Finite Group Databasebeta

← #28483921

Subgroups — #28483921

Order 15104.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434656C211yesmaximal, characteristic
2#21432842C211yesmaximal, characteristic
3#21422590C211yesmaximal, characteristic
4#21422499C211yesmaximal, characteristic, complemented
5#21421937C211yesmaximal, characteristic, complemented
6#21421934C211yesmaximal, characteristic, complemented
7C59×#1986C211yesmaximal, characteristic, complemented
8C2×#14273735—21no—
9#16837952C2211yescharacteristic
10#16837798C411yescharacteristic, complemented
11#16837776C411yescharacteristic, complemented
12#16837752C411yescharacteristic, complemented
13#16837750C411yescharacteristic, complemented
14#16837633—22no—
15C4×#11745671—22no—
16#16837481—21no—
17#16837475—22no—
18C8×D236—21no—
19#16837455C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21#16837370—21no—
22C2×C4×C59⋊C8C2211yescharacteristic
23#16837135—22no—
24Q8⋊(C59⋊C8)—21no—
25Q8⋊(C59⋊C8)—21no—
26C59×#533C2211yescharacteristic
27C4×M16×C59C2211yescharacteristic
28C4×C4⋊C4×C59C2211yescharacteristic
29C59×#339—21no—
30C59×#339—21no—
31C2×#11745685—21no—
32C2×#11745671—22no—
33#14273735—42no—
34#14273735—42no—
35C2×#11745659—22no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38C2×C8×D59—22no—
39#14273727C2×C411yescharacteristic
40#14273724C2×C411yescharacteristic
41#14273715—21no—
42#14273701—21no—
43#14273701D412yes—
44C4×D236—21no—
45C4×D236D412yes—
46#14273658—22no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)D411yescharacteristic
50C4⋊(C59⋊C8)D411yescharacteristic
51#14273651—22no—
52#14273647—22no—
53C22⋊(C59⋊C8)—21no—
54(C2×D59)⋊C8—22no—
55(C2×D59)⋊C8—22no—
56C4×C59⋊C8—21no—
57C4×C59⋊C8—21no—
58C4×C59⋊C8C2×C411yescharacteristic
59C4×C59⋊C8C2×C411yescharacteristic
60C2×M16×C59D411yescharacteristic
61C2×M16×C59D411yescharacteristic
62C4⋊Q8×C59C2×C411yescharacteristic
63C4⋊Q8×C59C2×C411yescharacteristic
64#125×C59—21no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59—21no—
67C4×Q8×C59—21no—
68C4×Q8×C59—21no—
69#117×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C42×C59—21no—
72C2×C42×C59C2311yescharacteristic
73#97×C59—21no—
74C4×C8×C59—21no—
75#95×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59—21no—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C2×C22⋊D59—22no—
83#11745685—41no—
84#11745685—41no—
85#11745685—41no—
86#11745685—41no—
87C2×D236—21no—
88C2×D236—21no—
89C2×D236C22⋊C412yes—
90C2×C4×D59—22no—
91C2×Dic118—21no—
92M16×C59—21no—
93M16×C59—21no—
94M16×C59—21no—
95M16×C59SD1612yescomplemented
96C2×C8×C59—21no—
97C2×C8×C59—21no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C42×C59—41no—
108C42×C59—41no—
109C42×C59—41no—
110C42×C59C22×C411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59C2×D411yescharacteristic
113C42×C59#4111yescharacteristic
114#11745671—44no—
115(C2×D59)⋊C4—22no—
116(C2×D59)⋊C4—22no—
117C4⋊Dic59—21no—
118C4⋊Dic59C22⋊C412yes—
119C2×C59⋊C8—21no—
120C2×C59⋊C8—21no—
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123C2×C59⋊C8—42no—
124C2×C59⋊C8—41no—
125C2×C59⋊C8—41no—
126C2×C59⋊C8—41no—
127#11745659—44no—
128C8×D59—44no—
129C23×C59#12111yescharacteristic
130C22×D59—22no—
131Q8×C59—42no—
132Q8×C59—41no—
133Q8×C59—41no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—41no—
137C2×C4×C59—42no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—81no—
141C2×C4×C59—81no—
142C2×C4×C59C2×C22⋊C411yescharacteristic
143C2×C4×C59C4×D411yescharacteristic
144C2×C4×C59C4×D411yescharacteristic
145C2×C4×C59#12011yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#12311yescharacteristic
148C2×C4×C59C2×SD1611yescharacteristic
149C2×C4×C59#13511yescharacteristic
150C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C59#13711yescharacteristic
152C22⋊D59—42no—
153C22⋊D59—42no—
154C2×Dic59—22no—
155D236—42no—
156D236—41no—
157D236—41no—
158D236—41no—
159D236—41no—
160C4×D59—42no—
161C4×D59—42no—
162Dic118—41no—
163Dic118—41no—
164C8×C59—22no—
165C8×C59—41no—
166C59⋊C8—22no—
167C59⋊C8—41no—
168C59⋊C8—41no—
169C59⋊C8—41no—
170C59⋊C8—81no—
171#9055—591nomaximal, Sylow
172C22×C59—21no—
173C22×C59—21no—
174C22×C59#38611yescharacteristic
175C22×C59#46111yescharacteristic
176C22×C59#46311yescharacteristic
177C2×D59—42no—
178C2×D59—42no—
179C2×D59—42no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—82no—
184C4×C59—81no—
185C4×C59—81no—
186C4×C59C4×SD1611yescharacteristic
187C4×C59#44011yescharacteristic
188C4×C59#44311yescharacteristic
189C4×C59#44411yescharacteristic
190C4×C59#45011yescharacteristic
191C4×C59#45211yescharacteristic
192Dic59—42no—
193#3167—591no—
194#2935—591no—
195#1986D5911yescharacteristic, complemented
196#1502—591no—
197#1481—591no—
198#1279—591no—
199#1278—591no—
200C2×C59—21no—
201C2×C59—21no—
202C2×C59#187611yescharacteristic
203C2×C59#187711yescharacteristic
204C2×C59#188411yescharacteristic
205D59—82no—
206C2×#131—1181no—
207#533C2×D5911yescharacteristic
208#532—591no—
209#498—591no—
210#492—591no—
211#487—591no—
212#486—591no—
213#465—1182no—
214C4×SD16—1182no—
215#436—1181no—
216#435—1182no—
217C8×D4—1181no—
218#431—591no—
219C2×C4⋊C8—591no—
220#408—1181no—
221C4×M16C2×D5911yescharacteristic
222C2×C4×C8—591no—
223C4×C4⋊C4C2×D5911yescharacteristic
224#339—21no—
225#339—21no—
226#327—1182no—
227Q8⋊C8—1181no—
228Q8⋊C8—1181no—
229C59#905511yescharacteristic, complemented, Sylow
230C2×#41—1181no—
231C2×SD16—1182no—
232#131—2362no—
233#131—2362no—
234C2×M16—1182no—
235C2×M16C22⋊D5911yescharacteristic
236C2×M16C22⋊D5911yescharacteristic
237C22×C8—1182no—
238C22×C8—1181no—
239C22×C8—1181no—
240C4⋊Q8—591no—
241C4⋊Q8C4×D5911yescharacteristic
242C4⋊Q8C4×D5911yescharacteristic
243#127—591no—
244#125—21no—
245#124—1181no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—592no—
249#121—1181no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—592no—
253C4×D4—1181no—
254#117—21no—
255C2×C4⋊C4—21no—
256C2×C42—21no—
257C2×C42C22×D5911yescharacteristic
258#106—1182no—
259C4⋊C8—591no—
260C4⋊C8—591no—
261C4⋊C8—1181no—
262C4⋊C8—1181no—
263#103—1182no—
264#102—1182no—
265C22⋊C8—1182no—
266C22⋊C8—1182no—
267C22⋊C8—1181no—
268#97—21no—
269C4×C8—21no—
270C4×C8—591no—
271C4×C8—591no—
272C4×C8—1181no—
273C4×C8—1181no—
274#95—21no—
275#41—2361no—
276#41—2361no—
277#41—2361no—
278#41—2361no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—1181no—
282C2×D4—592no—
283C2×D4—1182no—
284C2×D4—1181no—
285C2×D4—1181no—
286C22×C4—41no—
287C22×C4—1182no—
288C22×C4D4×D5911yescharacteristic
289C22×C4#1174568711yescharacteristic
290C22×C4C2×C22⋊D5911yescharacteristic
291SD16—2364no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—2364no—
296M16#1174567112yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—1181no—
300C2×C8—1181no—
301C2×C8—1181no—
302C2×C8—1181no—
303C2×C8—2364no—
304C2×C8—2362no—
305C2×C8—2361no—
306C2×C8—2361no—
307C2×C8—2361no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—592no—
317C4⋊C4—1181no—
318C22⋊C4—41no—
319C22⋊C4—1182no—
320C22⋊C4—1182no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42C2×C4×D5911yescharacteristic
325C42#1174568511yescharacteristic
326C42D4×D5911yescharacteristic
327C42D4×D5911yescharacteristic
328C23—1182no—
329C23#1427369811yescharacteristic
330Q8—42no—
331Q8—41no—
332Q8—41no—
333Q8—2361no—
334Q8—2361no—
335D4—2362no—
336D4—2362no—
337D4—2362no—
338D4—2361no—
339D4—2361no—
340D4—2361no—
341D4—2361no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—1182no—
351C2×C4—2362no—
352C2×C4—2362no—
353C2×C4C22⋊C4×D5911yescharacteristic
354C2×C4(C22⋊D59)⋊C411yescharacteristic
355C2×C4C4×C22⋊D5911yescharacteristic
356C2×C4#1427369511yescharacteristic
357C2×C4#1427370011yescharacteristic
358C2×C4#1427370911yescharacteristic
359C2×C4C2×#1174567111yescharacteristic
360C2×C4#1427374711yescharacteristic
361C2×C4#1427375211yescharacteristic
362C2×C4#1427375811yescharacteristic
363C8—22no—
364C8—41no—
365C8—1182no—
366C8—4721no—
367C8—2361no—
368C8—2361no—
369C8—2361no—
370C22—21no—
371C22—21no—
372C22—2362no—
373C22—2362no—
374C22—2362no—
375C22#1683727811yescharacteristic
376C22#1683760411yescharacteristic
377C22#1683761411yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—2362no—
385C4C4×#1174567111yescharacteristic
386C4#1683750011yescharacteristic
387C4#1683751611yescharacteristic
388C4#1683752111yescharacteristic
389C4#1683755211yescharacteristic
390C4#1683756211yescharacteristic
391C2—21no—
392C2—21no—
393C2—4722no—
394C2#2142442011yescharacteristic
395C2#2142443311yescharacteristic
396C2#2142447611yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1986C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1986D59
hypocenterC59#9055
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. #21422499C2semidirect
5. #21421937C2semidirect
6. #21421934C2semidirect
7. C59×#1986C2semidirect
10. #16837798C4semidirect
11. #16837776C4semidirect
12. #16837752C4semidirect
13. #16837750C4semidirect
95. M16×C59SD16semidirect
195. #1986D59semidirect
229. C59#9055semidirect
296. M16#11745671semidirect

Maximal subgroups

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

subgroupindex
#214346562
#214328422
#214225902
#214224992
#214219372
#214219342
C59×#19862
#905559