Finite Group Databasebeta

← #28492496

Subgroups — #28492496

Order 15104.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837603C211yesmaximal, characteristic
2#21425031C211yesmaximal, characteristic, complemented
3#21424901C211yesmaximal, characteristic
4#21424098C211yesmaximal, characteristic, complemented
5C2×#16837154C211yesmaximal, characteristic, complemented
6C2×#16837145C211yesmaximal, characteristic
7C59×#1296C211yesmaximal, characteristic, complemented
8C22×#11745660—21no—
9C2×#14273724C2211yescharacteristic
10C2×C4×Dic118C2211yescharacteristic
11#16837603—22no—
12#16837603—22no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273650—21no—
15C2×#14273649—21no—
16C2×#14273647—21no—
17C2×#14273647—21no—
18C2×C4×C59⋊C8C2211yescharacteristic
19#16837164—21no—
20#16837156—21no—
21#16837156—21no—
22#16837154—22no—
23#16837145—22no—
24C2×C4⋊C8×C59C2211yescharacteristic
25C2×C4⋊C8×C59C2211yescharacteristic
26C59×#384C2211yescharacteristic
27C22×Dic118D411yescharacteristic
28C22×D236—21no—
29C22×D236—21no—
30C22×D236—21no—
31C2×#11745660—24no—
32C2×#11745660—42no—
33C2×#11745660—42no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8D411yescharacteristic
36C22×C59⋊C8D411yescharacteristic
37#14273724—21no—
38#14273724—21no—
39#14273724—21no—
40#14273724D412yes—
41C4×Dic118—21no—
42C4×Dic118—21no—
43C4×Dic118—21no—
44C4×Dic118—21no—
45C2×C4⋊Dic59D411yescharacteristic
46C2×Dic59⋊C4—21no—
47C2×C4×Dic59—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)—21no—
50#14273650—42no—
51#14273649—24no—
52#14273647—42no—
53#14273647—42no—
54C4×C59⋊C8—21no—
55C4×C59⋊C8—21no—
56C22×C8×C59—21no—
57C22×C8×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C42×C59C2311yescharacteristic
62C4⋊C8×C59—21no—
63C4⋊C8×C59—21no—
64C4⋊C8×C59—21no—
65C4⋊C8×C59D412yes—
66C23×D59—41no—
67C22×C4×C59—41no—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59#4111yescharacteristic
71C22×Dic59—21no—
72C22×Dic59—21no—
73C2×D236—22no—
74C2×D236—22no—
75C2×D236—22no—
76C2×D236—41no—
77C2×D236—41no—
78C2×D236—41no—
79C2×D236—41no—
80C2×D236—41no—
81C2×D236—41no—
82C2×D236—41no—
83C2×D236—41no—
84C2×D236—41no—
85C2×D236—41no—
86C2×D236—41no—
87C2×D236—81no—
88C2×D236—81no—
89C2×Dic118—21no—
90C2×Dic118—21no—
91C2×Dic118—21no—
92C2×Dic118—21no—
93C2×Dic118—21no—
94C2×Dic118D812yes—
95C2×C8×C59—22no—
96C2×C8×C59—41no—
97C2×C8×C59—41no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C2×C8×C59—41no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—81no—
106C42×C59C2×D411yescharacteristic
107C42×C59C2×D411yescharacteristic
108C42×C59#4111yescharacteristic
109C42×C59#4111yescharacteristic
110C4⋊Dic59—21no—
111C4⋊Dic59—21no—
112Dic59⋊C4—41no—
113Dic59⋊C4—41no—
114C4×Dic59—41no—
115C4×Dic59—41no—
116C2×C59⋊C8—21no—
117C2×C59⋊C8—21no—
118C2×C59⋊C8—21no—
119C2×C59⋊C8—21no—
120C2×C59⋊C8—21no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—41no—
123C2×C59⋊C8—41no—
124C2×C59⋊C8SD1612yescomplemented
125#11745660—48no—
126C23×C59#12111yescharacteristic
127C22×D59—42no—
128C22×D59—81no—
129C22×D59—81no—
130C22×D59—81no—
131C22×D59—81no—
132C22×D59—81no—
133C22×D59—81no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#12311yescharacteristic
143C2×C4×C59#12311yescharacteristic
144C2×C4×C59#12411yescharacteristic
145C2×C4×C59C2×D811yescharacteristic
146C2×C4×C59C2×SD1611yescharacteristic
147C2×C4×C59#13511yescharacteristic
148C2×C4×C59#13511yescharacteristic
149C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C59#13711yescharacteristic
152C2×C4×C59#13711yescharacteristic
153C2×Dic59—22no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159D236—42no—
160D236—42no—
161D236—42no—
162D236—42no—
163D236—82no—
164D236—82no—
165D236—81no—
166D236—81no—
167D236—81no—
168D236—81no—
169Dic118—22no—
170Dic118—22no—
171Dic118—41no—
172Dic118—41no—
173C8×C59—42no—
174C8×C59—81no—
175C59⋊C8—22no—
176C59⋊C8—41no—
177C59⋊C8—81no—
178#10474—591nomaximal, Sylow
179C22×C59#39411yescharacteristic
180C22×C59#46011yescharacteristic
181C22×C59#46411yescharacteristic
182C22×C59#46611yescharacteristic
183C22×C59#46811yescharacteristic
184C22×C59#47111yescharacteristic
185C22×C59#47211yescharacteristic
186C2×D59—82no—
187C2×D59—82no—
188C2×D59—82no—
189C2×D59—161no—
190C2×D59—161no—
191C2×D59—161no—
192C2×D59—161no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—161no—
196C4×C59#44711yescharacteristic
197C4×C59#45211yescharacteristic
198C4×C59#46111yescharacteristic
199C4×C59#46311yescharacteristic
200C4×C59#46511yescharacteristic
201C4×C59#46811yescharacteristic
202C4×C59#48011yescharacteristic
203C4×C59#48211yescharacteristic
204Dic59—42no—
205Dic59—81no—
206C2×#460—591no—
207#1984—591no—
208#1957—591no—
209#1803—591no—
210C2×#335—591no—
211C2×#331—591no—
212#1296D5911yescharacteristic, complemented
213C2×C59#166811yescharacteristic
214C2×C59#168211yescharacteristic
215C2×C59#168611yescharacteristic
216C2×C59#169811yescharacteristic
217C2×C59#203311yescharacteristic
218C2×C59#205111yescharacteristic
219C2×C59#205511yescharacteristic
220D59—162no—
221C22×SD16—1181no—
222C2×#127—591no—
223C2×C4×Q8—591no—
224#460—1182no—
225#460—1182no—
226C2×C4⋊C8—591no—
227C2×C4⋊C8C2×D5911yescharacteristic
228C2×C4⋊C8C2×D5911yescharacteristic
229C2×#103—1181no—
230C2×#102—1181no—
231C2×#102—1181no—
232C2×#102—1181no—
233C2×C4×C8—591no—
234#384C2×D5911yescharacteristic
235#340—1181no—
236#336—1181no—
237#336—1181no—
238#335—1182no—
239#331—1182no—
240C59#1047411yescharacteristic, complemented, Sylow
241C22×Q8—591no—
242C22×D4—1181no—
243C22×D4—1181no—
244C22×D4—1181no—
245C2×SD16—1184no—
246C2×SD16—2362no—
247C2×SD16—2362no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—591no—
251C22×C8—591no—
252C22×C8—1181no—
253#127—592no—
254#127—1181no—
255#127—1181no—
256#127—1181no—
257C4×Q8—1181no—
258C4×Q8—1181no—
259C4×Q8—1181no—
260C4×Q8—1181no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—591no—
265C2×C4⋊C4—1181no—
266C2×C42—1181no—
267C2×C42C22×D5911yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1181no—
272C4⋊C8—1181no—
273C4⋊C8D23612yes—
274#103—2362no—
275#102—1184no—
276#102—2362no—
277#102—2362no—
278C4×C8—1181no—
279C4×C8—1181no—
280C24—2361no—
281C2×Q8—592no—
282C2×Q8—1181no—
283C2×Q8—1181no—
284C2×Q8—1181no—
285C2×Q8—1181no—
286C2×Q8—1181no—
287C2×D4—1182no—
288C2×D4—1182no—
289C2×D4—1182no—
290C2×D4—4721no—
291C2×D4—4721no—
292C2×D4—2361no—
293C2×D4—2361no—
294C2×D4—2361no—
295C2×D4—2361no—
296C2×D4—2361no—
297C2×D4—2361no—
298C2×D4—2361no—
299C2×D4—2361no—
300C2×D4—2361no—
301C2×D4—2361no—
302C2×D4—2361no—
303C22×C4—41no—
304C22×C4—1181no—
305C22×C4—1181no—
306C22×C4#1174568511yescharacteristic
307C22×C4D4×D5911yescharacteristic
308C22×C4D4×D5911yescharacteristic
309SD16—2368no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—592no—
317C2×C8—1181no—
318C2×C8—1181no—
319C2×C8—1181no—
320C2×C8—1181no—
321C2×C8—1181no—
322C2×C8—2361no—
323C2×C8—2361no—
324C2×C8—2361no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—81no—
330C4⋊C4—1181no—
331C4⋊C4—1181no—
332C4⋊C4—2361no—
333C4⋊C4—2361no—
334C42—2361no—
335C42—2361no—
336C42C2×D23611yescharacteristic
337C42#1174568511yescharacteristic
338C42D4×D5911yescharacteristic
339C42#1174568911yescharacteristic
340C23—4721no—
341C23—4721no—
342C23—4721no—
343C23—4721no—
344C23—4721no—
345C23—4721no—
346C23—2362no—
347C23#1427370011yescharacteristic
348Q8—1182no—
349Q8—1182no—
350Q8—2361no—
351Q8—2361no—
352D4—4722no—
353D4—4722no—
354D4—4721no—
355D4—4721no—
356D4—4721no—
357D4—4721no—
358D4—2362no—
359D4—2362no—
360D4—2362no—
361D4—2362no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—1182no—
368C2×C4—2361no—
369C2×C4—2361no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4#1427369711yescharacteristic
374C2×C4#1427370011yescharacteristic
375C2×C4#1427370211yescharacteristic
376C2×C4#1427371211yescharacteristic
377C2×C4#1427371211yescharacteristic
378C2×C4#1427371511yescharacteristic
379C2×C4D8×D5911yescharacteristic
380C2×C4SD16×D5911yescharacteristic
381C2×C4#1427374811yescharacteristic
382C2×C4#1427374811yescharacteristic
383C2×C4#1427375111yescharacteristic
384C2×C4#1427375411yescharacteristic
385C2×C4#1427375911yescharacteristic
386C2×C4#1427375911yescharacteristic
387C8—42no—
388C8—81no—
389C8—1182no—
390C8—4721no—
391C8—2361no—
392C22—9441no—
393C22—9441no—
394C22—9441no—
395C22—9441no—
396C22—4722no—
397C22—4722no—
398C22—4722no—
399C22#1683732011yescharacteristic
400C22#1683760111yescharacteristic
401C22#1683762711yescharacteristic
402C22#1683763711yescharacteristic
403C22#1683765111yescharacteristic
404C22#1683766811yescharacteristic
405C22#1683767111yescharacteristic
406C4—21no—
407C4—21no—
408C4—161no—
409C4—4721no—
410C4—2362no—
411C4#1683753711yescharacteristic
412C4#1683756711yescharacteristic
413C4#1683760611yescharacteristic
414C4#1683761611yescharacteristic
415C4#1683763211yescharacteristic
416C4#1683764911yescharacteristic
417C4#1683771711yescharacteristic
418C4#1683772811yescharacteristic
419C2—9442no—
420C2#2142345511yescharacteristic
421C2#2142354311yescharacteristic
422C2#2142357111yescharacteristic
423C2#2142363911yescharacteristic
424C2#2142528111yescharacteristic
425C2#2142539111yescharacteristic
426C2#2142541511yescharacteristic
subgroupG/H
centerC23#14273700
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1296C2
socleC23×C59#121
prime radicalC2×#14273724C22
hypercenter#1296D59
hypocenterC59#10474
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #21425031C2semidirect
4. #21424098C2semidirect
5. C2×#16837154C2semidirect
7. C59×#1296C2semidirect
124. C2×C59⋊C8SD16semidirect
212. #1296D59semidirect
240. C59#10474semidirect

Maximal subgroups

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

subgroupindex
C2×#168376032
#214250312
#214249012
#214240982
C2×#168371542
C2×#168371452
C59×#12962
#1047459