Finite Group Databasebeta

← #28484526

Subgroups — #28484526

Order 15104.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434195C211yesmaximal, characteristic
2#21423921C211yesmaximal, characteristic
3#21422712C211yesmaximal, characteristic, complemented
4#21422596C211yesmaximal, characteristic, complemented
5#21421996C211yesmaximal, characteristic, complemented
6#21421951C211yesmaximal, characteristic, complemented
7C59×#2974C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837737—21no—
10#16837682C411yescharacteristic, complemented
11#16837682C411yescharacteristic, complemented
12#16837607C411yescharacteristic, complemented
13#16837607C411yescharacteristic, complemented
14#16837553—21no—
15#16837502—21no—
16#16837502—21no—
17#16837459C2211yescharacteristic
18C2×#14273648—21no—
19C2×#14273638C2211yescharacteristic
20C4×C4⋊Dic59C2211yescharacteristic
21#16837164—21no—
22#16837143—22no—
23#16837140—21no—
24#16837140—21no—
25C2×C4×D4×C59C2211yescharacteristic
26C59×#436—21no—
27C59×#435—22no—
28C8×D4×C59—21no—
29C59×#432C2211yescharacteristic
30C59×#431C2211yescharacteristic
31C59×#407—21no—
32C2×C22⋊C8×C59—21no—
33C2×#11745670—21no—
34C2×#11745670—21no—
35C22×C59⋊C8—21no—
36#14273727C2×C411yescharacteristic
37#14273727C2×C411yescharacteristic
38#14273719—21no—
39#14273706—21no—
40#14273706—21no—
41C4×Dic118—21no—
42C4×Dic118—21no—
43#14273686—21no—
44C2×C4⋊Dic59—21no—
45C2×C4×Dic59—21no—
46#14273660—21no—
47#14273660—21no—
48C4⋊(C59⋊C8)C2×C411yescharacteristic
49C4⋊(C59⋊C8)C2×C411yescharacteristic
50#14273650—21no—
51#14273650—21no—
52#14273648—24no—
53#14273648—21no—
54#14273648—21no—
55C22⋊(C59⋊C8)—21no—
56#14273638—21no—
57#14273638—21no—
58#14273638—21no—
59C59⋊#95—21no—
60C22×D4×C59D411yescharacteristic
61C23×C4×C59—21no—
62C2×M16×C59—22no—
63C22×C8×C59—22no—
64C4×D4×C59—21no—
65C4×D4×C59—21no—
66C4×D4×C59C2×C411yescharacteristic
67C4×D4×C59C2×C411yescharacteristic
68C4×D4×C59D411yescharacteristic
69C4×D4×C59D411yescharacteristic
70C2×C4⋊C4×C59D411yescharacteristic
71C2×C22⋊C4×C59—21no—
72C2×C42×C59C2311yescharacteristic
73C4⋊C8×C59—21no—
74C4⋊C8×C59—21no—
75C22⋊C8×C59—22no—
76C22⋊C8×C59—22no—
77C22⋊C8×C59—21no—
78C22⋊C8×C59D411yescharacteristic
79C22⋊C8×C59D411yescharacteristic
80#97×C59D412yes—
81C4×C8×C59—21no—
82C24×C59—21no—
83C2×D4×C59—22no—
84C2×D4×C59—21no—
85C2×D4×C59—41no—
86C2×D4×C59C22⋊C411yescharacteristic
87C2×D4×C59C22⋊C411yescharacteristic
88C22×C4×C59—21no—
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—82no—
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59#4111yescharacteristic
97C22×Dic59—41no—
98C2×Dic118—21no—
99C2×Dic118—21no—
100M16×C59—82no—
101C2×C8×C59—21no—
102C2×C8×C59—21no—
103C2×C8×C59—22no—
104C2×C8×C59—82no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59C22⋊C411yescharacteristic
107C4⋊C4×C59C22⋊C411yescharacteristic
108C22⋊C4×C59—21no—
109C22⋊C4×C59—21no—
110C22⋊C4×C59—21no—
111C22⋊C4×C59—21no—
112C42×C59C22×C411yescharacteristic
113C42×C59C2×D411yescharacteristic
114C42×C59C2×D411yescharacteristic
115C42×C59#4111yescharacteristic
116C22⋊Dic59—41no—
117#11745670—42no—
118#11745670—42no—
119C4⋊Dic59—22no—
120C4⋊Dic59—21no—
121C4⋊Dic59—21no—
122C4⋊Dic59—21no—
123C4⋊Dic59—21no—
124C4⋊Dic59—41no—
125Dic59⋊C4—41no—
126Dic59⋊C4—41no—
127C4×Dic59—41no—
128C4×Dic59—81no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—41no—
133C2×C59⋊C8—41no—
134C23×C59—21no—
135C23×C59—21no—
136C23×C59—42no—
137C23×C59—41no—
138C23×C59—41no—
139C23×C59—81no—
140C23×C59#12111yescharacteristic
141D4×C59—21no—
142D4×C59—21no—
143D4×C59—21no—
144D4×C59—42no—
145D4×C59#10412yes—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—41no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—81no—
156C2×C4×C59—81no—
157C2×C4×C59—81no—
158C2×C4×C59C2×C22⋊C411yescharacteristic
159C2×C4×C59C4×D411yescharacteristic
160C2×C4×C59C4×D411yescharacteristic
161C2×C4×C59#12011yescharacteristic
162C2×C4×C59#12111yescharacteristic
163C2×C4×C59#12311yescharacteristic
164C2×Dic59—42no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—81no—
168C2×Dic59—81no—
169Dic118—41no—
170Dic118—41no—
171Dic118—81no—
172C8×C59—41no—
173C8×C59—42no—
174C8×C59—41no—
175C59⋊C8—42no—
176C59⋊C8—41no—
177C59⋊C8—41no—
178#9165—591nomaximal, Sylow
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—22no—
182C22×C59—41no—
183C22×C59—41no—
184C22×C59—41no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—82no—
188C22×C59—82no—
189C22×C59#38611yescharacteristic
190C22×C59#47111yescharacteristic
191C22×C59#47311yescharacteristic
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59C2×#10411yescharacteristic
200C4×C59#42111yescharacteristic
201Dic59—82no—
202Dic59—81no—
203Dic59—81no—
204#3110—591no—
205#2974D5911yescharacteristic
206#1762—591no—
207#1531—591no—
208#1503—591no—
209#1302—591no—
210#1285—591no—
211C2×C59—21no—
212C2×C59—21no—
213C2×C59—42no—
214C2×C59—81no—
215C2×C59#187111yescharacteristic
216C2×C59#190411yescharacteristic
217C2×C59#190811yescharacteristic
218#533—591no—
219C2×C4×D4C2×D5911yescharacteristic
220#484—1181no—
221#474—591no—
222#474—591no—
223#461—591no—
224#461—591no—
225#450—1181no—
226#440—1181no—
227#440—1181no—
228#436—21no—
229#435—22no—
230C8×D4—21no—
231#432—591no—
232#432C2×D5911yescharacteristic
233#431C2×D5911yescharacteristic
234C2×#102—1181no—
235#407—21no—
236C2×C22⋊C8—21no—
237C2×#97—591no—
238C4×C4⋊C4—591no—
239#340—1181no—
240#330—1182no—
241#328—1181no—
242#328—1181no—
243C59#916511yescharacteristic, complemented, Sylow
244C22×D4D23611yescharacteristic
245C23×C4—21no—
246C2×SD16—1181no—
247C2×SD16—1181no—
248C2×M16—22no—
249C22×C8—22no—
250C22×C8—1181no—
251C4⋊Q8—591no—
252C4⋊Q8—591no—
253#125—1181no—
254C22⋊Q8—1181no—
255C22⋊Q8—1181no—
256C4×Q8—1181no—
257C4×Q8—1181no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C4×D5911yescharacteristic
261C4×D4C4×D5911yescharacteristic
262C4×D4C22⋊D5911yescharacteristic
263C4×D4C22⋊D5911yescharacteristic
264#117—1181no—
265C2×C4⋊C4—1181no—
266C2×C4⋊C4D23611yescharacteristic
267C2×C22⋊C4—21no—
268C2×C42—1181no—
269C2×C42C22×D5911yescharacteristic
270#107—1181no—
271#107—1181no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—591no—
275C4⋊C8—591no—
276#103—1181no—
277#103—1181no—
278#102—1184no—
279#102—1181no—
280#102—1181no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—21no—
284C22⋊C8—1181no—
285C22⋊C8C22⋊D5911yescharacteristic
286C22⋊C8C22⋊D5911yescharacteristic
287#97—1181no—
288#97—1181no—
289#97—1181no—
290#97C22⋊D5912yes—
291C4×C8—21no—
292#95—1181no—
293C24—21no—
294C2×Q8—1181no—
295C2×Q8—1181no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4(C2×D59)⋊C411yescharacteristic
300C2×D4(C2×D59)⋊C411yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—82no—
307C22×C4—2361no—
308C22×C4C2×D23611yescharacteristic
309C22×C4#1174568511yescharacteristic
310C22×C4C2×C22⋊D5911yescharacteristic
311SD16—2362no—
312SD16—2362no—
313M16—82no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—82no—
318C2×C8—1182no—
319C2×C8—1181no—
320C2×C8—1181no—
321C2×C8—2361no—
322C2×C8—2361no—
323C4⋊C4—21no—
324C4⋊C4—1182no—
325C4⋊C4—1181no—
326C4⋊C4—1181no—
327C4⋊C4—1181no—
328C4⋊C4—1181no—
329C4⋊C4—2361no—
330C4⋊C4—2361no—
331C4⋊C4—2361no—
332C4⋊C4(C2×D59)⋊C411yescharacteristic
333C4⋊C4(C2×D59)⋊C411yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—2361no—
339C42—4721no—
340C42—2361no—
341C42C2×C4×D5911yescharacteristic
342C42#1174568511yescharacteristic
343C42C2×C22⋊D5911yescharacteristic
344C42C2×C22⋊D5911yescharacteristic
345C23—21no—
346C23—21no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—81no—
351C23#1427370111yescharacteristic
352Q8—4721no—
353Q8—2361no—
354Q8—2361no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4#1427365312yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—4721no—
373C2×C4—4721no—
374C2×C4—2362no—
375C2×C4—2361no—
376C2×C4—2361no—
377C2×C4C2×(C2×D59)⋊C411yescharacteristic
378C2×C4C4×C22⋊D5911yescharacteristic
379C2×C4C4×C22⋊D5911yescharacteristic
380C2×C4#1427369611yescharacteristic
381C2×C4#1427370111yescharacteristic
382C2×C4#1427371011yescharacteristic
383C8—41no—
384C8—42no—
385C8—41no—
386C8—2362no—
387C8—2361no—
388C8—2361no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—82no—
398C22—82no—
399C22#1683728111yescharacteristic
400C22#1683766511yescharacteristic
401C22#1683767711yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—4722no—
410C4—4721no—
411C4—4721no—
412C4C2×#1427365311yescharacteristic
413C4#1683741811yescharacteristic
414C2—21no—
415C2—21no—
416C2—42no—
417C2—81no—
418C2#2142439811yescharacteristic
419C2#2142458311yescharacteristic
420C2#2142461111yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2974C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#2974D59
hypocenterC59#9165
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
3. #21422712C2semidirect
4. #21422596C2semidirect
5. #21421996C2semidirect
6. #21421951C2semidirect
10. #16837682C4semidirect
11. #16837682C4semidirect
12. #16837607C4semidirect
13. #16837607C4semidirect
243. C59#9165semidirect

Maximal subgroups

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

subgroupindex
#214341952
#214239212
#214227122
#214225962
#214219962
#214219512
C59×#29742
#916559