Finite Group Databasebeta

← #28484596

Subgroups — #28484596

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#21422659C211yesmaximal, characteristic, complemented
3#21422594C211yesmaximal, characteristic, complemented
4#21422487C211yesmaximal, characteristic
5#21421999C211yesmaximal, characteristic, complemented
6#21421954C211yesmaximal, characteristic, complemented
7C59×#2349C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837702—22no—
10#16837692C412yescomplemented
11#16837682C412yescomplemented
12#16837455C2211yescharacteristic
13#16837455C2211yescharacteristic
14#16837442—21no—
15#16837426C2211yescharacteristic
16#16837407—21no—
17#16837138—21no—
18#16837138—21no—
19#16837138—22no—
20D4⋊(C59⋊C8)—22no—
21C2×C4×D4×C59—21no—
22C2×C4×D4×C59C2211yescharacteristic
23C59×#389—21no—
24C59×#385—22no—
25C59×#382C2211yescharacteristic
26C59×#380—21no—
27C4×C4⋊C4×C59C2211yescharacteristic
28C2×#95×C59—21no—
29C2×#11745664—21no—
30#14273727C2×C411yescharacteristic, complemented
31#14273727C2×C411yescharacteristic, complemented
32#14273719—21no—
33#14273706—21no—
34#14273706—21no—
35C4×Dic118—21no—
36C4×Dic118—21no—
37#14273686—21no—
38#14273660—22no—
39#14273658—22no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)C2×C412yescomplemented
44#14273648—22no—
45#14273648—22no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48C4×C59⋊C8—21no—
49C4×C59⋊C8—21no—
50C22×D4×C59D411yescharacteristic
51C23×C4×C59—21no—
52C23×C4×C59—22no—
53C4×D4×C59—21no—
54C4×D4×C59—21no—
55C4×D4×C59—42no—
56C4×D4×C59—42no—
57C4×D4×C59C2×C412yes—
58C4×D4×C59D412yes—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59—22no—
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C22⋊C4×C59—21no—
64C2×C22⋊C4×C59—22no—
65C2×C42×C59—21no—
66C2×C42×C59C2311yescharacteristic
67#95×C59—22no—
68#95×C59—21no—
69#95×C59D411yescharacteristic
70#95×C59D411yescharacteristic
71#95×C59D412yes—
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—21no—
81C22×C4×C59—22no—
82C22×C4×C59—22no—
83C22×C4×C59—22no—
84C22×C4×C59—21no—
85C22×C4×C59—44no—
86C22×C4×C59—44no—
87C22×C4×C59—82no—
88C22×C4×C59—82no—
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C2×Dic118—21no—
93C2×Dic118—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—42no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59—41no—
98C4⋊C4×C59—82no—
99C4⋊C4×C59C22⋊C412yes—
100C22⋊C4×C59—22no—
101C22⋊C4×C59—22no—
102C22⋊C4×C59—44no—
103C42×C59—41no—
104C42×C59—41no—
105C42×C59C22×C411yescharacteristic
106C42×C59C2×D411yescharacteristic
107C42×C59#4111yescharacteristic
108C42×C59#4111yescharacteristic
109C22⋊Dic59—41no—
110C4⋊Dic59—21no—
111C4⋊Dic59—21no—
112C4⋊Dic59—21no—
113C4⋊Dic59—21no—
114Dic59⋊C4—41no—
115Dic59⋊C4—41no—
116C4×Dic59—41no—
117#11745664—41no—
118#11745664—41no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8—41no—
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—22no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—42no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—41no—
144C2×C4×C59—42no—
145C2×C4×C59—82no—
146C2×C4×C59—82no—
147C2×C4×C59—88no—
148C2×C4×C59C2×C22⋊C411yescharacteristic
149C2×C4×C59C4×D411yescharacteristic
150C2×C4×C59C4×D411yescharacteristic
151C2×C4×C59#12111yescharacteristic
152C2×C4×C59#12111yescharacteristic
153C2×C4×C59#12311yescharacteristic
154C2×C4×C59C2×SD1611yescharacteristic
155C2×C4×C59C2×Q1611yescharacteristic
156C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C59#13711yescharacteristic
158C2×Dic59—41no—
159C2×Dic59—41no—
160Dic118—81no—
161Dic118—81no—
162C59⋊C8—42no—
163C59⋊C8—81no—
164C59⋊C8—81no—
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×C59—42no—
182C4×C59—81no—
183C4×C59—82no—
184C4×C59C2×#10311yescharacteristic
185C4×C59#41711yescharacteristic
186C4×C59#48311yescharacteristic
187C4×C59#48411yescharacteristic
188Dic59—81no—
189Dic59—81no—
190#3077—591no—
191#2349D5911yescharacteristic
192#1518—591no—
193#1503—591no—
194#1478—591no—
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—21no—
205C2×C4×D4C2×D5911yescharacteristic
206#478—1182no—
207#476—592no—
208#474—592no—
209#431—591no—
210#431—591no—
211#428—1181no—
212#423—591no—
213#418—1181no—
214#389—21no—
215#385—22no—
216#382C2×D5911yescharacteristic
217#380—21no—
218C4×C4⋊C4C2×D5911yescharacteristic
219C2×#95—21no—
220#328—1181no—
221#328—1181no—
222#328—1182no—
223D4⋊C8—1182no—
224C59#917511yescharacteristic, complemented, Sylow
225C22×D4C22⋊D5911yescharacteristic
226C23×C4—21no—
227C23×C4—22no—
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—42no—
239C4×D4—42no—
240C4×D4C4×D5912yes—
241C4×D4D23612yes—
242#117—1181no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C22⋊D5911yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C42—21no—
250C2×C42C22×D5911yescharacteristic
251#107—1182no—
252#106—1182no—
253C4⋊C8—592no—
254C4⋊C8—1181no—
255C4⋊C8—1181no—
256C4⋊C8—1181no—
257#102—1182no—
258#102—1182no—
259C22⋊C8—1181no—
260C22⋊C8—1181no—
261C4×C8—1181no—
262C4×C8—1181no—
263#95—22no—
264#95—21no—
265#95C22⋊D5911yescharacteristic
266#95C22⋊D5911yescharacteristic
267#95C22⋊D5912yes—
268C24—21no—
269C2×Q8—1181no—
270C2×Q8—1181no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4(C2×D59)⋊C412yes—
275C2×D4#1174567012yes—
276C2×D4#1174567212yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C22⋊D5911yescharacteristic
288C22×C4C2×C22⋊D5911yescharacteristic
289C22×C4C2×C22⋊D5911yescharacteristic
290M16—2361no—
291M16—2361no—
292C2×C8—1182no—
293C2×C8—2361no—
294C2×C8—2361no—
295C4⋊C4—21no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C4⋊C4—1181no—
301C4⋊C4—1181no—
302C4⋊C4—1181no—
303C4⋊C4—1181no—
304C4⋊C4—2361no—
305C4⋊C4—2361no—
306C4⋊C4(C2×D59)⋊C412yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—44no—
310C22⋊C4—2361no—
311C42—41no—
312C42—41no—
313C42—2361no—
314C42C2×C4×D5911yescharacteristic
315C42C2×D23611yescharacteristic
316C42#1174568511yescharacteristic
317C42#1174568511yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1427369611yescharacteristic
324Q8—4721no—
325Q8—4721no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1427365014yes—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—88no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4C2×(C2×D59)⋊C411yescharacteristic
349C2×C4C4×C22⋊D5911yescharacteristic
350C2×C4C4×C22⋊D5911yescharacteristic
351C2×C4#1427370111yescharacteristic
352C2×C4#1427370111yescharacteristic
353C2×C4#1427371011yescharacteristic
354C2×C4C2×#1174567011yescharacteristic
355C2×C4C2×#1174567211yescharacteristic
356C2×C4#1427375311yescharacteristic
357C2×C4#1427375811yescharacteristic
358C8—4721no—
359C8—4721no—
360C8—2362no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1683728111yescharacteristic
369C22#1683755311yescharacteristic
370C22#1683755811yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—82no—
378C4—81no—
379C4—4721no—
380C4—4721no—
381C4C2×#1427365011yescharacteristic
382C4#1683740111yescharacteristic
383C4#1683773211yescharacteristic
384C4#1683773711yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2142440311yescharacteristic
389C2#2142458311yescharacteristic
390C2#2142459711yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2349C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#2349D59
hypocenterC59#9175
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #21422659C2semidirect
3. #21422594C2semidirect
5. #21421999C2semidirect
6. #21421954C2semidirect
10. #16837692C4semidirect
11. #16837682C4semidirect
30. #14273727C2×C4semidirect
31. #14273727C2×C4semidirect
43. 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
#214226592
#214225942
#214224872
#214219992
#214219542
C59×#23492
#917559