Finite Group Databasebeta

← #28485234

Subgroups — #28485234

Order 15104.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435011C211yesmaximal, characteristic
2#21424936C211yesmaximal, characteristic, complemented
3#21422773C211yesmaximal, characteristic, complemented
4C2×#16837145C211yesmaximal, characteristic, complemented
5#21421993C211yesmaximal, characteristic, complemented
6#21421947C211yesmaximal, characteristic, complemented
7C59×#2935C211yesmaximal, characteristic, complemented
8C2×#14273724C2211yescharacteristic
9#16837778—21no—
10#16837750—21no—
11#16837597—22no—
12#16837535C414yes—
13#16837455C2211yescharacteristic
14#16837426C2211yescharacteristic
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×#14273643—21no—
17C2×#14273643—21no—
18#16837148—21no—
19#16837146—21no—
20#16837145—21no—
21#16837145—21no—
22C2×#131×C59—21no—
23C59×#532C2211yescharacteristic
24C59×#436—21no—
25C59×#435—22no—
26C8×D4×C59—21no—
27C2×C4⋊C8×C59C2211yescharacteristic
28C59×#408—21no—
29C4×M16×C59C2211yescharacteristic
30C22×D236—21no—
31C22×D236—21no—
32C22×D236C2×C412yes—
33C2×#11745669—24no—
34C2×#11745664—21no—
35C22×C59⋊C8—21no—
36#14273724—21no—
37#14273724—21no—
38#14273724—21no—
39#14273724—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45#14273647—24no—
46#14273643—42no—
47#14273643—42no—
48C22⋊(C59⋊C8)C2×C412yes—
49C4×C59⋊C8—21no—
50C2×#41×C59—21no—
51#131×C59—42no—
52#131×C59—42no—
53C2×M16×C59—22no—
54C2×M16×C59D411yescharacteristic
55C2×M16×C59D411yescharacteristic
56C22×C8×C59—22no—
57C22×C8×C59—21no—
58C4⋊Q8×C59D411yescharacteristic
59#127×C59D411yescharacteristic
60#124×C59—21no—
61#121×C59—21no—
62#121×C59C2×C412yes—
63C4×D4×C59—21no—
64C4×D4×C59D412yes—
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—21no—
67C4⋊C8×C59D411yescharacteristic
68C4⋊C8×C59D411yescharacteristic
69C22⋊C8×C59—22no—
70C22⋊C8×C59—22no—
71#97×C59—21no—
72C4×C8×C59—21no—
73C23×D59—22no—
74#41×C59—41no—
75#41×C59—41no—
76#41×C59—41no—
77#41×C59—41no—
78C2×Q8×C59—21no—
79C2×D4×C59—22no—
80C2×D4×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59C22⋊C412yes—
83C22×C4×C59—22no—
84C22×C4×C59C22×C411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59#4111yescharacteristic
87C2×D236—22no—
88C2×D236—22no—
89C2×D236—41no—
90C2×D236—41no—
91C2×D236—41no—
92C2×D236—41no—
93C2×D236—41no—
94C2×D236—41no—
95C2×D236—41no—
96C2×D236—41no—
97C2×D236—82no—
98C2×D236—81no—
99C2×D236—81no—
100M16×C59—21no—
101M16×C59—21no—
102M16×C59—21no—
103M16×C59—44no—
104M16×C59D812yescomplemented
105C2×C8×C59—21no—
106C2×C8×C59—21no—
107C2×C8×C59—21no—
108C2×C8×C59—21no—
109C2×C8×C59—44no—
110C2×C8×C59—41no—
111C2×C8×C59—41no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59C22⋊C412yes—
114C22⋊C4×C59—22no—
115C22⋊C4×C59—22no—
116C42×C59C2×D411yescharacteristic
117C42×C59C2×D411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59#4111yescharacteristic
120#11745669—84no—
121#11745664—41no—
122#11745664—41no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—41no—
125C2×C59⋊C8—41no—
126C2×C59⋊C8—81no—
127C23×C59—22no—
128C23×C59C4×D411yescharacteristic
129C22×D59—42no—
130C22×D59—42no—
131C22×D59—42no—
132C22×D59—82no—
133C22×D59—82no—
134Q8×C59—41no—
135Q8×C59—41no—
136D4×C59—42no—
137D4×C59—42no—
138D4×C59—42no—
139D4×C59—41no—
140D4×C59—41no—
141D4×C59—41no—
142D4×C59—41no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59C2×C22⋊C411yescharacteristic
150C2×C4×C59C4×D411yescharacteristic
151C2×C4×C59#12011yescharacteristic
152C2×C4×C59#12111yescharacteristic
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59#12311yescharacteristic
155C2×C4×C59C2×D811yescharacteristic
156C2×C4×C59#13511yescharacteristic
157C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
159D236—44no—
160D236—82no—
161D236—82no—
162D236—81no—
163D236—81no—
164D236—81no—
165D236—81no—
166D236—81no—
167D236—81no—
168C8×C59—22no—
169C8×C59—41no—
170C8×C59—41no—
171C8×C59—41no—
172C59⋊C8—42no—
173C59⋊C8—81no—
174C59⋊C8—81no—
175#9273—591nomaximal, Sylow
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—42no—
181C22×C59#38611yescharacteristic
182C22×C59C4×D811yescharacteristic
183C22×C59#44211yescharacteristic
184C2×D59—82no—
185C2×D59—82no—
186C2×D59—82no—
187C2×D59—162no—
188C2×D59—162no—
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—42no—
193C4×C59#44711yescharacteristic
194C4×C59#44911yescharacteristic
195C4×C59#45911yescharacteristic
196C4×C59#46311yescharacteristic
197C4×C59#47111yescharacteristic
198C4×C59#47211yescharacteristic
199#3205—591no—
200#2935D5911yescharacteristic, complemented
201#1965—591no—
202#1547—591no—
203C2×#331—591no—
204#1301—591no—
205#1283—591no—
206C2×C59—21no—
207C2×C59—21no—
208C2×C59—82no—
209C2×C59#187911yescharacteristic
210C2×C59#190711yescharacteristic
211C2×C59#191411yescharacteristic
212D59—162no—
213C2×#131—21no—
214#532C2×D5911yescharacteristic
215C2×#127—591no—
216#493—1181no—
217#486—1181no—
218#459—1182no—
219#447—594no—
220#436—21no—
221#435—22no—
222C8×D4—21no—
223#431—591no—
224#423—591no—
225C2×C4⋊C8—591no—
226C2×C4⋊C8C2×D5911yescharacteristic
227C2×#100—1181no—
228C2×#100—1181no—
229#408—21no—
230C4×M16C2×D5911yescharacteristic
231#332—1181no—
232#331—1181no—
233#331—1181no—
234#331—1181no—
235C59#927311yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C22×D4—592no—
238C22×D4—1181no—
239C22×D4—1181no—
240C2×D8—1184no—
241#131—42no—
242#131—42no—
243C2×M16—22no—
244C2×M16—1181no—
245C2×M16C22⋊D5911yescharacteristic
246C2×M16C22⋊D5911yescharacteristic
247C22×C8—22no—
248C22×C8—21no—
249C22×C8—1181no—
250C4⋊Q8D23611yescharacteristic
251#127—1181no—
252#127—1181no—
253#127—1181no—
254#127—1181no—
255#127D23611yescharacteristic
256#124—21no—
257#121—21no—
258#121C4×D5912yes—
259C4×D4—21no—
260C4×D4C22⋊D5912yes—
261C2×C42C22×D5911yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—1181no—
264C4⋊C8—1181no—
265C4⋊C8—1181no—
266C4⋊C8—1181no—
267C4⋊C8—1181no—
268C4⋊C8C22⋊D5911yescharacteristic
269C4⋊C8C22⋊D5911yescharacteristic
270#102—1184no—
271#100—2362no—
272#100—2362no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—592no—
276#97—21no—
277C4×C8—21no—
278C4×C8—1181no—
279C24—1182no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—1182no—
289C2×D4—1182no—
290C2×D4—4721no—
291C2×D4—4721no—
292C2×D4—4722no—
293C2×D4—2361no—
294C2×D4—2361no—
295C2×D4—2361no—
296C2×D4—2361no—
297C2×D4—2361no—
298C2×D4—2361no—
299C2×D4—2361no—
300C2×D4—2361no—
301C2×D4(C2×D59)⋊C412yes—
302C22×C4—22no—
303C22×C4C2×C4×D5911yescharacteristic
304C22×C4#1174568511yescharacteristic
305C22×C4C2×C22⋊D5911yescharacteristic
306D8—4724no—
307M16—21no—
308M16—21no—
309M16—21no—
310M16—44no—
311M16—2361no—
312M16—2361no—
313M16#1174566912yescomplemented
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—44no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—1182no—
322C2×C8—4721no—
323C2×C8—2361no—
324C2×C8—2361no—
325C4⋊C4—21no—
326C4⋊C4(C2×D59)⋊C412yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C42C2×D23611yescharacteristic
330C42#1174568511yescharacteristic
331C42C2×C22⋊D5911yescharacteristic
332C42C2×C22⋊D5911yescharacteristic
333C23—22no—
334C23—4722no—
335C23—4722no—
336C23—2362no—
337C23—2362no—
338C23—2362no—
339C23C4×C22⋊D5911yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—4722no—
350D4—4722no—
351D4—4721no—
352D4—4721no—
353D4—4721no—
354D4—4721no—
355D4—4721no—
356D4—4721no—
357D4—2364no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4C2×(C2×D59)⋊C411yescharacteristic
365C2×C4C4×C22⋊D5911yescharacteristic
366C2×C4#1427369611yescharacteristic
367C2×C4#1427370111yescharacteristic
368C2×C4#1427370111yescharacteristic
369C2×C4#1427371011yescharacteristic
370C2×C4C2×#1174566911yescharacteristic
371C2×C4#1427374711yescharacteristic
372C2×C4#1427375211yescharacteristic
373C2×C4#1427375311yescharacteristic
374C8—22no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—4721no—
379C8—4721no—
380C8—2362no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—9442no—
387C22—9442no—
388C22—4722no—
389C22—4722no—
390C22—4722no—
391C22#1683728111yescharacteristic
392C22C4×#1174566911yescharacteristic
393C22#1683751211yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4#1683753611yescharacteristic
399C4#1683754711yescharacteristic
400C4#1683759711yescharacteristic
401C4#1683761711yescharacteristic
402C4#1683766511yescharacteristic
403C4#1683767211yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—9442no—
408C2#2142444411yescharacteristic
409C2#2142460411yescharacteristic
410C2#2142464311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2935C2
socleC22×C59#386
prime radical#21435011C2
cosocleC2×C42×C59C23
hypercenter#2935D59
hypocenterC59#9273
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #21424936C2semidirect
3. #21422773C2semidirect
4. C2×#16837145C2semidirect
5. #21421993C2semidirect
6. #21421947C2semidirect
7. C59×#2935C2semidirect
104. M16×C59D8semidirect
200. #2935D59semidirect
235. C59#9273semidirect
313. M16#11745669semidirect

Maximal subgroups

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

subgroupindex
#214350112
#214249362
#214227732
C2×#168371452
#214219932
#214219472
C59×#29352
#927359