Finite Group Databasebeta

← #28483920

Subgroups — #28483920

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434651C211yesmaximal, characteristic
2#21433039C211yesmaximal, characteristic, complemented
3#21425016C211yesmaximal, characteristic
4#21422596C211yesmaximal, characteristic, complemented
5#21421935C211yesmaximal, characteristic, complemented
6#21421932C211yesmaximal, characteristic, complemented
7C59×#1480C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837798C411yescharacteristic, complemented
10#16837774C411yescharacteristic, complemented
11#16837754C411yescharacteristic, complemented
12#16837748C411yescharacteristic, complemented
13#16837629—22no—
14#16837531—22no—
15#16837529—21no—
16#16837526—21no—
17C4×#11745670—22no—
18#16837428—21no—
19C2×#14273654—21no—
20C2×#14273654—21no—
21#16837357—21no—
22C4×#11745664C2211yescharacteristic
23C2×C4×C59⋊C8C2211yescharacteristic
24C4×C4⋊Dic59C2211yescharacteristic
25#16837140—22no—
26C59×#532C2211yescharacteristic
27C59×#431C2211yescharacteristic
28C2×C4⋊C8×C59C2211yescharacteristic
29C59×D4⋊C8—21no—
30C59×D4⋊C8—21no—
31C2×#11745670—22no—
32C2×#11745664D411yescharacteristic
33C2×#11745664D411yescharacteristic
34C22×C59⋊C8—21no—
35#14273727C2×C411yescharacteristic
36#14273727C2×C411yescharacteristic
37#14273719—21no—
38#14273706—21no—
39#14273706D412yes—
40C4×Dic118—21no—
41C4×Dic118D412yes—
42#14273686—21no—
43C2×C4⋊Dic59—21no—
44C2×C4×Dic59—21no—
45#14273658—22no—
46Dic59⋊C8—22no—
47Dic59⋊C8—22no—
48#14273654—42no—
49#14273654—42no—
50#14273650—22no—
51#14273648—22no—
52#14273638—21no—
53#14273637—22no—
54C4×C59⋊C8—21no—
55C4×C59⋊C8—21no—
56C4×C59⋊C8C2×C411yescharacteristic
57C4×C59⋊C8C2×C411yescharacteristic
58C8×Dic59—22no—
59C59⋊#95—21no—
60C2×#41×C59—21no—
61C22×C8×C59—21no—
62C4⋊Q8×C59C2×C411yescharacteristic
63#127×C59C2×C411yescharacteristic
64#124×C59—21no—
65#121×C59—21no—
66#121×C59—21no—
67C4×D4×C59—21no—
68C4×D4×C59—21no—
69C2×C42×C59C2311yescharacteristic
70C4⋊C8×C59—21no—
71C4⋊C8×C59—21no—
72C4⋊C8×C59D411yescharacteristic
73C4⋊C8×C59D411yescharacteristic
74C22⋊C8×C59—21no—
75C4×C8×C59—21no—
76#41×C59—41no—
77#41×C59—41no—
78#41×C59—41no—
79#41×C59—41no—
80C2×Q8×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×Dic59—41no—
90C2×Dic118—21no—
91C2×Dic118C22⋊C412yes—
92C2×C8×C59—21no—
93C2×C8×C59—21no—
94C2×C8×C59—41no—
95C2×C8×C59—41no—
96C2×C8×C59—41no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C42×C59C22×C411yescharacteristic
102C42×C59C2×D411yescharacteristic
103C42×C59C2×D411yescharacteristic
104C42×C59#4111yescharacteristic
105C22⋊Dic59—22no—
106#11745670—44no—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110C4⋊Dic59—22no—
111C4⋊Dic59—41no—
112C4⋊Dic59C22⋊C412yes—
113Dic59⋊C4—22no—
114Dic59⋊C4—22no—
115C4×Dic59—22no—
116C4×Dic59—41no—
117C4×Dic59—41no—
118#11745664—21no—
119#11745664—21no—
120#11745664—21no—
121#11745664SD1612yescomplemented
122C2×C59⋊C8—21no—
123C2×C59⋊C8—21no—
124C2×C59⋊C8—21no—
125C2×C59⋊C8—21no—
126C2×C59⋊C8—42no—
127C23×C59—41no—
128C23×C59#12111yescharacteristic
129Q8×C59—41no—
130Q8×C59—41no—
131D4×C59—42no—
132D4×C59—41no—
133D4×C59—41no—
134D4×C59—41no—
135D4×C59—41no—
136D4×C59—81no—
137D4×C59—81no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—81no—
143C2×C4×C59—81no—
144C2×C4×C59C2×C22⋊C411yescharacteristic
145C2×C4×C59C4×D411yescharacteristic
146C2×C4×C59C4×D411yescharacteristic
147C2×C4×C59#12011yescharacteristic
148C2×C4×C59#12111yescharacteristic
149C2×C4×C59#12311yescharacteristic
150C2×C4×C59C2×SD1611yescharacteristic
151C2×C4×C59#13511yescharacteristic
152C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C59#13711yescharacteristic
154C2×Dic59—22no—
155C2×Dic59—22no—
156C2×Dic59—42no—
157C2×Dic59—81no—
158C2×Dic59—81no—
159Dic118—41no—
160Dic118—41no—
161Dic118—42no—
162C8×C59—41no—
163C8×C59—41no—
164C8×C59—81no—
165C59⋊C8—22no—
166C59⋊C8—22no—
167C59⋊C8—41no—
168C59⋊C8—41no—
169#9054—591nomaximal, Sylow
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59#38611yescharacteristic
176C22×C59#46011yescharacteristic
177C22×C59#46411yescharacteristic
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—81no—
182C4×C59C4×SD1611yescharacteristic
183C4×C59#44011yescharacteristic
184C4×C59#44311yescharacteristic
185C4×C59#44411yescharacteristic
186C4×C59#44811yescharacteristic
187C4×C59#44911yescharacteristic
188Dic59—42no—
189Dic59—42no—
190Dic59—82no—
191#3167—591no—
192#2962—591no—
193#1980—591no—
194#1503—591no—
195#1480D5911yescharacteristic
196#1279—591no—
197#1278—591no—
198C2×C59—21no—
199C2×C59—21no—
200C2×C59—161no—
201C2×C59#187611yescharacteristic
202C2×C59#188011yescharacteristic
203C2×C59#188411yescharacteristic
204#533—591no—
205#532C2×D5911yescharacteristic
206#498—591no—
207#492—591no—
208#487—591no—
209#486—591no—
210#465—1182no—
211#446—1182no—
212#446—1181no—
213C8×Q8—1181no—
214C4×SD16—1182no—
215#431C2×D5911yescharacteristic
216#424—1181no—
217C2×C4⋊C8C2×D5911yescharacteristic
218C2×#104—1181no—
219C2×#104—1181no—
220#405—1181no—
221C4×M16—591no—
222C2×C4×C8—591no—
223C4×C4⋊C4—591no—
224#328—1182no—
225D4⋊C8—21no—
226D4⋊C8—21no—
227C59#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—1182no—
230C2×M16—591no—
231C2×M16—591no—
232C22×C8—21no—
233C22×C8—1181no—
234C4⋊Q8—591no—
235C4⋊Q8—591no—
236C4⋊Q8C4×D5911yescharacteristic
237#127C4×D5911yescharacteristic
238#125—1181no—
239#124—21no—
240C22⋊Q8—592no—
241C22⋊Q8—1181no—
242#121—21no—
243#121—21no—
244C4×Q8—592no—
245C4×Q8—1181no—
246C4×D4—21no—
247C4×D4—21no—
248#117—1181no—
249C2×C4⋊C4—1181no—
250C2×C42—1181no—
251C2×C42C22×D5911yescharacteristic
252#106—1182no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—1182no—
256C4⋊C8—1182no—
257C4⋊C8C22⋊D5911yescharacteristic
258C4⋊C8C22⋊D5911yescharacteristic
259#104—2362no—
260#104—2362no—
261#103—1182no—
262#102—1182no—
263C22⋊C8—21no—
264#97—1182no—
265#97—1181no—
266C4×C8—21no—
267C4×C8—591no—
268C4×C8—591no—
269C4×C8—1182no—
270C4×C8—1181no—
271C4×C8—1181no—
272#95—1181no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—592no—
279C2×Q8—1181no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—2361no—
286C22×C4#1174568511yescharacteristic
287C22×C4D4×D5911yescharacteristic
288C22×C4D4×D5911yescharacteristic
289SD16—2364no—
290M16—592no—
291M16—1181no—
292M16—1181no—
293M16—1181no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—1181no—
300C2×C8—1181no—
301C2×C8—1181no—
302C2×C8—1181no—
303C2×C8—2362no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—592no—
307C4⋊C4—1182no—
308C4⋊C4—1181no—
309C4⋊C4—1181no—
310C4⋊C4—1181no—
311C4⋊C4—1182no—
312C4⋊C4—1182no—
313C4⋊C4—2361no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—1182no—
317C42—1182no—
318C42—2361no—
319C42—2361no—
320C42C2×C4×D5911yescharacteristic
321C42D4×D5911yescharacteristic
322C42#1174568711yescharacteristic
323C42C2×C22⋊D5911yescharacteristic
324C23—41no—
325C23#1427370011yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—2362no—
329Q8—2361no—
330Q8—2361no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—1182no—
345C2×C4—1182no—
346C2×C4—4721no—
347C2×C4—4721no—
348C2×C4—2362no—
349C2×C4C22⋊C4×D5911yescharacteristic
350C2×C4(C22⋊D59)⋊C411yescharacteristic
351C2×C4C4×C22⋊D5911yescharacteristic
352C2×C4#1427369511yescharacteristic
353C2×C4#1427369811yescharacteristic
354C2×C4#1427370911yescharacteristic
355C2×C4SD16×D5911yescharacteristic
356C2×C4#1427374611yescharacteristic
357C2×C4#1427375111yescharacteristic
358C2×C4#1427375611yescharacteristic
359C8—41no—
360C8—41no—
361C8—81no—
362C8—1182no—
363C8—1182no—
364C8—2361no—
365C8—2361no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1683727811yescharacteristic
372C22#1683760111yescharacteristic
373C22#1683762311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—4722no—
379C4—2362no—
380C4—2362no—
381C4#1683749311yescharacteristic
382C4#1683749911yescharacteristic
383C4#1683751711yescharacteristic
384C4#1683752211yescharacteristic
385C4#1683753911yescharacteristic
386C4#1683754411yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2142442611yescharacteristic
391C2#2142444811yescharacteristic
392C2#2142447611yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1480C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#1480D59
hypocenterC59#9054
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #21433039C2semidirect
4. #21422596C2semidirect
5. #21421935C2semidirect
6. #21421932C2semidirect
9. #16837798C4semidirect
10. #16837774C4semidirect
11. #16837754C4semidirect
12. #16837748C4semidirect
121. #11745664SD16semidirect
227. C59#9054semidirect

Maximal subgroups

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

subgroupindex
#214346512
#214330392
#214250162
#214225962
#214219352
#214219322
C59×#14802
#905459