Finite Group Databasebeta

← #28484514

Subgroups — #28484514

Order 15104.

508 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433131C211yesmaximal, characteristic, complemented
2#21423921C211yesmaximal, characteristic, complemented
3#21422707C211yesmaximal, characteristic, complemented
4#21422591C211yesmaximal, characteristic, complemented
5#21421994C211yesmaximal, characteristic, complemented
6#21421953C211yesmaximal, characteristic, complemented
7C59×#3109C211yesmaximal, characteristic
8#16837480—21no—
9#16837474—22no—
10D4×C59⋊C8—21no—
11#16837459C2211yescharacteristic
12#16837455C2211yescharacteristic
13C2×#14273648—21no—
14#16837366—21no—
15C2×C22⋊(C59⋊C8)—21no—
16C2×#14273638C2211yescharacteristic
17C4×C4⋊Dic59C2211yescharacteristic
18#16837164—21no—
19#16837142—22no—
20#16837136—21no—
21#16837136—21no—
22C59×#532C2211yescharacteristic
23C2×C4×D4×C59C2211yescharacteristic
24C59×#481—21no—
25C59×#476C411yescharacteristic, complemented
26C59×#476C411yescharacteristic, complemented
27C59×#459C411yescharacteristic, complemented
28C59×#459C411yescharacteristic, complemented
29C59×#447—21no—
30C59×#442—21no—
31C59×#442—21no—
32C59×#432C2211yescharacteristic
33C2×#11745664—22no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—22no—
36C2×C4⋊Dic59—21no—
37C2×C4×Dic59—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40#14273648—24no—
41C22⋊(C59⋊C8)—22no—
42C22⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)—21no—
44C22⋊(C59⋊C8)D411yescharacteristic
45C22⋊(C59⋊C8)D411yescharacteristic
46#14273638—21no—
47#14273638—21no—
48#14273638D412yes—
49C4×C59⋊C8—21no—
50C59⋊#95—21no—
51C2×#41×C59—21no—
52C22×D4×C59D411yescharacteristic
53C23×C4×C59—21no—
54C2×D8×C59—21no—
55C2×D8×C59—21no—
56C4⋊Q8×C59C2×C411yescharacteristic
57#127×C59C2×C411yescharacteristic
58#124×C59—21no—
59#121×C59—21no—
60#121×C59—21no—
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59C2×C411yescharacteristic
66C4×D4×C59C2×C411yescharacteristic
67C4×D4×C59D411yescharacteristic
68C4×D4×C59D411yescharacteristic
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C22⋊C4×C59—21no—
71C2×C42×C59C2311yescharacteristic
72#106×C59—21no—
73#106×C59—21no—
74C4⋊C8×C59C2×C411yescharacteristic
75C4⋊C8×C59C2×C411yescharacteristic
76#102×C59—21no—
77#102×C59—21no—
78#102×C59—21no—
79#102×C59—21no—
80C22⋊C8×C59—21no—
81#97×C59—21no—
82C24×C59—21no—
83#41×C59—81no—
84#41×C59—81no—
85C2×Q8×C59—21no—
86C2×D4×C59—22no—
87C2×D4×C59—21no—
88C2×D4×C59—21no—
89C2×D4×C59—21no—
90C2×D4×C59—21no—
91C2×D4×C59—41no—
92C2×D4×C59—41no—
93C2×D4×C59C22⋊C411yescharacteristic
94C2×D4×C59C22⋊C411yescharacteristic
95C22×C4×C59—21no—
96C22×C4×C59—21no—
97C22×C4×C59—21no—
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—41no—
101C22×C4×C59—82no—
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59C2×D411yescharacteristic
104C22×C4×C59#4111yescharacteristic
105C22×Dic59—41no—
106D8×C59—42no—
107D8×C59—42no—
108C2×C8×C59—21no—
109C2×C8×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59C22⋊C411yescharacteristic
114C4⋊C4×C59C22⋊C411yescharacteristic
115C22⋊C4×C59—21no—
116C22⋊C4×C59—21no—
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—41no—
120C22⋊C4×C59—41no—
121C42×C59C22×C411yescharacteristic
122C42×C59C2×D411yescharacteristic
123C42×C59C2×D411yescharacteristic
124C42×C59#4111yescharacteristic
125C4⋊Dic59—22no—
126C4⋊Dic59—41no—
127C4×Dic59—81no—
128#11745664—82no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8—82no—
136C23×C59—21no—
137C23×C59—21no—
138C23×C59—42no—
139C23×C59—41no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—81no—
143C23×C59#12111yescharacteristic
144Q8×C59—81no—
145D4×C59—21no—
146D4×C59—21no—
147D4×C59—21no—
148D4×C59—42no—
149D4×C59—41no—
150D4×C59—41no—
151D4×C59—81no—
152D4×C59—81no—
153D4×C59—81no—
154D4×C59—81no—
155D4×C59#10412yes—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—41no—
163C2×C4×C59—42no—
164C2×C4×C59—42no—
165C2×C4×C59—41no—
166C2×C4×C59—81no—
167C2×C4×C59—81no—
168C2×C4×C59—82no—
169C2×C4×C59—81no—
170C2×C4×C59C2×C22⋊C411yescharacteristic
171C2×C4×C59C4×D411yescharacteristic
172C2×C4×C59C4×D411yescharacteristic
173C2×C4×C59#12011yescharacteristic
174C2×C4×C59#12111yescharacteristic
175C2×C4×C59#12311yescharacteristic
176C2×Dic59—42no—
177C2×Dic59—81no—
178C2×Dic59—81no—
179C8×C59—41no—
180C8×C59—41no—
181C59⋊C8—42no—
182C59⋊C8—41no—
183C59⋊C8—42no—
184C59⋊C8—41no—
185#9163—591nomaximal, Sylow
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—22no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59—41no—
194C22×C59—82no—
195C22×C59—82no—
196C22×C59—81no—
197C22×C59—81no—
198C22×C59—81no—
199C22×C59#38611yescharacteristic
200C22×C59#47111yescharacteristic
201C22×C59#47211yescharacteristic
202C4×C59—21no—
203C4×C59—21no—
204C4×C59—21no—
205C4×C59—21no—
206C4×C59—21no—
207C4×C59—41no—
208C4×C59—41no—
209C4×C59—81no—
210C4×C59C2×#10411yescharacteristic
211C4×C59#42111yescharacteristic
212Dic59—82no—
213#3109D5911yescharacteristic
214#2974—591no—
215#1762—591no—
216#1530—591no—
217#1502—591no—
218#1302—591no—
219#1285—591no—
220C2×C59—21no—
221C2×C59—21no—
222C2×C59—42no—
223C2×C59—81no—
224C2×C59—161no—
225C2×C59#187111yescharacteristic
226C2×C59#190411yescharacteristic
227C2×C59#190711yescharacteristic
228#532C2×D5911yescharacteristic
229C2×C4×D4C2×D5911yescharacteristic
230#481—21no—
231#476Dic5911yescharacteristic, complemented
232#476Dic5911yescharacteristic, complemented
233#459Dic5911yescharacteristic, complemented
234#459Dic5911yescharacteristic, complemented
235#447—21no—
236#442—21no—
237#442—21no—
238#436—1181no—
239#435—1182no—
240C8×D4—1181no—
241#432—591no—
242#432C2×D5911yescharacteristic
243#431—591no—
244C2×#102—1181no—
245#407—1181no—
246C2×C22⋊C8—1181no—
247C2×#97—591no—
248C4×C4⋊C4—591no—
249#340—1181no—
250#329—1182no—
251#327—1181no—
252#327—1181no—
253C59#916311yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C22×D4C22⋊D5911yescharacteristic
256C23×C4—21no—
257C2×D8—21no—
258C2×D8—21no—
259C2×M16—1182no—
260C22×C8—1182no—
261C22×C8—1181no—
262C4⋊Q8C2×Dic5911yescharacteristic
263#127C2×Dic5911yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C2×Dic5911yescharacteristic
272C4×D4C2×Dic5911yescharacteristic
273C4×D4C22⋊D5911yescharacteristic
274C4×D4C22⋊D5911yescharacteristic
275C2×C4⋊C4—1181no—
276C2×C4⋊C4C22⋊D5911yescharacteristic
277C2×C22⋊C4—21no—
278C2×C42—1181no—
279C2×C42C22×D5911yescharacteristic
280#106—21no—
281#106—21no—
282C4⋊C8—1181no—
283C4⋊C8—1181no—
284C4⋊C8C2×Dic5911yescharacteristic
285C4⋊C8C2×Dic5911yescharacteristic
286#102—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—1184no—
291C22⋊C8—21no—
292C22⋊C8—591no—
293C22⋊C8—591no—
294C22⋊C8—1182no—
295C22⋊C8—1182no—
296C22⋊C8—1181no—
297#97—21no—
298#97—592no—
299#97—1181no—
300#97—1181no—
301C4×C8—1181no—
302#95—1181no—
303C24—21no—
304#41—81no—
305#41—81no—
306C2×Q8—21no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4C22⋊Dic5911yescharacteristic
315C2×D4C22⋊Dic5911yescharacteristic
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—82no—
323C22×C4—2361no—
324C22×C4D4×D5911yescharacteristic
325C22×C4#1174568711yescharacteristic
326C22×C4C2×C22⋊D5911yescharacteristic
327D8—42no—
328D8—42no—
329M16—4722no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—1182no—
333C2×C8—1181no—
334C2×C8—1181no—
335C2×C8—1182no—
336C2×C8—4722no—
337C2×C8—2361no—
338C2×C8—2361no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—1182no—
343C4⋊C4—2361no—
344C4⋊C4C22⋊Dic5911yescharacteristic
345C4⋊C4C22⋊Dic5911yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C42—4721no—
353C42D4×D5911yescharacteristic
354C42#1174568711yescharacteristic
355C42C22×Dic5911yescharacteristic
356C42C2×C22⋊D5911yescharacteristic
357C23—21no—
358C23—21no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—81no—
364C23#1427369911yescharacteristic
365Q8—81no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—41no—
371D4—41no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4#1427365412yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—4721no—
392C2×C4—4721no—
393C2×C4—2362no—
394C2×C4C2×C22⋊Dic5911yescharacteristic
395C2×C4D4×Dic5911yescharacteristic
396C2×C4D4×Dic5911yescharacteristic
397C2×C4#1427369511yescharacteristic
398C2×C4#1427369911yescharacteristic
399C2×C4#1427370811yescharacteristic
400C8—41no—
401C8—41no—
402C8—2361no—
403C8—2362no—
404C8—2362no—
405C8—2361no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—82no—
415C22—82no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22#1683728211yescharacteristic
420C22#1683766311yescharacteristic
421C22#1683767011yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—81no—
430C4—4722no—
431C4C2×#1427365411yescharacteristic
432C4#1683741911yescharacteristic
433C2—21no—
434C2—21no—
435C2—42no—
436C2—81no—
437C2—161no—
438C2#2142439911yescharacteristic
439C2#2142458511yescharacteristic
440C2#2142460611yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3109C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3109D59
hypocenterC59#9163
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #21433131C2semidirect
2. #21423921C2semidirect
3. #21422707C2semidirect
4. #21422591C2semidirect
5. #21421994C2semidirect
6. #21421953C2semidirect
25. C59×#476C4semidirect
26. C59×#476C4semidirect
27. C59×#459C4semidirect
28. C59×#459C4semidirect
231. #476Dic59semidirect
232. #476Dic59semidirect
233. #459Dic59semidirect
234. #459Dic59semidirect
253. C59#9163semidirect

Maximal subgroups

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

subgroupindex
#214331312
#214239212
#214227072
#214225912
#214219942
#214219532
C59×#31092
#916359