Finite Group Databasebeta

← #28492068

Subgroups — #28492068

Order 15104.

572 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429095C211yesmaximal, characteristic
2#21424901C211yesmaximal, characteristic
3#21424852C211yesmaximal, characteristic, complemented
4C2×#16837145C211yesmaximal, characteristic
5C2×Q8⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
6#21421935C211yesmaximal, characteristic, complemented
7C59×#1803C211yesmaximal, characteristic, complemented
8C2×#14273724C2211yescharacteristic
9C2×#14273716—21no—
10C2×#14273660—21no—
11C2×C4⋊(C59⋊C8)C2211yescharacteristic
12C2×#14273647—22no—
13C2×C4×C59⋊C8C2211yescharacteristic
14#16837319—22no—
15#16837304—21no—
16#16837304—21no—
17#16837303—21no—
18C4×C4⋊Dic59C2211yescharacteristic
19#16837164—21no—
20#16837145—22no—
21Q8⋊(C59⋊C8)—22no—
22C2×C4×Q8×C59C2211yescharacteristic
23C2×C4⋊C8×C59C2211yescharacteristic
24C2×#103×C59—21no—
25C59×#384C2211yescharacteristic
26C59×#340—21no—
27C22×D236—21no—
28C22×D236—21no—
29C22×D236D412yes—
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32#14273724—21no—
33#14273724—21no—
34#14273724—21no—
35#14273724—21no—
36#14273716—42no—
37#14273716—42no—
38C2×C4⋊Dic59D411yescharacteristic
39C2×C4⋊Dic59D411yescharacteristic
40C2×(C2×D59)⋊C4—22no—
41C2×(C2×D59)⋊C4—22no—
42C2×(C2×D59)⋊C4—22no—
43C2×(C2×D59)⋊C4—22no—
44C2×C4×Dic59—21no—
45#14273660—42no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)D412yes—
48#14273647—44no—
49C4×C59⋊C8—21no—
50C4×C59⋊C8—21no—
51C59⋊#95—21no—
52C22×Q8×C59D411yescharacteristic
53C22×C8×C59—21no—
54C4×Q8×C59—21no—
55C4×Q8×C59—21no—
56C4×Q8×C59—21no—
57C4×Q8×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C42×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67#103×C59—42no—
68C23×D59—22no—
69C2×Q8×C59—21no—
70C2×Q8×C59—21no—
71C2×Q8×C59—21no—
72C2×Q8×C59—21no—
73C2×Q8×C59—21no—
74C2×Q8×C59D812yes—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—41no—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×Dic59—21no—
82C22×Dic59—21no—
83C2×D236—22no—
84C2×D236—22no—
85C2×D236—42no—
86C2×D236—42no—
87C2×D236—41no—
88C2×D236—41no—
89C2×D236—41no—
90C2×D236—41no—
91C2×D236—41no—
92C2×D236—41no—
93C2×D236—41no—
94C2×D236—41no—
95C2×D236—41no—
96C2×D236—41no—
97C2×D236—41no—
98C2×D236—41no—
99C2×C8×C59—41no—
100C2×C8×C59—41no—
101C2×C8×C59—41no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—42no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C42×C59—41no—
112C42×C59—41no—
113C42×C59C2×D411yescharacteristic
114C42×C59#4111yescharacteristic
115C42×C59#4111yescharacteristic
116C42×C59#4111yescharacteristic
117(C2×D59)⋊C4—44no—
118(C2×D59)⋊C4—44no—
119(C2×D59)⋊C4—82no—
120(C2×D59)⋊C4—82no—
121C4⋊Dic59—21no—
122C4⋊Dic59—21no—
123C4⋊Dic59—21no—
124C4⋊Dic59—21no—
125C4×Dic59—41no—
126C4×Dic59—41no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—41no—
130C2×C59⋊C8—41no—
131C2×C59⋊C8—41no—
132C2×C59⋊C8—41no—
133C23×C59#12011yescharacteristic
134C22×D59—42no—
135C22×D59—42no—
136C22×D59—42no—
137C22×D59—42no—
138C22×D59—42no—
139C22×D59—42no—
140C22×D59—42no—
141Q8×C59—22no—
142Q8×C59—22no—
143Q8×C59—41no—
144Q8×C59—41no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—22no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—81no—
154C2×C4×C59—81no—
155C2×C4×C59—81no—
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12111yescharacteristic
158C2×C4×C59#12111yescharacteristic
159C2×C4×C59#12311yescharacteristic
160C2×C4×C59#12311yescharacteristic
161C2×C4×C59#12411yescharacteristic
162C2×C4×C59C2×D811yescharacteristic
163C2×C4×C59#13511yescharacteristic
164C2×C4×C59#13511yescharacteristic
165C2×C4×C59#13511yescharacteristic
166C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C59#13711yescharacteristic
169C2×C4×C59#13711yescharacteristic
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176D236—44no—
177D236—82no—
178D236—81no—
179D236—81no—
180D236—81no—
181D236—81no—
182D236—81no—
183D236—81no—
184D236—81no—
185D236—81no—
186C8×C59—81no—
187C59⋊C8—42no—
188C59⋊C8—42no—
189#10411—591nomaximal, Sylow
190C22×C59#39411yescharacteristic
191C22×C59#44711yescharacteristic
192C22×C59#44911yescharacteristic
193C22×C59#45211yescharacteristic
194C22×C59#45211yescharacteristic
195C22×C59#45311yescharacteristic
196C22×C59#45511yescharacteristic
197C2×D59—82no—
198C2×D59—82no—
199C2×D59—82no—
200C2×D59—82no—
201C2×D59—82no—
202C2×D59—82no—
203C2×D59—82no—
204C4×C59—21no—
205C4×C59—21no—
206C4×C59—42no—
207C4×C59—81no—
208C4×C59—161no—
209C4×C59#46311yescharacteristic
210C4×C59#46411yescharacteristic
211C4×C59#46611yescharacteristic
212C4×C59#46811yescharacteristic
213C4×C59#48011yescharacteristic
214C4×C59#48211yescharacteristic
215C4×C59#48811yescharacteristic
216C4×C59#48911yescharacteristic
217Dic59—81no—
218Dic59—81no—
219#2548—591no—
220#1957—591no—
221#1947—591no—
222#1803D5911yescharacteristic, complemented
223C2×#331—591no—
224C2×Q8⋊C8—591no—
225#1279—591no—
226C2×C59#162111yescharacteristic
227C2×C59#163011yescharacteristic
228C2×C59#164011yescharacteristic
229C2×C59#164111yescharacteristic
230C2×C59#203311yescharacteristic
231C2×C59#203711yescharacteristic
232C2×C59#204211yescharacteristic
233D59—162no—
234C2×#127—591no—
235C2×#124—1181no—
236C2×C4×Q8C2×D5911yescharacteristic
237C2×#107—1181no—
238C2×C4⋊C8—591no—
239C2×C4⋊C8C2×D5911yescharacteristic
240C2×#103—21no—
241C2×#102—1182no—
242C2×C4×C8—591no—
243#394—1182no—
244#390—1181no—
245#390—1181no—
246#390—1181no—
247#384C2×D5911yescharacteristic
248C4×C4⋊C4—591no—
249#340—21no—
250#340—1181no—
251#331—1182no—
252Q8⋊C8—1182no—
253C59#1041111yescharacteristic, complemented, Sylow
254C22×Q8C22⋊D5911yescharacteristic
255C22×D4—592no—
256C22×D4—1181no—
257C22×D4—1181no—
258C22×C8—21no—
259C22×C8—1181no—
260C22×C8—1181no—
261#127—1181no—
262#127—1181no—
263#127—1181no—
264#127—1181no—
265#124—2362no—
266#124—2362no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—591no—
276C2×C4⋊C4—591no—
277C2×C4⋊C4C22⋊D5911yescharacteristic
278C2×C22⋊C4—1182no—
279C2×C22⋊C4—1182no—
280C2×C22⋊C4—1182no—
281C2×C22⋊C4—1182no—
282C2×C42—21no—
283C2×C42—1181no—
284C2×C42C22×D5911yescharacteristic
285#107—2362no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—592no—
289C4⋊C8—1181no—
290#103—42no—
291#102—2364no—
292C4×C8—1181no—
293C4×C8—1181no—
294#95—1181no—
295C24—1182no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8#1174566912yes—
302C2×D4—1182no—
303C2×D4—1182no—
304C2×D4—2362no—
305C2×D4—2362no—
306C2×D4—2361no—
307C2×D4—2361no—
308C2×D4—2361no—
309C2×D4—2361no—
310C2×D4—2361no—
311C2×D4—2361no—
312C2×D4—2361no—
313C2×D4—2361no—
314C2×D4—2361no—
315C2×D4—2361no—
316C2×D4—2361no—
317C2×D4—2361no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—1181no—
322C22×C4—1181no—
323C22×C4D4×D5911yescharacteristic
324C22×C4D4×D5911yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—1182no—
330C2×C8—1182no—
331C2×C8—2361no—
332C2×C8—2361no—
333C2×C8—2361no—
334C2×C8—2361no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—1181no—
345C4⋊C4—1181no—
346C4⋊C4—1181no—
347C4⋊C4—1181no—
348C22⋊C4—4722no—
349C22⋊C4—4722no—
350C22⋊C4—2364no—
351C22⋊C4—2364no—
352C42—41no—
353C42—41no—
354C42—2361no—
355C42—2361no—
356C42#1174568511yescharacteristic
357C42#1174568511yescharacteristic
358C42D4×D5911yescharacteristic
359C42#1174568711yescharacteristic
360C23—2362no—
361C23—2362no—
362C23—2362no—
363C23—2362no—
364C23—2362no—
365C23—2362no—
366C23—2362no—
367C23#1427369511yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372D4—4722no—
373D4—4721no—
374D4—4721no—
375D4—4721no—
376D4—4721no—
377D4—4721no—
378D4—4721no—
379D4—4721no—
380D4—4721no—
381D4—2364no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2361no—
396C2×C4—2361no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4#1427369911yescharacteristic
400C2×C4#1427370011yescharacteristic
401C2×C4#1427370011yescharacteristic
402C2×C4#1427370911yescharacteristic
403C2×C4#1427371011yescharacteristic
404C2×C4#1427371311yescharacteristic
405C2×C4C2×#1174566911yescharacteristic
406C2×C4#1427374711yescharacteristic
407C2×C4#1427374811yescharacteristic
408C2×C4#1427375011yescharacteristic
409C2×C4#1427375211yescharacteristic
410C2×C4#1427375411yescharacteristic
411C2×C4#1427375811yescharacteristic
412C2×C4#1427375911yescharacteristic
413C8—81no—
414C8—2362no—
415C8—2362no—
416C22—4722no—
417C22—4722no—
418C22—4722no—
419C22—4722no—
420C22—4722no—
421C22—4722no—
422C22—4722no—
423C22#1683731711yescharacteristic
424C22#1683753511yescharacteristic
425C22#1683754611yescharacteristic
426C22#1683756211yescharacteristic
427C22#1683756311yescharacteristic
428C22#1683756811yescharacteristic
429C22#1683757811yescharacteristic
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—161no—
435C4—4721no—
436C4—4721no—
437C4#1683762011yescharacteristic
438C4#1683762711yescharacteristic
439C4#1683763611yescharacteristic
440C4#1683765011yescharacteristic
441C4#1683771511yescharacteristic
442C4#1683772611yescharacteristic
443C4#1683775811yescharacteristic
444C4#1683776311yescharacteristic
445C2—9442no—
446C2#2142315911yescharacteristic
447C2#2142321611yescharacteristic
448C2#2142328111yescharacteristic
449C2#2142328711yescharacteristic
450C2#2142527811yescharacteristic
451C2#2142530411yescharacteristic
452C2#2142533311yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1803C2
socleC23×C59#120
prime radicalC2×#14273724C22
hypercenter#1803D59
hypocenterC59#10411
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
3. #21424852C2semidirect
5. C2×Q8⋊(C59⋊C8)C2semidirect
6. #21421935C2semidirect
7. C59×#1803C2semidirect
222. #1803D59semidirect
253. C59#10411semidirect

Maximal subgroups

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

subgroupindex
#214290952
#214249012
#214248522
C2×#168371452
C2×Q8⋊(C59⋊C8)2
#214219352
C59×#18032
#1041159