Finite Group Databasebeta

← #28485388

Subgroups — #28485388

Order 15104.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435037C211yesmaximal, characteristic
2#21425077C211yesmaximal, characteristic, complemented
3#21422764C211yesmaximal, characteristic
4#21422659C211yesmaximal, characteristic, complemented
5#21421954C211yesmaximal, characteristic, complemented
6#21421925C211yesmaximal, characteristic, complemented
7C59×#2328C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837774C412yescomplemented
10#16837748C412yescomplemented
11#16837599—22no—
12#16837553—22no—
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16#16837162—21no—
17#16837162—21no—
18#16837146—22no—
19#16837138—22no—
20C2×#127×C59C2211yescharacteristic
21C2×C4×D4×C59—21no—
22C59×#390—21no—
23C59×#386—22no—
24C59×#386—22no—
25C59×#382C2211yescharacteristic
26C59×#376C2211yescharacteristic
27C2×#11745670—24no—
28C2×#11745664—21no—
29#14273727C2×C411yescharacteristic, complemented
30#14273727C2×C411yescharacteristic, complemented
31#14273719—21no—
32#14273706—21no—
33#14273706—21no—
34C4×Dic118—21no—
35C4×Dic118—21no—
36#14273686—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41#14273648—24no—
42C22⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C4×C59⋊C8—21no—
45C4×C59⋊C8C2×C412yescomplemented
46C22×D4×C59—21no—
47C22×D4×C59—21no—
48C22×D4×C59D412yes—
49C23×C4×C59—22no—
50#127×C59—21no—
51#127×C59—21no—
52#127×C59C2×C412yes—
53#127×C59D412yes—
54C4×D4×C59—42no—
55C4×D4×C59—42no—
56C2×C4⋊C4×C59—21no—
57C2×C22⋊C4×C59—22no—
58C2×C22⋊C4×C59—22no—
59C2×C22⋊C4×C59—22no—
60C2×C42×C59D411yescharacteristic
61C2×C42×C59D411yescharacteristic
62C2×C42×C59C2311yescharacteristic
63#95×C59—21no—
64#95×C59—21no—
65#95×C59—21no—
66#95×C59D412yes—
67C24×C59—22no—
68C2×D4×C59—22no—
69C2×D4×C59—22no—
70C2×D4×C59—22no—
71C2×D4×C59—22no—
72C2×D4×C59—22no—
73C2×D4×C59—41no—
74C2×D4×C59—41no—
75C2×D4×C59—41no—
76C2×D4×C59—41no—
77C2×D4×C59—82no—
78C2×D4×C59—82no—
79C2×D4×C59C22⋊C414yes—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—22no—
83C22×C4×C59—44no—
84C22×C4×C59—82no—
85C22×C4×C59—82no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C2×Dic118—21no—
90C2×Dic118—21no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—44no—
94C22⋊C4×C59—82no—
95C22⋊C4×C59—82no—
96C42×C59—21no—
97C42×C59—21no—
98C42×C59—21no—
99C42×C59SD1612yescomplemented
100C42×C59C22×C411yescharacteristic
101C42×C59C2×D411yescharacteristic
102C42×C59#4111yescharacteristic
103C42×C59#4111yescharacteristic
104C22⋊Dic59—41no—
105#11745670—84no—
106C4⋊Dic59—21no—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110Dic59⋊C4—41no—
111Dic59⋊C4—41no—
112C4×Dic59—41no—
113#11745664—41no—
114#11745664—41no—
115C2×C59⋊C8—22no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—41no—
118C23×C59—24no—
119C23×C59—42no—
120C23×C59—42no—
121C23×C59—42no—
122C23×C59—42no—
123C23×C59—42no—
124C23×C59—42no—
125C23×C59#12011yescharacteristic
126D4×C59—44no—
127D4×C59—42no—
128D4×C59—42no—
129D4×C59—42no—
130D4×C59—42no—
131D4×C59—82no—
132D4×C59—82no—
133D4×C59—82no—
134C2×C4×C59—22no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—42no—
144C2×C4×C59—41no—
145C2×C4×C59—88no—
146C2×C4×C59C2×C22⋊C411yescharacteristic
147C2×C4×C59C4×D411yescharacteristic
148C2×C4×C59C4×D411yescharacteristic
149C2×C4×C59#12111yescharacteristic
150C2×C4×C59#12111yescharacteristic
151C2×C4×C59#12311yescharacteristic
152C2×C4×C59C2×SD1611yescharacteristic
153C2×C4×C59#13511yescharacteristic
154C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C59#13711yescharacteristic
156C2×Dic59—41no—
157C2×Dic59—41no—
158Dic118—42no—
159Dic118—81no—
160C59⋊C8—42no—
161C59⋊C8—81no—
162C59⋊C8—81no—
163#9295—591nomaximal, Sylow
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—44no—
167C22×C59—44no—
168C22×C59—44no—
169C22×C59—82no—
170C22×C59—82no—
171C22×C59—82no—
172C22×C59—82no—
173C22×C59#38611yescharacteristic
174C22×C59#45011yescharacteristic
175C22×C59#45211yescharacteristic
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—42no—
181C4×C59—81no—
182C4×C59—82no—
183C4×C59C4×SD1611yescharacteristic
184C4×C59#44011yescharacteristic
185C4×C59#46111yescharacteristic
186C4×C59#46311yescharacteristic
187Dic59—81no—
188Dic59—81no—
189#3208—591no—
190#2328D5911yescharacteristic
191#1995—591no—
192#1545—591no—
193#1518—591no—
194#1286—591no—
195#1274—591no—
196C2×C59—21no—
197C2×C59—21no—
198C2×C59—84no—
199C2×C59#187711yescharacteristic
200C2×C59#190411yescharacteristic
201C2×C59#191111yescharacteristic
202#533—591no—
203C2×#127C2×D5911yescharacteristic
204C2×C4×D4—21no—
205#492—592no—
206#486—592no—
207#460—1182no—
208#450—1182no—
209#431—591no—
210#431—591no—
211#423—591no—
212#390—21no—
213#386—22no—
214#386—22no—
215#382C2×D5911yescharacteristic
216#376C2×D5911yescharacteristic
217#339—1181no—
218#339—1181no—
219#331—1182no—
220#328—1182no—
221C59#929511yescharacteristic, complemented, Sylow
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C22⋊D5912yes—
225C23×C4—22no—
226C2×SD16—1184no—
227C2×M16—1181no—
228C4⋊Q8—591no—
229C4⋊Q8—591no—
230#127—21no—
231#127—21no—
232#127C4×D5912yes—
233#127D23612yes—
234#125—1181no—
235C22⋊Q8—1181no—
236C22⋊Q8—1181no—
237C4×Q8—1181no—
238C4×Q8—1181no—
239C4×D4—42no—
240C4×D4—42no—
241#117—1181no—
242C2×C4⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C42C22⋊D5911yescharacteristic
247C2×C42C22⋊D5911yescharacteristic
248C2×C42C22×D5911yescharacteristic
249C4⋊C8—1181no—
250C4⋊C8—1181no—
251C4⋊C8—1181no—
252C4⋊C8—1181no—
253#102—1184no—
254C22⋊C8—1181no—
255C22⋊C8—1181no—
256C4×C8—592no—
257C4×C8—1181no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95C22⋊D5912yes—
262C24—22no—
263C2×Q8—1181no—
264C2×Q8—1181no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4(C2×D59)⋊C414yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×C22⋊D5911yescharacteristic
284C22×C4C2×C22⋊D5911yescharacteristic
285C22×C4C2×C22⋊D5911yescharacteristic
286SD16—4724no—
287M16—2361no—
288M16—2361no—
289C2×C8—1182no—
290C2×C8—2361no—
291C2×C8—2361no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—1181no—
295C4⋊C4—1181no—
296C4⋊C4—1181no—
297C4⋊C4—1181no—
298C4⋊C4—2361no—
299C4⋊C4—2361no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—2361no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—2361no—
308C42#1174567012yescomplemented
309C42C2×C4×D5911yescharacteristic
310C42C2×D23611yescharacteristic
311C42#1174568511yescharacteristic
312C42#1174568511yescharacteristic
313C23—24no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23#1427369611yescharacteristic
321Q8—4721no—
322Q8—2362no—
323D4—44no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—88no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4C2×(C2×D59)⋊C411yescharacteristic
346C2×C4C4×C22⋊D5911yescharacteristic
347C2×C4C4×C22⋊D5911yescharacteristic
348C2×C4#1427370111yescharacteristic
349C2×C4#1427370111yescharacteristic
350C2×C4#1427371011yescharacteristic
351C2×C4C2×#1174567011yescharacteristic
352C2×C4#1427374711yescharacteristic
353C2×C4#1427375311yescharacteristic
354C2×C4#1427375711yescharacteristic
355C8—4721no—
356C8—4721no—
357C8—2362no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22#1683728111yescharacteristic
368C22#1683755311yescharacteristic
369C22#1683756411yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—82no—
376C4—81no—
377C4—4721no—
378C4—4721no—
379C4C4×#1174567011yescharacteristic
380C4#1683750211yescharacteristic
381C4#1683760711yescharacteristic
382C4#1683761711yescharacteristic
383C2—21no—
384C2—21no—
385C2—84no—
386C2#2142443011yescharacteristic
387C2#2142458311yescharacteristic
388C2#2142462811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2328C2
socleC22×C59#386
prime radicalC2×#127×C59C22
cosocleC2×C42×C59C23
hypercenter#2328D59
hypocenterC59#9295
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #21425077C2semidirect
4. #21422659C2semidirect
5. #21421954C2semidirect
6. #21421925C2semidirect
9. #16837774C4semidirect
10. #16837748C4semidirect
29. #14273727C2×C4semidirect
30. #14273727C2×C4semidirect
45. C4×C59⋊C8C2×C4semidirect
99. C42×C59SD16semidirect
221. C59#9295semidirect
308. C42#11745670semidirect

Maximal subgroups

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

subgroupindex
#214350372
#214250772
#214227642
#214226592
#214219542
#214219252
C59×#23282
#929559