Finite Group Databasebeta

← #28485547

Subgroups — #28485547

Order 15104.

580 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#21433576C211yesmaximal, characteristic
2#21433207C211yesmaximal, characteristic, complemented
3#21425023C211yesmaximal, characteristic, complemented
4#21422765C211yesmaximal, characteristic, complemented
5#21421999C211yesmaximal, characteristic, complemented
6#21421970C211yesmaximal, characteristic, complemented
7C59×#1806C211yesmaximal, characteristic, complemented
8C2×#14273752—21no—
9C2×#14273724C2211yescharacteristic
10#16837602—21no—
11#16837602C412yescomplemented
12#16837597—21no—
13#16837597C412yescomplemented
14Q8×C59⋊C8—21no—
15D4×C59⋊C8—22no—
16D4×C59⋊C8—21no—
17#16837455—21no—
18#16837455C2211yescharacteristic
19#16837426C2211yescharacteristic
20C2×C4⋊(C59⋊C8)—21no—
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22#16837406—21no—
23C2×#14273647—22no—
24#16837145—21no—
25#16837145—21no—
26C4×#41×C59C2211yescharacteristic
27C59×#423C2211yescharacteristic
28C2×#104×C59—21no—
29C4×C4⋊C4×C59C2211yescharacteristic
30C59×#339—21no—
31C22×D236—21no—
32C22×D236—21no—
33C22×D236—21no—
34#14273752—42no—
35#14273752—42no—
36C2×#11745671—22no—
37C2×#11745669—22no—
38C2×#11745664—21no—
39C22×C59⋊C8—22no—
40C22×C59⋊C8D412yes—
41#14273724—21no—
42#14273724—21no—
43#14273724—21no—
44#14273724C2×C412yes—
45C4⋊(C59⋊C8)—22no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)C2×C412yes—
50C4⋊(C59⋊C8)D411yescharacteristic
51C4⋊(C59⋊C8)D411yescharacteristic
52#14273647—22no—
53#14273647—22no—
54#14273647—44no—
55C22⋊(C59⋊C8)—22no—
56C22⋊(C59⋊C8)—21no—
57C4×C59⋊C8—22no—
58C4×C59⋊C8—21no—
59C2×#41×C59D411yescharacteristic
60C2×M16×C59—21no—
61C4×Q8×C59C2×C411yescharacteristic
62C4×Q8×C59D411yescharacteristic
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59C2×C411yescharacteristic
66C4×D4×C59D411yescharacteristic
67#117×C59—21no—
68#117×C59D411yescharacteristic
69C2×C4⋊C4×C59—21no—
70C2×C42×C59—21no—
71C2×C42×C59—21no—
72C2×C42×C59C2311yescharacteristic
73C4⋊C8×C59—21no—
74C4⋊C8×C59—21no—
75#104×C59—42no—
76#95×C59—21no—
77C23×D59—41no—
78#41×C59—21no—
79#41×C59—21no—
80#41×C59D812yes—
81#41×C59SD1612yes—
82C2×Q8×C59C22⋊C411yescharacteristic
83C2×D4×C59—21no—
84C2×D4×C59C22⋊C411yescharacteristic
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—41no—
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59#4111yescharacteristic
92C2×D236—22no—
93C2×D236—22no—
94C2×D236—22no—
95C2×D236—42no—
96C2×D236—42no—
97C2×D236—41no—
98C2×D236—41no—
99C2×D236—41no—
100C2×D236—41no—
101C2×D236—41no—
102C2×D236—41no—
103C2×D236—41no—
104C2×D236—82no—
105M16×C59—41no—
106M16×C59—41no—
107C2×C8×C59—41no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—22no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59C22⋊C411yescharacteristic
113C4⋊C4×C59C22⋊C411yescharacteristic
114C22⋊C4×C59—21no—
115C22⋊C4×C59—21no—
116C22⋊C4×C59—21no—
117C42×C59—21no—
118C42×C59—21no—
119C42×C59—21no—
120C42×C59—41no—
121C42×C59—41no—
122C42×C59C22×C411yescharacteristic
123C42×C59C2×D411yescharacteristic
124C42×C59C2×D411yescharacteristic
125C42×C59#4111yescharacteristic
126#11745671—44no—
127#11745669—44no—
128#11745664—41no—
129#11745664—41no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8—42no—
135C2×C59⋊C8—82no—
136C23×C59—21no—
137C23×C59#12111yescharacteristic
138C22×D59—42no—
139C22×D59—82no—
140C22×D59—82no—
141C22×D59—81no—
142C22×D59—81no—
143Q8×C59—21no—
144Q8×C59#10212yescomplemented
145D4×C59—22no—
146D4×C59—22no—
147D4×C59—21no—
148D4×C59#10212yescomplemented
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—22no—
153C2×C4×C59—22no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—42no—
160C2×C4×C59—81no—
161C2×C4×C59—81no—
162C2×C4×C59—81no—
163C2×C4×C59C2×C22⋊C411yescharacteristic
164C2×C4×C59C4×D411yescharacteristic
165C2×C4×C59C4×D411yescharacteristic
166C2×C4×C59#12011yescharacteristic
167C2×C4×C59#12111yescharacteristic
168C2×C4×C59#12311yescharacteristic
169C2×C4×C59C2×D811yescharacteristic
170C2×C4×C59C2×SD1611yescharacteristic
171C2×C4×C59#13511yescharacteristic
172C2×C4×C59#13511yescharacteristic
173D236—44no—
174D236—42no—
175D236—42no—
176D236—82no—
177D236—82no—
178D236—81no—
179D236—81no—
180D236—81no—
181D236—81no—
182C8×C59—81no—
183C59⋊C8—42no—
184C59⋊C8—42no—
185C59⋊C8—41no—
186C59⋊C8—41no—
187#9318—591nomaximal, Sylow
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—22no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59#38611yescharacteristic
194C22×C59#46511yescharacteristic
195C22×C59#46611yescharacteristic
196C2×D59—82no—
197C2×D59—82no—
198C2×D59—82no—
199C2×D59—162no—
200C2×D59—162no—
201C4×C59—21no—
202C4×C59—22no—
203C4×C59—21no—
204C4×C59—21no—
205C4×C59—41no—
206C4×C59—41no—
207C4×C59—82no—
208C4×C59C2×#10211yescharacteristic
209C4×C59#41611yescharacteristic
210C4×C59#45311yescharacteristic
211C4×C59#45411yescharacteristic
212C4×C59#46511yescharacteristic
213C4×C59#46611yescharacteristic
214#3038—591no—
215#2988—591no—
216#1982—591no—
217#1806D5911yescharacteristic, complemented
218#1545—591no—
219#1303—591no—
220#1292—591no—
221C2×C59—21no—
222C2×C59—21no—
223C2×C59—42no—
224C2×C59#188311yescharacteristic
225C2×C59#189311yescharacteristic
226C2×C59#190311yescharacteristic
227D59—162no—
228C2×AGL(1,ℤ/8)—1181no—
229C2×#127—591no—
230C4×#41C2×D5911yescharacteristic
231#460—592no—
232#460—1181no—
233#459—592no—
234#459—1181no—
235C8×Q8—1181no—
236C8×D4—1182no—
237C8×D4—1181no—
238#431—591no—
239#431—1181no—
240#423—591no—
241#423C2×D5911yescharacteristic
242C2×C4⋊C8—591no—
243C2×C4⋊C8—1181no—
244C2×#104—21no—
245#418—1181no—
246C2×#102—1182no—
247C4×C4⋊C4C2×D5911yescharacteristic
248#339—21no—
249#331—1181no—
250#331—1181no—
251C59#931811yescharacteristic, complemented, Sylow
252C2×#41C22⋊D5911yescharacteristic
253C22×D4—1181no—
254C22×D4—1181no—
255C22×D4—1181no—
256AGL(1,ℤ/8)—2362no—
257AGL(1,ℤ/8)—2362no—
258C2×SD16—1182no—
259C2×D8—1182no—
260C2×M16—21no—
261C2×M16—1181no—
262C22×C8—592no—
263C22×C8—1182no—
264#127—592no—
265#127—1181no—
266#127—1181no—
267#127—1181no—
268C4×Q8C4×D5911yescharacteristic
269C4×Q8D23611yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D5911yescharacteristic
273C4×D4D23611yescharacteristic
274#117—21no—
275#117C22⋊D5911yescharacteristic
276C2×C4⋊C4—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C22×D5911yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—592no—
283C4⋊C8—591no—
284C4⋊C8—591no—
285C4⋊C8—1182no—
286C4⋊C8—1181no—
287C4⋊C8—1181no—
288C4⋊C8—1181no—
289#104—42no—
290#102—1182no—
291#102—1182no—
292#102—2364no—
293C22⋊C8—1182no—
294C22⋊C8—1181no—
295C4×C8—1182no—
296C4×C8—1181no—
297#95—21no—
298C24—2361no—
299#41—21no—
300#41—21no—
301#41#1174566912yes—
302#41#1174567112yes—
303C2×Q8(C2×D59)⋊C411yescharacteristic
304C2×D4—21no—
305C2×D4—1182no—
306C2×D4—1182no—
307C2×D4—1182no—
308C2×D4—4722no—
309C2×D4—2362no—
310C2×D4—2362no—
311C2×D4—2361no—
312C2×D4—2361no—
313C2×D4—2361no—
314C2×D4—2361no—
315C2×D4—2361no—
316C2×D4—2361no—
317C2×D4—2361no—
318C2×D4(C2×D59)⋊C411yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4D4×D5911yescharacteristic
324C22×C4#1174568711yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326SD16—2364no—
327D8—2364no—
328M16—41no—
329M16—41no—
330M16—2361no—
331M16—2361no—
332C2×C8—41no—
333C2×C8—1182no—
334C2×C8—1181no—
335C2×C8—1181no—
336C2×C8—1182no—
337C2×C8—4722no—
338C2×C8—2362no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—41no—
343C4⋊C4(C2×D59)⋊C411yescharacteristic
344C4⋊C4(C2×D59)⋊C411yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—41no—
352C42—41no—
353C42C2×C4×D5911yescharacteristic
354C42C2×D23611yescharacteristic
355C42D4×D5911yescharacteristic
356C42#1174568711yescharacteristic
357C23—21no—
358C23—4722no—
359C23—4722no—
360C23—4721no—
361C23—4721no—
362C23—2362no—
363C23#1427369911yescharacteristic
364Q8—21no—
365Q8#1427364712yescomplemented
366D4—22no—
367D4—22no—
368D4—21no—
369D4—4722no—
370D4—4722no—
371D4—4721no—
372D4—4721no—
373D4—4721no—
374D4—4721no—
375D4—2364no—
376D4—2362no—
377D4—2362no—
378D4#1427364712yescomplemented
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4C2×(C2×D59)⋊C411yescharacteristic
394C2×C4(C22⋊D59)⋊C411yescharacteristic
395C2×C4(C22⋊D59)⋊C411yescharacteristic
396C2×C4#1427369711yescharacteristic
397C2×C4#1427369911yescharacteristic
398C2×C4#1427371111yescharacteristic
399C2×C4C2×#1174566911yescharacteristic
400C2×C4C2×#1174567111yescharacteristic
401C2×C4#1427374711yescharacteristic
402C2×C4#1427374711yescharacteristic
403C8—81no—
404C8—2362no—
405C8—2361no—
406C8—2362no—
407C8—2361no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—9442no—
414C22—9442no—
415C22—4722no—
416C22—4722no—
417C22—4722no—
418C22#1683727911yescharacteristic
419C22#1683763311yescharacteristic
420C22#1683763611yescharacteristic
421C4—21no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4C2×#1427364711yescharacteristic
429C4#1683739611yescharacteristic
430C4#1683757111yescharacteristic
431C4#1683757611yescharacteristic
432C4#1683763311yescharacteristic
433C4#1683763611yescharacteristic
434C2—21no—
435C2—21no—
436C2—42no—
437C2—9442no—
438C2#2142447011yescharacteristic
439C2#2142452211yescharacteristic
440C2#2142457411yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1806C2
socleC22×C59#386
prime radical#21433576C2
cosocleC2×C42×C59C23
hypercenter#1806D59
hypocenterC59#9318
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #21433207C2semidirect
3. #21425023C2semidirect
4. #21422765C2semidirect
5. #21421999C2semidirect
6. #21421970C2semidirect
7. C59×#1806C2semidirect
11. #16837602C4semidirect
13. #16837597C4semidirect
144. Q8×C59#102semidirect
148. D4×C59#102semidirect
217. #1806D59semidirect
251. C59#9318semidirect
365. Q8#14273647semidirect
378. D4#14273647semidirect

Maximal subgroups

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

subgroupindex
#214335762
#214332072
#214250232
#214227652
#214219992
#214219702
C59×#18062
#931859