Finite Group Databasebeta

← #28559390

Subgroups — #28559390

Order 15104.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433245C212yesmaximal, complemented
2#21423010C211yesmaximal, characteristic, complemented
3#21423005C211yesmaximal, characteristic, complemented
4#21422987C212yesmaximal, complemented
5#21422920C211yesmaximal, characteristic, complemented
6#21422915C211yesmaximal, characteristic, complemented
7SD16×C59⋊C8C212yesmaximal, complemented
8#21422852C211yesmaximal, characteristic, complemented
9#21422561C212yesmaximal, complemented
10#21422379C211yesmaximal, characteristic, complemented
11C59×#3168C211yesmaximal, characteristic
12#16837533C2212yes—
13Q8×C59⋊C8C2212yes—
14#16837480—22no—
15#16837480C2212yes—
16#16837474—22no—
17D4×C59⋊C8C2212yes—
18#16837463—22no—
19#16837455C2211yescharacteristic
20#16837455C2211yescharacteristic
21#16837429—22no—
22#16837426C2212yes—
23#16837153C2212yes—
24#16837153C2211yescharacteristic
25#16837153C2211yescharacteristic
26Q8⋊(C59⋊C8)C2212yes—
27Q8⋊(C59⋊C8)C2212yes—
28D4⋊(C59⋊C8)C2212yes—
29D4⋊(C59⋊C8)C2212yes—
30#16837123C2211yescharacteristic
31#16837123C2211yescharacteristic
32C8×C59⋊C8C2211yescharacteristic
33C8×C59⋊C8C2211yescharacteristic
34C59×#529C2212yes—
35C59×#499C411yescharacteristic
36C59×#495C411yescharacteristic
37C59×#488C412yes—
38C59×#467C411yescharacteristic
39C59×#466C411yescharacteristic
40C59×#465C412yes—
41C4×Q16×C59C2211yescharacteristic
42C4×SD16×C59C2212yes—
43C4×D8×C59C2211yescharacteristic
44C2×C4×C8×C59C2211yescharacteristic
45C2×#11745664—22no—
46C2×#11745664—22no—
47C22×C59⋊C8—22no—
48C4⋊(C59⋊C8)—22no—
49C4⋊(C59⋊C8)—21no—
50C4⋊(C59⋊C8)—22no—
51C4⋊(C59⋊C8)—22no—
52C4⋊(C59⋊C8)D412yes—
53C4⋊(C59⋊C8)C2312yes—
54C4⋊(C59⋊C8)C2312yes—
55C22⋊(C59⋊C8)—21no—
56C22⋊(C59⋊C8)—22no—
57C22⋊(C59⋊C8)—22no—
58C22⋊(C59⋊C8)D412yes—
59#14273638—22no—
60C4×C59⋊C8—22no—
61C4×C59⋊C8C2311yescharacteristic
62C4×C59⋊C8C2311yescharacteristic
63C4×C59⋊C8C2311yescharacteristic
64C4×C59⋊C8C2311yescharacteristic
65C2×Q16×C59C2×C411yescharacteristic
66C2×SD16×C59C2×C412yes—
67C2×D8×C59C2×C411yescharacteristic
68C22×C8×C59C2×C411yescharacteristic
69C22×C8×C59C2×C411yescharacteristic
70#126×C59—22no—
71#125×C59C2×C412yes—
72#124×C59C2×C412yes—
73#123×C59—22no—
74C22⋊Q8×C59C2×C412yes—
75#121×C59C2×C412yes—
76C4×Q8×C59C2312yes—
77C4×D4×C59—22no—
78C4×D4×C59C2312yes—
79#117×C59—22no—
80C2×C42×C59C2311yescharacteristic
81#107×C59C2×C411yescharacteristic
82#107×C59C2×C411yescharacteristic
83#106×C59C2×C412yes—
84#103×C59C2×C412yes—
85#103×C59C2×C412yes—
86#102×C59C2×C412yes—
87#102×C59C2×C412yes—
88C4×C8×C59C2×C411yescharacteristic
89C4×C8×C59C2×C411yescharacteristic
90C4×C8×C59C2311yescharacteristic
91C4×C8×C59C2311yescharacteristic
92C2×Q8×C59C22×C412yes—
93C2×D4×C59—22no—
94C2×D4×C59C22×C412yes—
95C22×C4×C59—22no—
96C22×C4×C59C22×C411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59#4111yescharacteristic
99Q16×C59—22no—
100SD16×C59—24no—
101D8×C59—22no—
102C2×C8×C59—22no—
103C2×C8×C59—22no—
104C2×C8×C59C22×C411yescharacteristic
105C2×C8×C59C22×C411yescharacteristic
106C2×C8×C59C22×C411yescharacteristic
107C2×C8×C59C22×C411yescharacteristic
108C4⋊C4×C59—22no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—22no—
111C4⋊C4×C59—22no—
112C4⋊C4×C59C22×C412yes—
113C4⋊C4×C59C22×C412yes—
114C22⋊C4×C59—22no—
115C22⋊C4×C59—22no—
116C22⋊C4×C59—22no—
117C22⋊C4×C59—22no—
118C22⋊C4×C59—22no—
119C42×C59—22no—
120C42×C59C22×C411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59#4111yescharacteristic
123C42×C59C2411yescharacteristic
124#11745664—82no—
125#11745664—82no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—21no—
130C2×C59⋊C8—82no—
131C2×C59⋊C8C2×D412yes—
132C2×C59⋊C8C2×D412yes—
133C23×C59—22no—
134C23×C59C4×D411yescharacteristic
135Q8×C59—22no—
136Q8×C59—22no—
137D4×C59—22no—
138D4×C59—22no—
139D4×C59—82no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—22no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59C4×D411yescharacteristic
150C2×C4×C59C4×D411yescharacteristic
151C2×C4×C59C4×D411yescharacteristic
152C2×C4×C59C23×C411yescharacteristic
153C2×C4×C59C22×D411yescharacteristic
154C2×C4×C59C2×#4111yescharacteristic
155C8×C59—21no—
156C8×C59—21no—
157C8×C59#13112yes—
158C8×C59#13112yes—
159C59⋊C8—22no—
160C59⋊C8—22no—
161C59⋊C8—41no—
162C59⋊C8—41no—
163C59⋊C8—42no—
164C59⋊C8—42no—
165#17974—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—42no—
172C22×C59C2×C4×D411yescharacteristic
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—21no—
177C4×C59—42no—
178C4×C59—42no—
179C4×C59—42no—
180C4×C59C2×#13111yescharacteristic
181C4×C59C2×#13111yescharacteristic
182C4×C59#57712yes—
183#3168D5911yescharacteristic
184#2993—592no—
185#1594—591no—
186#1593—591no—
187#1590—592no—
188#1580—591no—
189#1579—591no—
190C8×SD16—592no—
191#1566—591no—
192#1495—592no—
193#1450—591no—
194C2×C59—41no—
195C2×C59—82no—
196C2×C59#293111yescharacteristic
197C2×C59#295211yescharacteristic
198C2×C59#295711yescharacteristic
199#529C2×D5912yes—
200#499Dic5911yescharacteristic
201#495Dic5911yescharacteristic
202#488Dic5912yes—
203#467Dic5911yescharacteristic
204#466Dic5911yescharacteristic
205#465Dic5912yes—
206#446—592no—
207C8×Q8—592no—
208C4×Q16C2×D5911yescharacteristic
209C4×SD16C2×D5912yes—
210C4×D8C2×D5911yescharacteristic
211#436—592no—
212#436—1182no—
213#435—1182no—
214C8×D4—592no—
215#433—1182no—
216#431—591no—
217#431—591no—
218#424—1182no—
219#423—592no—
220C2×C4×C8C2×D5911yescharacteristic
221#334—592no—
222#334—591no—
223#334—591no—
224Q8⋊C8—592no—
225Q8⋊C8—592no—
226D4⋊C8—592no—
227D4⋊C8—592no—
228#322—591no—
229#322—591no—
230C82—591no—
231C82—591no—
232C59#1797411yescharacteristic, complemented, Sylow
233C2×Q16C2×Dic5911yescharacteristic
234C2×SD16C2×Dic5912yes—
235C2×D8C2×Dic5911yescharacteristic
236C2×M16—1182no—
237C2×M16—1182no—
238C22×C8—1182no—
239C22×C8C2×Dic5911yescharacteristic
240C22×C8C2×Dic5911yescharacteristic
241#126—22no—
242#125C2×Dic5912yes—
243#124C2×Dic5912yes—
244#123—22no—
245C22⋊Q8C2×Dic5912yes—
246#121C2×Dic5912yes—
247C4×Q8C22×D5912yes—
248C4×D4—22no—
249C4×D4C22×D5912yes—
250#117—22no—
251C2×C42C22×D5911yescharacteristic
252#107C2×Dic5911yescharacteristic
253#107C2×Dic5911yescharacteristic
254#106C2×Dic5912yes—
255C4⋊C8—592no—
256C4⋊C8—592no—
257C4⋊C8—592no—
258C4⋊C8—1182no—
259C4⋊C8—1182no—
260C4⋊C8—1182no—
261C4⋊C8—1181no—
262#103C2×Dic5912yes—
263#103C2×Dic5912yes—
264#102C2×Dic5912yes—
265#102C2×Dic5912yes—
266C22⋊C8—592no—
267C22⋊C8—1182no—
268C22⋊C8—1182no—
269C22⋊C8—1181no—
270#97—1182no—
271C4×C8—591no—
272C4×C8—591no—
273C4×C8—591no—
274C4×C8—591no—
275C4×C8—1182no—
276C4×C8C2×Dic5911yescharacteristic
277C4×C8C2×Dic5911yescharacteristic
278C4×C8C22×D5911yescharacteristic
279C4×C8C22×D5911yescharacteristic
280C2×Q8C22×Dic5912yes—
281C2×D4—22no—
282C2×D4C22×Dic5912yes—
283C22×C4—22no—
284C22×C4D4×D5911yescharacteristic
285C22×C4#1174568711yescharacteristic
286C22×C4C22×Dic5911yescharacteristic
287Q16—22no—
288SD16—24no—
289D8—22no—
290M16—4722no—
291M16—4722no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—592no—
295C2×C8—592no—
296C2×C8—1182no—
297C2×C8—1182no—
298C2×C8—1181no—
299C2×C8—1181no—
300C2×C8—4722no—
301C2×C8C22×Dic5911yescharacteristic
302C2×C8C22×Dic5911yescharacteristic
303C2×C8C22×Dic5911yescharacteristic
304C2×C8C22×Dic5911yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4C22×Dic5912yes—
310C4⋊C4C22×Dic5912yes—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C42—22no—
317C42D4×D5911yescharacteristic
318C42#1174568711yescharacteristic
319C42C22×Dic5911yescharacteristic
320C42C23×D5911yescharacteristic
321C23—22no—
322C23D4×Dic5911yescharacteristic
323Q8—22no—
324Q8—22no—
325D4—22no—
326D4—22no—
327D4—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4D4×Dic5911yescharacteristic
338C2×C4D4×Dic5911yescharacteristic
339C2×C4D4×Dic5911yescharacteristic
340C2×C4C23×Dic5911yescharacteristic
341C2×C4C2×D4×D5911yescharacteristic
342C2×C4C2×#1174568711yescharacteristic
343C8—21no—
344C8—21no—
345C8—1182no—
346C8—1182no—
347C8—2361no—
348C8—2361no—
349C8—2362no—
350C8—2362no—
351C8#1427373412yes—
352C8#1427373412yes—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22C2×D4×Dic5911yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4C2×#1427373411yescharacteristic
368C4C2×#1427373411yescharacteristic
369C4#1683820912yes—
370C2—41no—
371C2—82no—
372C2#2143280811yescharacteristic
373C2#2143298011yescharacteristic
374C2#2143301311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#3168C2
socleC22×C59C2×C4×D4
prime radicalC59×#466C4
cosocleC42×C59C24
hypercenter#3168D59
hypocenterC59#17974
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #21433245C2semidirect
2. #21423010C2semidirect
3. #21423005C2semidirect
4. #21422987C2semidirect
5. #21422920C2semidirect
6. #21422915C2semidirect
7. SD16×C59⋊C8C2semidirect
8. #21422852C2semidirect
9. #21422561C2semidirect
10. #21422379C2semidirect
232. C59#17974semidirect

Maximal subgroups

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

subgroupindex
#214332452
#214230102
#214230052
#214229872
#214229202
#214229152
SD16×C59⋊C82
#214228522
#214225612
#214223792
C59×#31682
#1797459