Finite Group Databasebeta

← #27204492

Subgroups — #27204492

Order 13184.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#20268627C211yesmaximal, characteristic, complemented
2#20268589C212yesmaximal, complemented
3#20267973C212yesmaximal, complemented
4#20267968C212yesmaximal, complemented
5#20267963C212yesmaximal, complemented
6#20267958C212yesmaximal, complemented
7C2×#15692949C212yesmaximal, complemented
8#20267775C211yesmaximal, characteristic, complemented
9#532×C103C211yesmaximal, characteristic, complemented
10#15693069C2212yes—
11#15693068—22no—
12#15693067—22no—
13#15693066—22no—
14#15693066C2212yes—
15C2×#13110105—22no—
16C2×#13110105C2212yes—
17#41×D103—24no—
18#15693052C2214yescomplemented
19#15693051—22no—
20#15693047C2214yescomplemented
21#15693046—22no—
22#15693041—24no—
23C2×#13110089C2212yescomplemented
24#15692949C2214yescomplemented
25#15692949C2214yescomplemented
26#15692940C2212yescomplemented
27#15692938C2212yescomplemented
28C2×#41×C103C2211yescharacteristic
29C2×#41×C103C2211yescharacteristic, complemented
30C4⋊Q8×C103C2211yescharacteristic, complemented
31#127×C103C2211yescharacteristic, complemented
32#124×C103C2211yescharacteristic, complemented
33#124×C103C2211yescharacteristic, complemented
34#121×C103—22no—
35C4×D4×C103—22no—
36C2×C42×C103C2211yescharacteristic, complemented
37#13110106—24no—
38#13110105—24no—
39#13110105—24no—
40#13110105—22no—
41#13110105—24no—
42#13110105—22no—
43#13110105D414yescomplemented
44#13110105C2314yes—
45#13110104—28no—
46#13110104—24no—
47C2×Dic206—22no—
48C2×Dic206D412yescomplemented
49Q8×D103—24no—
50C2×D412—22no—
51C2×D412D412yescomplemented
52C2×C22⋊D103—24no—
53C2×C22⋊D103—22no—
54D4×D103—28no—
55D4×D103—24no—
56C2×C4×D103—24no—
57C2×C4×D103—22no—
58#13110095—24no—
59#13110093—24no—
60#13110092—24no—
61#13110090—24no—
62#13110089—22no—
63#13110089C2314yes—
64C2×C103⋊C8—22no—
65#41×C103—22no—
66#41×C103—22no—
67#41×C103—22no—
68#41×C103—22no—
69C2×Q8×C103C2311yescharacteristic
70C2×Q8×C103C2311yescharacteristic
71C2×D4×C103—22no—
72C2×D4×C103—21no—
73C2×D4×C103D412yes—
74C2×D4×C103C2311yescharacteristic
75C2×D4×C103C2311yescharacteristic
76C22×C4×C103—21no—
77C22×C4×C103—21no—
78C22×C4×C103D412yes—
79C22×C4×C103C2311yescharacteristic
80C4⋊C4×C103—22no—
81C22⋊C4×C103—22no—
82C22⋊C4×C103—22no—
83C42×C103—21no—
84C42×C103C2311yescharacteristic
85C42×C103C2311yescharacteristic
86C23×C103—21no—
87C23×C103C2×D412yes—
88C23×C103C2×D411yescharacteristic
89C22×D103—24no—
90C22×D103—22no—
91Q8×C103—22no—
92Q8×C103—22no—
93D4×C103—24no—
94D4×C103—22no—
95D4×C103—22no—
96D4×C103—44no—
97D4×C103—42no—
98C2×C4×C103—21no—
99C2×C4×C103—22no—
100C2×C4×C103—22no—
101C2×C4×C103—21no—
102C2×C4×C103—21no—
103C2×C4×C103—42no—
104C2×C4×C103—41no—
105C2×C4×C103—41no—
106C2×C4×C103C2×D412yes—
107C2×C4×C103C2×D412yes—
108C2×C4×C103C2×D412yes—
109C2×C4×C103C2×D411yescharacteristic
110C2×C4×C103C2411yescharacteristic
111C22⋊D103—28no—
112C22⋊D103—24no—
113C22⋊D103—24no—
114C2×Dic103—24no—
115C2×Dic103—22no—
116D412—24no—
117D412—22no—
118D412C2×D414yes—
119C4×D103—28no—
120C4×D103—24no—
121C4×D103—24no—
122Dic206—24no—
123Dic206—22no—
124Dic206C2×D414yes—
125C103⋊C8—24no—
126C22×C103—22no—
127C22×C103—41no—
128C22×C103—42no—
129C22×C103—42no—
130C22×C103—41no—
131C22×C103#12012yes—
132C22×C103C22×D411yescharacteristic
133C2×D103—24no—
134C2×D103—82no—
135C2×D103—84no—
136C4×C103—22no—
137C4×C103—22no—
138C4×C103—41no—
139C4×C103—41no—
140C4×C103#12011yescharacteristic
141C4×C103#12011yescharacteristic
142C4×C103C22×D412yes—
143Dic103—24no—
144C2×C103—21no—
145C2×C103—22no—
146C2×C103—42no—
147C2×C103—81no—
148C2×C103C2×#12011yescharacteristic
149D103—44no—
150#3018—1031nomaximal, Sylow
151C103#301811yescharacteristic, complemented, Sylow
152#585—1031no—
153#575—1032no—
154#532D10311yescharacteristic, complemented
155#456—1032no—
156#455—1032no—
157#454—1032no—
158#453—1032no—
159C2×#104—1032no—
160#413—1031no—
1612−1+4—1032no—
1622−1+4—2062no—
163H(5,𝔽2)—1032no—
164H(5,𝔽2)—2062no—
165H(5,𝔽2)—2062no—
166C2×#41—1032no—
167C2×#41—2064no—
168C2×#41—2062no—
169C2×#41C2×D10311yescharacteristic
170C2×#41C2×D10311yescharacteristic, complemented
171#137—1034no—
172#137—2062no—
173AGL(1,ℤ/8)—1034no—
174AGL(1,ℤ/8)—2062no—
175#135—2064no—
176C2×M16—1032no—
177C4⋊Q8C2×D10311yescharacteristic, complemented
178#127C2×D10311yescharacteristic, complemented
179#124C2×D10311yescharacteristic, complemented
180#124C2×D10311yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D10311yescharacteristic, complemented
184#104—1034no—
185#104—1034no—
186#101—1032no—
187#100—1032no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—1034no—
193#41—1034no—
194#41—2064no—
195#41—2064no—
196#41—2064no—
197#41—2068no—
198#41—2064no—
199#41—2062no—
200#41—2062no—
201#41—2064no—
202C2×Q8—1032no—
203C2×Q8—2062no—
204C2×Q8—2064no—
205C2×Q8C22×D10311yescharacteristic
206C2×Q8C22×D10311yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—1032no—
210C2×D4—2062no—
211C2×D4—2064no—
212C2×D4—2068no—
213C2×D4—2062no—
214C2×D4—2064no—
215C2×D4C22⋊D10312yes—
216C2×D4C22×D10311yescharacteristic
217C2×D4C22×D10311yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—2064no—
221C22×C4—2062no—
222C22×C4C22⋊D10312yes—
223C22×C4C22×D10311yescharacteristic
224Q16—2064no—
225SD16—2064no—
226SD16—2064no—
227D8—2064no—
228M16—1034no—
229M16—2062no—
230C2×C8—2062no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D10311yescharacteristic
236C42C22×D10311yescharacteristic
237C23—21no—
238C23—2064no—
239C23—2062no—
240C23C2×C22⋊D10312yes—
241C23C2×C22⋊D10311yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—1034no—
245Q8—2064no—
246Q8—2062no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—1034no—
253D4—2064no—
254D4—2068no—
255D4—2064no—
256D4—2064no—
257D4—2062no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—2068no—
267C2×C4—2064no—
268C2×C4—2062no—
269C2×C4—2064no—
270C2×C4—2064no—
271C2×C4D4×D10312yes—
272C2×C4D4×D10312yes—
273C2×C4C2×C22⋊D10312yes—
274C2×C4C2×C22⋊D10311yescharacteristic
275C2×C4C23×D10311yescharacteristic
276C8—2064no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—8242no—
283C22—8244no—
284C22—2064no—
285C22#1569298912yes—
286C22C22×C22⋊D10311yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—2064no—
292C4#1569298911yescharacteristic
293C4#1569298911yescharacteristic
294C4C2×D4×D10312yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—4124no—
300C2C2×#1569298911yescharacteristic
subgroupG/H
centerC4#15692989
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#532×C103C2
socleC2×C103C2×#120
hypercenter#532D103
hypocenterC103#3018
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20268627C2semidirect
2. #20268589C2semidirect
3. #20267973C2semidirect
4. #20267968C2semidirect
5. #20267963C2semidirect
6. #20267958C2semidirect
7. C2×#15692949C2semidirect
8. #20267775C2semidirect
9. #532×C103C2semidirect
18. #15693052C22semidirect
20. #15693047C22semidirect
23. C2×#13110089C22semidirect
24. #15692949C22semidirect
25. #15692949C22semidirect
26. #15692940C22semidirect
27. #15692938C22semidirect
29. C2×#41×C103C22semidirect
30. C4⋊Q8×C103C22semidirect
31. #127×C103C22semidirect
32. #124×C103C22semidirect
33. #124×C103C22semidirect
36. C2×C42×C103C22semidirect
43. #13110105D4semidirect
48. C2×Dic206D4semidirect
51. C2×D412D4semidirect
151. C103#3018semidirect
154. #532D103semidirect
170. C2×#41C2×D103semidirect
177. C4⋊Q8C2×D103semidirect
178. #127C2×D103semidirect
179. #124C2×D103semidirect
180. #124C2×D103semidirect
183. C2×C42C2×D103semidirect

Maximal subgroups

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

subgroupindex
#202686272
#202685892
#202679732
#202679682
#202679632
#202679582
C2×#156929492
#202677752
#532×C1032
#3018103