Finite Group Databasebeta

← #28593094

Subgroups — #28593094

Order 15104.

722 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
1C4×#14273747C211yesmaximal, characteristic
2#21432748C211yesmaximal, characteristic, complemented
3#21432747C211yesmaximal, characteristic, complemented
4#21424981C212yesmaximal
5#21424843C212yesmaximal, complemented
6#21424808C212yesmaximal, complemented
7#21424804C212yesmaximal, complemented
8C4×#14273662C211yesmaximal, characteristic, complemented
9#21423907C211yesmaximal, characteristic, complemented
10#21423783C211yesmaximal, characteristic, complemented
11C59×#1759C211yesmaximal, characteristic, complemented
12C2×#14273747C2211yescharacteristic
13C4×#11745685C2211yescharacteristic
14C4×#11745672C412yescomplemented
15C4×#11745671C412yescomplemented
16C4×#11745670C412yescomplemented
17C4×#11745669C412yescomplemented
18#16837463—22no—
19#16837461—22no—
20#16837459—22no—
21#16837457—22no—
22C2×#14273662C2212yes—
23#16837438C2211yescharacteristic
24#16837430C2212yes—
25#16837429C2212yes—
26C2×#14273655C2212yes—
27#16837398C2211yescharacteristic
28#16837396C2211yescharacteristic
29#16837370C2212yes—
30#16837369C2212yes—
31C4×#11745664C2211yescharacteristic
32C2×#14273638C2212yes—
33C2×#14273638C2211yescharacteristic
34C2×C4×C59⋊C8C2212yes—
35C2×C4×C59⋊C8C2211yescharacteristic
36#16837166C2212yes—
37#16837162C2212yes—
38C4×#41×C59C2211yescharacteristic
39C2×#104×C59C2212yes—
40C4×M16×C59C2211yescharacteristic
41C59×#376C2211yescharacteristic
42C59×#339C2212yes—
43C2×#11745685C2311yescharacteristic
44#14273747—24no—
45C2×#11745672C2×C411yescharacteristic
46C2×#11745670C2×C411yescharacteristic
47C2×#11745671C2×C411yescharacteristic
48C2×#11745669C2×C411yescharacteristic
49C2×#11745664C2312yes—
50C22×C59⋊C8C2311yescharacteristic
51C22×C59⋊C8C2312yes—
52C22×C59⋊C8C2311yescharacteristic
53C4×Dic118C2×C412yes—
54C4×D236C2×C412yes—
55C4×C22⋊D59—22no—
56#14273686C2311yescharacteristic
57#14273684—21no—
58C42×D59—21no—
59#14273662—24no—
60#14273660C2×C412yes—
61#14273658C2×C412yes—
62C4⋊(C59⋊C8)—24no—
63Dic59⋊C8—24no—
64#14273655—24no—
65#14273651C2×C412yes—
66#14273650C2×C412yes—
67#14273648C2×C412yes—
68#14273647C2×C412yes—
69C22⋊(C59⋊C8)—24no—
70(C2×D59)⋊C8—24no—
71#14273638—21no—
72#14273638—24no—
73#14273638D412yes—
74#14273638D412yes—
75C4×C59⋊C8—21no—
76C4×C59⋊C8—24no—
77C4×C59⋊C8C2×C412yes—
78C4×C59⋊C8C2×C412yes—
79C2×#41×C59C2311yescharacteristic
80C2×M16×C59C2312yes—
81C4×Q8×C59C2×C412yes—
82C4×D4×C59—22no—
83C4×D4×C59C2×C412yes—
84#117×C59—21no—
85#117×C59C2311yescharacteristic
86C2×C42×C59—21no—
87C2×C42×C59C2312yes—
88C2×C42×C59C2311yescharacteristic
89#104×C59—24no—
90#97×C59—21no—
91C4×C8×C59—21no—
92#95×C59—22no—
93#41×C59—22no—
94#41×C59—22no—
95C2×Q8×C59C22×C411yescharacteristic
96C2×D4×C59—21no—
97C2×D4×C59C22×C411yescharacteristic
98C22×C4×C59—22no—
99C22×C4×C59—21no—
100C22×C4×C59—22no—
101C22×C4×C59#4112yes—
102C22×C4×C59C2411yescharacteristic
103C2×C22⋊D59—21no—
104#11745685—22no—
105#11745685—22no—
106C2×D236C22×C411yescharacteristic
107C2×C4×D59—21no—
108C2×C4×D59—22no—
109C2×Dic118C22×C411yescharacteristic
110M16×C59—22no—
111M16×C59—22no—
112C2×C8×C59—22no—
113C4⋊C4×C59—22no—
114C4⋊C4×C59C22×C412yes—
115C22⋊C4×C59—21no—
116C22⋊C4×C59—22no—
117C42×C59—22no—
118C42×C59—21no—
119C42×C59—22no—
120C42×C59C22×C412yes—
121C42×C59C2×D411yescharacteristic
122C42×C59C2×D411yescharacteristic
123C42×C59#4114yes—
124C22⋊Dic59—21no—
125#11745672—22no—
126#11745671—22no—
127#11745670—22no—
128#11745669—22no—
129(C2×D59)⋊C4—22no—
130C4⋊Dic59C22×C412yes—
131Dic59⋊C4—22no—
132C4×Dic59—22no—
133C4×Dic59—21no—
134#11745664—22no—
135#11745664—22no—
136C2×C59⋊C8—22no—
137C2×C59⋊C8—22no—
138C2×C59⋊C8—22no—
139C2×C59⋊C8—22no—
140C2×C59⋊C8C22×C412yes—
141C2×C59⋊C8C22×C411yescharacteristic
142C2×C59⋊C8C22×C411yescharacteristic
143C2×C59⋊C8C2×D414yes—
144C2×C59⋊C8#4114yes—
145C23×C59—21no—
146C23×C59C2×#4111yescharacteristic
147C22×D59—21no—
148Q8×C59—21no—
149Q8×C59—21no—
150D4×C59—21no—
151D4×C59—21no—
152D4×C59—42no—
153C2×C4×C59—22no—
154C2×C4×C59—21no—
155C2×C4×C59—24no—
156C2×C4×C59—42no—
157C2×C4×C59—44no—
158C2×C4×C59—42no—
159C2×C4×C59—82no—
160C2×C4×C59C4×D412yes—
161C2×C4×C59C4×D412yes—
162C2×C4×C59C23×C411yescharacteristic
163C2×C4×C59C22×D411yescharacteristic
164C2×C4×C59C2×#4112yes—
165C2×C4×C59C2×#4112yes—
166C22⋊D59—42no—
167C2×Dic59—21no—
168C2×Dic59—22no—
169D236—21no—
170D236—21no—
171C4×D59—42no—
172C4×D59—82no—
173Dic118—21no—
174Dic118—21no—
175C8×C59—42no—
176C59⋊C8—22no—
177C59⋊C8—24no—
178C59⋊C8—42no—
179C59⋊C8#11712yes—
180C59⋊C8#11712yes—
181#21294—591nomaximal, Sylow
182C22×C59—22no—
183C22×C59—42no—
184C22×C59—41no—
185C22×C59C2×C4×D411yescharacteristic
186C22×C59#52912yes—
187C2×D59—42no—
188C2×D59—41no—
189C4×C59—22no—
190C4×C59—22no—
191C4×C59—44no—
192C4×C59—42no—
193C4×C59—41no—
194C4×C59C2×#11711yescharacteristic
195C4×C59C4×#4111yescharacteristic
196C4×C59#53212yes—
197Dic59—42no—
198Dic59—41no—
199C4×#135—591no—
200#2923—591no—
201#2923—591no—
202#1974—592no—
203#1945—592no—
204#1938—592no—
205#1938—592no—
206C4×#108—591no—
207#1759—591no—
208#1759D5911yescharacteristic, complemented
209#1726—591no—
210C2×C59—22no—
211C2×C59—81no—
212C2×C59#229311yescharacteristic
213C2×C59#295712yes—
214D59—81no—
215C2×#135—591no—
216C4×#41—591no—
217C4×#41C2×D5911yescharacteristic
218C4×Q16—592no—
219C4×SD16—592no—
220C4×SD16—592no—
221C4×D8—592no—
222#433—1182no—
223#433—1182no—
224#432—1182no—
225#432—1182no—
226C2×#108—592no—
227#427—591no—
228#424—592no—
229#424—592no—
230C2×#104—592no—
231C2×#104C2×D5912yes—
232#416—591no—
233#416—591no—
234#408—592no—
235#408—592no—
236C4×M16—591no—
237C4×M16C2×D5911yescharacteristic
238C2×#97—591no—
239C2×#97—592no—
240C2×C4×C8—591no—
241C2×C4×C8—592no—
242#376C2×D5911yescharacteristic
243#341—592no—
244#339—592no—
245#339C2×D5912yes—
246C59#2129411yescharacteristic, complemented, Sylow
247C2×#41—591no—
248C2×#41C22×D5911yescharacteristic
249#135—1184no—
250C2×Q16—591no—
251C2×SD16—591no—
252C2×SD16—591no—
253C2×D8—591no—
254C2×M16—592no—
255C2×M16C22×D5912yes—
256C22×C8—591no—
257C22×C8—591no—
258C22×C8—592no—
259C4×Q8—592no—
260C4×Q8C4×D5912yes—
261C4×D4—22no—
262C4×D4—592no—
263C4×D4—1182no—
264C4×D4C4×D5912yes—
265#117—21no—
266#117—591no—
267#117—1181no—
268#117C22×D5911yescharacteristic
269C2×C42—21no—
270C2×C42—1181no—
271C2×C42C22×D5912yes—
272C2×C42C22×D5911yescharacteristic
273#108—1184no—
274#107—592no—
275#106—592no—
276C4⋊C8—1184no—
277C4⋊C8—1184no—
278#104—24no—
279#104—1184no—
280#103—592no—
281#103—592no—
282#102—592no—
283#102—592no—
284C22⋊C8—1184no—
285C22⋊C8—1184no—
286#97—21no—
287#97—592no—
288#97—592no—
289#97—1184no—
290#97—1181no—
291C4×C8—21no—
292C4×C8—592no—
293C4×C8—592no—
294C4×C8—1184no—
295C4×C8—1181no—
296#95—22no—
297#41—22no—
298#41—22no—
299#41—1182no—
300#41—1182no—
301C2×Q8—591no—
302C2×Q8C2×C4×D5911yescharacteristic
303C2×D4—21no—
304C2×D4—591no—
305C2×D4—1181no—
306C2×D4C2×C4×D5911yescharacteristic
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—1182no—
311C22×C4—1181no—
312C22×C4#1174568712yes—
313C22×C4C23×D5911yescharacteristic
314Q16—1182no—
315SD16—1182no—
316SD16—1182no—
317D8—1182no—
318M16—22no—
319M16—22no—
320M16—1182no—
321M16—1182no—
322C2×C8—22no—
323C2×C8—591no—
324C2×C8—591no—
325C2×C8—594no—
326C2×C8—592no—
327C2×C8—594no—
328C2×C8—1182no—
329C2×C8—1182no—
330C2×C8—1182no—
331C2×C8—1182no—
332C4⋊C4—22no—
333C4⋊C4—592no—
334C4⋊C4—1182no—
335C4⋊C4C2×C4×D5912yes—
336C22⋊C4—21no—
337C22⋊C4—22no—
338C22⋊C4—1181no—
339C22⋊C4—1182no—
340C42—22no—
341C42—21no—
342C42—22no—
343C42—1182no—
344C42—1181no—
345C42C2×C4×D5912yes—
346C42#1174568514yes—
347C42D4×D5911yescharacteristic
348C42D4×D5911yescharacteristic
349C23—21no—
350C23—1181no—
351C23C2×#1174568711yescharacteristic
352Q8—21no—
353Q8—21no—
354Q8—1181no—
355Q8—1181no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—1181no—
360D4—1181no—
361D4—2362no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—1182no—
370C2×C4—1181no—
371C2×C4—4722no—
372C2×C4—2362no—
373C2×C4(C22⋊D59)⋊C412yes—
374C2×C4(C22⋊D59)⋊C412yes—
375C2×C4C22×C4×D5911yescharacteristic
376C2×C4C2×D4×D5911yescharacteristic
377C2×C4#41×D5912yes—
378C2×C4C2×#1174568512yes—
379C8—42no—
380C8—592no—
381C8—592no—
382C8—1184no—
383C8—1182no—
384C8—2362no—
385C22—22no—
386C22—41no—
387C22—42no—
388C22—2361no—
389C22—2362no—
390C22C2×(C22⋊D59)⋊C411yescharacteristic
391C22#1683793212yes—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—41no—
396C4—42no—
397C4—2361no—
398C4—2362no—
399C4#117×D5911yescharacteristic
400C4C4×#1174568511yescharacteristic
401C4#1683794912yes—
402C2—22no—
403C2—81no—
404C2—4721no—
405C2#2142655911yescharacteristic
406C2#2143300912yes—
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1759C2
socleC22×C59C2×C4×D4
prime radicalC2×#14273747C22
cosocleC22×C4×C59C24
hypercenter#1759D59
hypocenterC59#21294
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #21432748C2semidirect
3. #21432747C2semidirect
5. #21424843C2semidirect
6. #21424808C2semidirect
7. #21424804C2semidirect
8. C4×#14273662C2semidirect
9. #21423907C2semidirect
10. #21423783C2semidirect
11. C59×#1759C2semidirect
14. C4×#11745672C4semidirect
15. C4×#11745671C4semidirect
16. C4×#11745670C4semidirect
17. C4×#11745669C4semidirect
208. #1759D59semidirect
246. C59#21294semidirect

Maximal subgroups

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

subgroupindex
C4×#142737472
#214327482
#214327472
#214249812
#214248432
#214248082
#214248042
C4×#142736622
#214239072
#214237832
C59×#17592
#2129459