Finite Group Databasebeta

← #46820151

Subgroups — #46820151

Order 34432.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#22575499C211yesmaximal, characteristic, complemented
2C2×#22575479C211yesmaximal, characteristic, complemented
3C22×#16864387C211yesmaximal, characteristic, complemented
4#31770748C214yesmaximal, complemented
5#31770738C214yesmaximal, complemented
6#31770609C211yesmaximal, characteristic, complemented
7#31770599C211yesmaximal, characteristic, complemented
8#31770572C211yesmaximal, characteristic, complemented
9#535×C269C211yesmaximal, characteristic, complemented
10#22575499—24no—
11#22575499—24no—
12C2×#16864403—21no—
13C2×#16864402C2211yescharacteristic
14C2×#16864402C2212yescomplemented
15C22×D1076C2211yescharacteristic
16C22×D1076C2212yescomplemented
17C2×D4×D269—22no—
18C2×D4×D269—21no—
19#22575479—24no—
20C2×#16864390C2212yescomplemented
21C2×#16864387—24no—
22C2×#16864387C2212yescomplemented
23C2×#16864387C2214yescomplemented
24C2×#16864386C2211yescharacteristic, complemented
25C22×#14524130C2211yescharacteristic, complemented
26#22575376C2212yescomplemented
27#22575373C2212yescomplemented
28#22575373C2214yescomplemented
29C22⋊#14524130C2214yescomplemented
30C2×#41×C269C2211yescharacteristic
31C22×D4×C269C2211yescharacteristic, complemented
32#127×C269—21no—
33#124×C269—21no—
34#121×C269C2214yescomplemented
35#120×C269—22no—
36#117×C269C2211yescharacteristic, complemented
37C23×D269—21no—
38C23×D269—22no—
39#16864403—28no—
40#16864402—24no—
41#16864402—24no—
42#16864402—22no—
43#16864402D414yescomplemented
44C2×Dic538C2311yescharacteristic
45C2×D1076—24no—
46C2×D1076—24no—
47C2×D1076—21no—
48C2×D1076—22no—
49C2×D1076—22no—
50C2×D1076D414yescomplemented
51C2×D1076C2312yes—
52C2×D1076C2311yescharacteristic
53C2×C22⋊D269—21no—
54C2×C22⋊D269—22no—
55D4×D269—216no—
56D4×D269—28no—
57C2×C4×D269—21no—
58C2×C4×D269—22no—
59#16864390—24no—
60#16864387—28no—
61#16864387—24no—
62#16864386—22no—
63C2×#14524130—22no—
64C2×#14524130C2312yes—
65C2×#14524130C2312yes—
66C24×C269—21no—
67#41×C269—22no—
68#41×C269—22no—
69C2×Q8×C269D411yescharacteristic
70C2×D4×C269—24no—
71C2×D4×C269—22no—
72C2×D4×C269—22no—
73C2×D4×C269—21no—
74C2×D4×C269D412yes—
75C2×D4×C269D411yescharacteristic
76C2×D4×C269C2312yes—
77C22×C4×C269C2312yes—
78C22×C4×C269C2311yescharacteristic
79C4⋊C4×C269C2312yes—
80C22⋊C4×C269—21no—
81C22⋊C4×C269—24no—
82C42×C269—21no—
83C22×D269—21no—
84C22×D269—22no—
85C22×D269—28no—
86C22×D269—24no—
87C22×D269—82no—
88C22×D269—84no—
89Dic538—21no—
90Dic538C2×D412yes—
91D1076—28no—
92D1076—24no—
93D1076—22no—
94D1076—21no—
95D1076C2×D412yes—
96D1076C2×D414yes—
97C22⋊D269—28no—
98C22⋊D269—24no—
99C2×Dic269—21no—
100C4×D269—28no—
101C4×D269—24no—
102#14524130—22no—
103#14524130—22no—
104C23×C269—22no—
105C23×C269—42no—
106C23×C269—82no—
107C23×C269C2×D412yes—
108C23×C269C2×D411yescharacteristic
109Q8×C269—22no—
110D4×C269—24no—
111D4×C269—22no—
112D4×C269—22no—
113D4×C269—22no—
114D4×C269—42no—
115D4×C269—84no—
116C2×C4×C269—22no—
117C2×C4×C269—22no—
118C2×C4×C269—22no—
119C2×C4×C269C2×D412yes—
120C2×C4×C269C2×D412yes—
121C2×C4×C269C2×D412yes—
122C2×C4×C269C2×D411yescharacteristic
123C2×C4×C269C2411yescharacteristic
124C22×C269—22no—
125C22×C269—22no—
126C22×C269—24no—
127C22×C269—42no—
128C22×C269—44no—
129C22×C269—84no—
130C22×C269#12012yes—
131C22×C269C22×D411yescharacteristic
132C2×D269—24no—
133C2×D269—22no—
134C2×D269—44no—
135C2×D269—42no—
136C2×D269—84no—
137C2×D269—88no—
138C4×C269—22no—
139C4×C269—42no—
140C4×C269#12012yes—
141C4×C269C22×D411yescharacteristic
142C4×C269C22×D411yescharacteristic
143Dic269—22no—
144C2×C269—22no—
145C2×C269—42no—
146C2×C269—82no—
147C2×C269C2×#12011yescharacteristic
148C2×C269#57612yes—
149D269—44no—
150D269—42no—
151C269#300811yescharacteristic, complemented, Sylow
152#3008—2691nomaximal, Sylow
153C2×H(5,𝔽2)—2691no—
154C2×AGL(1,ℤ/8)—2691no—
155C22×D8—2691no—
156#535D26911yescharacteristic, complemented
157#449—2694no—
158#447—2694no—
159#418—2691no—
160#416—2691no—
161#408—2691no—
162H(5,𝔽2)—5384no—
163H(5,𝔽2)—5384no—
164C2×#41—2691no—
165C2×#41—2692no—
166C2×#41—5381no—
167C2×#41C2×D26911yescharacteristic
168C22×D4—2691no—
169C22×D4—2692no—
170C22×D4—5382no—
171C22×D4—5381no—
172C22×D4C2×D26911yescharacteristic, complemented
173AGL(1,ℤ/8)—5384no—
174C2×SD16—2692no—
175C2×D8—2694no—
176C2×D8—2692no—
177C2×D8—5384no—
178C2×M16—2691no—
179C22×C8—2691no—
180#127—21no—
181#124—21no—
182#121C2×D26914yescomplemented
183#120—22no—
184#117C2×D26911yescharacteristic, complemented
185#103—2692no—
186#102—2692no—
187#102—2694no—
188C22⋊C8—2694no—
189C24—21no—
190C24—5382no—
191C24—5381no—
192#41—22no—
193#41—22no—
194#41—2694no—
195#41—5384no—
196#41—5384no—
197#41—5388no—
198#41—5382no—
199C2×Q8—2691no—
200C2×Q8C22⋊D26911yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—2694no—
206C2×D4—2692no—
207C2×D4—2691no—
208C2×D4—5384no—
209C2×D4—5384no—
210C2×D4—53816no—
211C2×D4—5388no—
212C2×D4—5382no—
213C2×D4—5381no—
214C2×D4—5382no—
215C2×D4—5382no—
216C2×D4—5381no—
217C2×D4C22⋊D26912yes—
218C2×D4C22⋊D26911yescharacteristic
219C2×D4C22×D26912yes—
220C22×C4—5382no—
221C22×C4—5381no—
222C22×C4C22×D26912yes—
223C22×C4C22×D26911yescharacteristic
224SD16—5384no—
225D8—5388no—
226D8—5384no—
227M16—5382no—
228C2×C8—2692no—
229C2×C8—2692no—
230C2×C8—5382no—
231C4⋊C4C22×D26912yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—5388no—
239C23—5382no—
240C23—5381no—
241C23—5384no—
242C23—21522no—
243C23—21524no—
244C23C2×C22⋊D26912yes—
245C23C2×C22⋊D26911yescharacteristic
246Q8—22no—
247Q8—2692no—
248Q8—5381no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—2694no—
256D4—2692no—
257D4—5388no—
258D4—5388no—
259D4—5384no—
260D4—5384no—
261D4—5381no—
262D4—5382no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—5388no—
267C2×C4—5381no—
268C2×C4—5384no—
269C2×C4D4×D26912yes—
270C2×C4D4×D26912yes—
271C2×C4C2×C22⋊D26912yes—
272C2×C4C2×C22⋊D26911yescharacteristic
273C2×C4C23×D26911yescharacteristic
274C8—5382no—
275C8—5382no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—44no—
280C22—42no—
281C22—84no—
282C22—5384no—
283C22—5382no—
284C22—10764no—
285C22—10762no—
286C22—21524no—
287C22—21528no—
288C22#2257542112yes—
289C22C22×C22⋊D26911yescharacteristic
290C4—22no—
291C4—42no—
292C4—5382no—
293C4#2257542112yes—
294C4C2×D4×D26911yescharacteristic
295C4C2×D4×D26911yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—10764no—
300C2—10762no—
301C2C2×#2257542111yescharacteristic
302C2#3177140812yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#535×C269C2
socleC22×C269C22×D4
hypercenter#535D269
hypocenterC269#3008
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#22575499C2semidirect
2. C2×#22575479C2semidirect
3. C22×#16864387C2semidirect
4. #31770748C2semidirect
5. #31770738C2semidirect
6. #31770609C2semidirect
7. #31770599C2semidirect
8. #31770572C2semidirect
9. #535×C269C2semidirect
14. C2×#16864402C22semidirect
16. C22×D1076C22semidirect
20. C2×#16864390C22semidirect
22. C2×#16864387C22semidirect
23. C2×#16864387C22semidirect
24. C2×#16864386C22semidirect
25. C22×#14524130C22semidirect
26. #22575376C22semidirect
27. #22575373C22semidirect
28. #22575373C22semidirect
29. C22⋊#14524130C22semidirect
31. C22×D4×C269C22semidirect
34. #121×C269C22semidirect
36. #117×C269C22semidirect
43. #16864402D4semidirect
50. C2×D1076D4semidirect
151. C269#3008semidirect
156. #535D269semidirect
172. C22×D4C2×D269semidirect
182. #121C2×D269semidirect
184. #117C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754792
C22×#168643872
#317707482
#317707382
#317706092
#317705992
#317705722
#535×C2692
#3008269