Finite Group Databasebeta

← #27204402

Subgroups — #27204402

Order 13184.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#13110105C211yesmaximal, characteristic
2C2×#15693052C212yesmaximal, complemented
3#20267951C214yesmaximal, complemented
4#20267946C212yesmaximal, complemented
5#20267941C212yesmaximal, complemented
6#20267790C212yesmaximal, complemented
7#20267755C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C103C211yesmaximal, characteristic, complemented
9C2×#13110105—28no—
10C2×#13110105—21no—
11C2×#13110105—21no—
12C2×#13110105C2212yes—
13C2×#13110105C2212yes—
14C22×Dic206C2211yescharacteristic
15C22×D412C2211yescharacteristic
16C22×C22⋊D103—21no—
17C22×C4×D103—21no—
18#15693052—28no—
19C2×#13110095C2214yescomplemented
20C2×#13110092C2214yescomplemented
21C2×#13110089C2212yescomplemented
22#15692944C2214yescomplemented
23#15692941C2214yescomplemented
24C22⋊(C103⋊C8)C2212yes—
25C22⋊(C103⋊C8)C2212yescomplemented
26C22×Q8×C103C2211yescharacteristic, complemented
27C23×C4×C103C2211yescharacteristic
28C22⋊Q8×C103—21no—
29C22⋊Q8×C103—21no—
30C22⋊Q8×C103C2212yes—
31C22⋊Q8×C103C2212yescomplemented
32C2×C4⋊C4×C103—21no—
33C2×C4⋊C4×C103C2211yescharacteristic, complemented
34C2×C22⋊C4×C103—21no—
35C23×D103—21no—
36#13110105—28no—
37#13110105—28no—
38#13110105—24no—
39#13110105—24no—
40#13110105—24no—
41#13110105D418yes—
42C2×Dic206—21no—
43C2×Dic206—24no—
44C2×Dic206C2312yes—
45C2×D412—21no—
46C2×D412—24no—
47C2×D412C2312yes—
48C2×C22⋊D103—28no—
49C2×C22⋊D103—22no—
50C2×C22⋊D103—22no—
51C22×Dic103—21no—
52C2×C4×D103—28no—
53C2×C4×D103—22no—
54C2×C4×D103—22no—
55#13110095—28no—
56#13110092—28no—
57#13110089—24no—
58C2×C103⋊C8C2314yes—
59C24×C103D411yescharacteristic
60C2×Q8×C103—21no—
61C2×Q8×C103—24no—
62C2×Q8×C103C2312yes—
63C22×C4×C103—24no—
64C22×C4×C103—21no—
65C22×C4×C103—21no—
66C22×C4×C103D411yescharacteristic
67C22×C4×C103D411yescharacteristic
68C22×C4×C103D411yescharacteristic
69C22×C4×C103C2311yescharacteristic
70C22×C4×C103C2311yescharacteristic
71C22×C4×C103C2311yescharacteristic
72C4⋊C4×C103—22no—
73C4⋊C4×C103—22no—
74C4⋊C4×C103—21no—
75C4⋊C4×C103C2312yes—
76C22⋊C4×C103—22no—
77C22⋊C4×C103—22no—
78C23×C103—22no—
79C23×C103—22no—
80C23×C103—41no—
81C23×C103C2×D411yescharacteristic
82C23×C103C2×D411yescharacteristic
83C23×C103C2×D411yescharacteristic
84C22×D103—24no—
85C22×D103—22no—
86C22×D103—42no—
87C22×D103—42no—
88Q8×C103—24no—
89Q8×C103—42no—
90C2×C4×C103—22no—
91C2×C4×C103—22no—
92C2×C4×C103—22no—
93C2×C4×C103—22no—
94C2×C4×C103—22no—
95C2×C4×C103—22no—
96C2×C4×C103—22no—
97C2×C4×C103—42no—
98C2×C4×C103—42no—
99C2×C4×C103C2×D414yes—
100C2×C4×C103C2×D411yescharacteristic
101C2×C4×C103C2×D411yescharacteristic
102C2×C4×C103C2×D411yescharacteristic
103C2×C4×C103C2411yescharacteristic
104C22⋊D103—28no—
105C22⋊D103—28no—
106C2×Dic103—24no—
107C2×Dic103—22no—
108D412—24no—
109D412—22no—
110D412C2×D414yes—
111C4×D103—28no—
112C4×D103—28no—
113Dic206—24no—
114Dic206—22no—
115Dic206C2×D414yes—
116C103⋊C8—24no—
117C22×C103—21no—
118C22×C103—21no—
119C22×C103—21no—
120C22×C103—21no—
121C22×C103—21no—
122C22×C103—21no—
123C22×C103—21no—
124C22×C103—44no—
125C22×C103#12012yes—
126C22×C103#13712yescomplemented
127C22×C103C22×D411yescharacteristic
128C2×D103—24no—
129C2×D103—48no—
130C2×D103—42no—
131C2×D103—42no—
132C4×C103—21no—
133C4×C103—21no—
134C4×C103—42no—
135C4×C103—42no—
136C4×C103#12012yes—
137C4×C103C22×D412yes—
138Dic103—24no—
139C2×C103—22no—
140C2×C103—22no—
141C2×C103—41no—
142C2×C103C2×#12011yescharacteristic
143C2×C103C2×#13711yescharacteristic
144C2×C103#57511yescharacteristic
145D103—44no—
146#3007—1031nomaximal, Sylow
147C103#300711yescharacteristic, complemented, Sylow
148C22×#41—1031no—
149C2×#137—1032no—
150C2×C22⋊Q8D10311yescharacteristic, complemented
151#452—1034no—
152#451—1032no—
153#450—1032no—
154#417—1032no—
155#407—1031no—
156C2×#41—1032no—
157C2×#41—1032no—
158C2×#41—2068no—
159C2×#41—2061no—
160C2×#41—2061no—
161C22×Q8—1031no—
162C22×Q8C2×D10311yescharacteristic, complemented
163C22×D4—1031no—
164C22×D4—2061no—
165C23×C4—2061no—
166C23×C4C2×D10311yescharacteristic
167#137—2068no—
168C2×Q16—1034no—
169C2×SD16—1034no—
170C2×M16—1032no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D10312yes—
174C22⋊Q8C2×D10312yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D10311yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—1034no—
179#102—1034no—
180C22⋊C8—1032no—
181C22⋊C8—1032no—
182C24—2061no—
183C24C22⋊D10311yescharacteristic
184#41—1038no—
185#41—2064no—
186#41—2064no—
187#41—2068no—
188#41—2068no—
189#41—2064no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—1032no—
193C2×Q8—2064no—
194C2×Q8—2061no—
195C2×Q8C22×D10312yes—
196C2×D4—1032no—
197C2×D4—2064no—
198C2×D4—2068no—
199C2×D4—2062no—
200C2×D4—2062no—
201C2×D4—2061no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—2068no—
206C22×C4—2061no—
207C22×C4—2062no—
208C22×C4—2062no—
209C22×C4C22⋊D10311yescharacteristic
210C22×C4C22⋊D10311yescharacteristic
211C22×C4C22⋊D10311yescharacteristic
212C22×C4C22×D10311yescharacteristic
213C22×C4C22×D10311yescharacteristic
214C22×C4C22×D10311yescharacteristic
215Q16—2068no—
216SD16—2068no—
217M16—2064no—
218C2×C8—1034no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D10312yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—4122no—
229C23—4122no—
230C23—2064no—
231C23—2062no—
232C23C2×C22⋊D10311yescharacteristic
233C23C2×C22⋊D10311yescharacteristic
234C23C2×C22⋊D10311yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—1034no—
238Q8—2064no—
239Q8—2062no—
240D4—1034no—
241D4—2064no—
242D4—2068no—
243D4—2068no—
244D4—2062no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—2068no—
255C2×C4—2064no—
256C2×C4—2062no—
257C2×C4—2068no—
258C2×C4D4×D10314yes—
259C2×C4C2×C22⋊D10311yescharacteristic
260C2×C4C2×C22⋊D10311yescharacteristic
261C2×C4C2×C22⋊D10311yescharacteristic
262C2×C4C23×D10311yescharacteristic
263C8—2064no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—4128no—
273C22—4122no—
274C22—4122no—
275C22—2064no—
276C22#1569298912yes—
277C22#1569305212yescomplemented
278C22C22×C22⋊D10311yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—2064no—
284C4#1569298912yes—
285C4C2×D4×D10312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—4124no—
290C2C2×#1569298911yescharacteristic
291C2C2×#1569305211yescharacteristic
292C2#2026858911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
FittingC2×C22⋊Q8×C103C2
socleC22×C103C22×D4
prime radicalC22×#13110105C2
hypercenterC2×C22⋊Q8D103
hypocenterC103#3007
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15693052C2semidirect
3. #20267951C2semidirect
4. #20267946C2semidirect
5. #20267941C2semidirect
6. #20267790C2semidirect
7. #20267755C2semidirect
8. C2×C22⋊Q8×C103C2semidirect
19. C2×#13110095C22semidirect
20. C2×#13110092C22semidirect
21. C2×#13110089C22semidirect
22. #15692944C22semidirect
23. #15692941C22semidirect
25. C22⋊(C103⋊C8)C22semidirect
26. C22×Q8×C103C22semidirect
31. C22⋊Q8×C103C22semidirect
33. C2×C4⋊C4×C103C22semidirect
126. C22×C103#137semidirect
147. C103#3007semidirect
150. C2×C22⋊Q8D103semidirect
162. C22×Q8C2×D103semidirect
174. C22⋊Q8C2×D103semidirect
176. C2×C4⋊C4C2×D103semidirect
277. C22#15693052semidirect

Maximal subgroups

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

subgroupindex
C22×#131101052
C2×#156930522
#202679512
#202679462
#202679412
#202677902
#202677552
C2×C22⋊Q8×C1032
#3007103