Finite Group Databasebeta

← #28483917

Subgroups — #28483917

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#21434656C211yesmaximal, characteristic
2#21425015C211yesmaximal, characteristic
3#21422595C211yesmaximal, characteristic, complemented
4#21422494C211yesmaximal, characteristic
5#21421935C211yesmaximal, characteristic, complemented
6#21421934C211yesmaximal, characteristic, complemented
7C59×#2962C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837798C411yescharacteristic, complemented
10#16837776C411yescharacteristic, complemented
11#16837752C411yescharacteristic, complemented
12#16837750C411yescharacteristic, complemented
13#16837633—22no—
14C4×#11745671—22no—
15#16837455C2211yescharacteristic
16C2×#14273653—21no—
17C2×#14273653—21no—
18C2×C4×C59⋊C8C2211yescharacteristic
19C4×C4⋊Dic59C2211yescharacteristic
20#16837139—22no—
21#16837129—21no—
22#16837129—21no—
23C59×#533C2211yescharacteristic
24C59×#446—22no—
25C59×#446—21no—
26C8×Q8×C59—21no—
27C59×#424—21no—
28C2×C4⋊C8×C59C2211yescharacteristic
29C59×#405—21no—
30C4×M16×C59C2211yescharacteristic
31C2×#11745685—21no—
32C2×#11745671—22no—
33C22×C59⋊C8—21no—
34#14273727C2×C411yescharacteristic
35#14273724C2×C411yescharacteristic
36#14273715—21no—
37#14273701—21no—
38#14273701—21no—
39C4×D236—21no—
40C4×D236—21no—
41C2×C4⋊Dic59—21no—
42C2×C4×Dic59—21no—
43#14273658—22no—
44C4⋊(C59⋊C8)—21no—
45#14273653—42no—
46#14273653—42no—
47#14273651—22no—
48#14273647—22no—
49C22⋊(C59⋊C8)—21no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C4×C59⋊C8C2×C411yescharacteristic
53C4×C59⋊C8C2×C411yescharacteristic
54C59⋊#95—21no—
55C2×M16×C59D411yescharacteristic
56C2×M16×C59D411yescharacteristic
57C22×C8×C59—21no—
58C4⋊Q8×C59C2×C411yescharacteristic
59C4⋊Q8×C59C2×C411yescharacteristic
60#125×C59—21no—
61C22⋊Q8×C59—21no—
62C22⋊Q8×C59D412yes—
63C4×Q8×C59—21no—
64C4×Q8×C59D412yes—
65#117×C59—21no—
66C2×C42×C59C2311yescharacteristic
67C4⋊C8×C59—22no—
68C4⋊C8×C59—22no—
69C4⋊C8×C59—21no—
70C4⋊C8×C59D411yescharacteristic
71C4⋊C8×C59D411yescharacteristic
72#97×C59—22no—
73#97×C59—21no—
74C4×C8×C59—22no—
75C4×C8×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59C22⋊C412yes—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C2×C22⋊D59—41no—
82C22×Dic59—41no—
83#11745685—41no—
84#11745685—41no—
85#11745685—41no—
86#11745685—41no—
87C2×D236—21no—
88C2×D236—21no—
89C2×D236—21no—
90C2×C4×D59—41no—
91C2×Dic118—21no—
92M16×C59—21no—
93M16×C59—21no—
94M16×C59—21no—
95M16×C59SD1612yescomplemented
96C2×C8×C59—21no—
97C2×C8×C59—21no—
98C2×C8×C59—21no—
99C2×C8×C59—21no—
100C2×C8×C59—41no—
101C2×C8×C59—41no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—22no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59C22⋊C412yes—
108C22⋊C4×C59—22no—
109C42×C59—22no—
110C42×C59C22×C411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59C2×D411yescharacteristic
113C42×C59#4111yescharacteristic
114#11745671—44no—
115(C2×D59)⋊C4—41no—
116(C2×D59)⋊C4—41no—
117C4⋊Dic59—22no—
118C4⋊Dic59—21no—
119C4⋊Dic59—21no—
120C4⋊Dic59—41no—
121C4×Dic59—41no—
122C4×Dic59—41no—
123C2×C59⋊C8—21no—
124C2×C59⋊C8—21no—
125C2×C59⋊C8—42no—
126C2×C59⋊C8—41no—
127C23×C59#12111yescharacteristic
128C22×D59—41no—
129Q8×C59—42no—
130Q8×C59—41no—
131Q8×C59—41no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—41no—
137C2×C4×C59C2×C22⋊C411yescharacteristic
138C2×C4×C59C4×D411yescharacteristic
139C2×C4×C59C4×D411yescharacteristic
140C2×C4×C59#12011yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#12311yescharacteristic
143C2×C4×C59C2×SD1611yescharacteristic
144C2×C4×C59#13511yescharacteristic
145C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C59#13711yescharacteristic
147C22⋊D59—81no—
148C22⋊D59—81no—
149C2×Dic59—42no—
150C2×Dic59—41no—
151C2×Dic59—81no—
152C2×Dic59—81no—
153D236—42no—
154D236—41no—
155D236—41no—
156D236—41no—
157D236—41no—
158C4×D59—81no—
159C4×D59—81no—
160Dic118—41no—
161Dic118—41no—
162C8×C59—22no—
163C8×C59—41no—
164C8×C59—41no—
165C8×C59—41no—
166C59⋊C8—22no—
167C59⋊C8—41no—
168C59⋊C8—81no—
169#9054—591nomaximal, Sylow
170C22×C59—21no—
171C22×C59—21no—
172C22×C59#38611yescharacteristic
173C22×C59#46011yescharacteristic
174C22×C59#46411yescharacteristic
175C2×D59—81no—
176C2×D59—81no—
177C2×D59—81no—
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—42no—
182C4×C59—42no—
183C4×C59C4×SD1611yescharacteristic
184C4×C59#44011yescharacteristic
185C4×C59#44311yescharacteristic
186C4×C59#44411yescharacteristic
187C4×C59#44811yescharacteristic
188C4×C59#44911yescharacteristic
189Dic59—82no—
190Dic59—81no—
191#3167—591no—
192#2962D5911yescharacteristic, complemented
193#1980—591no—
194#1503—591no—
195#1480—591no—
196#1279—591no—
197#1278—591no—
198C2×C59—21no—
199C2×C59—21no—
200C2×C59#187611yescharacteristic
201C2×C59#188011yescharacteristic
202C2×C59#188411yescharacteristic
203D59—161no—
204#533C2×D5911yescharacteristic
205#532—591no—
206#498—591no—
207#492—591no—
208#487—591no—
209#486—591no—
210#465—1182no—
211#446—22no—
212#446—21no—
213C8×Q8—21no—
214C4×SD16—1182no—
215#431—591no—
216#424—21no—
217C2×C4⋊C8C2×D5911yescharacteristic
218C2×#104—1181no—
219C2×#104—1181no—
220#405—21no—
221C4×M16C2×D5911yescharacteristic
222C2×C4×C8—591no—
223C4×C4⋊C4—591no—
224#328—1182no—
225D4⋊C8—1181no—
226D4⋊C8—1181no—
227C59#905411yescharacteristic, complemented, Sylow
228C2×#41—1181no—
229C2×SD16—1182no—
230C2×M16C22⋊D5911yescharacteristic
231C2×M16C22⋊D5911yescharacteristic
232C22×C8—21no—
233C22×C8—1181no—
234C4⋊Q8—591no—
235C4⋊Q8C4×D5911yescharacteristic
236C4⋊Q8C4×D5911yescharacteristic
237#127—591no—
238#125—21no—
239#124—1181no—
240C22⋊Q8—21no—
241C22⋊Q8D23612yes—
242#121—1181no—
243#121—1181no—
244C4×Q8—21no—
245C4×Q8C22⋊D5912yes—
246C4×D4—1181no—
247C4×D4—1181no—
248#117—21no—
249C2×C4⋊C4—1181no—
250C2×C42—1181no—
251C2×C42C22×D5911yescharacteristic
252#106—1182no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—1181no—
257C4⋊C8C22⋊D5911yescharacteristic
258C4⋊C8C22⋊D5911yescharacteristic
259#104—2362no—
260#104—2362no—
261#103—1182no—
262#102—1182no—
263C22⋊C8—1181no—
264#97—22no—
265#97—21no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—591no—
269C4×C8—591no—
270C4×C8—1181no—
271C4×C8—1181no—
272#95—1181no—
273#41—2361no—
274#41—2361no—
275#41—2361no—
276#41—2361no—
277C2×Q8—21no—
278C2×Q8—1181no—
279C2×Q8(C2×D59)⋊C412yes—
280C2×D4—1181no—
281C2×D4—1181no—
282C2×D4—1181no—
283C2×D4—2361no—
284C22×C4—2361no—
285C22×C4—2361no—
286C22×C4C2×D23611yescharacteristic
287C22×C4#1174568511yescharacteristic
288C22×C4C2×C22⋊D5911yescharacteristic
289SD16—2364no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16#1174567112yescomplemented
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—1181no—
301C2×C8—1181no—
302C2×C8—2362no—
303C2×C8—2361no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—1182no—
310C4⋊C4—1181no—
311C4⋊C4—1181no—
312C4⋊C4—2361no—
313C4⋊C4(C2×D59)⋊C412yes—
314C22⋊C4—22no—
315C22⋊C4—2361no—
316C22⋊C4—2361no—
317C42—22no—
318C42—2361no—
319C42—2361no—
320C42C2×C4×D5911yescharacteristic
321C42#1174568511yescharacteristic
322C42C2×C22⋊D5911yescharacteristic
323C42C2×C22⋊D5911yescharacteristic
324C23—2361no—
325C23#1427370111yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—2361no—
330Q8—2361no—
331D4—4721no—
332D4—4721no—
333D4—2362no—
334D4—2361no—
335D4—2361no—
336D4—2361no—
337D4—2361no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—4721no—
344C2×C4—4721no—
345C2×C4—4721no—
346C2×C4—4721no—
347C2×C4—2362no—
348C2×C4—2361no—
349C2×C4C2×(C2×D59)⋊C411yescharacteristic
350C2×C4C4×C22⋊D5911yescharacteristic
351C2×C4C4×C22⋊D5911yescharacteristic
352C2×C4#1427369611yescharacteristic
353C2×C4#1427370111yescharacteristic
354C2×C4#1427371011yescharacteristic
355C2×C4C2×#1174567111yescharacteristic
356C2×C4#1427374711yescharacteristic
357C2×C4#1427375211yescharacteristic
358C2×C4#1427375811yescharacteristic
359C8—22no—
360C8—41no—
361C8—41no—
362C8—41no—
363C8—1182no—
364C8—4721no—
365C8—2361no—
366C22—21no—
367C22—21no—
368C22—4721no—
369C22—4721no—
370C22—4721no—
371C22#1683728111yescharacteristic
372C22#1683760211yescharacteristic
373C22#1683762411yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—4722no—
380C4—4721no—
381C4C4×#1174567111yescharacteristic
382C4#1683750011yescharacteristic
383C4#1683751711yescharacteristic
384C4#1683752211yescharacteristic
385C4#1683754111yescharacteristic
386C4#1683754711yescharacteristic
387C2—21no—
388C2—21no—
389C2—9441no—
390C2#2142442311yescharacteristic
391C2#2142445111yescharacteristic
392C2#2142447911yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2962C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#2962D59
hypocenterC59#9054
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
3. #21422595C2semidirect
5. #21421935C2semidirect
6. #21421934C2semidirect
7. C59×#2962C2semidirect
9. #16837798C4semidirect
10. #16837776C4semidirect
11. #16837752C4semidirect
12. #16837750C4semidirect
95. M16×C59SD16semidirect
192. #2962D59semidirect
227. C59#9054semidirect
293. M16#11745671semidirect

Maximal subgroups

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

subgroupindex
#214346562
#214250152
#214225952
#214224942
#214219352
#214219342
C59×#29622
#905459