Finite Group Databasebeta

← #28483000

Subgroups — #28483000

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432665C211yesmaximal, characteristic
2#21424113C211yesmaximal, characteristic
3#21422868C211yesmaximal, characteristic, complemented
4#21422479C211yesmaximal, characteristic
5#21421997C211yesmaximal, characteristic, complemented
6#21421936C211yesmaximal, characteristic, complemented
7C59×#1566C211yesmaximal, characteristic, complemented
8C2×#14273735—21no—
9C4×#11745685C2211yescharacteristic
10#16837455C2211yescharacteristic
11#16837430—21no—
12#16837427—22no—
13#16837424—21no—
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C4⋊C8×D59—21no—
16C2×#14273653—21no—
17C4×C4⋊Dic59C2211yescharacteristic
18#16837163—21no—
19#16837152—21no—
20#16837152—21no—
21#16837134C411yescharacteristic, complemented
22#16837134C411yescharacteristic, complemented
23#16837131C411yescharacteristic, complemented
24#16837131C411yescharacteristic, complemented
25C59×#423C2211yescharacteristic
26C59×#423C2211yescharacteristic
27C2×C4×C8×C59C2211yescharacteristic
28C59×#334—22no—
29C2×#11745685D411yescharacteristic
30#14273735—42no—
31#14273735—42no—
32C2×#11745659—22no—
33C22×C59⋊C8—21no—
34C2×C8×D59—22no—
35C4×Dic118C2×C411yescharacteristic
36C4×Dic118Q811yescharacteristic
37C4×D236C2×C411yescharacteristic
38C4×D236Q811yescharacteristic
39C4×C22⋊D59—21no—
40C4×C22⋊D59—21no—
41#14273686D411yescharacteristic
42#14273684—21no—
43C2×C4⋊Dic59—21no—
44C2×C4×Dic59—21no—
45C42×D59—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)C2×C411yescharacteristic
48C4⋊(C59⋊C8)C2×C411yescharacteristic
49C4⋊(C59⋊C8)D411yescharacteristic
50C4⋊(C59⋊C8)D411yescharacteristic
51Dic59⋊C8—22no—
52Dic59⋊C8—22no—
53#14273653—42no—
54C22⋊(C59⋊C8)—21no—
55C4×C59⋊C8—21no—
56C59⋊#95—21no—
57C2×M16×C59—21no—
58C2×M16×C59—21no—
59C22×C8×C59—21no—
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59—21no—
64C4⋊C8×C59D412yes—
65C4×C8×C59—21no—
66C4×C8×C59C2×C411yescharacteristic
67C4×C8×C59C2×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59#4111yescharacteristic
70C22×C4×C59#4111yescharacteristic
71C2×C22⋊D59—21no—
72C22×Dic59—41no—
73#11745685—21no—
74#11745685—21no—
75#11745685SD1612yes—
76#11745685SD1612yes—
77C2×D236C4⋊C411yescharacteristic
78C2×C4×D59—21no—
79C2×C4×D59—21no—
80C2×C4×D59—21no—
81C2×Dic118C4⋊C411yescharacteristic
82M16×C59—41no—
83M16×C59—41no—
84M16×C59—41no—
85M16×C59—41no—
86C2×C8×C59—21no—
87C2×C8×C59—21no—
88C2×C8×C59—22no—
89C2×C8×C59—42no—
90C2×C8×C59—41no—
91C42×C59C22×C411yescharacteristic
92C42×C59C2×D411yescharacteristic
93C42×C59C2×D411yescharacteristic
94C42×C59C2×Q811yescharacteristic
95C22⋊Dic59—21no—
96(C2×D59)⋊C4—21no—
97(C2×D59)⋊C4—21no—
98C4⋊Dic59—22no—
99C4⋊Dic59—41no—
100C4⋊Dic59C4⋊C411yescharacteristic
101C4⋊Dic59C4⋊C411yescharacteristic
102Dic59⋊C4—21no—
103Dic59⋊C4—21no—
104C4×Dic59—21no—
105C4×Dic59—21no—
106C4×Dic59—21no—
107C4×Dic59—41no—
108C4×Dic59—41no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115#11745659—44no—
116C8×D59—44no—
117C23×C59#12311yescharacteristic
118C22×D59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—41no—
122C2×C4×C59C2×C4⋊C411yescharacteristic
123C2×C4×C59C4×D411yescharacteristic
124C2×C4×C59C4×D411yescharacteristic
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59C22⋊Q811yescharacteristic
127C2×C4×C59C22⋊Q811yescharacteristic
128C2×C4×C59C2×SD1611yescharacteristic
129C2×C4×C59C2×SD1611yescharacteristic
130C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C59#13711yescharacteristic
132C22⋊D59—22no—
133C22⋊D59—22no—
134C2×Dic59—21no—
135C2×Dic59—21no—
136C2×Dic59—21no—
137C2×Dic59—42no—
138C2×Dic59—81no—
139C2×Dic59—81no—
140D236—21no—
141D236#10612yes—
142C4×D59—22no—
143C4×D59—22no—
144C4×D59—41no—
145C4×D59—41no—
146C4×D59—81no—
147Dic118—21no—
148Dic118#10612yes—
149C8×C59—22no—
150C8×C59—41no—
151C8×C59—42no—
152C8×C59—81no—
153C59⋊C8—41no—
154C59⋊C8—41no—
155C59⋊C8—41no—
156C59⋊C8—41no—
157#8904—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59#38511yescharacteristic
161C22×C59#48111yescharacteristic
162C22×C59#48311yescharacteristic
163C2×D59—22no—
164C2×D59—41no—
165C2×D59—41no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59C2×#10611yescharacteristic
170C4×C59#42811yescharacteristic
171C4×C59#44311yescharacteristic
172C4×C59#44411yescharacteristic
173C4×C59#44811yescharacteristic
174C4×C59#45011yescharacteristic
175Dic59—22no—
176Dic59—41no—
177Dic59—41no—
178Dic59—82no—
179#2906—591no—
180#1806—591no—
181#1570—591no—
182#1566D5911yescharacteristic, complemented
183#1476—591no—
184#1303—591no—
185#1279—591no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59#185311yescharacteristic
189C2×C59#186111yescharacteristic
190C2×C59#186611yescharacteristic
191D59—42no—
192C2×#131—1181no—
193C4×#41—591no—
194#431—591no—
195#424—1181no—
196#424—1182no—
197#423—1181no—
198#423C2×D5911yescharacteristic
199#423C2×D5911yescharacteristic
200C2×C4⋊C8—591no—
201C2×C4⋊C8—1181no—
202C2×#104—1181no—
203C2×C4×C8C2×D5911yescharacteristic
204C4×C4⋊C4—591no—
205#339—1181no—
206#334—22no—
207#334—1181no—
208#334—1181no—
209Q8⋊C8—591no—
210Q8⋊C8—591no—
211D4⋊C8—591no—
212D4⋊C8—591no—
213C59#890411yescharacteristic, complemented, Sylow
214C2×#41—591no—
215#131—2362no—
216#131—2362no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—1182no—
220C22×C8—21no—
221C22×C8—1182no—
222C22×C8—1181no—
223C4×Q8—591no—
224C4×Q8—591no—
225C4×D4—591no—
226C4×D4—591no—
227C4×D4—1181no—
228C4×D4—1181no—
229#117—591no—
230#117—1181no—
231C2×C4⋊C4—1181no—
232C2×C42—1181no—
233C2×C42—1181no—
234C2×C42C22×D5911yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—591no—
239C4⋊C8—591no—
240C4⋊C8—591no—
241C4⋊C8—591no—
242C4⋊C8—1182no—
243C4⋊C8—1182no—
244C4⋊C8—1181no—
245C4⋊C8C22⋊D5912yes—
246#104—2362no—
247C22⋊C8—1181no—
248C4×C8—21no—
249C4×C8—1181no—
250C4×C8C4×D5911yescharacteristic
251C4×C8C4×D5911yescharacteristic
252#95—1181no—
253#41—592no—
254#41—592no—
255#41—1181no—
256#41—1181no—
257C2×Q8—591no—
258C2×D4—591no—
259C2×D4—1181no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4—2361no—
264C22×C4#1174568511yescharacteristic
265C22×C4D4×D5911yescharacteristic
266C22×C4#1174568911yescharacteristic
267M16—41no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—2364no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—22no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—1181no—
278C2×C8—1181no—
279C2×C8—1181no—
280C2×C8—1181no—
281C2×C8—2364no—
282C2×C8—2361no—
283C2×C8—2361no—
284C4⋊C4—591no—
285C4⋊C4—591no—
286C4⋊C4—1182no—
287C4⋊C4—1181no—
288C4⋊C4—1181no—
289C4⋊C4—2361no—
290C22⋊C4—1181no—
291C22⋊C4—1181no—
292C22⋊C4—1181no—
293C42—1181no—
294C42—1181no—
295C42—1181no—
296C42—2361no—
297C42—2361no—
298C42C2×C4×D5911yescharacteristic
299C42D4×D5911yescharacteristic
300C42Q8×D5911yescharacteristic
301C42C2×C22⋊D5911yescharacteristic
302C23—1181no—
303C23#1427371211yescharacteristic
304Q8—592no—
305Q8—1181no—
306D4—592no—
307D4—1182no—
308D4—1182no—
309D4—1181no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—1181no—
314C2×C4—1182no—
315C2×C4—1182no—
316C2×C4—1181no—
317C2×C4—1181no—
318C2×C4—4721no—
319C2×C4—4721no—
320C2×C4—4721no—
321C2×C4—2362no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4C4⋊C4×D5911yescharacteristic
325C2×C4C4×C22⋊D5911yescharacteristic
326C2×C4#1427368911yescharacteristic
327C2×C4#1427369511yescharacteristic
328C2×C4#1427370311yescharacteristic
329C2×C4#1427370511yescharacteristic
330C2×C4SD16×D5911yescharacteristic
331C2×C4SD16×D5911yescharacteristic
332C2×C4#1427375411yescharacteristic
333C2×C4#1427375611yescharacteristic
334C8—22no—
335C8—41no—
336C8—42no—
337C8—81no—
338C8—2361no—
339C8—2361no—
340C8—2361no—
341C8—2361no—
342C22—21no—
343C22—21no—
344C22—1182no—
345C22—2361no—
346C22—2361no—
347C22#1683727311yescharacteristic
348C22#1683772211yescharacteristic
349C22#1683773411yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—1182no—
354C4—4722no—
355C4—2361no—
356C4—2361no—
357C4#106×D5911yescharacteristic
358C4#1683744111yescharacteristic
359C4#1683751811yescharacteristic
360C4#1683752311yescharacteristic
361C4#1683753911yescharacteristic
362C4#1683755111yescharacteristic
363C2—21no—
364C2—21no—
365C2—2362no—
366C2#2142428711yescharacteristic
367C2#2142433511yescharacteristic
368C2#2142436811yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1566C2
socleC22×C59#385
prime radicalC2×#11745685D4
cosocleC2×C42×C59C23
hypercenter#1566D59
hypocenterC59#8904
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1566

Complements

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

NH ≅ G/Ntype
3. #21422868C2semidirect
5. #21421997C2semidirect
6. #21421936C2semidirect
7. C59×#1566C2semidirect
21. #16837134C4semidirect
22. #16837134C4semidirect
23. #16837131C4semidirect
24. #16837131C4semidirect
182. #1566D59semidirect
213. C59#8904semidirect

Maximal subgroups

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

subgroupindex
#214326652
#214241132
#214228682
#214224792
#214219972
#214219362
C59×#15662
#890459