Finite Group Databasebeta

← #15830668

Subgroups — #15830668

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#13117757C211yesmaximal, characteristic, complemented
3#13113397C216yesmaximal, complemented
4#13113332C216yesmaximal, complemented
5C13×#2982C211yesmaximal, characteristic, complemented
6C2×#88254—21no—
7C2×C4×Q8×C13C2211yescharacteristic
8C2×Q8×Dic13—21no—
9#11719282C414yescomplemented
10#11719281—24no—
11Q8×D52C414yescomplemented
12#11719215—23no—
13#11719182C2216yes—
14C8×Q8×C13—24no—
15C13×#431C2216yes—
16#11718646—23no—
17#11718597C2213yes—
18C4×C4⋊Dic13C2213yes—
19#11718062C22112yes—
20#11718062C2214yes—
21C22×Q8×C13D411yescharacteristic
22#88254—44no—
23#88254—44no—
24C2×#6131—24no—
25C2×Q8×D13—24no—
26C2×#6127—26no—
27C4×Q8×C13C2×C414yes—
28C4×Q8×C13D414yes—
29C2×C13×C4⋊C4D413yes—
30C2×C42×C13C2313yes—
31Q8×Dic13—44no—
32#88184—26no—
33#88184C2314yes—
34C2×C4⋊Dic13—23no—
35C2×C4⋊Dic13—23no—
36C2×C4×Dic13—23no—
37#88151—212no—
38#88149—212no—
39#88129—26no—
40#88128C2×C416yes—
41C4×D52—26no—
42C4×D52C2×C4112yes—
43#88123C2×C416yes—
44C13×C4⋊C8—26no—
45C13×C22⋊C8C2314yes—
46C4×C8×C13—26no—
47#88071—23no—
48#88071C2311yescharacteristic
49#88071C2313yes—
50#17618—131nomaximal, Sylow
51C2×Q8×C13C22⋊C414yes—
52C2×Q8×C13D814yes—
53C2×Q8×C13SD1614yes—
54C22×C4×C13C2×D413yes—
55C22×C4×C13C2×D413yes—
56C22×C4×C13C2411yescharacteristic
57C2×C22⋊D13—24no—
58C22×Dic13—21no—
59C22×Dic13—23no—
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—
69C2×C8×C13—24no—
70C13×C4⋊C4C22⋊C4112yes—
71C42×C13C22×C416yes—
72C42×C13C2×D416yes—
73#6103—212no—
74#6103—24no—
75C4⋊Dic13—26no—
76C4⋊Dic13—46no—
77C4⋊Dic13—46no—
78C4⋊Dic13C22×C414yes—
79C4×Dic13—46no—
80#3497—131no—
81#2982D1311yescharacteristic, complemented
82#2351—131no—
83#1517—136no—
84#1502—136no—
85C23×C13C22×D411yescharacteristic
86C22×D13—24no—
87Q8×C13#102116yes—
88C2×C4×C13—43no—
89C2×C4×C13—41no—
90C2×C4×C13C2×C22⋊C416yes—
91C2×C4×C13C2×C22⋊C416yes—
92C2×C4×C13C2×D816yes—
93C2×C4×C13C2×SD1616yes—
94C2×C4×C13C23×C411yescharacteristic
95C2×C4×C13C22×D411yescharacteristic
96C22⋊D13—44no—
97C22⋊D13—44no—
98C2×Dic13—22no—
99C2×Dic13—24no—
100C2×Dic13—41no—
101C2×Dic13—41no—
102C2×Dic13—43no—
103C2×Dic13—43no—
104C2×Dic13—43no—
105D52—44no—
106D52—412no—
107D52—86no—
108C4×D13—424no—
109Dic26—412no—
110C8×C13—44no—
111C2×2−1+4—261no—
112#550—134no—
113D4×Q8—134no—
114#548—264no—
115#537—263no—
116#532—136no—
117C2×C4×Q8—261no—
118C2×C4×Q8C2×D1311yescharacteristic
119C8×Q8—24no—
120#431C2×D1316yes—
121#389—263no—
122#382—133no—
123C4×C4⋊C4—133no—
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×C13C2×#102112yes—
138Dic13—42no—
139Dic13—44no—
140Dic13—83no—
1412−1+4—524no—
1422−1+4—524no—
143C2×#41—264no—
144C2×#41—266no—
145C22×Q8—264no—
146C22×Q8D5211yescharacteristic
147C4⋊Q8—136no—
148#127—136no—
149#124—266no—
150C22⋊Q8—2612no—
151#121—134no—
152#121—266no—
153#121—2612no—
154C4×Q8—524no—
155C4×Q8C4×D1314yes—
156C4×Q8C22⋊D1314yes—
157C4×D4—1312no—
158C4×D4—266no—
159C2×C4⋊C4—263no—
160C2×C4⋊C4—263no—
161C2×C4⋊C4D5213yes—
162C2×C42—263no—
163C2×C42C22×D1313yes—
164C4⋊C8—26no—
165C22⋊C8C22×D1314yes—
166C4×C8—26no—
167#95—131no—
168#95—133no—
169#95—263no—
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#609514yes—
182C2×Q8D10414yes—
183C2×Q8#610314yes—
184C2×D4—134no—
185C2×D4—264no—
186C2×D4—266no—
187C2×D4—266no—
188C22×C4—261no—
189C22×C4—2612no—
190C22×C4—263no—
191C22×C4C2×D5213yes—
192C22×C4C2×D5213yes—
193C22×C4C23×D1311yescharacteristic
194C2×C8—24no—
195C4⋊C4—134no—
196C4⋊C4—266no—
197C4⋊C4—526no—
198C4⋊C4—526no—
199C4⋊C4#6103112yes—
200C22⋊C4—2612no—
201C22⋊C4—264no—
202C42—526no—
203C42C2×C4×D1316yes—
204C42C2×C22⋊D1316yes—
205C13#1761811yescharacteristic, complemented, Sylow
206C23—264no—
207C23C22×D5211yescharacteristic
208Q8—5212no—
209Q8#88061116yes—
210D4—524no—
211D4—524no—
212D4—524no—
213D4—5212no—
214D4—1046no—
215C2×C4—43no—
216C2×C4—41no—
217C2×C4—262no—
218C2×C4—264no—
219C2×C4—521no—
220C2×C4—521no—
221C2×C4—5224no—
222C2×C4—523no—
223C2×C4—523no—
224C2×C4—523no—
225C2×C4C2×#609516yes—
226C2×C4C2×D10416yes—
227C2×C4C2×#610316yes—
228C2×C4C2×#610316yes—
229C2×C4C22×C4×D1311yescharacteristic
230C2×C4C22×C22⋊D1311yescharacteristic
231C8—44no—
232C22—21no—
233C22—21no—
234C22—524no—
235C22—524no—
236C22—524no—
237C22C22×#609511yescharacteristic
238C22C22×D10411yescharacteristic
239C22C22×#610311yescharacteristic
240C4—26no—
241C4—21no—
242C4—21no—
243C4—522no—
244C4—524no—
245C4—1043no—
246C4C2×#88061112yes—
247C2—21no—
248C2—21no—
249C2—1044no—
250C2#1312329911yescharacteristic
251C2C22×#8806111yescharacteristic
252C2#1312338111yescharacteristic
subgroupG/H
centerC22C22×#6103
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2982C2
socleC22×C13C22×C22⋊C4
prime radical#13128607C2
cosocleC22×C4×C13C24
hypercenter#2982D13
hypocenterC13#17618
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #13117757C2semidirect
3. #13113397C2semidirect
4. #13113332C2semidirect
5. C13×#2982C2semidirect
9. #11719282C4semidirect
11. Q8×D52C4semidirect
81. #2982D13semidirect
205. C13#17618semidirect

Maximal subgroups

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

subgroupindex
#131286072
#131177572
#131133972
#131133322
C13×#29822
#1761813