
51C22×C4×C197—21no—
52C22×C4×C197—21no—
53C22×C4×C197D411yescharacteristic
54C22×C4×C197C2311yescharacteristic
55C22⋊C4×C197—21no—
56C22⋊C4×C197—21no—
57C22⋊C4×C197—42no—
58C22×D197—21no—
59Dic394—41no—
60D788—41no—
61C22⋊D197—41no—
62C22⋊D197—41no—
63C22⋊D197—81no—
64C22⋊D197—81no—
65C2×Dic197—21no—
66C2×Dic197—21no—
67C2×Dic197—21no—
68C4×D197—81no—
69#13096802—41no—
70#13096802—41no—
71C23×C197—21no—
72C23×C197—21no—
73C23×C197—21no—
74C23×C197—41no—
75C23×C197—41no—
76C23×C197C2×D411yescharacteristic
77Q8×C197—22no—
78Q8×C197—22no—
79Q8×C197—41no—
80Q8×C197—41no—
81C2×C4×C197—22no—
82C2×C4×C197—21no—
83C2×C4×C197—41no—
84C2×C4×C197—41no—
85C2×C4×C197—41no—
86C2×C4×C197—41no—
87C2×C4×C197—41no—
88C2×C4×C197—41no—
89C2×C4×C197—41no—
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—81no—
99C22×C197#12011yescharacteristic
100C22×C197#13511yescharacteristic
101C22×C197#13711yescharacteristic
102C2×D197—41no—
103C2×D197—41no—
104C2×D197—41no—
105C4×C197—21no—
106C4×C197—21no—
107C4×C197—41no—
108C4×C197—42no—
109C4×C197—81no—
110Dic197—41no—
111Dic197—41no—
112Dic197—41no—
113C2×C197—21no—
114C2×C197—21no—
115C2×C197—81no—
116C2×C197#45211yescharacteristic
117C2×C197#45511yescharacteristic
118C2×C197B(4,𝔽2)11yescharacteristic
119D197—81no—
120C197#160611yescharacteristic, complemented, Sylow
121#1606—1971nomaximal, Sylow
122#538—1971no—
123#451—1971no—
124#448—1971no—
125#393D19711yescharacteristic, complemented
126#328—1971no—
127#327—1971no—
128#323—1971no—
129C22×Q8C2×D19711yescharacteristic
130C2×Q16—3941no—
131C2×SD16—3941no—
132#126—3941no—
133#124—3941no—
134#123—3941no—
135C22⋊Q8—1971no—
136#121—3941no—
137#121—1971no—
138#120—3941no—
139C4×D4—3941no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4C2×D19711yescharacteristic
142#103—3941no—
143#103—3941no—
144C22⋊C8—1971no—
145C22⋊C8—1971no—
146#95—21no—
147#95C2×D19711yescharacteristic
148C24C22⋊D19711yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—1971no—
153C2×D4—3941no—
154C2×D4—3941no—
155C2×D4—3941no—
156C2×D4—1971no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—3941no—
160C22×C4C22⋊D19711yescharacteristic
161C22×C4C22×D19711yescharacteristic
162Q16—7882no—
163SD16—7882no—
164C2×C8—3941no—
165C2×C8—3941no—
166C4⋊C4—3941no—
167C4⋊C4—3941no—
168C4⋊C4—1971no—
169C4⋊C4—1971no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—42no—
173C22⋊C4—3941no—
174C22⋊C4—3941no—
175C22⋊C4—3941no—
176C22⋊C4—3941no—
177C22⋊C4—3941no—
178C42—3941no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—3941no—
185C23C2×C22⋊D19711yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—7881no—
191D4—7881no—
192D4—7881no—
193D4—7881no—
194D4—15761no—
195D4—15761no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—15761no—
206C2×C4—3941no—
207C2×C4—3941no—
208C2×C4—3941no—
209C2×C4D4×D19711yescharacteristic
210C2×C4D4×D19711yescharacteristic
211C8—7881no—
212C8—7881no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—7881no—
221C22—7881no—
222C22—7881no—
223C22#2026329411yescharacteristic
224C22#2026334611yescharacteristic
225C22#2026335711yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—81no—
231C4—7881no—
232C4—7881no—
233C4—7881no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—15761no—
238C2#2716123911yescharacteristic
239C2#2716125611yescharacteristic
240C2#2716126511yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#393×C197C2
socleC22×C197#120
prime radical#27161667C2
hypercenter#393D197
hypocenterC197#1606
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #27161235C2semidirect
3. #27161218C2semidirect
4. #27160815C2semidirect
5. #27160812C2semidirect
6. #27160803C2semidirect
7. #393×C197C2semidirect
120. C197#1606semidirect
125. #393D197semidirect

Maximal subgroups

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

subgroupindex
#271616672
#271612352
#271612182
#271608152
#271608122
#271608032
#393×C1972
#1606197