Finite Group Databasebeta

← #28483915

Subgroups — #28483915

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#21433045C211yesmaximal, characteristic, complemented
3#21425014C211yesmaximal, characteristic
4#21422496C211yesmaximal, characteristic
5#21421937C211yesmaximal, characteristic, complemented
6#21421932C211yesmaximal, characteristic, complemented
7C59×#1503C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837798C411yescharacteristic, complemented
10#16837776C411yescharacteristic, complemented
11#16837752C411yescharacteristic, complemented
12#16837750C411yescharacteristic, complemented
13#16837633—22no—
14#16837533—21no—
15#16837533—22no—
16Q8×C59⋊C8—21no—
17C4×#11745671—22no—
18#16837429—21no—
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273655—21no—
21C2×#14273655—21no—
22#16837358—21no—
23C4×#11745664C2211yescharacteristic
24C2×C4×C59⋊C8C2211yescharacteristic
25#16837131—21no—
26#16837131—21no—
27C59×#533C2211yescharacteristic
28C59×#431C2211yescharacteristic
29C4×C4⋊C4×C59C2211yescharacteristic
30C59×#328—22no—
31C2×#11745685—21no—
32C2×#11745671—22no—
33C2×#11745664D411yescharacteristic
34C2×#11745664D411yescharacteristic
35C22×C59⋊C8—21no—
36C22×C59⋊C8—21no—
37#14273727C2×C411yescharacteristic
38#14273724C2×C411yescharacteristic
39#14273715—21no—
40#14273701—21no—
41#14273701—21no—
42C4×D236—21no—
43C4×D236—21no—
44#14273658—22no—
45C4⋊(C59⋊C8)—22no—
46C4⋊(C59⋊C8)—22no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)D411yescharacteristic
49C4⋊(C59⋊C8)D411yescharacteristic
50#14273655—42no—
51#14273655—42no—
52#14273651—22no—
53#14273647—22no—
54#14273638—22no—
55#14273638—21no—
56C4×C59⋊C8—22no—
57C4×C59⋊C8—21no—
58C4×C59⋊C8—21no—
59C4×C59⋊C8C2×C411yescharacteristic
60C4×C59⋊C8C2×C411yescharacteristic
61C4⋊Q8×C59C2×C411yescharacteristic
62C4⋊Q8×C59C2×C411yescharacteristic
63#125×C59—21no—
64C22⋊Q8×C59—21no—
65C22⋊Q8×C59D412yes—
66C4×Q8×C59—21no—
67C4×Q8×C59D412yes—
68#117×C59—21no—
69C2×C4⋊C4×C59—21no—
70C2×C42×C59—21no—
71C2×C42×C59C2311yescharacteristic
72C4⋊C8×C59—21no—
73C22⋊C8×C59—21no—
74C4×C8×C59—21no—
75#95×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59C22⋊C412yes—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C2×C22⋊D59—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—
92C2×C8×C59—41no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—22no—
97C4⋊C4×C59—22no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—41no—
100C4⋊C4×C59C22⋊C412yes—
101C22⋊C4×C59—22no—
102C42×C59—22no—
103C42×C59—41no—
104C42×C59—41no—
105C42×C59C22×C411yescharacteristic
106C42×C59C2×D411yescharacteristic
107C42×C59C2×D411yescharacteristic
108C42×C59#4111yescharacteristic
109#11745671—44no—
110(C2×D59)⋊C4—41no—
111(C2×D59)⋊C4—41no—
112C4⋊Dic59—21no—
113C4⋊Dic59—21no—
114#11745664—21no—
115#11745664—21no—
116#11745664—21no—
117#11745664SD1612yescomplemented
118C2×C59⋊C8—21no—
119C2×C59⋊C8—21no—
120C2×C59⋊C8—21no—
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123C2×C59⋊C8—21no—
124C2×C59⋊C8—42no—
125C2×C59⋊C8—41no—
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×C59—42no—
138C2×C4×C59—81no—
139C2×C4×C59—81no—
140C2×C4×C59C2×C22⋊C411yescharacteristic
141C2×C4×C59C4×D411yescharacteristic
142C2×C4×C59C4×D411yescharacteristic
143C2×C4×C59#12011yescharacteristic
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12311yescharacteristic
146C2×C4×C59C2×SD1611yescharacteristic
147C2×C4×C59#13511yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C59#13711yescharacteristic
150C22⋊D59—81no—
151C22⋊D59—81no—
152C2×Dic59—41no—
153D236—42no—
154D236—41no—
155D236—41no—
156D236—41no—
157D236—41no—
158C4×D59—81no—
159C4×D59—81no—
160Dic118—41no—
161Dic118—41no—
162C8×C59—81no—
163C59⋊C8—22no—
164C59⋊C8—22no—
165C59⋊C8—41no—
166C59⋊C8—41no—
167C59⋊C8—41no—
168C59⋊C8—41no—
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×C59—82no—
184C4×C59C4×SD1611yescharacteristic
185C4×C59#44011yescharacteristic
186C4×C59#44311yescharacteristic
187C4×C59#44411yescharacteristic
188C4×C59#44811yescharacteristic
189C4×C59#44911yescharacteristic
190Dic59—81no—
191#3167—591no—
192#2962—591no—
193#1980—591no—
194#1503D5911yescharacteristic, complemented
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—1181no—
212#446—1182no—
213C8×Q8—1181no—
214C4×SD16—1182no—
215#431C2×D5911yescharacteristic
216#424—1181no—
217C2×C4⋊C8—591no—
218C2×#104—1181no—
219C2×#104—1181no—
220#405—1181no—
221C4×M16—591no—
222C2×C4×C8—591no—
223C4×C4⋊C4C2×D5911yescharacteristic
224#328—22no—
225D4⋊C8—1181no—
226D4⋊C8—1181no—
227C59#905411yescharacteristic, complemented, Sylow
228C2×#41—1181no—
229C2×SD16—1182no—
230C2×M16—591no—
231C2×M16—591no—
232C22×C8—1181no—
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⋊Q8C22⋊D5912yes—
242#121—1181no—
243#121—1181no—
244C4×Q8—21no—
245C4×Q8D23612yes—
246C4×D4—1181no—
247C4×D4—1181no—
248#117—21no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42C22×D5911yescharacteristic
252#106—1182no—
253C4⋊C8—21no—
254C4⋊C8—591no—
255C4⋊C8—591no—
256C4⋊C8—1182no—
257C4⋊C8—1182no—
258C4⋊C8—1181no—
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—21no—
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—41no—
285C22×C4—2361no—
286C22×C4D4×D5911yescharacteristic
287C22×C4#1174568711yescharacteristic
288C22×C4C2×C22⋊D5911yescharacteristic
289SD16—2364no—
290M16—592no—
291M16—1181no—
292M16—1181no—
293M16—1181no—
294C2×C8—41no—
295C2×C8—1181no—
296C2×C8—1181no—
297C2×C8—1181no—
298C2×C8—1181no—
299C2×C8—1181no—
300C2×C8—1181no—
301C2×C8—2362no—
302C2×C8—2361no—
303C2×C8—2361no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—1181no—
312C4⋊C4—1181no—
313C4⋊C4(C2×D59)⋊C412yes—
314C22⋊C4—22no—
315C22⋊C4—2361no—
316C22⋊C4—2361no—
317C42—22no—
318C42—41no—
319C42—41no—
320C42C2×C4×D5911yescharacteristic
321C42C2×D23611yescharacteristic
322C42D4×D5911yescharacteristic
323C42#1174568711yescharacteristic
324C23—2361no—
325C23#1427369911yescharacteristic
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—42no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—4721no—
347C2×C4—4721no—
348C2×C4—2361no—
349C2×C4C2×(C2×D59)⋊C411yescharacteristic
350C2×C4(C22⋊D59)⋊C411yescharacteristic
351C2×C4(C22⋊D59)⋊C411yescharacteristic
352C2×C4#1427369711yescharacteristic
353C2×C4#1427369911yescharacteristic
354C2×C4#1427371111yescharacteristic
355C2×C4SD16×D5911yescharacteristic
356C2×C4#1427375011yescharacteristic
357C2×C4#1427375411yescharacteristic
358C2×C4#1427375911yescharacteristic
359C8—81no—
360C8—1182no—
361C8—1182no—
362C8—2361no—
363C8—2361no—
364C8—2361no—
365C8—2361no—
366C22—21no—
367C22—21no—
368C22—4721no—
369C22—4721no—
370C22—4721no—
371C22#1683727911yescharacteristic
372C22#1683760011yescharacteristic
373C22#1683762611yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—82no—
380C4—4721no—
381C4#1683748711yescharacteristic
382C4#1683750511yescharacteristic
383C4#1683751611yescharacteristic
384C4#1683752111yescharacteristic
385C4#1683754211yescharacteristic
386C4#1683755011yescharacteristic
387C2—21no—
388C2—21no—
389C2—9441no—
390C2#2142442111yescharacteristic
391C2#2142444911yescharacteristic
392C2#2142447711yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1503C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1503D59
hypocenterC59#9054
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #21433045C2semidirect
5. #21421937C2semidirect
6. #21421932C2semidirect
7. C59×#1503C2semidirect
9. #16837798C4semidirect
10. #16837776C4semidirect
11. #16837752C4semidirect
12. #16837750C4semidirect
117. #11745664SD16semidirect
194. #1503D59semidirect
227. C59#9054semidirect

Maximal subgroups

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

subgroupindex
#214346562
#214330452
#214250142
#214224962
#214219372
#214219322
C59×#15032
#905459