Finite Group Databasebeta

← #28492565

Subgroups — #28492565

Order 15104.

490 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837683C211yesmaximal, characteristic
2#21424898C211yesmaximal, characteristic, complemented
3#21424890C211yesmaximal, characteristic
4#21424097C211yesmaximal, characteristic
5C2×#16837147C211yesmaximal, characteristic, complemented
6#21421980C211yesmaximal, characteristic, complemented
7C2×C59×#334C211yesmaximal, characteristic, complemented
8C2×#14273727C2211yescharacteristic
9C2×C4×D236C2211yescharacteristic
10#16837683—22no—
11#16837683—22no—
12C2×#14273661—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273652—21no—
15C2×#14273652—21no—
16C2×#14273649—21no—
17C2×#14273649—21no—
18C2×(C2×D59)⋊C8—21no—
19C2×(C2×D59)⋊C8—21no—
20#16837272C2211yescharacteristic
21#16837165—21no—
22#16837147—22no—
23C2×C4⋊C8×C59C2211yescharacteristic
24C2×C4⋊C8×C59C2211yescharacteristic
25C2×C4×C8×C59C2211yescharacteristic
26C59×#334—22no—
27C22×Dic118—21no—
28C22×D236D411yescharacteristic
29C22×C4×D59—21no—
30C22×C59⋊C8—21no—
31#14273727—21no—
32#14273727—21no—
33#14273727—21no—
34#14273727D412yes—
35C4×D236—21no—
36C4×D236—21no—
37C4×D236—21no—
38C4×D236—21no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—21no—
44C2×C4⋊Dic59D411yescharacteristic
45C2×(C2×D59)⋊C4—21no—
46#14273661—24no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49#14273652—42no—
50#14273652—42no—
51#14273649—24no—
52#14273649—42no—
53(C2×D59)⋊C8—42no—
54(C2×D59)⋊C8—42no—
55C22×C8×C59—21no—
56C22×C8×C59—21no—
57C22×C8×C59D411yescharacteristic
58C22×C8×C59D411yescharacteristic
59C2×C42×C59C2311yescharacteristic
60C4⋊C8×C59—21no—
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59D412yes—
64C4×C8×C59—21no—
65C4×C8×C59—21no—
66C23×D59—21no—
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59#4111yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—41no—
72C22×Dic59—41no—
73C2×D236—21no—
74C2×D236—21no—
75C2×D236—21no—
76C2×D236—21no—
77C2×D236—21no—
78C2×D236SD1612yes—
79C2×C4×D59—41no—
80C2×C4×D59—41no—
81C2×C4×D59—41no—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84C2×C4×D59—41no—
85C2×Dic118—22no—
86C2×Dic118—41no—
87C2×Dic118—41no—
88C2×Dic118—41no—
89C2×Dic118—41no—
90C2×Dic118—41no—
91C2×C8×C59—21no—
92C2×C8×C59—21no—
93C2×C8×C59—22no—
94C2×C8×C59—21no—
95C2×C8×C59—21no—
96C2×C8×C59—21no—
97C2×C8×C59—41no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C2×C8×C59—41no—
101C2×C8×C59—41no—
102C2×C8×C59D812yes—
103C42×C59C2×D411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C42×C59#4111yescharacteristic
106C42×C59#4111yescharacteristic
107(C2×D59)⋊C4—41no—
108(C2×D59)⋊C4—41no—
109C4⋊Dic59—22no—
110C4⋊Dic59—21no—
111C4⋊Dic59—22no—
112C4⋊Dic59—21no—
113C4⋊Dic59—41no—
114C4⋊Dic59—41no—
115C4⋊Dic59—41no—
116C4⋊Dic59—41no—
117C4⋊Dic59—41no—
118C4⋊Dic59—41no—
119C4⋊Dic59—81no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—41no—
123C23×C59#12111yescharacteristic
124C22×D59—22no—
125C22×D59—41no—
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—41no—
129C22×D59—41no—
130C22×D59—41no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59#12011yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59#12111yescharacteristic
136C2×C4×C59#12311yescharacteristic
137C2×C4×C59#12311yescharacteristic
138C2×C4×C59#12411yescharacteristic
139C2×C4×C59C2×D811yescharacteristic
140C2×C4×C59C2×SD1611yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C2×C4×C59#13511yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C59#13711yescharacteristic
146C2×C4×C59#13711yescharacteristic
147C2×Dic59—42no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—81no—
152C2×Dic59—81no—
153C2×Dic59—81no—
154C2×Dic59—81no—
155C2×Dic59—81no—
156D236—22no—
157D236—22no—
158D236—41no—
159D236—41no—
160C4×D59—81no—
161C4×D59—81no—
162C4×D59—81no—
163C4×D59—81no—
164Dic118—42no—
165Dic118—42no—
166Dic118—81no—
167Dic118—81no—
168C8×C59—22no—
169C8×C59—42no—
170C8×C59—41no—
171C8×C59—81no—
172C59⋊C8—81no—
173#10484—591nomaximal, Sylow
174C22×C59#39411yescharacteristic
175C22×C59#45911yescharacteristic
176C22×C59#46311yescharacteristic
177C22×C59#46511yescharacteristic
178C22×C59#47011yescharacteristic
179C22×C59#47111yescharacteristic
180C22×C59#47311yescharacteristic
181C2×D59—42no—
182C2×D59—42no—
183C2×D59—42no—
184C2×D59—81no—
185C2×D59—81no—
186C2×D59—81no—
187C2×D59—81no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59#44811yescharacteristic
191C4×C59#45211yescharacteristic
192C4×C59#45911yescharacteristic
193C4×C59#46311yescharacteristic
194C4×C59#46611yescharacteristic
195C4×C59#46911yescharacteristic
196C4×C59#48311yescharacteristic
197C4×C59#48511yescharacteristic
198Dic59—82no—
199Dic59—81no—
200Dic59—161no—
201C2×#474—591no—
202#1956—591no—
203#1954—591no—
204#1802—591no—
205C2×#334D5911yescharacteristic, complemented
206C2×#332—591no—
207#1296—591no—
208C2×C59#166511yescharacteristic
209C2×C59#167911yescharacteristic
210C2×C59#168911yescharacteristic
211C2×C59#170311yescharacteristic
212C2×C59#203511yescharacteristic
213C2×C59#205211yescharacteristic
214C2×C59#205411yescharacteristic
215D59—82no—
216C2×C4⋊Q8—591no—
217C2×C4×D4—591no—
218#474—1182no—
219#474—1182no—
220C2×#107—1181no—
221C2×C4⋊C8—591no—
222C2×C4⋊C8C2×D5911yescharacteristic
223C2×C4⋊C8C2×D5911yescharacteristic
224C2×#103—1181no—
225C2×#103—1181no—
226C2×#102—1181no—
227C2×#102—1181no—
228C2×C22⋊C8—1181no—
229C2×C22⋊C8—1181no—
230C2×C4×C8C2×D5911yescharacteristic
231#384—591no—
232#340—1181no—
233#334—22no—
234#332—1182no—
235C59#1048411yescharacteristic, complemented, Sylow
236C22×Q8—1181no—
237C22×D4—591no—
238C23×C4—1181no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—1181no—
242C22×C8D23611yescharacteristic
243C22×C8D23611yescharacteristic
244C4⋊Q8—592no—
245C4⋊Q8—1181no—
246C4⋊Q8—1181no—
247C4⋊Q8—1181no—
248C4×D4—1181no—
249C4×D4—1181no—
250C4×D4—1181no—
251C4×D4—1181no—
252C2×C4⋊C4—591no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4—1181no—
255C2×C4⋊C4—1181no—
256C2×C4⋊C4—1181no—
257C2×C4⋊C4—1181no—
258C2×C22⋊C4—1181no—
259C2×C42C22×D5911yescharacteristic
260#107—1184no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—1181no—
265C4⋊C8—1181no—
266C4⋊C8C22⋊D5912yes—
267#103—2362no—
268#103—2362no—
269#102—1184no—
270#102—2362no—
271C22⋊C8—2362no—
272C22⋊C8—2362no—
273C4×C8—21no—
274C4×C8—21no—
275C24—1181no—
276C2×Q8—1182no—
277C2×Q8—2361no—
278C2×Q8—2361no—
279C2×Q8—2361no—
280C2×Q8—2361no—
281C2×Q8—2361no—
282C2×D4—592no—
283C2×D4—1181no—
284C2×D4—1181no—
285C2×D4—1181no—
286C2×D4—1181no—
287C2×D4—1181no—
288C22×C4—1181no—
289C22×C4—2361no—
290C22×C4—2361no—
291C22×C4—2361no—
292C22×C4—2361no—
293C22×C4—2361no—
294C22×C4—2361no—
295C22×C4—2361no—
296C22×C4—2361no—
297C22×C4C2×D23611yescharacteristic
298C22×C4D4×D5911yescharacteristic
299C22×C4#1174568911yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—2361no—
312C2×C8—2361no—
313C2×C8—2361no—
314C2×C8D47212yes—
315C4⋊C4—1182no—
316C4⋊C4—1181no—
317C4⋊C4—1181no—
318C4⋊C4—1182no—
319C4⋊C4—4721no—
320C4⋊C4—2361no—
321C4⋊C4—2361no—
322C4⋊C4—2361no—
323C4⋊C4—2361no—
324C4⋊C4—2361no—
325C4⋊C4—2361no—
326C22⋊C4—2361no—
327C22⋊C4—2361no—
328C42#1174568511yescharacteristic
329C42D4×D5911yescharacteristic
330C42#1174568911yescharacteristic
331C42C2×C22⋊D5911yescharacteristic
332C23—1182no—
333C23—2361no—
334C23—2361no—
335C23—2361no—
336C23—2361no—
337C23—2361no—
338C23—2361no—
339C23#1427370211yescharacteristic
340Q8—4721no—
341Q8—4721no—
342Q8—2362no—
343Q8—2362no—
344D4—1182no—
345D4—1182no—
346D4—2361no—
347D4—2361no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—4721no—
351C2×C4—4721no—
352C2×C4—4721no—
353C2×C4—4721no—
354C2×C4—4721no—
355C2×C4—4721no—
356C2×C4—4721no—
357C2×C4—4721no—
358C2×C4—4721no—
359C2×C4—2361no—
360C2×C4—2362no—
361C2×C4—2361no—
362C2×C4—2361no—
363C2×C4#1427369511yescharacteristic
364C2×C4#1427370111yescharacteristic
365C2×C4#1427370211yescharacteristic
366C2×C4#1427371211yescharacteristic
367C2×C4#1427371211yescharacteristic
368C2×C4#1427371611yescharacteristic
369C2×C4C2×D47211yescharacteristic
370C2×C4SD16×D5911yescharacteristic
371C2×C4#1427374811yescharacteristic
372C2×C4#1427374911yescharacteristic
373C2×C4#1427375511yescharacteristic
374C2×C4#1427375511yescharacteristic
375C2×C4#1427375611yescharacteristic
376C2×C4#1427375611yescharacteristic
377C8—22no—
378C8—41no—
379C8—42no—
380C8—81no—
381C8—4721no—
382C22—4721no—
383C22—4721no—
384C22—4721no—
385C22—4721no—
386C22—2362no—
387C22—2362no—
388C22—2362no—
389C22#1683731911yescharacteristic
390C22#1683759811yescharacteristic
391C22#1683761811yescharacteristic
392C22#1683763211yescharacteristic
393C22#1683766111yescharacteristic
394C22#1683766611yescharacteristic
395C22#1683767811yescharacteristic
396C4—21no—
397C4—21no—
398C4—9441no—
399C4—4721no—
400C4—4722no—
401C4#1683753911yescharacteristic
402C4#1683756111yescharacteristic
403C4#1683759811yescharacteristic
404C4#1683761811yescharacteristic
405C4#1683763811yescharacteristic
406C4#1683765511yescharacteristic
407C4#1683773411yescharacteristic
408C4#1683774611yescharacteristic
409C2—4722no—
410C2#2142343811yescharacteristic
411C2#2142352011yescharacteristic
412C2#2142358611yescharacteristic
413C2#2142367211yescharacteristic
414C2#2142529411yescharacteristic
415C2#2142539711yescharacteristic
416C2#2142540711yescharacteristic
subgroupG/H
centerC23#14273702
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#334C2
socleC23×C59#121
prime radicalC22×D236D4
hypercenterC2×#334D59
hypocenterC59#10484
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. #21424898C2semidirect
5. C2×#16837147C2semidirect
6. #21421980C2semidirect
7. C2×C59×#334C2semidirect
205. C2×#334D59semidirect
235. C59#10484semidirect

Maximal subgroups

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

subgroupindex
C2×#168376832
#214248982
#214248902
#214240972
C2×#168371472
#214219802
C2×C59×#3342
#1048459