Finite Group Databasebeta

← #28483576

Subgroups — #28483576

Order 15104.

530 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433551C211yesmaximal, characteristic
2#21422546C211yesmaximal, characteristic, complemented
3#21422494C211yesmaximal, characteristic, complemented
4#21422478C211yesmaximal, characteristic, complemented
5#21421991C211yesmaximal, characteristic, complemented
6#21421930C211yesmaximal, characteristic, complemented
7C59×#2906C211yesmaximal, characteristic, complemented
8C2×#14273747—21no—
9#16837952C2211yescharacteristic
10#16837624—21no—
11#16837617—21no—
12#16837612C411yescharacteristic
13#16837607C411yescharacteristic
14#16837602C411yescharacteristic
15#16837597C411yescharacteristic
16#16837455C2211yescharacteristic
17C2×C4⋊(C59⋊C8)C2211yescharacteristic
18#16837396—21no—
19C2×C4×C59⋊C8C2211yescharacteristic
20Q8⋊(C59⋊C8)—21no—
21D4⋊(C59⋊C8)—21no—
22#16837129—21no—
23#16837129—21no—
24#16837129—22no—
25C2×#131×C59—21no—
26C4×#41×C59C2211yescharacteristic
27C59×#424—21no—
28C59×#424—22no—
29C59×#423—21no—
30C59×#423C2211yescharacteristic
31C2×C4⋊C8×C59—21no—
32C2×C4⋊C8×C59C2211yescharacteristic
33C2×#11745685—21no—
34#14273747—42no—
35#14273747—42no—
36C2×#11745672—21no—
37C2×#11745670—21no—
38C2×#11745671—21no—
39C2×#11745669—21no—
40C22×C59⋊C8—21no—
41C22×C59⋊C8—21no—
42#14273727C2×C411yescharacteristic
43#14273724C2×C411yescharacteristic
44#14273715—21no—
45#14273701—21no—
46#14273701—21no—
47C4×D236—21no—
48C4×D236—21no—
49C4⋊(C59⋊C8)—21no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)C2×C411yescharacteristic
52C4⋊(C59⋊C8)C2×C411yescharacteristic
53#14273650—21no—
54#14273650—21no—
55#14273647—21no—
56#14273647—21no—
57C22⋊(C59⋊C8)—21no—
58C4×C59⋊C8—21no—
59C4×C59⋊C8—21no—
60C4×C59⋊C8—21no—
61C2×#41×C59D411yescharacteristic
62#131×C59—42no—
63#131×C59—42no—
64C2×M16×C59—22no—
65C2×M16×C59—21no—
66C22×C8×C59—22no—
67C22×C8×C59—21no—
68C4×Q8×C59C2×C411yescharacteristic
69C4×Q8×C59D411yescharacteristic
70C4×D4×C59—21no—
71C4×D4×C59—21no—
72C4×D4×C59C2×C411yescharacteristic
73C4×D4×C59D411yescharacteristic
74#117×C59—21no—
75#117×C59D411yescharacteristic
76C2×C42×C59—21no—
77C2×C42×C59C2311yescharacteristic
78C4⋊C8×C59—22no—
79C4⋊C8×C59—22no—
80C4⋊C8×C59—21no—
81C4⋊C8×C59—21no—
82C4⋊C8×C59D411yescharacteristic
83C4⋊C8×C59D411yescharacteristic
84C4⋊C8×C59D412yes—
85#41×C59—21no—
86#41×C59—21no—
87#41×C59D812yes—
88#41×C59SD1612yes—
89C2×Q8×C59C22⋊C411yescharacteristic
90C2×D4×C59—21no—
91C2×D4×C59C22⋊C411yescharacteristic
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—21no—
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59#4111yescharacteristic
97C22×C4×C59#4111yescharacteristic
98C2×C22⋊D59—41no—
99#11745685—41no—
100#11745685—41no—
101#11745685—41no—
102#11745685—41no—
103C2×D236—21no—
104C2×D236—21no—
105C2×D236—21no—
106C2×C4×D59—41no—
107C2×Dic118—21no—
108M16×C59—44no—
109M16×C59—41no—
110M16×C59—41no—
111C2×C8×C59—21no—
112C2×C8×C59—22no—
113C2×C8×C59—21no—
114C2×C8×C59—44no—
115C2×C8×C59—41no—
116C2×C8×C59—41no—
117C4⋊C4×C59—21no—
118C4⋊C4×C59—21no—
119C4⋊C4×C59C22⋊C411yescharacteristic
120C4⋊C4×C59C22⋊C411yescharacteristic
121C22⋊C4×C59—21no—
122C22⋊C4×C59—21no—
123C22⋊C4×C59—21no—
124C42×C59—21no—
125C42×C59—21no—
126C42×C59—21no—
127C42×C59C22×C411yescharacteristic
128C42×C59C2×D411yescharacteristic
129C42×C59C2×D411yescharacteristic
130C42×C59C2×D411yescharacteristic
131#11745672—42no—
132#11745671—42no—
133#11745670—42no—
134#11745669—42no—
135(C2×D59)⋊C4—41no—
136(C2×D59)⋊C4—41no—
137C4⋊Dic59—21no—
138C4⋊Dic59—21no—
139C2×C59⋊C8—22no—
140C2×C59⋊C8—21no—
141C2×C59⋊C8—21no—
142C2×C59⋊C8—41no—
143C2×C59⋊C8—41no—
144C2×C59⋊C8—41no—
145C2×C59⋊C8—41no—
146C2×C59⋊C8—41no—
147C23×C59—21no—
148C23×C59#12311yescharacteristic
149C22×D59—41no—
150Q8×C59—21no—
151Q8×C59#10212yes—
152D4×C59—22no—
153D4×C59—22no—
154D4×C59—21no—
155D4×C59#10212yes—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—22no—
160C2×C4×C59—22no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—81no—
167C2×C4×C59C2×C22⋊C411yescharacteristic
168C2×C4×C59C4×D411yescharacteristic
169C2×C4×C59C4×D411yescharacteristic
170C2×C4×C59#12011yescharacteristic
171C2×C4×C59#12111yescharacteristic
172C2×C4×C59#12111yescharacteristic
173C2×C4×C59C2×D811yescharacteristic
174C2×C4×C59C2×SD1611yescharacteristic
175C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C59#13711yescharacteristic
177C22⋊D59—81no—
178C22⋊D59—81no—
179C2×Dic59—41no—
180D236—42no—
181D236—41no—
182D236—41no—
183D236—41no—
184D236—41no—
185C4×D59—81no—
186C4×D59—81no—
187Dic118—41no—
188Dic118—41no—
189C8×C59—41no—
190C8×C59—42no—
191C8×C59—41no—
192C59⋊C8—42no—
193C59⋊C8—41no—
194C59⋊C8—41no—
195C59⋊C8—81no—
196#8995—591nomaximal, Sylow
197C22×C59—21no—
198C22×C59—21no—
199C22×C59—22no—
200C22×C59—41no—
201C22×C59—41no—
202C22×C59#38611yescharacteristic
203C22×C59#48011yescharacteristic
204C22×C59#48111yescharacteristic
205C2×D59—81no—
206C2×D59—81no—
207C2×D59—81no—
208C4×C59—22no—
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—41no—
213C4×C59—41no—
214C4×C59C2×#10211yescharacteristic
215C4×C59#41711yescharacteristic
216C4×C59#44311yescharacteristic
217C4×C59#44411yescharacteristic
218C4×C59#44711yescharacteristic
219C4×C59#44811yescharacteristic
220Dic59—81no—
221#3035—591no—
222#2906D5911yescharacteristic, complemented
223#1492—591no—
224#1480—591no—
225#1476—591no—
226#1301—591no—
227#1277—591no—
228C2×C59—21no—
229C2×C59—21no—
230C2×C59—42no—
231C2×C59#187411yescharacteristic
232C2×C59#187611yescharacteristic
233C2×C59#188111yescharacteristic
234D59—161no—
235C2×#135—1181no—
236C2×#131—21no—
237#532—591no—
238C4×#41C2×D5911yescharacteristic
239#464—1181no—
240#463—1181no—
241#462—591no—
242#461—591no—
243#460—591no—
244#459—591no—
245#431—591no—
246#424—21no—
247#424—22no—
248#423—21no—
249#423C2×D5911yescharacteristic
250C2×C4⋊C8—21no—
251C2×C4⋊C8—591no—
252C2×C4⋊C8C2×D5911yescharacteristic
253#416—1181no—
254C2×C4×C8—591no—
255Q8⋊C8—1181no—
256D4⋊C8—1181no—
257D4⋊C8—1181no—
258D4⋊C8—1181no—
259D4⋊C8—1182no—
260C59#899511yescharacteristic, complemented, Sylow
261C2×#41—1181no—
262C2×#41C22⋊D5911yescharacteristic
263#135—2362no—
264#135—2362no—
265C2×Q16—1181no—
266C2×SD16—1181no—
267C2×SD16—1181no—
268C2×D8—1181no—
269#131—42no—
270#131—42no—
271C2×M16—22no—
272C2×M16—21no—
273C22×C8—22no—
274C22×C8—21no—
275C22×C8—1181no—
276C22×C8—1181no—
277C4⋊Q8—591no—
278#127—591no—
279#124—1181no—
280#121—1181no—
281#121—1181no—
282C4×Q8C4×D5911yescharacteristic
283C4×Q8D23611yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—1181no—
287C4×D4—1181no—
288C4×D4C4×D5911yescharacteristic
289C4×D4D23611yescharacteristic
290#117—21no—
291#117C22⋊D5911yescharacteristic
292C2×C42—21no—
293C2×C42C22×D5911yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—591no—
299C4⋊C8—591no—
300C4⋊C8—1181no—
301C4⋊C8—1181no—
302C4⋊C8C22⋊D5911yescharacteristic
303C4⋊C8C22⋊D5911yescharacteristic
304C4⋊C8C22⋊D5912yes—
305#103—1181no—
306#103—1181no—
307#102—1181no—
308#102—1181no—
309C22⋊C8—1181no—
310C4×C8—1181no—
311C4×C8—1181no—
312C4×C8—1181no—
313#41—21no—
314#41—21no—
315#41—2361no—
316#41—2361no—
317#41—2361no—
318#41—2361no—
319#41#1174566912yes—
320#41#1174567112yes—
321C2×Q8—1181no—
322C2×Q8(C2×D59)⋊C411yescharacteristic
323C2×D4—21no—
324C2×D4—1181no—
325C2×D4—1181no—
326C2×D4—1181no—
327C2×D4—2361no—
328C2×D4(C2×D59)⋊C411yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—2361no—
333C22×C4#1174568511yescharacteristic
334C22×C4#1174568511yescharacteristic
335C22×C4C2×C22⋊D5911yescharacteristic
336Q16—2362no—
337SD16—2362no—
338SD16—2362no—
339D8—2362no—
340M16—44no—
341M16—41no—
342M16—41no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—44no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—1182no—
350C2×C8—1181no—
351C2×C8—1181no—
352C2×C8—2361no—
353C2×C8—2361no—
354C2×C8—2361no—
355C2×C8—2361no—
356C2×C8—2361no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—1181no—
360C4⋊C4—1181no—
361C4⋊C4(C2×D59)⋊C411yescharacteristic
362C4⋊C4(C2×D59)⋊C411yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—2361no—
367C22⋊C4—2361no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42C2×C4×D5911yescharacteristic
372C42C2×D23611yescharacteristic
373C42C2×C22⋊D5911yescharacteristic
374C42C2×C22⋊D5911yescharacteristic
375C23—21no—
376C23—2361no—
377C23#1427371011yescharacteristic
378Q8—21no—
379Q8—2361no—
380Q8—2361no—
381Q8#1427364712yes—
382D4—22no—
383D4—22no—
384D4—21no—
385D4—4721no—
386D4—4721no—
387D4—2362no—
388D4—2361no—
389D4—2361no—
390D4—2361no—
391D4—2361no—
392D4#1427364712yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—4721no—
405C2×C4—4721no—
406C2×C4—2361no—
407C2×C4C2×(C2×D59)⋊C411yescharacteristic
408C2×C4C4×C22⋊D5911yescharacteristic
409C2×C4C4×C22⋊D5911yescharacteristic
410C2×C4#1427369611yescharacteristic
411C2×C4#1427370111yescharacteristic
412C2×C4#1427370111yescharacteristic
413C2×C4C2×#1174566911yescharacteristic
414C2×C4C2×#1174567111yescharacteristic
415C2×C4#1427375311yescharacteristic
416C2×C4#1427375811yescharacteristic
417C8—41no—
418C8—42no—
419C8—41no—
420C8—4721no—
421C8—2362no—
422C8—2361no—
423C8—2361no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—4721no—
430C22—4721no—
431C22—4721no—
432C22#1683728111yescharacteristic
433C22#1683771511yescharacteristic
434C22#1683772011yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—4721no—
442C4C2×#1427364711yescharacteristic
443C4#1683740111yescharacteristic
444C4#1683751711yescharacteristic
445C4#1683752211yescharacteristic
446C4#1683753611yescharacteristic
447C4#1683754111yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—9441no—
452C2#2142441311yescharacteristic
453C2#2142442311yescharacteristic
454C2#2142445811yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2906C2
socleC22×C59#386
prime radical#21433551C2
cosocleC2×C42×C59C23
hypercenter#2906D59
hypocenterC59#8995
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #21422546C2semidirect
3. #21422494C2semidirect
4. #21422478C2semidirect
5. #21421991C2semidirect
6. #21421930C2semidirect
7. C59×#2906C2semidirect
222. #2906D59semidirect
260. C59#8995semidirect

Maximal subgroups

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

subgroupindex
#214335512
#214225462
#214224942
#214224782
#214219912
#214219302
C59×#29062
#899559