Finite Group Databasebeta

← #15830665

Subgroups — #15830665

Order 3328.

1202 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 355 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#13128607C211yesmaximal, characteristic
2#13123917C211yesmaximal, characteristic, complemented
3#13113395C216yesmaximal, complemented
4#13113330C216yesmaximal, complemented
5C13×#2351C211yesmaximal, characteristic, complemented
6C2×#88254—21no—
7C2×C4×Q8×C13—21no—
8C2×C4×Q8×C13C2211yescharacteristic
9#11719282C414yescomplemented
10#11719281—24no—
11Q8×D52C414yescomplemented
12#11719215—23no—
13#11719182C2216yes—
14C13×#389—23no—
15C13×#382C2213yes—
16C4×C13×C4⋊C4C2213yes—
17Q8×#1055—24no—
18#11718586C2216yes—
19#11718043C22112yes—
20#11718043C2214yes—
21C22×Q8×C13D411yescharacteristic
22#88254—44no—
23#88254—44no—
24C2×#6131—24no—
25C2×Q8×D13—24no—
26C2×#6127—26no—
27C4×Q8×C13—44no—
28C4×Q8×C13C2×C414yes—
29C4×Q8×C13D414yes—
30C2×C13×C4⋊C4—23no—
31C2×C13×C4⋊C4—23no—
32C2×C13×C4⋊C4D413yes—
33C2×C42×C13—23no—
34C2×C42×C13C2313yes—
35#88184—26no—
36#88184C2314yes—
37#88151—212no—
38#88149—212no—
39#88129—26no—
40#88128C2×C416yes—
41C4×D52—26no—
42C4×D52C2×C4112yes—
43#88123C2×C416yes—
44C13×#95—23no—
45C13×#95C2311yescharacteristic
46C13×#95C2313yes—
47C22⋊#1055C2314yes—
48C4⋊#1055—26no—
49C4×#1055—26no—
50#17618—131nomaximal, Sylow
51C2×Q8×C13C22⋊C414yes—
52C2×Q8×C13D814yes—
53C2×Q8×C13SD1614yes—
54C22×C4×C13—21no—
55C22×C4×C13—23no—
56C22×C4×C13C2×D413yes—
57C22×C4×C13C2×D413yes—
58C22×C4×C13C2411yescharacteristic
59C2×C22⋊D13—24no—
60#6131—416no—
61Q8×D13—416no—
62#6127—412no—
63#6127—412no—
64C2×D52—26no—
65C2×D52—26no—
66C2×D52C22×C414yes—
67C2×C4×D13—212no—
68C2×Dic26—26no—
69C13×C4⋊C4—46no—
70C13×C4⋊C4—46no—
71C13×C4⋊C4C22⋊C4112yes—
72C42×C13—46no—
73C42×C13C22×C416yes—
74C42×C13C2×D416yes—
75#6103—212no—
76#6103—24no—
77C4⋊Dic13—26no—
78C4⋊Dic13C22×C414yes—
79C2×#1055—24no—
80#3497—131no—
81#2982—131no—
82#2351D1311yescharacteristic, complemented
83#1517—136no—
84#1502—136no—
85C23×C13C22×D411yescharacteristic
86C22×D13—24no—
87Q8×C13#102116yes—
88C2×C4×C13—22no—
89C2×C4×C13—43no—
90C2×C4×C13—41no—
91C2×C4×C13—41no—
92C2×C4×C13—41no—
93C2×C4×C13—43no—
94C2×C4×C13—43no—
95C2×C4×C13—43no—
96C2×C4×C13C2×C22⋊C416yes—
97C2×C4×C13C2×C22⋊C416yes—
98C2×C4×C13C2×D816yes—
99C2×C4×C13C2×SD1616yes—
100C2×C4×C13C23×C411yescharacteristic
101C2×C4×C13C22×D411yescharacteristic
102C22⋊D13—44no—
103C22⋊D13—44no—
104C2×Dic13—24no—
105D52—44no—
106D52—412no—
107D52—86no—
108C4×D13—424no—
109Dic26—412no—
110#1055—44no—
111C2×2−1+4—261no—
112#550—134no—
113D4×Q8—134no—
114#548—264no—
115#537—263no—
116#532—136no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2×D1311yescharacteristic
119C8×Q8—264no—
120#431—136no—
121#389—23no—
122#382C2×D1313yes—
123C4×C4⋊C4C2×D1313yes—
124#327—134no—
125#327—1312no—
126C22×C13—21no—
127C22×C13—21no—
128C22×C13C22×C22⋊C411yescharacteristic
129C22×C13C22×D811yescharacteristic
130C22×C13C22×SD1611yescharacteristic
131C2×D13—44no—
132C2×D13—44no—
133C2×D13—44no—
134C4×C13—21no—
135C4×C13—21no—
136C4×C13—26no—
137C4×C13—42no—
138C4×C13—83no—
139C4×C13C2×#102112yes—
140Dic13—44no—
1412−1+4—524no—
1422−1+4—524no—
143C2×#41—264no—
144C2×#41—266no—
145C22×Q8—264no—
146C22×Q8C22⋊D1311yescharacteristic
147C4⋊Q8—136no—
148#127—136no—
149#124—266no—
150C22⋊Q8—2612no—
151#121—134no—
152#121—266no—
153#121—2612no—
154C4×Q8—44no—
155C4×Q8C4×D1314yes—
156C4×Q8D5214yes—
157C4×D4—1312no—
158C4×D4—266no—
159C2×C4⋊C4—23no—
160C2×C4⋊C4—23no—
161C2×C4⋊C4C22⋊D1313yes—
162C2×C42—23no—
163C2×C42C22×D1313yes—
164C4⋊C8—266no—
165C22⋊C8—134no—
166C4×C8—266no—
167#95—23no—
168#95C22×D1311yescharacteristic
169#95C22×D1313yes—
170C2×C13—21no—
171C2×C13—21no—
172C2×C13#288411yescharacteristic
173C2×C13C22×#10211yescharacteristic
174C2×C13#290111yescharacteristic
175D13—84no—
176#41—5216no—
177#41—5212no—
178#41—5212no—
179C2×Q8—266no—
180C2×Q8—5216no—
181C2×Q8#610314yes—
182C2×Q8#610414yes—
183C2×Q8#610614yes—
184C2×D4—134no—
185C2×D4—264no—
186C2×D4—266no—
187C2×D4—266no—
188C22×C4—21no—
189C22×C4—23no—
190C22×C4—2612no—
191C22×C4C2×C22⋊D1313yes—
192C22×C4C2×C22⋊D1313yes—
193C22×C4C23×D1311yescharacteristic
194C2×C8—264no—
195C4⋊C4—46no—
196C4⋊C4—46no—
197C4⋊C4—134no—
198C4⋊C4—266no—
199C4⋊C4#6103112yes—
200C22⋊C4—2612no—
201C22⋊C4—264no—
202C42—46no—
203C42C2×C4×D1316yes—
204C42C2×D5216yes—
205C13#1761811yescharacteristic, complemented, Sylow
206C23—264no—
207C23C22×C22⋊D1311yescharacteristic
208Q8—5212no—
209Q8#88049116yes—
210D4—524no—
211D4—524no—
212D4—524no—
213D4—5212no—
214D4—1046no—
215C2×C4—22no—
216C2×C4—43no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—43no—
221C2×C4—43no—
222C2×C4—43no—
223C2×C4—264no—
224C2×C4—5224no—
225C2×C4C2×#610316yes—
226C2×C4C2×#610316yes—
227C2×C4C2×#610416yes—
228C2×C4C2×#610616yes—
229C2×C4C22×C4×D1311yescharacteristic
230C2×C4C22×D5211yescharacteristic
231C8—524no—
232C22—21no—
233C22—21no—
234C22—524no—
235C22—524no—
236C22—524no—
237C22C22×#610311yescharacteristic
238C22C22×#610411yescharacteristic
239C22C22×#610611yescharacteristic
240C4—26no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4—83no—
245C4—524no—
246C4C2×#88049112yes—
247C2—21no—
248C2—21no—
249C2—1044no—
250C2#1312329911yescharacteristic
251C2C22×#8804911yescharacteristic
252C2#1312338411yescharacteristic
subgroupG/H
centerC22C22×#6103
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2351C2
socleC22×C13C22×C22⋊C4
prime radical#13128607C2
cosocleC22×C4×C13C24
hypercenter#2351D13
hypocenterC13#17618
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #13123917C2semidirect
3. #13113395C2semidirect
4. #13113330C2semidirect
5. C13×#2351C2semidirect
9. #11719282C4semidirect
11. Q8×D52C4semidirect
82. #2351D13semidirect
205. C13#17618semidirect

Maximal subgroups

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

subgroupindex
#131286072
#131239172
#131133952
#131133302
C13×#23512
#1761813