Finite Group Databasebeta

← #28559145

Subgroups — #28559145

Order 15104.

598 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433181C211yesmaximal, characteristic, complemented
2#21433172C211yesmaximal, characteristic
3#21422999C212yesmaximal, complemented
4#21422987C212yesmaximal, complemented
5#21422954C212yesmaximal, complemented
6#21422933C212yesmaximal, complemented
7#21422876C211yesmaximal, characteristic, complemented
8#21422543C211yesmaximal, characteristic, complemented
9#21422513C211yesmaximal, characteristic
10#21422391C211yesmaximal, characteristic, complemented
11C59×#3103C211yesmaximal, characteristic
12#16837533—22no—
13#16837533C2212yes—
14#16837533C2212yes—
15#16837480C2212yes—
16#16837474—22no—
17D4×C59⋊C8C2212yes—
18#16837459—22no—
19#16837459C2211yescharacteristic
20#16837455C2211yescharacteristic
21#16837455C2211yescharacteristic
22#16837369—22no—
23C4×#11745664C2211yescharacteristic
24#16837155C2212yes—
25#16837153C2212yes—
26Q8⋊(C59⋊C8)C2212yes—
27Q8⋊(C59⋊C8)C2212yes—
28D4⋊(C59⋊C8)C2212yes—
29D4⋊(C59⋊C8)C2212yes—
30#16837124C2211yescharacteristic
31#16837124C2211yescharacteristic
32#16837123C2211yescharacteristic
33C8×C59⋊C8C2211yescharacteristic
34C59×#532C2211yescharacteristic
35C2×C4×Q8×C59C2211yescharacteristic
36C59×#481C412yes—
37C59×#475C412yes—
38C59×#460C412yes—
39C59×#448C412yes—
40C4×SD16×C59C2212yes—
41C4×SD16×C59C2212yes—
42C59×#431C2211yescharacteristic
43C2×#11745664—21no—
44C2×#11745664—22no—
45C22×C59⋊C8—22no—
46C4⋊(C59⋊C8)—22no—
47C4⋊(C59⋊C8)—22no—
48C4⋊(C59⋊C8)D412yes—
49C4⋊(C59⋊C8)C2312yes—
50C4⋊(C59⋊C8)C2312yes—
51C22⋊(C59⋊C8)—21no—
52C22⋊(C59⋊C8)—21no—
53C22⋊(C59⋊C8)—22no—
54C22⋊(C59⋊C8)—22no—
55C22⋊(C59⋊C8)D412yes—
56#14273638—21no—
57#14273638—21no—
58#14273638—22no—
59#14273638—22no—
60C4×C59⋊C8—21no—
61C4×C59⋊C8C2311yescharacteristic
62C4×C59⋊C8C2311yescharacteristic
63C4×C59⋊C8C2311yescharacteristic
64C4×C59⋊C8C2311yescharacteristic
65C2×#41×C59—21no—
66C22×Q8×C59C2×C411yescharacteristic
67C2×SD16×C59C2×C412yes—
68C2×SD16×C59C2×C412yes—
69C4⋊Q8×C59C2×C411yescharacteristic
70#127×C59C2×C411yescharacteristic
71#124×C59—21no—
72#121×C59—21no—
73#121×C59C2×C412yes—
74C4×Q8×C59—21no—
75C4×Q8×C59—21no—
76C4×Q8×C59C2×C412yes—
77C4×Q8×C59C2312yes—
78C4×D4×C59—21no—
79C4×D4×C59C2312yes—
80C2×C4⋊C4×C59—21no—
81C2×C4⋊C4×C59C2×C411yescharacteristic
82C2×C42×C59—21no—
83C2×C42×C59C2311yescharacteristic
84#106×C59C2×C412yes—
85#106×C59C2×C412yes—
86C4⋊C8×C59C2×C412yes—
87#103×C59C2×C412yes—
88#103×C59C2×C412yes—
89#102×C59C2×C412yes—
90#102×C59C2×C412yes—
91C22⋊C8×C59C2×C412yes—
92C4×C8×C59C2311yescharacteristic
93C4×C8×C59C2311yescharacteristic
94#41×C59—82no—
95C2×Q8×C59—21no—
96C2×Q8×C59—21no—
97C2×Q8×C59—42no—
98C2×Q8×C59C22×C412yes—
99C2×D4×C59—22no—
100C2×D4×C59—21no—
101C2×D4×C59—21no—
102C2×D4×C59C22×C412yes—
103C22×C4×C59—21no—
104C22×C4×C59—21no—
105C22×C4×C59—22no—
106C22×C4×C59C22×C411yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59#4111yescharacteristic
109SD16×C59—42no—
110SD16×C59—42no—
111C2×C8×C59C22×C412yes—
112C2×C8×C59C22×C412yes—
113C4⋊C4×C59—22no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59—21no—
116C4⋊C4×C59—22no—
117C4⋊C4×C59C22×C412yes—
118C4⋊C4×C59C22×C412yes—
119C22⋊C4×C59—22no—
120C22⋊C4×C59—22no—
121C42×C59—22no—
122C42×C59—22no—
123C42×C59C22×C411yescharacteristic
124C42×C59C2×D411yescharacteristic
125C42×C59#4111yescharacteristic
126C42×C59C2411yescharacteristic
127#11745664—42no—
128#11745664—82no—
129C2×C59⋊C8—21no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—82no—
134C2×C59⋊C8C2×D412yes—
135C2×C59⋊C8C2×D412yes—
136C23×C59—22no—
137C23×C59C4×D411yescharacteristic
138Q8×C59—22no—
139Q8×C59—22no—
140Q8×C59—42no—
141Q8×C59—81no—
142D4×C59—22no—
143D4×C59—22no—
144D4×C59—81no—
145D4×C59—81no—
146D4×C59—82no—
147C2×C4×C59—22no—
148C2×C4×C59—22no—
149C2×C4×C59—22no—
150C2×C4×C59—22no—
151C2×C4×C59—22no—
152C2×C4×C59—41no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—42no—
156C2×C4×C59—81no—
157C2×C4×C59C4×D411yescharacteristic
158C2×C4×C59C4×D411yescharacteristic
159C2×C4×C59C4×D411yescharacteristic
160C2×C4×C59C23×C411yescharacteristic
161C2×C4×C59C22×D411yescharacteristic
162C2×C4×C59C2×#4111yescharacteristic
163C8×C59—22no—
164C8×C59—22no—
165C59⋊C8—22no—
166C59⋊C8—22no—
167C59⋊C8—22no—
168C59⋊C8—41no—
169C59⋊C8—42no—
170C59⋊C8—42no—
171#17951—591nomaximal, Sylow
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—42no—
176C22×C59—42no—
177C22×C59—42no—
178C22×C59C2×C4×D411yescharacteristic
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—42no—
184C4×C59—42no—
185C4×C59—42no—
186C4×C59#56812yes—
187C4×C59#57712yes—
188#3103D5911yescharacteristic
189#2983—591no—
190#2981—591no—
191#1592—592no—
192#1590—592no—
193#1585—592no—
194#1582—592no—
195#1572—591no—
196#1491—591no—
197#1484—591no—
198#1453—591no—
199C2×C59—41no—
200C2×C59—82no—
201C2×C59#293011yescharacteristic
202C2×C59#294911yescharacteristic
203C2×C59#295611yescharacteristic
204#532C2×D5911yescharacteristic
205C2×C4×Q8C2×D5911yescharacteristic
206#481Dic5912yes—
207#475Dic5912yes—
208#460Dic5912yes—
209#448Dic5912yes—
210#446—592no—
211#446—592no—
212#446—1182no—
213C4×SD16C2×D5912yes—
214C4×SD16C2×D5912yes—
215#436—592no—
216#435—1182no—
217C8×D4—592no—
218#432—591no—
219#432—1182no—
220#431—591no—
221#431—591no—
222#431C2×D5911yescharacteristic
223#408—1182no—
224C4×M16—591no—
225#335—592no—
226#334—592no—
227Q8⋊C8—592no—
228Q8⋊C8—592no—
229D4⋊C8—592no—
230D4⋊C8—592no—
231#322—591no—
232#322—591no—
233#322—591no—
234C82—591no—
235C59#1795111yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C22×Q8C2×Dic5911yescharacteristic
238C2×SD16C2×Dic5912yes—
239C2×SD16C2×Dic5912yes—
240C2×M16—1182no—
241C2×M16—1181no—
242C22×C8—1182no—
243C4⋊Q8C2×Dic5911yescharacteristic
244#127C2×Dic5911yescharacteristic
245#124—21no—
246#121—21no—
247#121C2×Dic5912yes—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8C2×Dic5912yes—
251C4×Q8C22×D5912yes—
252C4×D4—21no—
253C4×D4C22×D5912yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4C2×Dic5911yescharacteristic
256C2×C42—21no—
257C2×C42C22×D5911yescharacteristic
258#106C2×Dic5912yes—
259#106C2×Dic5912yes—
260C4⋊C8—592no—
261C4⋊C8—592no—
262C4⋊C8—592no—
263C4⋊C8—1182no—
264C4⋊C8—1182no—
265C4⋊C8C2×Dic5912yes—
266#103C2×Dic5912yes—
267#103C2×Dic5912yes—
268#102C2×Dic5912yes—
269#102C2×Dic5912yes—
270C22⋊C8—592no—
271C22⋊C8—1182no—
272C22⋊C8—1182no—
273C22⋊C8—1181no—
274C22⋊C8—1181no—
275C22⋊C8C2×Dic5912yes—
276#97—1182no—
277#97—1182no—
278#97—1181no—
279#97—1181no—
280C4×C8—591no—
281C4×C8—591no—
282C4×C8—591no—
283C4×C8—591no—
284C4×C8—1181no—
285C4×C8C22×D5911yescharacteristic
286C4×C8C22×D5911yescharacteristic
287#41—82no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—42no—
291C2×Q8C22×Dic5912yes—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4C22×Dic5912yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4D4×D5911yescharacteristic
300C22×C4#1174568711yescharacteristic
301C22×C4C22×Dic5911yescharacteristic
302SD16—42no—
303SD16—42no—
304M16—4722no—
305M16—2362no—
306C2×C8—592no—
307C2×C8—592no—
308C2×C8—1181no—
309C2×C8—1182no—
310C2×C8—1182no—
311C2×C8—1181no—
312C2×C8—4722no—
313C2×C8C22×Dic5912yes—
314C2×C8C22×Dic5912yes—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4C22×Dic5912yes—
320C4⋊C4C22×Dic5912yes—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C42—22no—
324C42—22no—
325C42D4×D5911yescharacteristic
326C42#1174568711yescharacteristic
327C42C22×Dic5911yescharacteristic
328C42C23×D5911yescharacteristic
329C23—22no—
330C23D4×Dic5911yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—42no—
334Q8—81no—
335D4—22no—
336D4—22no—
337D4—81no—
338D4—81no—
339D4—82no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4D4×Dic5911yescharacteristic
351C2×C4D4×Dic5911yescharacteristic
352C2×C4D4×Dic5911yescharacteristic
353C2×C4C23×Dic5911yescharacteristic
354C2×C4C2×D4×D5911yescharacteristic
355C2×C4C2×#1174568711yescharacteristic
356C8—22no—
357C8—22no—
358C8—1182no—
359C8—1182no—
360C8—1182no—
361C8—2361no—
362C8—2362no—
363C8—2362no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22C2×D4×Dic5911yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4#1683816912yes—
379C4#1683820812yes—
380C2—41no—
381C2—82no—
382C2#2143280011yescharacteristic
383C2#2143295611yescharacteristic
384C2#2143300611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#3103C2
socleC22×C59C2×C4×D4
prime radicalC59×#532C22
cosocleC42×C59C24
hypercenter#3103D59
hypocenterC59#17951
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. #21433181C2semidirect
3. #21422999C2semidirect
4. #21422987C2semidirect
5. #21422954C2semidirect
6. #21422933C2semidirect
7. #21422876C2semidirect
8. #21422543C2semidirect
10. #21422391C2semidirect
235. C59#17951semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214331722
#214229992
#214229872
#214229542
#214229332
#214228762
#214225432
#214225132
#214223912
C59×#31032
#1795159