Finite Group Databasebeta

← #29452638

Subgroups — #29452638

Order 15104.

2706 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838168C212yesmaximal, complemented
2#21437429C211yesmaximal, characteristic
3#21432820C2116yesmaximal, complemented
4#21432813C214yesmaximal, complemented
5#21432807C214yesmaximal, complemented
6C2×#16837370C212yesmaximal, complemented
7C2×#16837358C211yesmaximal, characteristic, complemented
8C59×#2905C211yesmaximal, characteristic, complemented
9C22×#11745685C2212yes—
10#16838168—216no—
11C2×#14273735C2218yescomplemented
12C2×#14273733C2218yes—
13C22×#11745664C2211yescharacteristic
14C2×M16×D59C2218yescomplemented
15C23×C59⋊C8C2211yescharacteristic
16#16837983C414yes—
17#16837978C414yes—
18#16837971C414yes—
19#16837964C414yes—
20#16837952C2214yes—
21#16837914C2218yes—
22#16837479C22116yescomplemented
23#16837472C22132yescomplemented
24#16837467C22116yescomplemented
25#16837370C2218yes—
26#16837370C2214yescomplemented
27#16837370C2214yescomplemented
28C2×(C2×D59)⋊C8C2218yescomplemented
29#16837358C2218yescomplemented
30C2×#14273638C2212yescomplemented
31C2×C4×C59⋊C8C2212yescomplemented
32C22×M16×C59C2212yes—
33C2×#117×C59C2211yescharacteristic
34C59×#424C2218yescomplemented
35C59×#423C2214yescomplemented
36C2×#11745685—28no—
37C2×#11745685—28no—
38C2×#11745685C2×C414yes—
39C2×#11745685C2×C414yes—
40C2×#11745685C2314yes—
41C2×#11745685C2314yes—
42C22×Dic118C2×C412yes—
43C22×D236C2×C412yes—
44C22×C22⋊D59C2×C414yes—
45C22×C4×D59C2314yes—
46#14273735—232no—
47#14273733—232no—
48C2×#11745664D418yescomplemented
49C2×#11745664C2314yes—
50C2×#11745659C23116yes—
51M16×D59—232no—
52C22×C59⋊C8—24no—
53C22×C59⋊C8C2311yescharacteristic
54C22×C59⋊C8C2311yescharacteristic
55C22×C59⋊C8C2314yes—
56C22×C59⋊C8C2312yes—
57C2×C8×D59C23116yes—
58#14273727C2×C414yes—
59#14273724C2×C414yes—
60#14273715C2×C418yes—
61#14273711C2×C4116yes—
62#14273707C2×C4116yes—
63#14273702C2×C4116yes—
64#14273701C2×C4116yes—
65#14273697C2×C4116yes—
66C4×D236C23116yes—
67(C2×D59)⋊C8C23116yes—
68(C2×D59)⋊C8C23116yes—
69#14273638C2318yes—
70C4×C59⋊C8C2318yes—
71C23×C4×C59C2311yescharacteristic
72C2×M16×C59—28no—
73C2×M16×C59C2314yes—
74C2×M16×C59C2314yes—
75C22×C8×C59C2314yes—
76#117×C59C2×C414yes—
77#117×C59C2314yes—
78C2×C4⋊C4×C59C2×C412yes—
79C2×C22⋊C4×C59C2×C412yes—
80C2×C42×C59C2312yes—
81C4⋊C8×C59C23116yes—
82C24×C59C22×C411yescharacteristic
83C23×D59C22×C414yes—
84C22×C4×C59C22×C414yes—
85C22×C4×C59C22×C411yescharacteristic
86C22×C4×C59C22×C412yes—
87C22×C4×C59C2×D414yes—
88C22×C4×C59#4114yes—
89C22×C4×C59C2411yescharacteristic
90C22×C4×C59C2412yes—
91C2×C22⋊D59—216no—
92C2×C22⋊D59C22×C418yes—
93C2×C22⋊D59C22×C418yes—
94C22×Dic59C22×C414yes—
95#11745685—216no—
96#11745685—216no—
97#11745685—28no—
98#11745685—28no—
99#11745685—28no—
100#11745685—28no—
101C2×D236—28no—
102C2×D236C22×C414yes—
103C2×D236C22×C414yes—
104C2×D236C22×C4116yes—
105C2×C4×D59—28no—
106C2×C4×D59—28no—
107C2×C4×D59C22×C418yes—
108C2×C4×D59C2418yes—
109C2×Dic118—28no—
110C2×Dic118C22×C414yes—
111C2×Dic118C22×C414yes—
112M16×C59—28no—
113M16×C59—28no—
114C2×C8×C59—28no—
115C2×C8×C59C2418yes—
116C4⋊C4×C59C22×C418yes—
117C22⋊C4×C59C22×C418yes—
118C42×C59C22×C414yes—
119C42×C59C2414yes—
120(C2×D59)⋊C4C22×C4132yes—
121C4⋊Dic59C22×C4116yes—
122#11745664C2×D4116yes—
123C2×C59⋊C8—28no—
124C2×C59⋊C8—22no—
125C2×C59⋊C8—22no—
126C2×C59⋊C8C2×D418yes—
127C2×C59⋊C8C2412yes—
128C2×C59⋊C8C2412yes—
129C2×C59⋊C8C2414yes—
130#11745659—232no—
131C8×D59—232no—
132C23×C59—42no—
133C23×C59C4×D414yes—
134C23×C59C23×C412yes—
135C23×C59C23×C411yescharacteristic
136C22×D59—28no—
137C22×D59—48no—
138C22×D59C23×C418yes—
139C2×C4×C59—44no—
140C2×C4×C59C4×D418yes—
141C2×C4×C59C4×D414yes—
142C2×C4×C59C23×C418yes—
143C2×C4×C59C23×C411yescharacteristic
144C2×C4×C59C23×C412yes—
145C2×C4×C59C22×D412yes—
146C2×C4×C59C22×D414yes—
147C2×C4×C59C2×#4114yes—
148C2×C4×C59C2×#4112yes—
149C2×C4×C59C2511yescharacteristic
150C22⋊D59—216no—
151C22⋊D59—216no—
152C2×Dic59—28no—
153C2×Dic59C23×C418yes—
154D236—28no—
155D236—28no—
156D236—416no—
157C4×D59—216no—
158C4×D59—28no—
159C4×D59—28no—
160Dic118—28no—
161Dic118—28no—
162C8×C59—28no—
163C59⋊C8—22no—
164C59⋊C8—22no—
165C59⋊C8C22×D418yes—
166#60143—591nomaximal, Sylow
167C22×C59—22no—
168C22×C59—24no—
169C22×C59—44no—
170C22×C59C2×C4×D414yes—
171C22×C59C2×C4×D412yes—
172C22×C59C24×C411yescharacteristic
173C2×D59—28no—
174C2×D59—216no—
175C2×D59—88no—
176C4×C59—28no—
177C4×C59C2×C4×D412yes—
178C4×C59C2×C4×D414yes—
179C4×C59C23×D411yescharacteristic
180C4×C59C22×#4111yescharacteristic
181Dic59—28no—
182C2×#568—592no—
183#3449—591no—
184#2933—5916no—
185#2932—594no—
186#2931—594no—
187#2905D5911yescharacteristic, complemented
188C2×#408—592no—
189C2×#405—591no—
190C2×C59—24no—
191C2×C59—22no—
192C2×C59C22×C4×D411yescharacteristic
193C2×C59#357312yes—
194D59—48no—
195C22×#41—592no—
196#568—11816no—
197C2×#131—598no—
198C2×#131—598no—
199C22×M16—591no—
200C22×M16—598no—
201C22×M16C2×D5912yes—
202C23×C8—591no—
203#537—594no—
204#536—594no—
205#535—594no—
206#534—594no—
207#532—594no—
208#525—598no—
209C2×#117C2×D5911yescharacteristic
210#436—5916no—
211#435—5932no—
212C8×D4—5916no—
213#424C2×D5918yescomplemented
214#423C2×D5914yescomplemented
215#408—594no—
216#408—594no—
217#408—598no—
218C2×C22⋊C8—598no—
219#405—598no—
220C2×#97—592no—
221C2×C4×C8—592no—
222C59#6014311yescharacteristic, complemented, Sylow
223C2×#41—594no—
224C2×#41—594no—
225C2×#41—594no—
226C2×#41—594no—
227C2×#41—1188no—
228C2×#41—1188no—
229C22×Q8—592no—
230C22×D4—594no—
231C22×D4—592no—
232C23×C4—594no—
233C23×C4C22×D5911yescharacteristic
234#131—11832no—
235#131—11832no—
236C2×M16—28no—
237C2×M16—598no—
238C2×M16—594no—
239C2×M16—5916no—
240C2×M16—11832no—
241C2×M16C22×D5914yes—
242C2×M16C22×D5914yes—
243C22×C8—592no—
244C22×C8—591no—
245C22×C8—591no—
246C22×C8—594no—
247C22×C8—5916no—
248C22×C8—1184no—
249C22×C8C22×D5914yes—
250C4⋊Q8—594no—
251#127—594no—
252#124—598no—
253#123—5916no—
254C22⋊Q8—5916no—
255#121—5916no—
256#121—5916no—
257#120—5916no—
258C4×D4—5916no—
259#117C4×D5914yes—
260#117C22×D5914yes—
261C2×C4⋊C4C4×D5912yes—
262C2×C22⋊C4C4×D5912yes—
263C2×C42C22×D5912yes—
264C4⋊C8C22×D59116yes—
265C22⋊C8—5916no—
266C22⋊C8—5916no—
267#97—598no—
268C4×C8—598no—
269C24—594no—
270C24C2×C4×D5911yescharacteristic
271#41—11816no—
272#41—11816no—
273#41—1188no—
274#41—1188no—
275#41—1188no—
276#41—1188no—
277C2×Q8—594no—
278C2×Q8—594no—
279C2×Q8—1188no—
280C2×D4—598no—
281C2×D4—598no—
282C2×D4—594no—
283C2×D4—594no—
284C2×D4—5916no—
285C2×D4—11816no—
286C2×D4—1188no—
287C22×C4—594no—
288C22×C4—598no—
289C22×C4—598no—
290C22×C4—1188no—
291C22×C4—1188no—
292C22×C4C2×C4×D5914yes—
293C22×C4C2×C4×D5911yescharacteristic
294C22×C4C2×C4×D5912yes—
295C22×C4D4×D5914yes—
296C22×C4#1174568914yes—
297C22×C4C23×D5911yescharacteristic
298C22×C4C23×D5912yes—
299M16—28no—
300M16—28no—
301M16—5916no—
302M16—11832no—
303C2×C8—28no—
304C2×C8—598no—
305C2×C8—594no—
306C2×C8—592no—
307C2×C8—592no—
308C2×C8—11832no—
309C2×C8—1188no—
310C2×C8—1182no—
311C2×C8—1182no—
312C2×C8C23×D5918yes—
313C4⋊C4—5916no—
314C4⋊C4C2×C4×D5918yes—
315C22⋊C4—5932no—
316C22⋊C4C2×C4×D5918yes—
317C42C2×C4×D5914yes—
318C42C23×D5914yes—
319C23—42no—
320C23—598no—
321C23—1188no—
322C23—2368no—
323C23#1427368914yes—
324C23C22×C4×D5912yes—
325C23C22×C4×D5911yescharacteristic
326Q8—1188no—
327Q8—1188no—
328D4—11816no—
329D4—11816no—
330D4—1188no—
331D4—1188no—
332D4—23616no—
333C2×C4—44no—
334C2×C4—598no—
335C2×C4—1188no—
336C2×C4—1188no—
337C2×C4—1188no—
338C2×C4—11816no—
339C2×C4#1427368918yes—
340C2×C4#1427368914yes—
341C2×C4C22×C4×D5918yes—
342C2×C4C22×C4×D5911yescharacteristic
343C2×C4C22×C4×D5912yes—
344C2×C4C2×D4×D5914yes—
345C2×C4C2×D4×D5912yes—
346C2×C4C2×#1174568914yes—
347C2×C4C2×#1174568912yes—
348C2×C4C24×D5911yescharacteristic
349C8—28no—
350C8—598no—
351C8—1182no—
352C8—1182no—
353C22—22no—
354C22—24no—
355C22—44no—
356C22—1188no—
357C22—11816no—
358C22—4728no—
359C22C2×#1427368914yes—
360C22C2×#1427368912yes—
361C22C23×C4×D5911yescharacteristic
362C4—28no—
363C4—1188no—
364C4C2×#1427368914yes—
365C4C2×#1427368912yes—
366C4C22×D4×D5911yescharacteristic
367C4C22×#1174568911yescharacteristic
368C2—24no—
369C2—22no—
370C2—2368no—
371C2C22×#1427368911yescharacteristic
372C2#2143880912yes—
subgroupG/H
center, FrattiniC2×C4C24×D59
commutator, socleC22×C59C24×C4
FittingC59×#2905C2
prime radical#21437429C2
cosocleC2×C4×C59C25
hypercenter#2905D59
hypocenterC59#60143
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. C2×#16838168C2semidirect
3. #21432820C2semidirect
4. #21432813C2semidirect
5. #21432807C2semidirect
6. C2×#16837370C2semidirect
7. C2×#16837358C2semidirect
8. C59×#2905C2semidirect
11. C2×#14273735C22semidirect
14. C2×M16×D59C22semidirect
22. #16837479C22semidirect
23. #16837472C22semidirect
24. #16837467C22semidirect
26. #16837370C22semidirect
27. #16837370C22semidirect
28. C2×(C2×D59)⋊C8C22semidirect
29. #16837358C22semidirect
30. C2×#14273638C22semidirect
31. C2×C4×C59⋊C8C22semidirect
34. C59×#424C22semidirect
35. C59×#423C22semidirect
48. C2×#11745664D4semidirect
187. #2905D59semidirect
213. #424C2×D59semidirect
214. #423C2×D59semidirect
222. C59#60143semidirect

Maximal subgroups

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

subgroupindex
C2×#168381682
#214374292
#214328202
#214328132
#214328072
C2×#168373702
C2×#168373582
C59×#29052
#6014359