Finite Group Databasebeta

← #28504167

Subgroups — #28504167

Order 15104.

1168 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837189C214yesmaximal, complemented
2#21424024C211yesmaximal, characteristic, complemented
3#21424014C211yesmaximal, characteristic, complemented
4C2×C59×#535C211yesmaximal, characteristic
5C2×#14273646—21no—
6C2×#14273644C2212yes—
7C2×#14273642—21no—
8C2×#14273642C2212yes—
9#16837366—21no—
10#16837265—21no—
11#16837189—28no—
12C22×#41×C59C2211yescharacteristic
13C23×D4×C59C411yescharacteristic, complemented
14C59×#535—28no—
15C2×#127×C59—21no—
16C2×#124×C59—21no—
17C2×#121×C59—21no—
18C2×#121×C59—21no—
19C2×#120×C59C412yescomplemented
20C2×#120×C59C2212yes—
21C2×#117×C59C411yescharacteristic, complemented
22C2×#11745664—22no—
23C2×C22⋊Dic59—22no—
24#14273646—42no—
25#14273644—24no—
26#14273642—24no—
27#14273642—42no—
28C22⋊(C59⋊C8)—42no—
29C59⋊#95—42no—
30C25×C59C2×C412yes—
31C2×#41×C59—24no—
32C2×#41×C59—42no—
33C2×#41×C59—42no—
34C22×Q8×C59D411yescharacteristic
35C22×D4×C59—24no—
36C22×D4×C59—22no—
37C22×D4×C59—22no—
38C22×D4×C59—21no—
39C22×D4×C59—21no—
40C22×D4×C59—42no—
41C22×D4×C59—42no—
42C22×D4×C59C2×C411yescharacteristic
43C22×D4×C59D411yescharacteristic
44C22×D4×C59C2311yescharacteristic
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C23×C4×C59D411yescharacteristic
48#127×C59—28no—
49#124×C59—28no—
50#121×C59—28no—
51#121×C59—28no—
52#120×C59—24no—
53#120×C59—24no—
54#120×C59—24no—
55#120×C59D418yes—
56#117×C59—24no—
57C2×C4⋊C4×C59—21no—
58C2×C22⋊C4×C59—22no—
59C2×C22⋊C4×C59—22no—
60C2×C22⋊C4×C59C2×C412yes—
61C2×C42×C59—21no—
62C24×C59—22no—
63C24×C59—22no—
64C24×C59—42no—
65C24×C59—42no—
66C24×C59—84no—
67C24×C59C22⋊C411yescharacteristic
68C24×C59C22⋊C414yes—
69C24×C59C22×C411yescharacteristic
70C24×C59C2×D411yescharacteristic
71#41×C59—416no—
72C2×Q8×C59—22no—
73C2×Q8×C59—22no—
74C2×Q8×C59—41no—
75C2×D4×C59—24no—
76C2×D4×C59—24no—
77C2×D4×C59—22no—
78C2×D4×C59—22no—
79C2×D4×C59—28no—
80C2×D4×C59—28no—
81C2×D4×C59—22no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—416no—
88C2×D4×C59—42no—
89C2×D4×C59—42no—
90C2×D4×C59—42no—
91C2×D4×C59—42no—
92C2×D4×C59—84no—
93C2×D4×C59—84no—
94C2×D4×C59C2×D414yes—
95C22×C4×C59—22no—
96C22×C4×C59—22no—
97C22×C4×C59—22no—
98C22×C4×C59—22no—
99C22×C4×C59—22no—
100C22×C4×C59—22no—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—41no—
104C22×C4×C59—41no—
105C22×C4×C59C22⋊C411yescharacteristic
106C22×C4×C59C22⋊C411yescharacteristic
107C22×C4×C59C22⋊C411yescharacteristic
108C22×C4×C59C2×D411yescharacteristic
109C22×Dic59—42no—
110C4⋊C4×C59—24no—
111C22⋊C4×C59—28no—
112C22⋊C4×C59—22no—
113C22⋊C4×C59—28no—
114C22⋊C4×C59C22⋊C414yes—
115C42×C59—24no—
116C22⋊Dic59—42no—
117C22⋊Dic59—42no—
118#11745664—42no—
119#11745664—42no—
120C2×C59⋊C8—42no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—24no—
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—21no—
129C23×C59—42no—
130C23×C59—42no—
131C23×C59—42no—
132C23×C59—44no—
133C23×C59—44no—
134C23×C59—44no—
135C23×C59—44no—
136C23×C59—42no—
137C23×C59—42no—
138C23×C59—84no—
139C23×C59—84no—
140C23×C59—88no—
141C23×C59—88no—
142C23×C59C2×C22⋊C411yescharacteristic
143C23×C59C2×C22⋊C412yes—
144C23×C59#12012yes—
145Q8×C59—44no—
146D4×C59—44no—
147D4×C59—44no—
148D4×C59—48no—
149D4×C59—48no—
150D4×C59—44no—
151D4×C59—88no—
152D4×C59—88no—
153C2×C4×C59—28no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—24no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—44no—
165C2×C4×C59—44no—
166C2×C4×C59—44no—
167C2×C4×C59#9912yes—
168C2×C4×C59#9912yes—
169C2×C4×C59#12012yes—
170C2×Dic59—82no—
171C2×Dic59—84no—
172C59⋊C8—82no—
173#12691—591nomaximal, Sylow
174C22×C59—22no—
175C22×C59—22no—
176C22×C59—22no—
177C22×C59—42no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—42no—
181C22×C59—44no—
182C22×C59—44no—
183C22×C59—42no—
184C22×C59—42no—
185C22×C59—42no—
186C22×C59—41no—
187C22×C59—44no—
188C22×C59—81no—
189C22×C59—81no—
190C22×C59—84no—
191C22×C59—88no—
192C22×C59—88no—
193C22×C59—84no—
194C22×C59#37911yescharacteristic
195C22×C59C2×#9911yescharacteristic
196C22×C59C2×#9911yescharacteristic
197C4×C59—22no—
198C4×C59—22no—
199C4×C59—22no—
200C4×C59—44no—
201C4×C59—41no—
202Dic59—162no—
203C2×#535D5911yescharacteristic
204C2×AΣL(1,𝔽16)—594no—
205#1785—591no—
206#1782—591no—
207C2×C59—21no—
208C2×C59—21no—
209C2×C59—42no—
210C2×C59—42no—
211C2×C59—81no—
212C2×C59—84no—
213C2×C59#178111yescharacteristic
214C2×C59#212212yes—
215C22×#41C2×D5911yescharacteristic
216C23×D4Dic5911yescharacteristic, complemented
217#535—28no—
218C2×#127—21no—
219C2×#124—21no—
220C2×#121—21no—
221C2×#121—21no—
222C2×#120Dic5912yescomplemented
223C2×#120C2×D5912yes—
224C2×#117Dic5911yescharacteristic, complemented
225C2×#101—1181no—
226C2×#100—592no—
227C2×#99—592no—
228C2×#99—1181no—
229#407—1181no—
230#381—1181no—
231AΣL(1,𝔽16)—1188no—
232C59#1269111yescharacteristic, complemented, Sylow
233C25C2×Dic5912yes—
234C2×#41—24no—
235C2×#41—42no—
236C2×#41—42no—
237C22×Q8C22⋊D5911yescharacteristic
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—42no—
244C22×D4—42no—
245C22×D4C2×Dic5911yescharacteristic
246C22×D4C22⋊D5911yescharacteristic
247C22×D4C22×D5911yescharacteristic
248C23×C4C2×Dic5911yescharacteristic
249C23×C4C22⋊D5911yescharacteristic
250C23×C4C22⋊D5911yescharacteristic
251C2×M16—1182no—
252#127—28no—
253#124—28no—
254#121—28no—
255#121—28no—
256#120—24no—
257#120—24no—
258#120—24no—
259#120C22⋊D5918yes—
260#117—24no—
261C2×C4⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—1182no—
265C2×C22⋊C4C2×Dic5912yes—
266C2×C42—21no—
267#101—2362no—
268#100—1184no—
269#99—1184no—
270#99—2362no—
271C22⋊C8—2362no—
272#95—2362no—
273C24—22no—
274C24—22no—
275C24—42no—
276C24—42no—
277C24—84no—
278C24C22⋊Dic5911yescharacteristic
279C24C22⋊Dic5914yes—
280C24C22×Dic5911yescharacteristic
281C24C2×C22⋊D5911yescharacteristic
282#41—416no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—28no—
291C2×D4—28no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—416no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—84no—
304C2×D4—84no—
305C2×D4C2×C22⋊D5914yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—2362no—
317C22×C4C22⋊Dic5911yescharacteristic
318C22×C4C22⋊Dic5911yescharacteristic
319C22×C4C22⋊Dic5911yescharacteristic
320C22×C4C2×C22⋊D5911yescharacteristic
321M16—2362no—
322M16—2362no—
323C2×C8—2362no—
324C4⋊C4—24no—
325C22⋊C4—28no—
326C22⋊C4—22no—
327C22⋊C4—28no—
328C22⋊C4—2362no—
329C22⋊C4—2362no—
330C22⋊C4C22⋊Dic5914yes—
331C42—24no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—24no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—42no—
348C23—42no—
349C23—84no—
350C23—84no—
351C23—88no—
352C23—88no—
353C23C2×C22⋊Dic5911yescharacteristic
354C23C2×C22⋊Dic5912yes—
355C23#1427369612yes—
356Q8—44no—
357D4—44no—
358D4—44no—
359D4—48no—
360D4—48no—
361D4—44no—
362D4—88no—
363D4—88no—
364C2×C4—28no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—4722no—
379C2×C4—4724no—
380C2×C4#1427364212yes—
381C2×C4#1427364212yes—
382C2×C4#1427369612yes—
383C8—4722no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—44no—
392C22—44no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—44no—
398C22—81no—
399C22—81no—
400C22—84no—
401C22—88no—
402C22—88no—
403C22—84no—
404C22#1683725811yescharacteristic
405C22C2×#1427364211yescharacteristic
406C22C2×#1427364211yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—44no—
411C4—41no—
412C4—9442no—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—42no—
417C2—81no—
418C2—84no—
419C2#2142401111yescharacteristic
420C2#2142574612yes—
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
Fitting, prime radicalC2×C59×#535C2
socleC22×C59C2×#99
cosocleC22×D4×C59C23
hypercenterC2×#535D59
hypocenterC59#12691
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#16837189C2semidirect
2. #21424024C2semidirect
3. #21424014C2semidirect
13. C23×D4×C59C4semidirect
19. C2×#120×C59C4semidirect
21. C2×#117×C59C4semidirect
216. C23×D4Dic59semidirect
222. C2×#120Dic59semidirect
224. C2×#117Dic59semidirect
232. C59#12691semidirect

Maximal subgroups

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

subgroupindex
C2×#168371892
#214240242
#214240142
C2×C59×#5352
#1269159