Finite Group Databasebeta

← #28562612

Subgroups — #28562612

Order 15104.

940 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#21434868C212yesmaximal
2#21429909C211yesmaximal, characteristic
3#21423138C214yesmaximal, complemented
4#21423070C214yesmaximal, complemented
5#21422668C212yesmaximal
6C2×#16837128C211yesmaximal, characteristic
7C2×C59×#544C211yesmaximal, characteristic
8C2×#14273706C2211yescharacteristic
9#16837737—22no—
10#16837735C2214yes—
11#16837641—24no—
12#16837558—22no—
13#16837557C2214yes—
14C2×C22⋊(C59⋊C8)C2212yes—
15#16837341C2212yes—
16#16837335—22no—
17#16837322C2212yes—
18#16837275—21no—
19#16837265—21no—
20#16837262C2211yescharacteristic
21#16837139C2218yes—
22#16837128C2214yes—
23C59×#544—24no—
24C59×#544—22no—
25C59×#544C2214yes—
26C2×C4⋊Q8×C59—22no—
27C2×#125×C59—22no—
28C2×C22⋊Q8×C59—21no—
29C2×C22⋊Q8×C59C2212yes—
30C2×#117×C59—21no—
31C22×C4⋊C4×C59C2211yescharacteristic
32C22×Dic118C2311yescharacteristic
33C2×#11745672—24no—
34C22×C59⋊C8—22no—
35#14273706—22no—
36#14273706—22no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59C2311yescharacteristic
39C2×Dic59⋊C4—21no—
40C2×Dic59⋊C4—22no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43#14273660—24no—
44#14273651—24no—
45#14273650—24no—
46C22⋊(C59⋊C8)—22no—
47C22⋊(C59⋊C8)C2314yes—
48C59⋊#95—22no—
49C59⋊#95—22no—
50C59⋊#95—22no—
51C59⋊#95C2312yes—
52C22×Q8×C59—21no—
53C22×Q8×C59D412yes—
54C23×C4×C59D411yescharacteristic
55C23×C4×C59D411yescharacteristic
56C23×C4×C59C2311yescharacteristic
57C4⋊Q8×C59—28no—
58C4⋊Q8×C59—28no—
59#125×C59—28no—
60#125×C59—28no—
61C22⋊Q8×C59—24no—
62C22⋊Q8×C59—24no—
63C22⋊Q8×C59—22no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59D414yes—
66C22⋊Q8×C59C2314yes—
67#117×C59—24no—
68#117×C59—24no—
69C2×C4⋊C4×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59—21no—
72C2×C4⋊C4×C59—22no—
73C2×C4⋊C4×C59—22no—
74C2×C4⋊C4×C59—22no—
75C2×C4⋊C4×C59—22no—
76C2×C4⋊C4×C59D412yes—
77C2×C4⋊C4×C59D412yes—
78C2×C4⋊C4×C59C2312yes—
79C2×C22⋊C4×C59—22no—
80C2×C42×C59—22no—
81C24×C59C2×D411yescharacteristic
82C2×Q8×C59—24no—
83C2×Q8×C59—22no—
84C2×Q8×C59—82no—
85C2×Q8×C59—82no—
86C2×Q8×C59C2×D414yes—
87C22×C4×C59—21no—
88C22×C4×C59—21no—
89C22×C4×C59—22no—
90C22×C4×C59—22no—
91C22×C4×C59—22no—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—81no—
95C22×C4×C59—81no—
96C22×C4×C59Q1614yescomplemented
97C22×C4×C59C2×D412yes—
98C22×C4×C59C2×D412yes—
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59C2×D412yes—
101C22×C4×C59C2×D412yes—
102C22×C4×C59C2411yescharacteristic
103C22×Dic59—21no—
104C22×Dic59—22no—
105C22×Dic59—21no—
106C2×Dic118—22no—
107C2×Dic118—81no—
108C4⋊C4×C59—24no—
109C4⋊C4×C59—24no—
110C4⋊C4×C59—24no—
111C4⋊C4×C59—28no—
112C4⋊C4×C59—24no—
113C4⋊C4×C59—24no—
114C4⋊C4×C59—22no—
115C4⋊C4×C59—24no—
116C4⋊C4×C59—24no—
117C4⋊C4×C59C2×D414yes—
118C22⋊C4×C59—24no—
119C22⋊C4×C59—24no—
120C42×C59—28no—
121C22⋊Dic59—41no—
122C22⋊Dic59—41no—
123C22⋊Dic59—42no—
124#11745672—84no—
125C4⋊Dic59—22no—
126C4⋊Dic59—81no—
127Dic59⋊C4—42no—
128Dic59⋊C4—82no—
129C2×C59⋊C8—24no—
130C2×C59⋊C8—82no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59#12011yescharacteristic
134C23×C59#12011yescharacteristic
135C23×C59C2×Q1612yes—
136C23×C59#13712yes—
137C23×C59C22×D411yescharacteristic
138Q8×C59—44no—
139Q8×C59—82no—
140Q8×C59—84no—
141C2×C4×C59—22no—
142C2×C4×C59—22no—
143C2×C4×C59—24no—
144C2×C4×C59—24no—
145C2×C4×C59—24no—
146C2×C4×C59—24no—
147C2×C4×C59—22no—
148C2×C4×C59—42no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—42no—
152C2×C4×C59—44no—
153C2×C4×C59—42no—
154C2×C4×C59—44no—
155C2×C4×C59—42no—
156C2×C4×C59—81no—
157C2×C4×C59#12012yes—
158C2×C4×C59C2×Q1614yes—
159C2×C4×C59C22×D412yes—
160C2×Dic59—42no—
161C2×Dic59—42no—
162C2×Dic59—42no—
163C2×Dic59—42no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—42no—
167Dic118—44no—
168C59⋊C8—44no—
169#18295—591nomaximal, Sylow
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—42no—
177C22×C59—42no—
178C22×C59#45112yes—
179C22×C59#45112yes—
180C22×C59C2×#12011yescharacteristic
181C22×C59C22×Q1611yescharacteristic
182C22×C59C2×#13711yescharacteristic
183C4×C59—24no—
184C4×C59—22no—
185C4×C59—22no—
186C4×C59—44no—
187C4×C59—42no—
188C4×C59—44no—
189Dic59—82no—
190Dic59—81no—
191Dic59—81no—
192C2×#544D5911yescharacteristic
193#3189—592no—
194#2627—591no—
195#1617—594no—
196#1605—594no—
197#1520—592no—
198C2×#324—591no—
199C2×C59—21no—
200C2×C59—21no—
201C2×C59—22no—
202C2×C59—22no—
203C2×C59C2×#45111yescharacteristic
204C2×C59#302011yescharacteristic
205C2×C59#302611yescharacteristic
206#544—24no—
207#544—22no—
208#544C2×D5914yes—
209C2×C4⋊Q8—22no—
210C2×#125—22no—
211C2×C22⋊Q8—21no—
212C2×C22⋊Q8—591no—
213C2×C22⋊Q8C2×D5912yes—
214C2×#117—21no—
215C22×C4⋊C4C2×D5911yescharacteristic
216#484—594no—
217#484—1182no—
218#467—1184no—
219#451—594no—
220#451—1182no—
221C2×C22⋊C8—592no—
222#400—592no—
223#398—1182no—
224#395—592no—
225#385—1181no—
226#381—1181no—
227#380—591no—
228#328—598no—
229#324—594no—
230C59#1829511yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8—591no—
233C22×Q8C22⋊D5912yes—
234C23×C4C22⋊D5911yescharacteristic
235C23×C4C22⋊D5911yescharacteristic
236C23×C4C22×D5911yescharacteristic
237C2×Q16—1184no—
238C22×C8—1182no—
239C4⋊Q8—28no—
240C4⋊Q8—28no—
241#125—28no—
242#125—28no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—1182no—
248C22⋊Q8—1182no—
249C22⋊Q8C22⋊D5914yes—
250C22⋊Q8C22×D5914yes—
251#117—24no—
252#117—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—591no—
261C2×C4⋊C4—1181no—
262C2×C4⋊C4—1182no—
263C2×C4⋊C4—1181no—
264C2×C4⋊C4C22⋊D5912yes—
265C2×C4⋊C4C22⋊D5912yes—
266C2×C4⋊C4C22×D5912yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—1181no—
269C2×C22⋊C4—1181no—
270C2×C42—22no—
271#107—1184no—
272#103—1184no—
273#103—1184no—
274C22⋊C8—594no—
275C22⋊C8—1182no—
276#95—592no—
277#95—1182no—
278#95—1182no—
279#95—1182no—
280C24C2×C22⋊D5911yescharacteristic
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—1182no—
286C2×Q8—4721no—
287C2×Q8C2×C22⋊D5914yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—1182no—
298C22×C4—1181no—
299C22×C4—1181no—
300C22×C4#1174567214yescomplemented
301C22×C4C2×C22⋊D5912yes—
302C22×C4C2×C22⋊D5912yes—
303C22×C4C2×C22⋊D5911yescharacteristic
304C22×C4C2×C22⋊D5912yes—
305C22×C4C2×C22⋊D5912yes—
306C22×C4C23×D5911yescharacteristic
307Q16—4724no—
308C2×C8—1184no—
309C2×C8—4722no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—1182no—
320C4⋊C4—4722no—
321C4⋊C4—4721no—
322C4⋊C4—2362no—
323C4⋊C4C2×C22⋊D5914yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—2361no—
327C22⋊C4—2361no—
328C22⋊C4—2362no—
329C42—28no—
330C23—41no—
331C23—41no—
332C23#1427369611yescharacteristic
333C23#1427369611yescharacteristic
334C23C2×#1174567212yes—
335C23#1427375812yes—
336C23C22×C22⋊D5911yescharacteristic
337Q8—44no—
338Q8—82no—
339Q8—84no—
340Q8—2364no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—2362no—
358C2×C4—2362no—
359C2×C4—2362no—
360C2×C4—2362no—
361C2×C4—2361no—
362C2×C4—2361no—
363C2×C4—2362no—
364C2×C4#1427369612yes—
365C2×C4C2×#1174567214yes—
366C2×C4C22×C22⋊D5912yes—
367C8—2364no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1683755812yes—
377C22#1683755812yes—
378C22C2×#1427369611yescharacteristic
379C22C22×#1174567211yescharacteristic
380C22C2×#1427375811yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—44no—
385C4—42no—
386C4—44no—
387C4—4722no—
388C4—4721no—
389C4—4721no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—22no—
394C2C2×#1683755811yescharacteristic
395C2#2143346011yescharacteristic
396C2#2143349311yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#544C2
socleC22×C59C2×#120
prime radicalC24×C59C2×D4
hypercenterC2×#544D59
hypocenterC59#18295
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #21423138C2semidirect
4. #21423070C2semidirect
96. C22×C4×C59Q16semidirect
230. C59#18295semidirect
300. C22×C4#11745672semidirect

Maximal subgroups

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

subgroupindex
#214348682
#214299092
#214231382
#214230702
#214226682
C2×#168371282
C2×C59×#5442
#1829559