Finite Group Databasebeta

← #28483919

Subgroups — #28483919

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#21433045C211yesmaximal, characteristic, complemented
2#21425016C211yesmaximal, characteristic
3#21422595C211yesmaximal, characteristic, complemented
4#21422495C211yesmaximal, characteristic
5#21421936C211yesmaximal, characteristic, complemented
6#21421933C211yesmaximal, characteristic, complemented
7C59×#3167C211yesmaximal, characteristic
8#16837533—21no—
9#16837533—22no—
10Q8×C59⋊C8—21no—
11#16837455C2211yescharacteristic
12#16837429—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273654—21no—
15C2×#14273654—21no—
16#16837358—21no—
17C4×#11745664C2211yescharacteristic
18C4×C4⋊Dic59C2211yescharacteristic
19#16837139—22no—
20D4⋊(C59⋊C8)—21no—
21D4⋊(C59⋊C8)—21no—
22C59×#533C2211yescharacteristic
23C59×#532C2211yescharacteristic
24C59×#498C411yescharacteristic, complemented
25C59×#492C411yescharacteristic, complemented
26C59×#487C411yescharacteristic, complemented
27C59×#486C411yescharacteristic, complemented
28C59×#465—22no—
29C4×SD16×C59—22no—
30C2×C4×C8×C59C2211yescharacteristic
31C2×#11745664D411yescharacteristic
32C2×#11745664D411yescharacteristic
33C22×C59⋊C8—21no—
34C2×C4⋊Dic59—21no—
35C2×C4×Dic59—21no—
36C4⋊(C59⋊C8)—22no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)D411yescharacteristic
41C4⋊(C59⋊C8)D411yescharacteristic
42#14273654—42no—
43#14273654—42no—
44C22⋊(C59⋊C8)—21no—
45#14273638—22no—
46#14273638—21no—
47C4×C59⋊C8—22no—
48C4×C59⋊C8—21no—
49C4×C59⋊C8—21no—
50C59⋊#95—21no—
51C2×#41×C59—21no—
52C2×SD16×C59—22no—
53C22×C8×C59—21no—
54C4⋊Q8×C59C2×C411yescharacteristic
55C4⋊Q8×C59C2×C411yescharacteristic
56C4⋊Q8×C59C2×C411yescharacteristic
57#127×C59C2×C411yescharacteristic
58#125×C59—21no—
59#124×C59—21no—
60C22⋊Q8×C59—21no—
61C22⋊Q8×C59D412yes—
62#121×C59—21no—
63#121×C59—21no—
64C4×Q8×C59—21no—
65C4×Q8×C59D412yes—
66C4×D4×C59—21no—
67C4×D4×C59—21no—
68#117×C59—21no—
69C2×C42×C59C2311yescharacteristic
70#106×C59—22no—
71#103×C59—22no—
72#102×C59—22no—
73C4×C8×C59—21no—
74C4×C8×C59C2×C411yescharacteristic
75C4×C8×C59C2×C411yescharacteristic
76#41×C59—41no—
77#41×C59—41no—
78#41×C59—41no—
79#41×C59—41no—
80C2×Q8×C59—21no—
81C2×Q8×C59—21no—
82C2×Q8×C59C22⋊C412yes—
83C2×D4×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59#4111yescharacteristic
91C22×Dic59—41no—
92SD16×C59—44no—
93C2×C8×C59—21no—
94C2×C8×C59—21no—
95C2×C8×C59—42no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59C22⋊C412yes—
104C22⋊C4×C59—22no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C42×C59—22no—
108C42×C59C22×C411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59#4111yescharacteristic
112C4⋊Dic59—22no—
113C4⋊Dic59—41no—
114C4×Dic59—41no—
115C4×Dic59—41no—
116#11745664—21no—
117#11745664—21no—
118#11745664—21no—
119#11745664SD1612yescomplemented
120C2×C59⋊C8—21no—
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123C2×C59⋊C8—21no—
124C2×C59⋊C8—41no—
125C2×C59⋊C8—41no—
126C2×C59⋊C8—41no—
127C23×C59—41no—
128C23×C59#12111yescharacteristic
129Q8×C59—42no—
130Q8×C59—41no—
131Q8×C59—41no—
132Q8×C59—41no—
133Q8×C59—41no—
134D4×C59—42no—
135D4×C59—41no—
136D4×C59—41no—
137D4×C59—41no—
138D4×C59—41no—
139D4×C59—81no—
140D4×C59—81no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—22no—
144C2×C4×C59—22no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—81no—
148C2×C4×C59—81no—
149C2×C4×C59C2×C22⋊C411yescharacteristic
150C2×C4×C59C4×D411yescharacteristic
151C2×C4×C59C4×D411yescharacteristic
152C2×C4×C59#12011yescharacteristic
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59#12311yescharacteristic
155C2×C4×C59C2×SD1611yescharacteristic
156C2×C4×C59#13511yescharacteristic
157C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C59#13711yescharacteristic
159C2×Dic59—42no—
160C2×Dic59—81no—
161C2×Dic59—81no—
162C8×C59—22no—
163C8×C59—41no—
164C59⋊C8—22no—
165C59⋊C8—41no—
166C59⋊C8—41no—
167C59⋊C8—41no—
168C59⋊C8—81no—
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—42no—
182C4×C59—42no—
183C4×C59—81no—
184C4×C59C4×SD1611yescharacteristic
185C4×C59#44011yescharacteristic
186C4×C59#44311yescharacteristic
187C4×C59#44411yescharacteristic
188C4×C59#44811yescharacteristic
189C4×C59#44911yescharacteristic
190Dic59—82no—
191#3167D5911yescharacteristic
192#2962—591no—
193#1980—591no—
194#1503—591no—
195#1480—591no—
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#533C2×D5911yescharacteristic
205#532C2×D5911yescharacteristic
206#498Dic5911yescharacteristic, complemented
207#492Dic5911yescharacteristic, complemented
208#487Dic5911yescharacteristic, complemented
209#486Dic5911yescharacteristic, complemented
210#465—22no—
211#446—1181no—
212#446—1182no—
213C8×Q8—1181no—
214C4×SD16—22no—
215#431—591no—
216#424—1181no—
217C2×C4⋊C8—591no—
218C2×#104—1181no—
219C2×#104—1181no—
220#405—1181no—
221C4×M16—591no—
222C2×C4×C8C2×D5911yescharacteristic
223C4×C4⋊C4—591no—
224#328—1182no—
225D4⋊C8—1181no—
226D4⋊C8—1181no—
227C59#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—22no—
230C2×M16—591no—
231C2×M16—591no—
232C22×C8—21no—
233C22×C8—1181no—
234C4⋊Q8C2×Dic5911yescharacteristic
235C4⋊Q8C2×Dic5911yescharacteristic
236C4⋊Q8C2×Dic5911yescharacteristic
237#127C2×Dic5911yescharacteristic
238#125—21no—
239#124—21no—
240C22⋊Q8—21no—
241C22⋊Q8C22⋊D5912yes—
242#121—21no—
243#121—21no—
244C4×Q8—21no—
245C4×Q8C22⋊D5912yes—
246C4×D4—21no—
247C4×D4—21no—
248#117—21no—
249C2×C4⋊C4—1181no—
250C2×C42—1181no—
251C2×C42C22×D5911yescharacteristic
252#106—22no—
253C4⋊C8—591no—
254C4⋊C8—591no—
255C4⋊C8—1182no—
256C4⋊C8—1182no—
257C4⋊C8—1181no—
258C4⋊C8—1181no—
259#104—2362no—
260#104—2362no—
261#103—22no—
262#102—22no—
263C22⋊C8—1181no—
264#97—1182no—
265#97—1181no—
266C4×C8—21no—
267C4×C8—1182no—
268C4×C8—1181no—
269C4×C8—1181no—
270C4×C8C2×Dic5911yescharacteristic
271C4×C8C2×Dic5911yescharacteristic
272#95—1181no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C22⋊Dic5912yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—2361no—
286C22×C4D4×D5911yescharacteristic
287C22×C4#1174568711yescharacteristic
288C22×C4C2×C22⋊D5911yescharacteristic
289SD16—44no—
290M16—592no—
291M16—1181no—
292M16—1181no—
293M16—1181no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—42no—
297C2×C8—1181no—
298C2×C8—1181no—
299C2×C8—1181no—
300C2×C8—1181no—
301C2×C8—2361no—
302C2×C8—2361no—
303C2×C8—2361no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—1182no—
312C4⋊C4—2361no—
313C4⋊C4C22⋊Dic5912yes—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—2361no—
319C42—2361no—
320C42D4×D5911yescharacteristic
321C42#1174568711yescharacteristic
322C42C22×Dic5911yescharacteristic
323C42C2×C22⋊D5911yescharacteristic
324C23—41no—
325C23#1427369911yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—4721no—
347C2×C4—4721no—
348C2×C4—2362no—
349C2×C4C2×C22⋊Dic5911yescharacteristic
350C2×C4D4×Dic5911yescharacteristic
351C2×C4D4×Dic5911yescharacteristic
352C2×C4#1427369511yescharacteristic
353C2×C4#1427369911yescharacteristic
354C2×C4#1427370811yescharacteristic
355C2×C4SD16×D5911yescharacteristic
356C2×C4#1427374811yescharacteristic
357C2×C4#1427375411yescharacteristic
358C2×C4#1427375611yescharacteristic
359C8—22no—
360C8—41no—
361C8—1182no—
362C8—4721no—
363C8—2361no—
364C8—2361no—
365C8—2361no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1683728211yescharacteristic
372C22#1683760011yescharacteristic
373C22#1683762211yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—4722no—
381C4SD16×Dic5911yescharacteristic
382C4#1683750311yescharacteristic
383C4#1683751911yescharacteristic
384C4#1683752411yescharacteristic
385C4#1683753911yescharacteristic
386C4#1683754611yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2142442511yescharacteristic
391C2#2142445311yescharacteristic
392C2#2142448011yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3167C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#3167D59
hypocenterC59#9054
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21433045C2semidirect
3. #21422595C2semidirect
5. #21421936C2semidirect
6. #21421933C2semidirect
24. C59×#498C4semidirect
25. C59×#492C4semidirect
26. C59×#487C4semidirect
27. C59×#486C4semidirect
119. #11745664SD16semidirect
206. #498Dic59semidirect
207. #492Dic59semidirect
208. #487Dic59semidirect
209. #486Dic59semidirect
227. C59#9054semidirect

Maximal subgroups

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

subgroupindex
#214330452
#214250162
#214225952
#214224952
#214219362
#214219332
C59×#31672
#905459