Finite Group Databasebeta

← #29458042

Subgroups — #29458042

Order 15104.

2042 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437915C211yesmaximal, characteristic, complemented
2#21437881C211yesmaximal, characteristic
3#21434490C213yesmaximal, complemented
4#21434477C213yesmaximal, complemented
5#21434463C2112yesmaximal, complemented
6#21434419C213yesmaximal, complemented
7C2×#16837753C213yesmaximal, complemented
8#21432630C214yesmaximal, complemented
9C4×#131×C59C211yesmaximal, characteristic, complemented
10C2×#14273774C2212yes—
11Q8×Dic118—21no—
12#16838131—21no—
13#16838109—23no—
14Q8×D236—21no—
15Q8×D236—21no—
16D4×Dic118—23no—
17#16838056—23no—
18#16838056—23no—
19#16838030C2216yes—
20#16837978C2216yes—
21#16837959C2213yes—
22#16837952C2213yes—
23C2×#14273727C2213yes—
24C2×#14273727C2213yes—
25#16837885C2212yes—
26#16837771C22112yes—
27#16837762C22112yes—
28#16837757C22112yes—
29#16837753C2213yes—
30#16837753C22112yes—
31#16837753C2216yes—
32#16837753C2213yes—
33#16837749C2211yescharacteristic
34#16837749C2213yes—
35#16837412C22112yes—
36#16837404C22112yes—
37#16837399C22112yes—
38C2×#14273652C22112yes—
39C2×#131×C59C2212yes—
40C4×#41×C59C2211yescharacteristic
41C59×#405C2216yes—
42C4×M16×C59C2213yes—
43C2×C4×C8×C59C2213yes—
44#14273774—216no—
45C2×#11745685C2316yes—
46C2×#11745687—26no—
47C22×Dic118—23no—
48C22×Dic118C2316yes—
49C2×Q8×D59—22no—
50#14273727—23no—
51#14273727—23no—
52#14273727—23no—
53#14273727—23no—
54#14273727C2313yes—
55#14273727C2316yes—
56#14273727C2313yes—
57#14273727C2316yes—
58#14273727C2311yescharacteristic
59#14273726—26no—
60#14273724C2311yescharacteristic
61#14273719C2316yes—
62#14273718—26no—
63#14273715C2316yes—
64#14273711—212no—
65#14273707—26no—
66#14273707—26no—
67#14273706—23no—
68#14273706—23no—
69#14273704—212no—
70#14273701—23no—
71#14273701—23no—
72Q8×Dic59—22no—
73C4×Dic118—23no—
74C4×Dic118—23no—
75C4×D236—23no—
76C4×D236—23no—
77D4×Dic59—26no—
78#14273686C2316yes—
79C2×C4⋊Dic59—23no—
80C2×C4⋊Dic59—23no—
81C2×C4⋊Dic59—23no—
82C2×C4⋊Dic59C2316yes—
83#14273652C23112yes—
84#14273652C23124yes—
85#14273652C23112yes—
86#14273649C2314yes—
87#14273649C23112yes—
88C2×#41×C59C2311yescharacteristic
89#131×C59—28no—
90C2×M16×C59C2316yes—
91C22×C8×C59C2316yes—
92C4×Q8×C59D411yescharacteristic
93C4×Q8×C59D411yescharacteristic
94C4×D4×C59D413yes—
95C4×D4×C59D413yes—
96#117×C59C2313yes—
97C2×C42×C59C2313yes—
98#97×C59C2316yes—
99C4×C8×C59C2311yescharacteristic
100C4×C8×C59C2313yes—
101C4×C8×C59C2316yes—
102#41×C59#4118yes—
103C2×Q8×C59C2×D411yescharacteristic
104C2×D4×C59C2×D413yes—
105C22×C4×C59C2×D413yes—
106C22×C4×C59C2×D413yes—
107C22×C4×C59C2413yes—
108C2×C22⋊D59—26no—
109C22×Dic59—26no—
110C22×Dic59—26no—
111Q8×D59—48no—
112#11745687—424no—
113#11745685—224no—
114C2×D236—21no—
115C2×D236—21no—
116C2×D236C2412yes—
117C2×C4×D59—26no—
118C2×Dic118—224no—
119C2×Dic118—21no—
120C2×Dic118—23no—
121C2×Dic118—26no—
122C2×Dic118—26no—
123C2×Dic118—86no—
124C2×Dic118C2416yes—
125M16×C59—212no—
126C2×C8×C59—212no—
127C2×C8×C59C2412yes—
128C2×C8×C59C2416yes—
129C4⋊C4×C59C2×D413yes—
130C4⋊C4×C59C2×D413yes—
131C22⋊C4×C59C2×D416yes—
132C42×C59C2×D413yes—
133C42×C59C2×D413yes—
134C42×C59C2411yescharacteristic
135C42×C59C2413yes—
136C22⋊Dic59—26no—
137(C2×D59)⋊C4—26no—
138(C2×D59)⋊C4—26no—
139C4⋊Dic59—23no—
140C4⋊Dic59—23no—
141C4⋊Dic59—23no—
142C4⋊Dic59—26no—
143C4⋊Dic59—26no—
144C4⋊Dic59—26no—
145C4⋊Dic59—26no—
146C4⋊Dic59—21no—
147C4⋊Dic59C2416yes—
148C4⋊Dic59C2412yes—
149Dic59⋊C4—26no—
150Dic59⋊C4—26no—
151C4×Dic59—26no—
152C23×C59C22×D413yes—
153C22×D59—22no—
154Q8×C59#12414yes—
155D4×C59#124112yes—
156C2×C4×C59—26no—
157C2×C4×C59—26no—
158C2×C4×C59C22×D413yes—
159C2×C4×C59C22×D413yes—
160C2×C4×C59C22×D413yes—
161C2×C4×C59C22×D411yescharacteristic
162C2×C4×C59C22×D411yescharacteristic
163C2×C4×C59C2×#41112yes—
164C2×C4×C59C2511yescharacteristic
165C22⋊D59—412no—
166C2×Dic59—22no—
167C2×Dic59—26no—
168C2×Dic59—26no—
169C2×Dic59—412no—
170C2×Dic59—86no—
171D236—24no—
172D236—42no—
173D236—42no—
174C4×D59—412no—
175Dic118—212no—
176Dic118—412no—
177Dic118—46no—
178Dic118—42no—
179Dic118—86no—
180C8×C59—22no—
181C8×C59—26no—
182#60623—591nomaximal, Sylow
183C22×C59—23no—
184C22×C59—23no—
185C22×C59C2×#12416yes—
186C22×C59C23×D411yescharacteristic
187C2×D59—42no—
188C2×D59—42no—
189C2×D59—42no—
190C4×C59—23no—
191C4×C59—23no—
192C4×C59C2×#12416yes—
193C4×C59#57712yes—
194C4×C59#57812yes—
195C4×C59C22×#4112yes—
196Dic59—46no—
197Dic59—46no—
198Dic59—42no—
199#3503—591no—
200#3499—591no—
201#3141—593no—
202#3139—593no—
203#3137—5912no—
204#3131—593no—
205C2×#487—593no—
206#2898—594no—
207C4×#131D5911yescharacteristic, complemented
208C2×C59—26no—
209C2×C59C22×#12411yescharacteristic
210C2×C59C2×#57711yescharacteristic
211C2×C59C2×#57811yescharacteristic
212D59—82no—
213C2×2−1+4—592no—
214C2×#131C2×D5912yes—
215Q82—1181no—
216#557—1181no—
217#554—1183no—
218D4×Q8—1181no—
219D4×Q8—1181no—
220D4×Q8—1183no—
221#547—1183no—
222#547—1183no—
223#544—596no—
224#536—596no—
225#533—593no—
226#532—593no—
227C2×C4⋊Q8—593no—
228C2×C4⋊Q8—593no—
229#520—592no—
230C4×#41C2×D5911yescharacteristic
231#491—5912no—
232#489—5912no—
233#488—5912no—
234#487—593no—
235#487—5912no—
236#487—596no—
237#487—593no—
238#486—591no—
239#486—593no—
240#419—5912no—
241#417—5912no—
242#416—5912no—
243C2×#103—5912no—
244#405C2×D5916yes—
245C4×M16C2×D5913yes—
246C2×C4×C8C2×D5913yes—
247C59#6062311yescharacteristic, complemented, Sylow
2482−1+4—11816no—
249C2×#41—596no—
250C2×#41—1186no—
251C2×#41C22×D5911yescharacteristic
252C22×Q8—596no—
253C22×Q8—1182no—
254C22×Q8—1183no—
255#131—28no—
256C2×M16C22×D5916yes—
257C22×C8C22×D5916yes—
258C4⋊Q8—593no—
259C4⋊Q8—596no—
260C4⋊Q8—591no—
261C4⋊Q8—593no—
262C4⋊Q8—596no—
263C4⋊Q8—1186no—
264C4⋊Q8—1183no—
265C4⋊Q8—1183no—
266C4⋊Q8—1183no—
267C4⋊Q8—1183no—
268#127—591no—
269#125—596no—
270#125—1186no—
271#124—596no—
272#123—11812no—
273C22⋊Q8—1183no—
274C22⋊Q8—1183no—
275C22⋊Q8—1186no—
276C22⋊Q8—1186no—
277C22⋊Q8—11812no—
278#121—1183no—
279#121—1183no—
280C4×Q8—1182no—
281C4×Q8—1183no—
282C4×Q8—1183no—
283C4×Q8D23611yescharacteristic
284C4×Q8D23611yescharacteristic
285C4×D4—1186no—
286C4×D4—1183no—
287C4×D4—1183no—
288C4×D4D23613yes—
289C4×D4D23613yes—
290#117—596no—
291#117C22×D5913yes—
292C2×C4⋊C4—596no—
293C2×C4⋊C4—1183no—
294C2×C4⋊C4—1183no—
295C2×C4⋊C4—1183no—
296C2×C42C22×D5913yes—
297#103—5912no—
298#103—5924no—
299#103—5912no—
300#102—594no—
301#102—5912no—
302#97C22×D5916yes—
303C4×C8C22×D5911yescharacteristic
304C4×C8C22×D5913yes—
305C4×C8C22×D5916yes—
306#41—11824no—
307#41—23624no—
308#41#1174568518yes—
309C2×Q8—596no—
310C2×Q8—1186no—
311C2×Q8—1181no—
312C2×Q8—1183no—
313C2×Q8—1186no—
314C2×Q8—11824no—
315C2×Q8—4726no—
316C2×Q8—2368no—
317C2×Q8C2×D23611yescharacteristic
318C2×D4—592no—
319C2×D4—1186no—
320C2×D4—1181no—
321C2×D4—1181no—
322C2×D4C2×D23613yes—
323C22×C4—1186no—
324C22×C4—1186no—
325C22×C4—1186no—
326C22×C4C2×D23613yes—
327C22×C4C2×D23613yes—
328C22×C4C23×D5913yes—
329M16—212no—
330C2×C8—212no—
331C2×C8C23×D5912yes—
332C2×C8C23×D5916yes—
333C4⋊C4—596no—
334C4⋊C4—592no—
335C4⋊C4—1186no—
336C4⋊C4—1183no—
337C4⋊C4—1183no—
338C4⋊C4—1183no—
339C4⋊C4—1186no—
340C4⋊C4—1186no—
341C4⋊C4—1186no—
342C4⋊C4—1181no—
343C4⋊C4—1186no—
344C4⋊C4—1186no—
345C4⋊C4C2×D23613yes—
346C4⋊C4C2×D23613yes—
347C22⋊C4—1186no—
348C22⋊C4—1186no—
349C22⋊C4—1186no—
350C22⋊C4C2×D23616yes—
351C42—1186no—
352C42C2×D23613yes—
353C42C2×D23613yes—
354C42C23×D5911yescharacteristic
355C42C23×D5913yes—
356C23—1182no—
357C23C22×D23613yes—
358Q8—11812no—
359Q8—4726no—
360Q8—23612no—
361Q8—2366no—
362Q8—2362no—
363Q8#1427371514yes—
364D4—1184no—
365D4—23612no—
366D4—2362no—
367D4—2362no—
368D4#14273715112yes—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—1182no—
372C2×C4—1186no—
373C2×C4—1186no—
374C2×C4—4726no—
375C2×C4—23612no—
376C2×C4—23612no—
377C2×C4C22×D23613yes—
378C2×C4C22×D23613yes—
379C2×C4C22×D23613yes—
380C2×C4C22×D23611yescharacteristic
381C2×C4C22×D23611yescharacteristic
382C2×C4C2×#11745685112yes—
383C2×C4C24×D5911yescharacteristic
384C8—22no—
385C8—26no—
386C22—23no—
387C22—23no—
388C22—2362no—
389C22—2362no—
390C22—2362no—
391C22C2×#1427371516yes—
392C22C23×D23611yescharacteristic
393C4—23no—
394C4—23no—
395C4—2366no—
396C4—2362no—
397C4—2366no—
398C4C2×#1427371516yes—
399C4#1683821412yes—
400C4#1683821912yes—
401C4C22×#1174568512yes—
402C2—26no—
403C2—4722no—
404C2C22×#1427371511yescharacteristic
405C2C2×#1683821411yescharacteristic
406C2C2×#1683821911yescharacteristic
subgroupG/H
centerC22C23×D236
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC4×#131×C59C2
socleC22×C59C23×D4
prime radical#21437881C2
hypercenterC4×#131D59
hypocenterC59#60623
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131

Complements

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

NH ≅ G/Ntype
1. #21437915C2semidirect
3. #21434490C2semidirect
4. #21434477C2semidirect
5. #21434463C2semidirect
6. #21434419C2semidirect
7. C2×#16837753C2semidirect
8. #21432630C2semidirect
9. C4×#131×C59C2semidirect
207. C4×#131D59semidirect
247. C59#60623semidirect

Maximal subgroups

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

subgroupindex
#214379152
#214378812
#214344902
#214344772
#214344632
#214344192
C2×#168377532
#214326302
C4×#131×C592
#6062359