Finite Group Databasebeta

← #29458383

Subgroups — #29458383

Order 15104.

1834 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437909C212yesmaximal
2#21434640C216yesmaximal, complemented
3#21434636C213yesmaximal, complemented
4#21434634C216yesmaximal, complemented
5#21434632C213yesmaximal, complemented
6#21434608C213yesmaximal, complemented
7C2×#16837806C213yesmaximal, complemented
8#21432696C211yesmaximal, characteristic
9#21432694C212yesmaximal
10#131×Dic59C211yesmaximal, characteristic
11C59×#2913C211yesmaximal, characteristic
12C2×#14273734C2211yescharacteristic
13Q8×Dic118—22no—
14#16838128—22no—
15#16838111—26no—
16D4×Dic118—26no—
17#16838033C2216yes—
18#16838030C2216yes—
19#16837955C2216yes—
20C2×#14273726C2216yes—
21#16837885C2212yes—
22#41×Dic59C2211yescharacteristic
23#16837810C22112yes—
24#16837809C22112yes—
25#16837806C2216yes—
26#16837806C2216yes—
27#16837806C2216yes—
28#16837806C2213yes—
29#16837806C2211yescharacteristic
30#16837803C2213yes—
31#16837803C2213yes—
32#16837802C2216yes—
33#16837799C2216yes—
34#16837445C2216yes—
35#16837444C2216yes—
36#16837442C2216yes—
37#16837440C2213yes—
38#16837438C2216yes—
39C2×#14273661C2213yes—
40C2×#14273660C2216yes—
41#16837360C2213yes—
42#16837357C2213yes—
43M16×Dic59C2213yes—
44C2×C8×Dic59C2213yes—
45C2×#131×C59C2211yescharacteristic
46C59×#520C2212yes—
47C59×#428C2216yes—
48C59×#427C2213yes—
49C2×#107×C59C2213yes—
50C22×Dic118—26no—
51#14273734Q818yescomplemented
52C2×#11745664C2313yes—
53C22×C59⋊C8C2313yes—
54#14273727—26no—
55#14273726—26no—
56#14273726—26no—
57#14273726—26no—
58#14273726C2316yes—
59#14273726C23112yes—
60#14273726C2312yes—
61#14273719—26no—
62#14273718—26no—
63#14273718C2316yes—
64#14273718C2316yes—
65#14273706—212no—
66#14273704—26no—
67#14273704—26no—
68#14273704—212no—
69Q8×Dic59—22no—
70Q8×Dic59D411yescharacteristic
71Q8×Dic59D411yescharacteristic
72C4×Dic118—26no—
73C4×Dic118—26no—
74D4×Dic59—26no—
75D4×Dic59D413yes—
76D4×Dic59D413yes—
77#14273686C2316yes—
78#14273686C2313yes—
79C2×C4⋊Dic59C2316yes—
80C2×Dic59⋊C4—26no—
81C2×C4×Dic59C2313yes—
82#14273661C2313yes—
83#14273661C2316yes—
84#14273661C2311yescharacteristic
85#14273660C2316yes—
86#14273660C23112yes—
87#14273660C2312yes—
88#14273659C2316yes—
89#14273658C2316yes—
90#14273658C2316yes—
91#14273637C2316yes—
92C8×Dic59C2313yes—
93C8×Dic59C2313yes—
94C8×Dic59C2313yes—
95C8×Dic59C2311yescharacteristic
96C2×#41×C59C2311yescharacteristic
97#131×C59Q818yescomplemented
98C2×M16×C59C2313yes—
99C22×C8×C59C2313yes—
100C4×Q8×C59—22no—
101C4×D4×C59—26no—
102#117×C59C2316yes—
103C2×C4⋊C4×C59C2316yes—
104#107×C59C2313yes—
105#107×C59C2316yes—
106#107×C59C2311yescharacteristic
107#106×C59C2316yes—
108#41×C59C2×Q814yes—
109#41×C59C2×Q814yes—
110C2×Q8×C59C2×D411yescharacteristic
111C2×D4×C59C2×D413yes—
112C22×C4×C59—26no—
113C22×C4×C59C2413yes—
114C22×Dic59—26no—
115C22×Dic59C2×D413yes—
116C22×Dic59C2×D413yes—
117C2×Dic118—22no—
118C2×Dic118—26no—
119C2×Dic118—212no—
120C2×Dic118—812no—
121M16×C59C2×Q8112yescomplemented
122C2×C8×C59C2×Q8112yescomplemented
123C2×C8×C59C2413yes—
124C2×C8×C59C2411yescharacteristic
125C4⋊C4×C59—26no—
126C4⋊C4×C59C2412yes—
127C4⋊C4×C59C2416yes—
128C22⋊C4×C59—26no—
129C42×C59—26no—
130C22⋊Dic59—26no—
131C22⋊Dic59C2×D416yes—
132C4⋊Dic59—26no—
133C4⋊Dic59—26no—
134C4⋊Dic59—26no—
135C4⋊Dic59C2×D413yes—
136C4⋊Dic59C2×D413yes—
137C4⋊Dic59C2412yes—
138C4⋊Dic59C2416yes—
139Dic59⋊C4—26no—
140Dic59⋊C4—212no—
141Dic59⋊C4—22no—
142Dic59⋊C4—26no—
143Dic59⋊C4—26no—
144C4×Dic59—26no—
145C4×Dic59C2×D413yes—
146C4×Dic59C2×D413yes—
147C4×Dic59C2411yescharacteristic
148C4×Dic59C2413yes—
149#11745664C2×Q8112yescomplemented
150C2×C59⋊C8C2×Q8112yescomplemented
151C2×C59⋊C8C2411yescharacteristic
152C2×C59⋊C8C2413yes—
153C23×C59C22×D413yes—
154Q8×C59C4⋊Q814yes—
155D4×C59C4⋊Q8112yes—
156C2×C4×C59—22no—
157C2×C4×C59—26no—
158C2×C4×C59—86no—
159C2×C4×C59C22×D413yes—
160C2×C4×C59C22×Q816yes—
161C2×C4×C59C22×Q816yes—
162C2×C4×C59C2511yescharacteristic
163C2×Dic59—26no—
164C2×Dic59—26no—
165C2×Dic59—22no—
166C2×Dic59—26no—
167C2×Dic59—86no—
168C2×Dic59C22×D413yes—
169C2×Dic59C22×D413yes—
170C2×Dic59C22×D411yescharacteristic
171C2×Dic59C22×D411yescharacteristic
172Dic118—412no—
173Dic118—44no—
174Dic118—812no—
175C8×C59C22×Q812yes—
176C8×C59C22×Q816yes—
177C59⋊C8C22×Q812yes—
178C59⋊C8C22×Q816yes—
179#60667—591nomaximal, Sylow
180C22×C59—23no—
181C22×C59—23no—
182C22×C59C2×C4⋊Q816yes—
183C22×C59C23×D411yescharacteristic
184C4×C59—46no—
185C4×C59—42no—
186C4×C59C2×C4⋊Q816yes—
187C4×C59C23×Q811yescharacteristic
188C4×C59C23×Q811yescharacteristic
189Dic59—23no—
190Dic59—23no—
191Dic59—46no—
192Dic59—42no—
193Dic59#57612yes—
194Dic59#57812yes—
195#3503—592no—
196#3165—596no—
197#3164—596no—
198#3164—593no—
199#3164—593no—
200#3160—593no—
201C2×#500—593no—
202#2913—591no—
203#2913—592no—
204#2913D5911yescharacteristic
205C4×#131—591no—
206C2×C59—26no—
207C2×C59C22×C4⋊Q811yescharacteristic
208C2×C59C2×#57611yescharacteristic
209C2×C59C2×#57811yescharacteristic
210C2×#131—591no—
211C2×#131C2×D5911yescharacteristic
212Q82—1182no—
213#557—1182no—
214#554—1186no—
215D4×Q8—1186no—
216#544—596no—
217#544—596no—
218#533—596no—
219C2×C4⋊Q8—596no—
220#520—592no—
221#520C2×D5912yes—
222C4×#41—591no—
223#501—5912no—
224#501—5912no—
225#500—593no—
226#500—596no—
227#500—596no—
228#500—596no—
229#500—591no—
230#499—593no—
231#499—593no—
232#499—596no—
233#498—596no—
234#428—596no—
235#428—596no—
236#428—596no—
237#428C2×D5916yes—
238#427—593no—
239#427—596no—
240#427C2×D5913yes—
241C2×#107—593no—
242C2×#107—596no—
243C2×#107C2×D5913yes—
244#405—593no—
245#405—593no—
246C4×M16—593no—
247C2×C4×C8—593no—
248C59#6066711yescharacteristic, complemented, Sylow
249C2×#41C22×D5911yescharacteristic
250C22×Q8—1186no—
251#131—598no—
252#131Dic11818yescomplemented
253C2×M16—593no—
254C2×M16C22×D5913yes—
255C22×C8—593no—
256C22×C8C22×D5913yes—
257C4⋊Q8—596no—
258C4⋊Q8—5912no—
259C4⋊Q8—592no—
260C4⋊Q8—1186no—
261C4⋊Q8—1186no—
262C4⋊Q8—1186no—
263C4⋊Q8—1186no—
264#125—596no—
265#125—596no—
266#125—1186no—
267#125—1186no—
268C22⋊Q8—11812no—
269C22⋊Q8—1186no—
270C22⋊Q8—1186no—
271C22⋊Q8—11812no—
272C4×Q8—22no—
273C4×Q8—591no—
274C4×Q8—591no—
275C4×Q8—1186no—
276C4×Q8—1186no—
277C4×Q8—1182no—
278C4×D4—26no—
279C4×D4—593no—
280C4×D4—593no—
281C4×D4—1186no—
282#117—593no—
283#117—596no—
284#117C22×D5916yes—
285C2×C4⋊C4—596no—
286C2×C4⋊C4—1186no—
287C2×C4⋊C4C22×D5916yes—
288C2×C42—593no—
289#107—593no—
290#107—596no—
291#107—591no—
292#107—596no—
293#107—5912no—
294#107—592no—
295#107C22×D5913yes—
296#107C22×D5916yes—
297#107C22×D5911yescharacteristic
298#106—596no—
299#106—596no—
300#106—596no—
301#106C22×D5916yes—
302#97—596no—
303C4×C8—591no—
304C4×C8—593no—
305C4×C8—593no—
306C4×C8—593no—
307#41C2×Dic11814yes—
308#41Q8×D5914yes—
309C2×Q8—11812no—
310C2×Q8—1182no—
311C2×Q8—1186no—
312C2×Q8—47212no—
313C2×Q8D4×D5911yescharacteristic
314C2×D4D4×D5913yes—
315C22×C4—26no—
316C22×C4—593no—
317C22×C4—593no—
318C22×C4—1186no—
319C22×C4C23×D5913yes—
320M16—5912no—
321M16C2×Dic118112yescomplemented
322C2×C8—5912no—
323C2×C8—591no—
324C2×C8—593no—
325C2×C8C2×Dic118112yescomplemented
326C2×C8C23×D5911yescharacteristic
327C2×C8C23×D5913yes—
328C4⋊C4—26no—
329C4⋊C4—593no—
330C4⋊C4—593no—
331C4⋊C4—592no—
332C4⋊C4—596no—
333C4⋊C4—1186no—
334C4⋊C4—11812no—
335C4⋊C4—1182no—
336C4⋊C4—1186no—
337C4⋊C4—1186no—
338C4⋊C4—1186no—
339C4⋊C4—1186no—
340C4⋊C4—1186no—
341C4⋊C4C23×D5912yes—
342C4⋊C4C23×D5916yes—
343C22⋊C4—26no—
344C22⋊C4—596no—
345C22⋊C4—1186no—
346C42—26no—
347C42—593no—
348C42—593no—
349C42—591no—
350C42—593no—
351C42—1186no—
352C23C2×D4×D5913yes—
353Q8—47212no—
354Q8—2364no—
355Q8—23612no—
356Q8#1427372614yes—
357D4#14273726112yes—
358C2×C4—22no—
359C2×C4—26no—
360C2×C4—86no—
361C2×C4—591no—
362C2×C4—591no—
363C2×C4—593no—
364C2×C4—593no—
365C2×C4—1186no—
366C2×C4—1186no—
367C2×C4—1182no—
368C2×C4—1186no—
369C2×C4—4726no—
370C2×C4C2×D4×D5913yes—
371C2×C4C2×Q8×D5916yes—
372C2×C4C22×Dic11816yes—
373C2×C4C24×D5911yescharacteristic
374C8—592no—
375C8—596no—
376C8C22×Dic11812yes—
377C8C22×Dic11816yes—
378C22—23no—
379C22—23no—
380C22C2×#1427372616yes—
381C22C22×D4×D5911yescharacteristic
382C4—46no—
383C4—42no—
384C4—592no—
385C4—592no—
386C4—1183no—
387C4—1183no—
388C4—2366no—
389C4—2362no—
390C4C2×#1427372616yes—
391C4C22×Q8×D5911yescharacteristic
392C4C23×Dic11811yescharacteristic
393C2—26no—
394C2C22×#1427372611yescharacteristic
395C2#576×D5911yescharacteristic
396C2#578×D5911yescharacteristic
subgroupG/H
centerC22C22×D4×D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC59×#2913C2
socleC22×C59C23×D4
prime radicalC2×#41×C59C23
hypercenter#2913D59
hypocenterC59#60667
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2913

Complements

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

NH ≅ G/Ntype
2. #21434640C2semidirect
3. #21434636C2semidirect
4. #21434634C2semidirect
5. #21434632C2semidirect
6. #21434608C2semidirect
7. C2×#16837806C2semidirect
51. #14273734Q8semidirect
97. #131×C59Q8semidirect
121. M16×C59C2×Q8semidirect
122. C2×C8×C59C2×Q8semidirect
149. #11745664C2×Q8semidirect
150. C2×C59⋊C8C2×Q8semidirect
248. C59#60667semidirect
252. #131Dic118semidirect
321. M16C2×Dic118semidirect
325. C2×C8C2×Dic118semidirect

Maximal subgroups

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

subgroupindex
#214379092
#214346402
#214346362
#214346342
#214346322
#214346082
C2×#168378062
#214326962
#214326942
#131×Dic592
C59×#29132
#6066759