Finite Group Databasebeta

← #28484442

Subgroups — #28484442

Order 15104.

946 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432383C211yesmaximal, characteristic
2C2×#16837262C211yesmaximal, characteristic
3C2×#16837140C212yesmaximal, complemented
4#21421976C212yesmaximal, complemented
5C59×#1779C211yesmaximal, characteristic
6C2×#14273706C412yescomplemented
7C2×#14273706C2211yescharacteristic
8C22×C4⋊Dic59—21no—
9C22×C22⋊Dic59—21no—
10#16837275—21no—
11#16837275—22no—
12#16837262—24no—
13#16837262C412yescomplemented
14#16837262C2211yescharacteristic
15#16837258—21no—
16C2×C59⋊#95C2212yes—
17#16837140—24no—
18C24×C4×C59C2211yescharacteristic
19C2×C22⋊C8×C59—22no—
20C2×C22⋊C8×C59C2212yes—
21C22×Dic118C2×C411yescharacteristic
22C23×Dic59—22no—
23#14273706—22no—
24#14273706—22no—
25#14273706—44no—
26C2×C4⋊Dic59—24no—
27C2×C4⋊Dic59—44no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×C4⋊Dic59C2×C412yes—
30C2×Dic59⋊C4—22no—
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59—24no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59—22no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—42no—
38C2×C22⋊Dic59—42no—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C59⋊#95—21no—
42C59⋊#95D414yes—
43C25×C59D411yescharacteristic
44C23×C4×C59—22no—
45C23×C4×C59—22no—
46C23×C4×C59—44no—
47C23×C4×C59C2×C412yes—
48C23×C4×C59D411yescharacteristic
49C23×C4×C59D412yes—
50C23×C4×C59C2311yescharacteristic
51C22×C8×C59—22no—
52C22⋊C8×C59—22no—
53C22⋊C8×C59—22no—
54C22⋊C8×C59—44no—
55C24×C59—21no—
56C24×C59—21no—
57C24×C59—21no—
58C24×C59—21no—
59C24×C59—42no—
60C24×C59—42no—
61C24×C59C22⋊C412yes—
62C24×C59SD1612yes—
63C24×C59Q1612yes—
64C24×C59C2×D411yescharacteristic
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—24no—
68C22×C4×C59—24no—
69C22×C4×C59—24no—
70C22×C4×C59—24no—
71C22×C4×C59—24no—
72C22×C4×C59—42no—
73C22×C4×C59—42no—
74C22×C4×C59—42no—
75C22×C4×C59—48no—
76C22×C4×C59—48no—
77C22×C4×C59C22⋊C412yes—
78C22×C4×C59C22×C411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D412yes—
81C22×C4×C59#4112yes—
82C22×Dic59—21no—
83C22×Dic59—22no—
84C22×Dic59—21no—
85C22×Dic59—24no—
86C22×Dic59—44no—
87C22×Dic59—44no—
88C22×Dic59—42no—
89C22×Dic59—42no—
90C2×Dic118—22no—
91C2×Dic118—42no—
92C2×C8×C59—44no—
93C2×C8×C59—42no—
94C22⋊Dic59—44no—
95C22⋊Dic59—48no—
96C22⋊Dic59—44no—
97C22⋊Dic59—41no—
98C22⋊Dic59—41no—
99C22⋊Dic59—41no—
100C22⋊Dic59—41no—
101C4⋊Dic59—22no—
102C4⋊Dic59—48no—
103C4⋊Dic59—42no—
104Dic59⋊C4—44no—
105Dic59⋊C4—42no—
106C23×C59—22no—
107C23×C59—22no—
108C23×C59—22no—
109C23×C59—22no—
110C23×C59—22no—
111C23×C59—22no—
112C23×C59—22no—
113C23×C59—22no—
114C23×C59—22no—
115C23×C59—22no—
116C23×C59—22no—
117C23×C59—22no—
118C23×C59—44no—
119C23×C59—44no—
120C23×C59—42no—
121C23×C59—42no—
122C23×C59—42no—
123C23×C59—42no—
124C23×C59—44no—
125C23×C59—44no—
126C23×C59#10314yes—
127C23×C59C2×C22⋊C411yescharacteristic
128C23×C59#12011yescharacteristic
129C23×C59#12311yescharacteristic
130C23×C59C2×SD1611yescharacteristic
131C23×C59C2×Q1611yescharacteristic
132C23×C59AGL(1,ℤ/8)11yescharacteristic
133C23×C59#13711yescharacteristic
134C2×C4×C59—24no—
135C2×C4×C59—24no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—22no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—48no—
143C2×C4×C59—48no—
144C2×C4×C59#9914yes—
145C2×C4×C59C4×D412yes—
146C2×C4×C59#12112yes—
147C2×Dic59—44no—
148C2×Dic59—44no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—42no—
152C2×Dic59—42no—
153C2×Dic59—42no—
154C2×Dic59—42no—
155C2×Dic59—42no—
156C2×Dic59—44no—
157C2×Dic59—84no—
158C2×Dic59—84no—
159Dic118—82no—
160C8×C59—82no—
161#9151—591nomaximal, Sylow
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—22no—
167C22×C59—22no—
168C22×C59—22no—
169C22×C59—22no—
170C22×C59—24no—
171C22×C59—24no—
172C22×C59—24no—
173C22×C59—44no—
174C22×C59—44no—
175C22×C59—44no—
176C22×C59—44no—
177C22×C59—42no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—42no—
181C22×C59#38611yescharacteristic
182C22×C59C2×#9912yes—
183C22×C59C2×#10311yescharacteristic
184C22×C59#41711yescharacteristic
185C22×C59#42112yes—
186C22×C59#45011yescharacteristic
187C22×C59#45111yescharacteristic
188C22×C59#48311yescharacteristic
189C22×C59#48411yescharacteristic
190C4×C59—22no—
191C4×C59—22no—
192C4×C59—24no—
193Dic59—84no—
194Dic59—81no—
195Dic59—81no—
196Dic59—82no—
197#2848—591no—
198C2×#380—591no—
199#1779D5911yescharacteristic
200C2×#328—592no—
201#1294—592no—
202C2×C59—21no—
203C2×C59—21no—
204C2×C59—21no—
205C2×C59—21no—
206C2×C59—44no—
207C2×C59#151312yes—
208C2×C59#152012yes—
209C2×C59#187211yescharacteristic
210C2×C59#189711yescharacteristic
211C2×C59#189811yescharacteristic
212C24×C4C2×D5911yescharacteristic
213C2×C22⋊Q8—591no—
214C2×C22⋊Q8—592no—
215C22×C4⋊C4—1181no—
216C22×C22⋊C4—1181no—
217C2×C22⋊C8—22no—
218C2×C22⋊C8C2×D5912yes—
219#385—1181no—
220#385—1182no—
221#380—592no—
222#380—591no—
223#380—1184no—
224#379—1181no—
225C2×#95—592no—
226#328—1184no—
227C59#915111yescharacteristic, complemented, Sylow
228C25D23611yescharacteristic
229C22×Q8—591no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—44no—
233C23×C4—1182no—
234C23×C4C4×D5912yes—
235C23×C4D23611yescharacteristic
236C23×C4C22⋊D5912yes—
237C23×C4C22×D5911yescharacteristic
238C22×C8—22no—
239C22⋊Q8—1182no—
240C22⋊Q8—1182no—
241C22⋊Q8—2364no—
242C2×C4⋊C4—591no—
243C2×C4⋊C4—592no—
244C2×C4⋊C4—1184no—
245C2×C4⋊C4—1182no—
246C2×C4⋊C4—1181no—
247C2×C4⋊C4—2364no—
248C2×C22⋊C4—1184no—
249C2×C22⋊C4—1182no—
250C2×C22⋊C4—1182no—
251C2×C22⋊C4—1181no—
252C2×C22⋊C4—1181no—
253C2×C22⋊C4—2362no—
254C2×C22⋊C4—2362no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—44no—
258#95—594no—
259#95—1182no—
260#95—1182no—
261#95—1181no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—42no—
267C24—42no—
268C24#1174566012yes—
269C24Dic23612yes—
270C24(C2×D59)⋊C412yes—
271C24C2×D23611yescharacteristic
272C2×Q8—1182no—
273C2×Q8—2362no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—1182no—
287C22×C4—1181no—
288C22×C4—1181no—
289C22×C4—1184no—
290C22×C4—2364no—
291C22×C4—2364no—
292C22×C4—2362no—
293C22×C4—2362no—
294C22×C4(C2×D59)⋊C412yes—
295C22×C4C2×C4×D5911yescharacteristic
296C22×C4D4×D5912yes—
297C22×C4#1174568712yes—
298C22×C4C2×C22⋊D5911yescharacteristic
299C2×C8—44no—
300C2×C8—42no—
301C4⋊C4—1182no—
302C4⋊C4—2362no—
303C4⋊C4—2368no—
304C4⋊C4—2364no—
305C4⋊C4—2362no—
306C22⋊C4—2368no—
307C22⋊C4—2364no—
308C22⋊C4—2364no—
309C22⋊C4—2361no—
310C22⋊C4—2361no—
311C22⋊C4—2361no—
312C22⋊C4—2361no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—44no—
326C23—44no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23#1427365214yes—
334C23C2×(C2×D59)⋊C411yescharacteristic
335C23#1427369711yescharacteristic
336C23#1427371111yescharacteristic
337C23C2×#1174566011yescharacteristic
338C23C2×Dic23611yescharacteristic
339C23#1427375511yescharacteristic
340C23#1427376011yescharacteristic
341Q8—4722no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—4724no—
353C2×C4—4724no—
354C2×C4—2364no—
355C2×C4—2364no—
356C2×C4—2362no—
357C2×C4—2362no—
358C2×C4—2362no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2364no—
362C2×C4—2362no—
363C2×C4—2362no—
364C2×C4#1427364114yes—
365C2×C4(C22⋊D59)⋊C412yes—
366C2×C4#1427369912yes—
367C8—82no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22#1683727911yescharacteristic
388C22C2×#1427364112yes—
389C22C2×#1427365211yescharacteristic
390C22#1683740411yescharacteristic
391C22#1683741812yes—
392C22#1683755511yescharacteristic
393C22#1683756011yescharacteristic
394C22#1683773311yescharacteristic
395C22#1683773811yescharacteristic
396C4—22no—
397C4—24no—
398C4—22no—
399C4—4724no—
400C4—4722no—
401C4—4721no—
402C4—4721no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—44no—
408C2#2142264012yes—
409C2#2142266912yes—
410C2#2142440111yescharacteristic
411C2#2142454211yescharacteristic
412C2#2142454711yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1779C2
socleC23×C59C2×C22⋊C4
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#1779D59
hypocenterC59#9151
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. C2×#16837140C2semidirect
4. #21421976C2semidirect
6. C2×#14273706C4semidirect
13. #16837262C4semidirect
227. C59#9151semidirect

Maximal subgroups

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

subgroupindex
#214323832
C2×#168372622
C2×#168371402
#214219762
C59×#17792
#915159