Finite Group Databasebeta

← #28492545

Subgroups — #28492545

Order 15104.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837693C211yesmaximal, characteristic
2#21425056C211yesmaximal, characteristic, complemented
3#21424889C211yesmaximal, characteristic
4#21424093C211yesmaximal, characteristic
5C2×#16837154C211yesmaximal, characteristic, complemented
6C2×#16837150C211yesmaximal, characteristic, complemented
7C59×#1296C211yesmaximal, characteristic, complemented
8C2×#14273727C2211yescharacteristic
9C2×C4×D236C2211yescharacteristic
10#16837693—22no—
11#16837693—22no—
12C2×#14273659—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273649—21no—
15C2×#14273647—21no—
16C2×(C2×D59)⋊C8—21no—
17C2×(C2×D59)⋊C8—21no—
18C2×C4×C59⋊C8C2211yescharacteristic
19#16837164—21no—
20#16837164—21no—
21#16837164—21no—
22#16837154—22no—
23#16837150—22no—
24C2×C4⋊C8×C59C2211yescharacteristic
25C2×C4⋊C8×C59C2211yescharacteristic
26C59×#384C2211yescharacteristic
27C22×Dic118—21no—
28C22×D236D411yescharacteristic
29C22×C4×D59—21no—
30C22×C59⋊C8—21no—
31C22×C59⋊C8D411yescharacteristic
32C22×C59⋊C8D411yescharacteristic
33#14273727—21no—
34#14273727—21no—
35#14273727—21no—
36#14273727D412yes—
37C4×D236—21no—
38C4×D236—21no—
39C4×D236—21no—
40C4×D236—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59D411yescharacteristic
44C2×(C2×D59)⋊C4—21no—
45#14273659—24no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48#14273649—24no—
49#14273647—42no—
50(C2×D59)⋊C8—42no—
51(C2×D59)⋊C8—42no—
52C4×C59⋊C8—21no—
53C4×C59⋊C8—21no—
54C22×C8×C59—21no—
55C22×C8×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C42×C59C2311yescharacteristic
60C4⋊C8×C59—21no—
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59D412yes—
64C23×D59—21no—
65C22×C4×C59—41no—
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59#4111yescharacteristic
69C22×Dic59—21no—
70C22×Dic59—41no—
71C2×D236—21no—
72C2×D236—21no—
73C2×D236—21no—
74C2×D236—21no—
75C2×D236—21no—
76C2×D236Q1612yes—
77C2×C4×D59—41no—
78C2×C4×D59—41no—
79C2×C4×D59—41no—
80C2×C4×D59—41no—
81C2×C4×D59—41no—
82C2×C4×D59—41no—
83C2×Dic118—22no—
84C2×Dic118—41no—
85C2×Dic118—81no—
86C2×Dic118—81no—
87C2×C8×C59—22no—
88C2×C8×C59—41no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C2×C8×C59—41no—
92C2×C8×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59—81no—
98C42×C59C2×D411yescharacteristic
99C42×C59C2×D411yescharacteristic
100C42×C59#4111yescharacteristic
101C42×C59#4111yescharacteristic
102(C2×D59)⋊C4—41no—
103(C2×D59)⋊C4—41no—
104C4⋊Dic59—22no—
105C4⋊Dic59—21no—
106C4⋊Dic59—22no—
107C4⋊Dic59—21no—
108C4⋊Dic59—41no—
109C4⋊Dic59—41no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—21no—
114C2×C59⋊C8—21no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—41no—
118C2×C59⋊C8SD1612yes—
119C23×C59#12111yescharacteristic
120C22×D59—22no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—41no—
124C22×D59—41no—
125C22×D59—41no—
126C22×D59—41no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—81no—
130C2×C4×C59—81no—
131C2×C4×C59—81no—
132C2×C4×C59#12011yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59#12311yescharacteristic
136C2×C4×C59#12311yescharacteristic
137C2×C4×C59#12411yescharacteristic
138C2×C4×C59C2×SD1611yescharacteristic
139C2×C4×C59C2×Q1611yescharacteristic
140C2×C4×C59#13511yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C59#13711yescharacteristic
145C2×C4×C59#13711yescharacteristic
146C2×Dic59—42no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—81no—
151C2×Dic59—81no—
152D236—22no—
153D236—22no—
154D236—41no—
155D236—41no—
156C4×D59—81no—
157C4×D59—81no—
158C4×D59—81no—
159C4×D59—81no—
160Dic118—82no—
161Dic118—82no—
162C8×C59—42no—
163C8×C59—81no—
164C59⋊C8—22no—
165C59⋊C8—41no—
166C59⋊C8—81no—
167#10481—591nomaximal, Sylow
168C22×C59#39411yescharacteristic
169C22×C59#46111yescharacteristic
170C22×C59#46311yescharacteristic
171C22×C59#46711yescharacteristic
172C22×C59#46811yescharacteristic
173C22×C59#47111yescharacteristic
174C22×C59#47211yescharacteristic
175C2×D59—42no—
176C2×D59—42no—
177C2×D59—42no—
178C2×D59—81no—
179C2×D59—81no—
180C2×D59—81no—
181C2×D59—81no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—161no—
185C4×C59#44911yescharacteristic
186C4×C59#45111yescharacteristic
187C4×C59#46011yescharacteristic
188C4×C59#46411yescharacteristic
189C4×C59#46511yescharacteristic
190C4×C59#46811yescharacteristic
191C4×C59#48411yescharacteristic
192C4×C59#48511yescharacteristic
193Dic59—82no—
194Dic59—81no—
195C2×#476—591no—
196#1990—591no—
197#1954—591no—
198#1802—591no—
199C2×#335—591no—
200C2×#333—591no—
201#1296D5911yescharacteristic, complemented
202C2×C59#167511yescharacteristic
203C2×C59#168311yescharacteristic
204C2×C59#169111yescharacteristic
205C2×C59#169711yescharacteristic
206C2×C59#203611yescharacteristic
207C2×C59#205111yescharacteristic
208C2×C59#205511yescharacteristic
209D59—82no—
210C2×C4⋊Q8—591no—
211C2×C4×D4—591no—
212#476—1182no—
213#476—1182no—
214C2×#106—1181no—
215C2×C4⋊C8—591no—
216C2×C4⋊C8C2×D5911yescharacteristic
217C2×C4⋊C8C2×D5911yescharacteristic
218C2×#102—1181no—
219C2×#102—1181no—
220C2×C22⋊C8—1181no—
221C2×C22⋊C8—1181no—
222C2×C4×C8—591no—
223#384C2×D5911yescharacteristic
224#340—1181no—
225#340—1181no—
226#340—1181no—
227#335—1182no—
228#333—1182no—
229C59#1048111yescharacteristic, complemented, Sylow
230C22×Q8—1181no—
231C22×D4—591no—
232C23×C4—1181no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—591no—
236C22×C8—591no—
237C22×C8—1181no—
238C4⋊Q8—592no—
239C4⋊Q8—1181no—
240C4⋊Q8—1181no—
241C4⋊Q8—1181no—
242C4×D4—1181no—
243C4×D4—1181no—
244C4×D4—1181no—
245C4×D4—1181no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—591no—
250C2×C4⋊C4—1181no—
251C2×C4⋊C4—1181no—
252C2×C22⋊C4—1181no—
253C2×C42C22×D5911yescharacteristic
254#106—1184no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—1181no—
259C4⋊C8—1181no—
260C4⋊C8D23612yes—
261#102—1184no—
262#102—2362no—
263C22⋊C8—2362no—
264C22⋊C8—2362no—
265C4×C8—1181no—
266C4×C8—1181no—
267C24—1181no—
268C2×Q8—1182no—
269C2×Q8—4721no—
270C2×Q8—4721no—
271C2×Q8—2361no—
272C2×D4—592no—
273C2×D4—1181no—
274C2×D4—1181no—
275C2×D4—1181no—
276C2×D4—1181no—
277C2×D4—1181no—
278C22×C4—41no—
279C22×C4—1181no—
280C22×C4—2361no—
281C22×C4—2361no—
282C22×C4—2361no—
283C22×C4—2361no—
284C22×C4—2361no—
285C22×C4—2361no—
286C22×C4—2361no—
287C22×C4#1174568511yescharacteristic
288C22×C4D4×D5911yescharacteristic
289C22×C4D4×D5911yescharacteristic
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—592no—
297C2×C8—1181no—
298C2×C8—1181no—
299C2×C8—1181no—
300C2×C8—1181no—
301C2×C8—1181no—
302C2×C8—2361no—
303C2×C8—2361no—
304C2×C8—2361no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—81no—
310C4⋊C4—1182no—
311C4⋊C4—1181no—
312C4⋊C4—1182no—
313C4⋊C4—1181no—
314C4⋊C4—2361no—
315C4⋊C4—2361no—
316C22⋊C4—2361no—
317C22⋊C4—2361no—
318C42C2×D23611yescharacteristic
319C42#1174568511yescharacteristic
320C42D4×D5911yescharacteristic
321C42#1174568911yescharacteristic
322C23—1182no—
323C23—2361no—
324C23—2361no—
325C23—2361no—
326C23—2361no—
327C23—2361no—
328C23—2361no—
329C23#1427370011yescharacteristic
330Q8—4722no—
331Q8—4722no—
332D4—1182no—
333D4—1182no—
334D4—2361no—
335D4—2361no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—4721no—
342C2×C4—4721no—
343C2×C4—4721no—
344C2×C4—4721no—
345C2×C4—4721no—
346C2×C4—4721no—
347C2×C4—2361no—
348C2×C4—2362no—
349C2×C4—2361no—
350C2×C4—2361no—
351C2×C4#1427369711yescharacteristic
352C2×C4#1427370011yescharacteristic
353C2×C4#1427370211yescharacteristic
354C2×C4#1427371211yescharacteristic
355C2×C4#1427371211yescharacteristic
356C2×C4#1427371511yescharacteristic
357C2×C4SD16×D5911yescharacteristic
358C2×C4Q16×D5911yescharacteristic
359C2×C4#1427374811yescharacteristic
360C2×C4#1427374811yescharacteristic
361C2×C4#1427375111yescharacteristic
362C2×C4#1427375111yescharacteristic
363C2×C4#1427375611yescharacteristic
364C2×C4#1427375911yescharacteristic
365C8—42no—
366C8—81no—
367C8—1182no—
368C8—4721no—
369C8—2361no—
370C22—4721no—
371C22—4721no—
372C22—4721no—
373C22—4721no—
374C22—2362no—
375C22—2362no—
376C22—2362no—
377C22#1683732011yescharacteristic
378C22#1683760611yescharacteristic
379C22#1683761611yescharacteristic
380C22#1683764211yescharacteristic
381C22#1683764711yescharacteristic
382C22#1683766811yescharacteristic
383C22#1683767111yescharacteristic
384C4—21no—
385C4—21no—
386C4—161no—
387C4—4721no—
388C4—4722no—
389C4#1683754811yescharacteristic
390C4#1683755911yescharacteristic
391C4#1683760111yescharacteristic
392C4#1683762711yescharacteristic
393C4#1683763211yescharacteristic
394C4#1683764911yescharacteristic
395C4#1683773911yescharacteristic
396C4#1683774611yescharacteristic
397C2—4722no—
398C2#2142349411yescharacteristic
399C2#2142354611yescharacteristic
400C2#2142359411yescharacteristic
401C2#2142362811yescharacteristic
402C2#2142530211yescharacteristic
403C2#2142539111yescharacteristic
404C2#2142541511yescharacteristic
subgroupG/H
centerC23#14273700
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1296C2
socleC23×C59#121
prime radicalC22×D236D4
hypercenter#1296D59
hypocenterC59#10481
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. #21425056C2semidirect
5. C2×#16837154C2semidirect
6. C2×#16837150C2semidirect
7. C59×#1296C2semidirect
201. #1296D59semidirect
229. C59#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#168376932
#214250562
#214248892
#214240932
C2×#168371542
C2×#168371502
C59×#12962
#1048159