Finite Group Databasebeta

← #28484594

Subgroups — #28484594

Order 15104.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433893C211yesmaximal, characteristic
2#21427006C211yesmaximal, characteristic
3#21422661C211yesmaximal, characteristic, complemented
4#21422596C211yesmaximal, characteristic, complemented
5#21421998C211yesmaximal, characteristic, complemented
6#21421955C211yesmaximal, characteristic, complemented
7C59×#1478C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9C2×D4×Dic59—21no—
10#16837702—22no—
11#16837692C412yescomplemented
12#16837682C412yescomplemented
13#16837442—21no—
14#16837426C2211yescharacteristic
15#16837407—21no—
16#16837301—21no—
17#16837277—22no—
18#16837268C2211yescharacteristic
19#16837262—21no—
20C4×C4⋊Dic59C2211yescharacteristic
21C2×C59⋊#95—21no—
22#16837140—21no—
23#16837140—21no—
24#16837140—22no—
25C2×C4×D4×C59C2211yescharacteristic
26C59×#431C2211yescharacteristic
27C59×#431C2211yescharacteristic
28C59×D4⋊C8—22no—
29C23×Dic59—22no—
30C2×#11745664—21no—
31#14273727C2×C411yescharacteristic, complemented
32#14273727C2×C411yescharacteristic, complemented
33#14273719—21no—
34#14273706—21no—
35#14273706—21no—
36C4×Dic118—21no—
37C4×Dic118—21no—
38D4×Dic59—42no—
39D4×Dic59—42no—
40#14273686—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—22no—
44C2×C22⋊Dic59—22no—
45C2×C4×Dic59—21no—
46#14273660—22no—
47#14273658—22no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)C2×C412yescomplemented
50#14273648—22no—
51#14273648—22no—
52C59⋊#95—22no—
53C59⋊#95—21no—
54C59⋊#95D411yescharacteristic
55C59⋊#95D411yescharacteristic
56C59⋊#95D412yes—
57C22×D4×C59D411yescharacteristic
58C23×C4×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59C2×C412yes—
62C4×D4×C59D412yes—
63C2×C4⋊C4×C59D411yescharacteristic
64C2×C22⋊C4×C59—21no—
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C22⋊C8×C59—21no—
69C22⋊C8×C59—21no—
70C4×C8×C59—21no—
71C4×C8×C59—21no—
72C24×C59—21no—
73C2×D4×C59—21no—
74C2×D4×C59—21no—
75C2×D4×C59—21no—
76C2×D4×C59C22⋊C412yes—
77C2×D4×C59SD1612yes—
78C2×D4×C59Q1612yes—
79C22×C4×C59—21no—
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—82no—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×Dic59—21no—
87C22×Dic59—22no—
88C22×Dic59—21no—
89C22×Dic59—44no—
90C22×Dic59—44no—
91C22×Dic59—82no—
92C2×Dic118—21no—
93C2×Dic118—21no—
94C2×C8×C59—41no—
95C2×C8×C59—41no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59C22⋊C412yes—
98C22⋊C4×C59—22no—
99C22⋊C4×C59—22no—
100C42×C59C22×C411yescharacteristic
101C42×C59C2×D411yescharacteristic
102C42×C59#4111yescharacteristic
103C42×C59#4111yescharacteristic
104C22⋊Dic59—44no—
105C22⋊Dic59—41no—
106C4⋊Dic59—21no—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110C4⋊Dic59—42no—
111C4⋊Dic59—41no—
112C4⋊Dic59—41no—
113C4⋊Dic59—82no—
114Dic59⋊C4—41no—
115Dic59⋊C4—41no—
116C4×Dic59—41no—
117C4×Dic59—41no—
118C4×Dic59—41no—
119#11745664—41no—
120#11745664—41no—
121C2×C59⋊C8—22no—
122C23×C59—22no—
123C23×C59—22no—
124C23×C59—22no—
125C23×C59—42no—
126C23×C59—42no—
127C23×C59#12011yescharacteristic
128D4×C59—22no—
129D4×C59—22no—
130D4×C59—22no—
131D4×C59#10314yes—
132C2×C4×C59—22no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—42no—
136C2×C4×C59—82no—
137C2×C4×C59—82no—
138C2×C4×C59C2×C22⋊C411yescharacteristic
139C2×C4×C59C4×D411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12311yescharacteristic
144C2×C4×C59C2×SD1611yescharacteristic
145C2×C4×C59C2×Q1611yescharacteristic
146C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C59#13711yescharacteristic
148C2×Dic59—22no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—42no—
154C2×Dic59—42no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—42no—
159C2×Dic59—88no—
160Dic118—81no—
161Dic118—81no—
162C8×C59—81no—
163C8×C59—81no—
164C59⋊C8—42no—
165#9175—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—24no—
169C22×C59—44no—
170C22×C59—44no—
171C22×C59—42no—
172C22×C59—42no—
173C22×C59#38611yescharacteristic
174C22×C59#45011yescharacteristic
175C22×C59#45111yescharacteristic
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—42no—
181C4×C59C2×#10311yescharacteristic
182C4×C59#41711yescharacteristic
183C4×C59#48311yescharacteristic
184C4×C59#48411yescharacteristic
185Dic59—42no—
186Dic59—81no—
187Dic59—82no—
188Dic59—81no—
189Dic59—81no—
190#3077—591no—
191#2349—591no—
192#1518—591no—
193#1503—591no—
194#1478D5911yescharacteristic
195#1303—591no—
196#1286—591no—
197C2×C59—21no—
198C2×C59—21no—
199C2×C59—44no—
200C2×C59#187211yescharacteristic
201C2×C59#190411yescharacteristic
202C2×C59#190611yescharacteristic
203#533—591no—
204C2×C4×D4—1181no—
205C2×C4×D4C2×D5911yescharacteristic
206#478—1182no—
207#476—592no—
208#474—592no—
209#431C2×D5911yescharacteristic
210#431C2×D5911yescharacteristic
211#428—1181no—
212#423—591no—
213#418—1181no—
214#389—1181no—
215#385—1182no—
216#382—591no—
217#380—1181no—
218C4×C4⋊C4—591no—
219C2×#95—1181no—
220#328—1181no—
221#328—1181no—
222#328—1182no—
223D4⋊C8—22no—
224C59#917511yescharacteristic, complemented, Sylow
225C22×D4D23611yescharacteristic
226C23×C4—21no—
227C23×C4—1182no—
228C2×M16—1181no—
229C4⋊Q8—591no—
230C4⋊Q8—591no—
231#125—1181no—
232C22⋊Q8—1181no—
233C22⋊Q8—1181no—
234C4×Q8—1181no—
235C4×Q8—1181no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—2362no—
239C4×D4—2362no—
240C4×D4C4×D5912yes—
241C4×D4C22⋊D5912yes—
242#117—1181no—
243C2×C4⋊C4—1182no—
244C2×C4⋊C4—1181no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4D23611yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1182no—
249C2×C42—1181no—
250C2×C42C22×D5911yescharacteristic
251#107—1182no—
252#106—1182no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—592no—
256C4⋊C8—1181no—
257#102—1182no—
258#102—1182no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263#95—591no—
264#95—591no—
265#95—592no—
266#95—1182no—
267#95—1181no—
268C24—21no—
269C2×Q8—1181no—
270C2×Q8—1181no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#1174566012yes—
275C2×D4Dic23612yes—
276C2×D4(C2×D59)⋊C412yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—1181no—
282C22×C4—1182no—
283C22×C4—1181no—
284C22×C4—4722no—
285C22×C4—2364no—
286C22×C4—2364no—
287C22×C4C2×D23611yescharacteristic
288C22×C4D4×D5911yescharacteristic
289C22×C4D4×D5911yescharacteristic
290M16—2361no—
291M16—2361no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—1182no—
295C4⋊C4—21no—
296C4⋊C4—1181no—
297C4⋊C4—1181no—
298C4⋊C4—1181no—
299C4⋊C4—1181no—
300C4⋊C4—4722no—
301C4⋊C4—2362no—
302C4⋊C4—2361no—
303C4⋊C4—2361no—
304C4⋊C4—2361no—
305C4⋊C4—2361no—
306C4⋊C4(C2×D59)⋊C412yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—2364no—
310C22⋊C4—2361no—
311C42—2361no—
312C42—2361no—
313C42—2361no—
314C42C2×C4×D5911yescharacteristic
315C42#1174568711yescharacteristic
316C42#1174568711yescharacteristic
317C42C2×C22⋊D5911yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1427369711yescharacteristic
324Q8—4721no—
325Q8—4721no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1427365214yes—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—1182no—
337C2×C4—4728no—
338C2×C4—2361no—
339C2×C4—2362no—
340C2×C4—2361no—
341C2×C4—2362no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4—2362no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4C2×(C2×D59)⋊C411yescharacteristic
349C2×C4(C22⋊D59)⋊C411yescharacteristic
350C2×C4(C22⋊D59)⋊C411yescharacteristic
351C2×C4#1427369911yescharacteristic
352C2×C4#1427369911yescharacteristic
353C2×C4#1427371111yescharacteristic
354C2×C4C2×#1174566011yescharacteristic
355C2×C4C2×Dic23611yescharacteristic
356C2×C4#1427375511yescharacteristic
357C2×C4#1427376011yescharacteristic
358C8—81no—
359C8—81no—
360C8—2362no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1683727911yescharacteristic
369C22#1683755511yescharacteristic
370C22#1683756011yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—4722no—
377C4—4721no—
378C4—4721no—
379C4—4721no—
380C4—2362no—
381C4C2×#1427365211yescharacteristic
382C4#1683740411yescharacteristic
383C4#1683773311yescharacteristic
384C4#1683773811yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2142440111yescharacteristic
389C2#2142458111yescharacteristic
390C2#2142459511yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1478C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#1478D59
hypocenterC59#9175
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
3. #21422661C2semidirect
4. #21422596C2semidirect
5. #21421998C2semidirect
6. #21421955C2semidirect
11. #16837692C4semidirect
12. #16837682C4semidirect
31. #14273727C2×C4semidirect
32. #14273727C2×C4semidirect
49. C4⋊(C59⋊C8)C2×C4semidirect
224. C59#9175semidirect

Maximal subgroups

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

subgroupindex
#214338932
#214270062
#214226612
#214225962
#214219982
#214219552
C59×#14782
#917559