Finite Group Databasebeta

← #28491717

Subgroups — #28491717

Order 15104.

500 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429335C211yesmaximal, characteristic, complemented
2#21425023C211yesmaximal, characteristic
3#21425022C211yesmaximal, characteristic
4#21424902C211yesmaximal, characteristic
5#21421980C211yesmaximal, characteristic, complemented
6#21421936C211yesmaximal, characteristic, complemented
7C59×#1279C211yesmaximal, characteristic, complemented
8C2×#14273724C2211yescharacteristic
9C2×C4⋊(C59⋊C8)C2211yescharacteristic
10C2×#14273649—22no—
11C2×#14273649—21no—
12C2×#14273649—21no—
13C2×#14273647—21no—
14C2×#14273647—21no—
15#16837344—22no—
16#16837300—21no—
17#16837298—21no—
18#16837272C2211yescharacteristic
19#16837268—21no—
20#16837267—21no—
21C4×C4⋊Dic59C2211yescharacteristic
22C2×C4⋊C8×C59C2211yescharacteristic
23C2×C4×C8×C59C2211yescharacteristic
24C4×C4⋊C4×C59C2211yescharacteristic
25C22×D236—21no—
26C22×D236—21no—
27C22×D236—21no—
28C22×C59⋊C8—21no—
29#14273724—21no—
30#14273724—21no—
31#14273724—21no—
32#14273724Q812yes—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—21no—
35C2×C4⋊Dic59D411yescharacteristic
36C2×C4⋊Dic59D411yescharacteristic
37C2×C4⋊Dic59D412yes—
38C2×Dic59⋊C4—22no—
39C2×C4×Dic59—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42#14273649—42no—
43#14273649—42no—
44#14273649—44no—
45#14273647—42no—
46#14273647—42no—
47C59⋊#95—22no—
48C59⋊#95—22no—
49C59⋊#95—22no—
50C59⋊#95—21no—
51C22×C8×C59—21no—
52C22×C8×C59—21no—
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C4⋊C4×C59D411yescharacteristic
55C2×C42×C59—21no—
56C2×C42×C59C2311yescharacteristic
57C4⋊C8×C59—21no—
58C4⋊C8×C59—21no—
59C4×C8×C59—21no—
60C4×C8×C59—21no—
61#95×C59—21no—
62C23×D59—41no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—22no—
71C2×D236—22no—
72C2×D236—22no—
73C2×D236—22no—
74C2×D236—42no—
75C2×D236—42no—
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×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—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C42×C59—41no—
98C42×C59—41no—
99C42×C59C2×Q811yescharacteristic
100C42×C59#4111yescharacteristic
101C42×C59#4111yescharacteristic
102C42×C59#4111yescharacteristic
103C4⋊Dic59—21no—
104C4⋊Dic59—21no—
105C4⋊Dic59—21no—
106C4⋊Dic59—21no—
107C4⋊Dic59—41no—
108C4⋊Dic59—41no—
109C4⋊Dic59—41no—
110C4⋊Dic59—41no—
111C4⋊Dic59—42no—
112Dic59⋊C4—82no—
113C4×Dic59—41no—
114C4×Dic59—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—41no—
118C23×C59#12011yescharacteristic
119C22×D59—42no—
120C22×D59—81no—
121C22×D59—81no—
122C22×D59—81no—
123C22×D59—81no—
124C22×D59—81no—
125C22×D59—81no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59C22⋊Q811yescharacteristic
135C2×C4×C59C22⋊Q811yescharacteristic
136C2×C4×C59C22⋊Q811yescharacteristic
137C2×C4×C59#12411yescharacteristic
138C2×C4×C59#12411yescharacteristic
139C2×C4×C59#12411yescharacteristic
140C2×C4×C59#13511yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C2×C4×C59#13511yescharacteristic
143C2×C4×C59#13511yescharacteristic
144C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—42no—
153C2×Dic59—42no—
154C2×Dic59—41no—
155C2×Dic59—42no—
156C2×Dic59—41no—
157D236—44no—
158D236—42no—
159D236—42no—
160D236—42no—
161D236—42no—
162D236—82no—
163D236—81no—
164D236—81no—
165D236—81no—
166D236—81no—
167C8×C59—42no—
168C8×C59—81no—
169C59⋊C8—81no—
170#10351—591nomaximal, Sylow
171C22×C59#39311yescharacteristic
172C22×C59#44911yescharacteristic
173C22×C59#44911yescharacteristic
174C22×C59#44911yescharacteristic
175C22×C59#44911yescharacteristic
176C22×C59#45311yescharacteristic
177C22×C59#45411yescharacteristic
178C2×D59—82no—
179C2×D59—82no—
180C2×D59—82no—
181C2×D59—161no—
182C2×D59—161no—
183C2×D59—161no—
184C2×D59—161no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—81no—
188C4×C59—81no—
189C4×C59#47811yescharacteristic
190C4×C59#47811yescharacteristic
191C4×C59#47811yescharacteristic
192C4×C59#47811yescharacteristic
193C4×C59#48811yescharacteristic
194C4×C59#48811yescharacteristic
195C4×C59#49011yescharacteristic
196C4×C59#49011yescharacteristic
197Dic59—82no—
198Dic59—81no—
199Dic59—81no—
200#2571—591no—
201#1982—591no—
202#1982—591no—
203#1957—591no—
204#1296—591no—
205#1279—591no—
206#1279D5911yescharacteristic, complemented
207C2×C59#162011yescharacteristic
208C2×C59#162411yescharacteristic
209C2×C59#162811yescharacteristic
210C2×C59#162811yescharacteristic
211C2×C59#202511yescharacteristic
212C2×C59#202511yescharacteristic
213C2×C59#202711yescharacteristic
214D59—162no—
215C2×#127—591no—
216C2×C4⋊C8—591no—
217C2×C4⋊C8C2×D5911yescharacteristic
218C2×#102—1182no—
219C2×#102—1181no—
220C2×#102—1181no—
221C2×#102—1181no—
222C2×#102—1181no—
223C2×C4×C8C2×D5911yescharacteristic
224#400—1182no—
225#389—1181no—
226#389—1181no—
227#384—591no—
228#382—1181no—
229#382—1181no—
230C4×C4⋊C4—591no—
231C4×C4⋊C4C2×D5911yescharacteristic
232C59#1035111yescharacteristic, complemented, Sylow
233C22×D4—1181no—
234C22×D4—1181no—
235C22×D4—1181no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—1181no—
239#127—592no—
240#127—1181no—
241#127—1181no—
242#127—1181no—
243C2×C4⋊C4—591no—
244C2×C4⋊C4—591no—
245C2×C4⋊C4—592no—
246C2×C4⋊C4—1182no—
247C2×C4⋊C4—1181no—
248C2×C4⋊C4—1181no—
249C2×C4⋊C4D23611yescharacteristic
250C2×C4⋊C4D23611yescharacteristic
251C2×C42—21no—
252C2×C42—1181no—
253C2×C42C22×D5911yescharacteristic
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—1181no—
257C4⋊C8—1181no—
258#102—2362no—
259#102—2362no—
260#102—2362no—
261#102—2362no—
262#102—2364no—
263C4×C8—21no—
264C4×C8—21no—
265#95—21no—
266#95—1182no—
267#95—1182no—
268#95—1182no—
269#95—1181no—
270C24—2361no—
271C2×D4—1182no—
272C2×D4—1182no—
273C2×D4—1182no—
274C2×D4—2362no—
275C2×D4—2362no—
276C2×D4—2361no—
277C2×D4—2361no—
278C2×D4—2361no—
279C2×D4—2361no—
280C2×D4—2361no—
281C2×D4—2361no—
282C2×D4—2361no—
283C2×D4—2361no—
284C2×D4—2361no—
285C2×D4—2361no—
286C2×D4—2361no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1181no—
290C22×C4—1181no—
291C22×C4—1182no—
292C22×C4C2×D23611yescharacteristic
293C22×C4D4×D5911yescharacteristic
294C22×C4D4×D5911yescharacteristic
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—2361no—
302C2×C8—2361no—
303C2×C8—2361no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—1181no—
309C4⋊C4—1181no—
310C4⋊C4—1181no—
311C4⋊C4—1181no—
312C4⋊C4—4722no—
313C4⋊C4—2362no—
314C4⋊C4—2361no—
315C4⋊C4—2361no—
316C4⋊C4—2361no—
317C4⋊C4—2361no—
318C42—41no—
319C42—41no—
320C42—2361no—
321C42—2361no—
322C42#1174568511yescharacteristic
323C42#1174568511yescharacteristic
324C42#1174568711yescharacteristic
325C42Q8×D5911yescharacteristic
326C23—4721no—
327C23—4721no—
328C23—4721no—
329C23—4721no—
330C23—4721no—
331C23—4721no—
332C23—2362no—
333C23#1427369711yescharacteristic
334D4—4722no—
335D4—4721no—
336D4—4721no—
337D4—4721no—
338D4—4721no—
339D4—2364no—
340D4—2362no—
341D4—2362no—
342D4—2362no—
343D4—2362no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—2361no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4—2362no—
356C2×C4—2361no—
357C2×C4—2362no—
358C2×C4—2361no—
359C2×C4—2362no—
360C2×C4—2361no—
361C2×C4#1427370511yescharacteristic
362C2×C4#1427370511yescharacteristic
363C2×C4#1427370711yescharacteristic
364C2×C4#1427371311yescharacteristic
365C2×C4#1427371311yescharacteristic
366C2×C4#1427371511yescharacteristic
367C2×C4#1427374811yescharacteristic
368C2×C4#1427374911yescharacteristic
369C2×C4#1427374911yescharacteristic
370C2×C4#1427375011yescharacteristic
371C2×C4#1427375111yescharacteristic
372C2×C4#1427375411yescharacteristic
373C2×C4#1427375511yescharacteristic
374C2×C4#1427375511yescharacteristic
375C8—42no—
376C8—81no—
377C8—4721no—
378C22—9441no—
379C22—9441no—
380C22—9441no—
381C22—9441no—
382C22—4722no—
383C22—4722no—
384C22—4722no—
385C22#1683731511yescharacteristic
386C22#1683754811yescharacteristic
387C22#1683754911yescharacteristic
388C22#1683754911yescharacteristic
389C22#1683755011yescharacteristic
390C22#1683757111yescharacteristic
391C22#1683757611yescharacteristic
392C4—21no—
393C4—21no—
394C4—81no—
395C4—81no—
396C4—4722no—
397C4—4721no—
398C4—4721no—
399C4#1683770111yescharacteristic
400C4#1683770311yescharacteristic
401C4#1683770311yescharacteristic
402C4#1683770511yescharacteristic
403C4#1683775711yescharacteristic
404C4#1683775811yescharacteristic
405C4#1683776511yescharacteristic
406C4#1683776711yescharacteristic
407C2—9442no—
408C2#2142315711yescharacteristic
409C2#2142318111yescharacteristic
410C2#2142320811yescharacteristic
411C2#2142320911yescharacteristic
412C2#2142524111yescharacteristic
413C2#2142524311yescharacteristic
414C2#2142525111yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1279C2
socleC23×C59#120
prime radicalC2×#14273724C22
hypercenter#1279D59
hypocenterC59#10351
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #21429335C2semidirect
5. #21421980C2semidirect
6. #21421936C2semidirect
7. C59×#1279C2semidirect
206. #1279D59semidirect
232. C59#10351semidirect

Maximal subgroups

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

subgroupindex
#214293352
#214250232
#214250222
#214249022
#214219802
#214219362
C59×#12792
#1035159