Finite Group Databasebeta

← #46813513

Subgroups — #46813513

Order 34432.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575413C211yesmaximal, characteristic
2#31770497C211yesmaximal, characteristic, complemented
3#31770495C211yesmaximal, characteristic, complemented
4#31770492C211yesmaximal, characteristic, complemented
5#31770486C211yesmaximal, characteristic, complemented
6#31770484C211yesmaximal, characteristic
7#31770482C211yesmaximal, characteristic
8#31770480C211yesmaximal, characteristic, complemented
9#31770472C211yesmaximal, characteristic, complemented
10#31770470C211yesmaximal, characteristic, complemented
11#31770467C211yesmaximal, characteristic, complemented
12#31770462C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic269C211yesmaximal, characteristic, complemented
14C2×#22575360C211yesmaximal, characteristic, complemented
15#385×C269C211yesmaximal, characteristic, complemented
16C22×C22⋊D269C2211yescharacteristic
17C23×Dic269C2211yescharacteristic
18C23×Dic269C2211yescharacteristic
19C22×C4×D269C2211yescharacteristic
20#22575413C418yescomplemented
21C2×C4⋊Dic269C2211yescharacteristic
22C2×C4⋊Dic269C2211yescharacteristic
23C2×#16864381C2211yescharacteristic
24C2×C22⋊Dic269C2211yescharacteristic
25C2×#16864379C2211yescharacteristic
26C2×#16864379C2211yescharacteristic
27C2×#16864379C2211yescharacteristic
28C2×#16864379C2211yescharacteristic
29C2×#16864379C2211yescharacteristic
30C2×C4×Dic269C2211yescharacteristic
31C2×C4×Dic269C2211yescharacteristic
32C2×C4×Dic269C2211yescharacteristic
33#22575360C2211yescharacteristic
34#22575360C2211yescharacteristic
35#22575360C2211yescharacteristic
36#22575360C2211yescharacteristic
37#22575360C2211yescharacteristic
38#22575360C2211yescharacteristic
39#22575360C2211yescharacteristic
40#22575360C2211yescharacteristic
41#22575360C2211yescharacteristic
42#22575360C2211yescharacteristic
43#22575360C2211yescharacteristic
44#22575360C2211yescharacteristic
45C23×C4×C269C2211yescharacteristic
46C2×C4⋊C4×C269C2211yescharacteristic
47C2×C4⋊C4×C269C2211yescharacteristic
48C2×C22⋊C4×C269C2211yescharacteristic
49C2×C22⋊C4×C269C2211yescharacteristic
50#95×C269C2211yescharacteristic
51#95×C269C2211yescharacteristic
52C23×D269C2311yescharacteristic
53C2×C22⋊D269—22no—
54C2×C22⋊D269—22no—
55C2×C22⋊D269C2×C414yes—
56C22×Dic269—22no—
57C22×Dic269—41no—
58C22×Dic269—41no—
59C22×Dic269—41no—
60C22×Dic269—41no—
61C22×Dic269C2×C414yes—
62C22×Dic269C2311yescharacteristic
63C22×Dic269C2311yescharacteristic
64C22×Dic269C2311yescharacteristic
65C22×Dic269C2311yescharacteristic
66C22×Dic269C2311yescharacteristic
67C22×Dic269C2311yescharacteristic
68C22×Dic269C2311yescharacteristic
69C2×C4×D269—41no—
70C2×C4×D269—41no—
71C2×C4×D269C2×C414yescomplemented
72C4⋊Dic269—41no—
73C4⋊Dic269—41no—
74#16864381C2×C414yescomplemented
75C22⋊Dic269—22no—
76#16864379—41no—
77#16864379—41no—
78#16864379—41no—
79#16864379—41no—
80#16864379C2×C414yescomplemented
81C4×Dic269—41no—
82C4×Dic269—41no—
83C4×Dic269C2×C414yescomplemented
84C24×C269C2311yescharacteristic
85C22×C4×C269—22no—
86C22×C4×C269—41no—
87C22×C4×C269—41no—
88C22×C4×C269C2311yescharacteristic
89C22×C4×C269C2311yescharacteristic
90C22×C4×C269C2311yescharacteristic
91C22×C4×C269C2311yescharacteristic
92C22×C4×C269C2311yescharacteristic
93C22×C4×C269C2311yescharacteristic
94C4⋊C4×C269—41no—
95C4⋊C4×C269—41no—
96C22⋊C4×C269—22no—
97C22⋊C4×C269C2×C414yes—
98C22×D269—21no—
99C22×D269—21no—
100C22×D269—21no—
101C22×D269—21no—
102C22×D269—21no—
103C22×D269—21no—
104C22×D269C22×C412yes—
105C22⋊D269—28no—
106C2×Dic269—21no—
107C2×Dic269—21no—
108C2×Dic269—21no—
109C2×Dic269—21no—
110C2×Dic269—21no—
111C2×Dic269—21no—
112C2×Dic269—21no—
113C2×Dic269—21no—
114C2×Dic269—21no—
115C2×Dic269—21no—
116C2×Dic269—21no—
117C2×Dic269—21no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269—44no—
125C2×Dic269—44no—
126C2×Dic269C22×C412yes—
127C2×Dic269C22×C412yes—
128C2×Dic269C22×C412yes—
129C4×D269—44no—
130C23×C269—21no—
131C23×C269—21no—
132C23×C269—21no—
133C23×C269—21no—
134C23×C269C22×C412yes—
135C23×C269#4112yes—
136C23×C269#4112yes—
137C23×C269C2411yescharacteristic
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—21no—
149C2×C4×C269—21no—
150C2×C4×C269—21no—
151C2×C4×C269—21no—
152C2×C4×C269—21no—
153C2×C4×C269—21no—
154C2×C4×C269—44no—
155C2×C4×C269C22×C412yes—
156C2×C4×C269C22×C412yes—
157C22×C269—22no—
158C22×C269—22no—
159C22×C269—22no—
160C22×C269—22no—
161C22×C269—22no—
162C22×C269—22no—
163C22×C269#11714yes—
164C22×C269C23×C411yescharacteristic
165C22×C269C2×#4111yescharacteristic
166C22×C269C2×#4111yescharacteristic
167C22×C269H(5,𝔽2)11yescharacteristic
168C22×C269H(5,𝔽2)11yescharacteristic
169C22×C269H(5,𝔽2)11yescharacteristic
170C22×C2692−1+411yescharacteristic
171C2×D269—22no—
172C2×D269—22no—
173C2×D269—22no—
174C2×D269—41no—
175C2×D269—41no—
176C2×D269—41no—
177C2×D269—41no—
178C4×C269—22no—
179C4×C269—22no—
180C4×C269—41no—
181C4×C269—41no—
182C4×C269—41no—
183C4×C269—41no—
184Dic269—22no—
185Dic269—22no—
186Dic269—22no—
187Dic269—41no—
188Dic269—41no—
189Dic269—41no—
190Dic269—41no—
191C2×C269—24no—
192C2×C269C2×#11711yescharacteristic
193C2×C269#51811yescharacteristic
194C2×C269#52011yescharacteristic
195C2×C269#53811yescharacteristic
196C2×C269#53811yescharacteristic
197C2×C269#53911yescharacteristic
198C2×C269#53911yescharacteristic
199D269—42no—
200C269#231811yescharacteristic, complemented, Sylow
201#2318—2691nomaximal, Sylow
202C2×C4×D4—2691no—
203#388—2691no—
204#388—2691no—
205#387—2691no—
206#387—2691no—
207#386—2691no—
208#386—2691no—
209#385—2691no—
210#385D26911yescharacteristic, complemented
211#383—2691no—
212#382—2691no—
213#381—2691no—
214#380—2691no—
215C4×C22⋊C4—2691no—
216C2×#95—2691no—
217C22×D4—2691no—
218C23×C4—2691no—
219C23×C4—2691no—
220C23×C4—2691no—
221C23×C4C2×D26911yescharacteristic
222C4×D4—2698no—
223C2×C4⋊C4—2691no—
224C2×C4⋊C4—2691no—
225C2×C4⋊C4—2691no—
226C2×C4⋊C4C2×D26911yescharacteristic
227C2×C4⋊C4C2×D26911yescharacteristic
228C2×C22⋊C4—2691no—
229C2×C22⋊C4—2691no—
230C2×C22⋊C4—2691no—
231C2×C22⋊C4—2691no—
232C2×C22⋊C4—2691no—
233C2×C22⋊C4—2691no—
234C2×C22⋊C4C2×D26911yescharacteristic
235C2×C22⋊C4C2×D26911yescharacteristic
236C2×C42—2691no—
237C2×C42—2691no—
238C2×C42—2691no—
239#95—2691no—
240#95—2691no—
241#95—2691no—
242#95—2691no—
243#95—2691no—
244#95—2691no—
245#95—2691no—
246#95—2691no—
247#95—2691no—
248#95—2691no—
249#95—2691no—
250#95—2691no—
251#95C2×D26911yescharacteristic
252#95C2×D26911yescharacteristic
253C24—2691no—
254C24C22×D26911yescharacteristic
255C2×D4—2694no—
256C2×D4—5382no—
257C2×D4—5382no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—2691no—
262C22×C4—2691no—
263C22×C4—2691no—
264C22×C4—2694no—
265C22×C4—2694no—
266C22×C4—2691no—
267C22×C4—2691no—
268C22×C4—2691no—
269C22×C4—2691no—
270C22×C4—5382no—
271C22×C4—10761no—
272C22×C4—10761no—
273C22×C4—10761no—
274C22×C4—10761no—
275C22×C4—10761no—
276C22×C4—10761no—
277C22×C4C22×D26911yescharacteristic
278C22×C4C22×D26911yescharacteristic
279C22×C4C22×D26911yescharacteristic
280C22×C4C22×D26911yescharacteristic
281C22×C4C22×D26911yescharacteristic
282C22×C4C22×D26911yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—2694no—
286C4⋊C4—10761no—
287C4⋊C4—10761no—
288C22⋊C4—22no—
289C22⋊C4—2694no—
290C22⋊C4—5382no—
291C22⋊C4—10761no—
292C22⋊C4—10761no—
293C22⋊C4—10761no—
294C22⋊C4—10761no—
295C22⋊C4C4×D26914yes—
296C42—2694no—
297C42—10761no—
298C42—10761no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—2692no—
304C23—5381no—
305C23—5381no—
306C23—5381no—
307C23—5381no—
308C23—5381no—
309C23—5381no—
310C23C2×C4×D26912yes—
311C23#1686440112yes—
312C23#1686440312yes—
313C23C23×D26911yescharacteristic
314D4—5388no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—2692no—
333C2×C4—2692no—
334C2×C4—2692no—
335C2×C4—5381no—
336C2×C4—5381no—
337C2×C4—5381no—
338C2×C4—5381no—
339C2×C4—5381no—
340C2×C4—5381no—
341C2×C4—5381no—
342C2×C4—5381no—
343C2×C4—5381no—
344C2×C4—5381no—
345C2×C4—5381no—
346C2×C4—5381no—
347C2×C4—5381no—
348C2×C4—5381no—
349C2×C4—5381no—
350C2×C4—5381no—
351C2×C4—5381no—
352C2×C4—5381no—
353C2×C4—10764no—
354C2×C4—10764no—
355C2×C4—10764no—
356C2×C4C2×C4×D26912yes—
357C2×C4C2×C4×D26912yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—5382no—
365C22—5382no—
366C22—5382no—
367C22—10761no—
368C22—10761no—
369C22—10761no—
370C22—10761no—
371C22#2257541114yes—
372C22C22×C4×D26911yescharacteristic
373C22C2×#1686440111yescharacteristic
374C22C2×#1686440311yescharacteristic
375C22#2257549811yescharacteristic
376C22#2257549811yescharacteristic
377C22#2257549911yescharacteristic
378C22#2257550011yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—5382no—
386C4—5382no—
387C4—5382no—
388C4—10761no—
389C4—10761no—
390C4—10761no—
391C4—10761no—
392C2—24no—
393C2—10762no—
394C2C2×#2257541111yescharacteristic
395C2#3177107711yescharacteristic
396C2#3177108411yescharacteristic
397C2#3177119011yescharacteristic
398C2#3177119511yescharacteristic
399C2#3177119911yescharacteristic
400C2#3177120411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#385×C269C2
socle, cosocleC23×C269C24
prime radicalC22×C22⋊D269C22
hypercenter#385D269
hypocenterC269#2318
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #31770497C2semidirect
3. #31770495C2semidirect
4. #31770492C2semidirect
5. #31770486C2semidirect
8. #31770480C2semidirect
9. #31770472C2semidirect
10. #31770470C2semidirect
11. #31770467C2semidirect
12. #31770462C2semidirect
13. C22⋊C4×Dic269C2semidirect
14. C2×#22575360C2semidirect
15. #385×C269C2semidirect
20. #22575413C4semidirect
71. C2×C4×D269C2×C4semidirect
74. #16864381C2×C4semidirect
80. #16864379C2×C4semidirect
83. C4×Dic269C2×C4semidirect
200. C269#2318semidirect
210. #385D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754132
#317704972
#317704952
#317704922
#317704862
#317704842
#317704822
#317704802
#317704722
#317704702
#317704672
#317704622
C22⋊C4×Dic2692
C2×#225753602
#385×C2692
#2318269