Finite Group Databasebeta

← #28488032

Subgroups — #28488032

Order 15104.

514 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434522C211yesmaximal, characteristic
2C4⋊C4×C59⋊C8C211yesmaximal, characteristic, complemented
3#21422842C211yesmaximal, characteristic, complemented
4#21422590C211yesmaximal, characteristic, complemented
5#21422496C211yesmaximal, characteristic, complemented
6#21422495C211yesmaximal, characteristic, complemented
7C59×#1502C211yesmaximal, characteristic, complemented
8C2×#14273747—21no—
9C2×#14273747—21no—
10#16837952C2211yescharacteristic
11#16837784—21no—
12#16837784—21no—
13#16837781C411yescharacteristic, complemented
14#16837778C411yescharacteristic, complemented
15#16837776C411yescharacteristic, complemented
16#16837774C411yescharacteristic, complemented
17#16837455C2211yescharacteristic
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×C4×C59⋊C8—21no—
20C2×C4×C59⋊C8C2211yescharacteristic
21#16837157—21no—
22#16837154—21no—
23#16837152—21no—
24#16837135—22no—
25#16837131—22no—
26D4⋊(C59⋊C8)—22no—
27C59×#532C2211yescharacteristic
28C59×#431C2211yescharacteristic
29C4×C4⋊C4×C59C2211yescharacteristic
30C59×#327—22no—
31C2×#11745685—21no—
32#14273747—42no—
33#14273747—42no—
34#14273747—42no—
35#14273747—42no—
36C2×#11745672—21no—
37C2×#11745672—21no—
38C2×#11745670—21no—
39C2×#11745670—21no—
40C2×#11745671—21no—
41C2×#11745671—21no—
42C2×#11745669—21no—
43C2×#11745669—21no—
44C22×C59⋊C8—21no—
45C22×C59⋊C8D411yescharacteristic
46C22×C59⋊C8D411yescharacteristic
47#14273727C2×C411yescharacteristic
48#14273724C2×C411yescharacteristic
49#14273715—21no—
50#14273701—21no—
51#14273701—21no—
52C4×D236—21no—
53C4×D236—21no—
54C4⋊(C59⋊C8)—21no—
55C4⋊(C59⋊C8)—21no—
56C4⋊(C59⋊C8)D412yes—
57C22⋊(C59⋊C8)—21no—
58C4×C59⋊C8—21no—
59C4×C59⋊C8—42no—
60C4×C59⋊C8C2×C411yescharacteristic
61C4×C59⋊C8C2×C411yescharacteristic
62C4×C59⋊C8D411yescharacteristic
63C4×C59⋊C8D411yescharacteristic
64C2×#41×C59—21no—
65C4⋊Q8×C59C2×C411yescharacteristic
66#127×C59C2×C411yescharacteristic
67#124×C59—21no—
68#121×C59—21no—
69#121×C59—21no—
70C4×D4×C59—21no—
71C4×D4×C59—21no—
72C2×C4⋊C4×C59—21no—
73C2×C42×C59—21no—
74C2×C42×C59C2311yescharacteristic
75C4⋊C8×C59—21no—
76C22⋊C8×C59—21no—
77C4×C8×C59—21no—
78#95×C59D412yes—
79#41×C59—41no—
80#41×C59—41no—
81#41×C59—41no—
82#41×C59—41no—
83C2×Q8×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—21no—
87C2×D4×C59—41no—
88C22×C4×C59—22no—
89C22×C4×C59—41no—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C2×C22⋊D59—41no—
94#11745685—41no—
95#11745685—41no—
96#11745685—41no—
97#11745685—41no—
98C2×D236—21no—
99C2×D236—21no—
100C2×D236—21no—
101C2×C4×D59—41no—
102C2×Dic118—21no—
103C2×C8×C59—41no—
104C4⋊C4×C59—24no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C22⋊C4×C59—41no—
108C22⋊C4×C59—41no—
109C42×C59—41no—
110C42×C59—41no—
111C42×C59C22×C411yescharacteristic
112C42×C59C2×D411yescharacteristic
113C42×C59C2×D411yescharacteristic
114C42×C59#4111yescharacteristic
115#11745672—42no—
116#11745672—42no—
117#11745671—42no—
118#11745671—42no—
119#11745670—42no—
120#11745670—42no—
121#11745669—42no—
122#11745669—42no—
123(C2×D59)⋊C4—41no—
124(C2×D59)⋊C4—41no—
125C4⋊Dic59—21no—
126C4⋊Dic59—21no—
127C2×C59⋊C8—21no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—41no—
131C2×C59⋊C8—41no—
132C2×C59⋊C8—41no—
133C2×C59⋊C8C22⋊C411yescharacteristic
134C2×C59⋊C8C22⋊C411yescharacteristic
135C2×C59⋊C8C22⋊C411yescharacteristic
136C2×C59⋊C8C22⋊C411yescharacteristic
137C2×C59⋊C8D812yes—
138C2×C59⋊C8SD1612yes—
139C23×C59—41no—
140C23×C59#12111yescharacteristic
141C22×D59—41no—
142Q8×C59—41no—
143Q8×C59—41no—
144D4×C59—41no—
145D4×C59—41no—
146D4×C59—42no—
147D4×C59—81no—
148D4×C59—81no—
149D4×C59—81no—
150C2×C4×C59—22no—
151C2×C4×C59—24no—
152C2×C4×C59—41no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—41no—
156C2×C4×C59—81no—
157C2×C4×C59—81no—
158C2×C4×C59C2×C22⋊C411yescharacteristic
159C2×C4×C59C4×D411yescharacteristic
160C2×C4×C59C4×D411yescharacteristic
161C2×C4×C59#12111yescharacteristic
162C2×C4×C59#12411yescharacteristic
163C2×C4×C59#12711yescharacteristic
164C2×C4×C59C2×D811yescharacteristic
165C2×C4×C59C2×SD1611yescharacteristic
166C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
168C22⋊D59—81no—
169C22⋊D59—81no—
170C2×Dic59—41no—
171D236—41no—
172D236—41no—
173D236—42no—
174D236—81no—
175C4×D59—81no—
176C4×D59—81no—
177Dic118—41no—
178Dic118—41no—
179C8×C59—81no—
180C59⋊C8—21no—
181C59⋊C8—21no—
182C59⋊C8—42no—
183C59⋊C8—81no—
184C59⋊C8#10212yes—
185C59⋊C8#10212yes—
186#9750—591nomaximal, Sylow
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59—81no—
192C22×C59#39011yescharacteristic
193C22×C59#45911yescharacteristic
194C22×C59#46011yescharacteristic
195C2×D59—81no—
196C2×D59—81no—
197C2×D59—81no—
198C4×C59—21no—
199C4×C59—21no—
200C4×C59—21no—
201C4×C59—44no—
202C4×C59—81no—
203C4×C59C2×#10211yescharacteristic
204C4×C59#41811yescharacteristic
205C4×C59#44312yes—
206C4×C59#45911yescharacteristic
207C4×C59#46011yescharacteristic
208Dic59—81no—
209#3147—591no—
210C8×C4⋊C4—591no—
211#1564—591no—
212#1502—591no—
213#1502D5911yescharacteristic, complemented
214#1480—591no—
215#1480—591no—
216C2×C59—21no—
217C2×C59—21no—
218C2×C59—161no—
219C2×C59#197411yescharacteristic
220C2×C59#197611yescharacteristic
221C2×C59#198211yescharacteristic
222D59—161no—
223C2×#135—1181no—
224C2×#135—1181no—
225#532—591no—
226#532C2×D5911yescharacteristic
227#495—1181no—
228#495—1181no—
229#494—591no—
230#493—591no—
231#492—591no—
232#492—591no—
233#431—591no—
234#431C2×D5911yescharacteristic
235C2×C4⋊C8—591no—
236C2×C4×C8—591no—
237C2×C4×C8—1181no—
238C4×C4⋊C4C2×D5911yescharacteristic
239#336—1181no—
240#335—1181no—
241#334—1181no—
242#327—22no—
243#327—1182no—
244D4⋊C8—1182no—
245D4⋊C8—1182no—
246C59#975011yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C2×#41—1181no—
249#135—2362no—
250#135—2362no—
251#135—2362no—
252#135—2362no—
253C2×Q16—1181no—
254C2×Q16—1181no—
255C2×SD16—1181no—
256C2×SD16—1181no—
257C2×SD16—1181no—
258C2×SD16—1181no—
259C2×D8—1181no—
260C2×D8—1181no—
261C22×C8—591no—
262C22×C8—591no—
263C22×C8—1181no—
264C4⋊Q8—591no—
265C4⋊Q8C4×D5911yescharacteristic
266#127—591no—
267#127C4×D5911yescharacteristic
268#124—21no—
269#124—1181no—
270#121—21no—
271#121—21no—
272#121—1181no—
273#121—1181no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—1181no—
277C4×D4—1181no—
278C2×C4⋊C4—21no—
279C2×C42—21no—
280C2×C42C22×D5911yescharacteristic
281C4⋊C8—21no—
282C4⋊C8—592no—
283C4⋊C8—1181no—
284C4⋊C8—1181no—
285C22⋊C8—21no—
286C22⋊C8—1181no—
287C4×C8—21no—
288C4×C8—591no—
289C4×C8—591no—
290C4×C8—591no—
291C4×C8—591no—
292C4×C8—1181no—
293C4×C8—2362no—
294#95C22⋊D5912yes—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—2361no—
300#41—2361no—
301#41—2361no—
302#41—2361no—
303C2×Q8—21no—
304C2×Q8—1181no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—1181no—
310C2×D4—1181no—
311C2×D4—1181no—
312C2×D4—2361no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—2361no—
316C22×C4D4×D5911yescharacteristic
317C22×C4#1174568711yescharacteristic
318C22×C4C2×C22⋊D5911yescharacteristic
319Q16—2362no—
320Q16—2362no—
321SD16—2362no—
322SD16—2362no—
323SD16—2362no—
324SD16—2362no—
325D8—2362no—
326D8—2362no—
327C2×C8—41no—
328C2×C8—591no—
329C2×C8—591no—
330C2×C8—592no—
331C2×C8—592no—
332C2×C8—591no—
333C2×C8—591no—
334C2×C8—1181no—
335C2×C8—1181no—
336C2×C8—1182no—
337C2×C8—2361no—
338C2×C8—2361no—
339C2×C8—2361no—
340C4⋊C4—24no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—1181no—
344C4⋊C4—1181no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—2361no—
348C22⋊C4—2361no—
349C42—41no—
350C42—41no—
351C42C2×C4×D5911yescharacteristic
352C42#1174568511yescharacteristic
353C42D4×D5911yescharacteristic
354C42D4×D5911yescharacteristic
355C23—41no—
356C23—2361no—
357C23#1427369911yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—2361no—
361Q8—2361no—
362D4—41no—
363D4—41no—
364D4—42no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—4721no—
369D4—4721no—
370D4—4721no—
371D4—2361no—
372D4—2361no—
373D4—2362no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—4721no—
383C2×C4—4721no—
384C2×C4—2361no—
385C2×C4C22⋊C4×D5911yescharacteristic
386C2×C4(C22⋊D59)⋊C411yescharacteristic
387C2×C4C4×C22⋊D5911yescharacteristic
388C2×C4#1427370011yescharacteristic
389C2×C4#1427371311yescharacteristic
390C2×C4#1427372311yescharacteristic
391C2×C4D8×D5911yescharacteristic
392C2×C4SD16×D5911yescharacteristic
393C2×C4#1427375111yescharacteristic
394C2×C4#1427375411yescharacteristic
395C8—81no—
396C8—592no—
397C8—592no—
398C8—1181no—
399C8—1181no—
400C8—4721no—
401C8—2362no—
402C22—21no—
403C22—21no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—4721no—
408C22—4721no—
409C22—4721no—
410C22#1683730211yescharacteristic
411C22#1683759511yescharacteristic
412C22#1683760011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—81no—
418C4—4721no—
419C4#102×D5911yescharacteristic
420C4#1683740511yescharacteristic
421C4#1683751612yes—
422C4#1683759611yescharacteristic
423C4#1683760111yescharacteristic
424C2—21no—
425C2—21no—
426C2—161no—
427C2—9441no—
428C2#2142498111yescharacteristic
429C2#2142499111yescharacteristic
430C2#2142502111yescharacteristic
subgroupG/H
centerC2×C4#14273700
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1502C2
socleC22×C59#390
prime radical#21434522C2
cosocleC2×C42×C59C23
hypercenter#1502D59
hypocenterC59#9750
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C59⋊C8C2semidirect
3. #21422842C2semidirect
4. #21422590C2semidirect
5. #21422496C2semidirect
6. #21422495C2semidirect
7. C59×#1502C2semidirect
13. #16837781C4semidirect
14. #16837778C4semidirect
15. #16837776C4semidirect
16. #16837774C4semidirect
213. #1502D59semidirect
246. C59#9750semidirect

Maximal subgroups

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

subgroupindex
#214345222
C4⋊C4×C59⋊C82
#214228422
#214225902
#214224962
#214224952
C59×#15022
#975059