Finite Group Databasebeta

← #28560688

Subgroups — #28560688

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434666C211yesmaximal, characteristic
2#21433252C212yesmaximal, complemented
3#21422992C211yesmaximal, characteristic, complemented
4#21422980C211yesmaximal, characteristic, complemented
5#21422973C212yesmaximal, complemented
6#21422952C212yesmaximal, complemented
7#21422902C211yesmaximal, characteristic, complemented
8#21422890C211yesmaximal, characteristic, complemented
9#21422507C212yesmaximal, complemented
10C8×#11745664C211yesmaximal, characteristic, complemented
11C2×C59×#335C211yesmaximal, characteristic, complemented
12#16837936C2212yes—
13#16837804C411yescharacteristic
14#16837786C411yescharacteristic
15#16837757C412yes—
16#16837643C411yescharacteristic
17#16837638C411yescharacteristic
18#16837631C412yes—
19#16837529C2212yes—
20#16837526C2212yes—
21C4×Dic236C2211yescharacteristic
22C4×#11745660C2212yes—
23C4×D472C2211yescharacteristic
24#16837479C2212yes—
25#16837473—22no—
26#16837467C2212yes—
27C8×C22⋊D59—22no—
28#16837462—22no—
29#16837455C2211yescharacteristic
30#16837427—22no—
31C4×#11745664C2211yescharacteristic
32#16837132C2212yes—
33#16837132C2212yes—
34#16837129C2212yes—
35#16837129C2212yes—
36#16837124C2211yescharacteristic
37#16837124C2211yescharacteristic
38C8×C59⋊C8C2211yescharacteristic
39C8×C59⋊C8C2211yescharacteristic
40C2×C4⋊C8×C59C2212yes—
41C2×C4×C8×C59C2211yescharacteristic
42C59×#335C2211yescharacteristic
43C59×#335C2212yes—
44C59×#335C2211yescharacteristic
45C2×Dic236C2×C411yescharacteristic
46C2×#11745660C2×C412yes—
47C2×D472C2×C411yescharacteristic
48C2×#11745664—21no—
49C2×#11745659—22no—
50C2×C8×D59—22no—
51#14273721—22no—
52#14273719C2×C412yes—
53#14273715C2×C412yes—
54#14273710—22no—
55#14273706C2×C412yes—
56#14273701C2×C412yes—
57C4×Dic118C2312yes—
58C4×D236C2312yes—
59C4×C22⋊D59—22no—
60#14273684—22no—
61#14273661C2×C411yescharacteristic
62#14273661C2×C411yescharacteristic
63#14273659C2×C412yes—
64C4⋊(C59⋊C8)D411yescharacteristic
65C4⋊(C59⋊C8)D411yescharacteristic
66Dic59⋊C8—22no—
67Dic59⋊C8—22no—
68#14273652C2×C412yes—
69#14273652C2×C412yes—
70#14273649C2×C412yes—
71#14273649C2×C412yes—
72C22⋊(C59⋊C8)D411yescharacteristic
73C22⋊(C59⋊C8)D411yescharacteristic
74(C2×D59)⋊C8—22no—
75(C2×D59)⋊C8—22no—
76#14273638—21no—
77#14273637—22no—
78C4×C59⋊C8C2311yescharacteristic
79C4×C59⋊C8C2311yescharacteristic
80C4×C59⋊C8C2311yescharacteristic
81C4×C59⋊C8C2311yescharacteristic
82C8×Dic59—22no—
83C22×C8×C59—22no—
84C22×C8×C59C2×C411yescharacteristic
85C22×C8×C59C2×C411yescharacteristic
86C2×C42×C59C2311yescharacteristic
87C4⋊C8×C59—22no—
88C4⋊C8×C59C2312yes—
89C4⋊C8×C59C2312yes—
90C4×C8×C59C2×C411yescharacteristic
91C4×C8×C59C2×C411yescharacteristic
92C4×C8×C59C2311yescharacteristic
93C4×C8×C59C2311yescharacteristic
94C22×C4×C59C22×C411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59#4111yescharacteristic
97C2×C22⋊D59—22no—
98C2×D236C22×C412yes—
99C2×C4×D59—22no—
100C2×Dic118C22×C412yes—
101C2×C8×C59—22no—
102C2×C8×C59—22no—
103C2×C8×C59—22no—
104C2×C8×C59—41no—
105C2×C8×C59—82no—
106C2×C8×C59C22×C411yescharacteristic
107C2×C8×C59C22×C411yescharacteristic
108C2×C8×C59C22×C411yescharacteristic
109C2×C8×C59C22×C411yescharacteristic
110C42×C59C22×C411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59#4111yescharacteristic
113C42×C59C2411yescharacteristic
114C22⋊Dic59—22no—
115(C2×D59)⋊C4—22no—
116(C2×D59)⋊C4—22no—
117(C2×D59)⋊C4—22no—
118(C2×D59)⋊C4—22no—
119C4⋊Dic59C22×C412yes—
120C4⋊Dic59C22×C412yes—
121Dic59⋊C4—22no—
122Dic59⋊C4—22no—
123Dic59⋊C4—22no—
124Dic59⋊C4—22no—
125C4×Dic59—22no—
126#11745664—24no—
127C2×C59⋊C8—21no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8C2×D411yescharacteristic
130C2×C59⋊C8C2×D411yescharacteristic
131C2×C59⋊C8C2×D411yescharacteristic
132C2×C59⋊C8C2×D411yescharacteristic
133Dic236—22no—
134D472—22no—
135#11745660—42no—
136#11745659—82no—
137C8×D59—82no—
138C23×C59C4×D411yescharacteristic
139C22×D59—22no—
140C2×C4×C59—22no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59C4×D411yescharacteristic
146C2×C4×C59C4×D411yescharacteristic
147C2×C4×C59C4×D411yescharacteristic
148C2×C4×C59C23×C411yescharacteristic
149C2×C4×C59C22×D411yescharacteristic
150C2×C4×C59C2×#4111yescharacteristic
151C22⋊D59—44no—
152C2×Dic59—22no—
153C2×Dic59—22no—
154C2×Dic59—22no—
155D236—22no—
156D236—22no—
157C4×D59—42no—
158C4×D59—42no—
159Dic118—22no—
160Dic118—22no—
161C8×C59—21no—
162C8×C59—21no—
163C8×C59—21no—
164C8×C59—42no—
165C8×C59—42no—
166C8×C59#13112yes—
167C59⋊C8—22no—
168C59⋊C8—22no—
169C59⋊C8—21no—
170C59⋊C8—21no—
171C59⋊C8—21no—
172C59⋊C8—21no—
173#18087—591nomaximal, Sylow
174C22×C59—21no—
175C22×C59—21no—
176C22×C59#44312yes—
177C22×C59C2×C4×D411yescharacteristic
178C2×D59—42no—
179C2×D59—42no—
180C2×D59—42no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59C2×#13111yescharacteristic
186C4×C59#56811yescharacteristic
187C4×C59#57611yescharacteristic
188C4×C59#57811yescharacteristic
189Dic59—42no—
190Dic59—42no—
191Dic59—42no—
192#3168—591no—
193#2994—592no—
194#1591—591no—
195#1589—591no—
196#1588—592no—
197#1585—592no—
198C8×Q16—591no—
199C8×D8—591no—
200C2×#335D5911yescharacteristic, complemented
201#1483—592no—
202C8×M16—591no—
203C2×C59—22no—
204C2×C59#293511yescharacteristic
205C2×C59#295111yescharacteristic
206C2×C59C2×#44311yescharacteristic
207D59—82no—
208#529—592no—
209#499—591no—
210#495—591no—
211#488—592no—
212#467—591no—
213#466—591no—
214#465—592no—
215#446—592no—
216C8×Q8—592no—
217C4×Q16—591no—
218C4×SD16—592no—
219C4×D8—591no—
220#436—592no—
221#435—1182no—
222C8×D4—592no—
223C8×D4—1182no—
224#433—1182no—
225#431—591no—
226#424—1182no—
227C2×C4⋊C8C2×D5912yes—
228C4×M16—591no—
229C2×C4×C8C2×D5911yescharacteristic
230#335C2×D5911yescharacteristic
231#335C2×D5912yes—
232#335C2×D5911yescharacteristic
233Q8⋊C8—592no—
234Q8⋊C8—592no—
235D4⋊C8—592no—
236D4⋊C8—592no—
237#322—591no—
238#322—591no—
239C82—591no—
240C82—591no—
241C59#1808711yescharacteristic, complemented, Sylow
242C2×Q16—591no—
243C2×SD16—592no—
244C2×D8—591no—
245C2×M16—1182no—
246C2×M16—1181no—
247C22×C8—22no—
248C22×C8—1182no—
249C22×C8C4×D5911yescharacteristic
250C22×C8C4×D5911yescharacteristic
251#126—1182no—
252#125—592no—
253#124—592no—
254#123—1182no—
255C22⋊Q8—592no—
256#121—592no—
257C4×Q8—592no—
258C4×D4—592no—
259C4×D4—1182no—
260#117—1182no—
261C2×C42C22×D5911yescharacteristic
262#107—591no—
263#107—591no—
264#106—592no—
265C4⋊C8—22no—
266C4⋊C8—591no—
267C4⋊C8—591no—
268C4⋊C8—1182no—
269C4⋊C8—1182no—
270C4⋊C8C22×D5912yes—
271C4⋊C8C22×D5912yes—
272#103—592no—
273#103—592no—
274#102—592no—
275#102—592no—
276C22⋊C8—591no—
277C22⋊C8—591no—
278C22⋊C8—1182no—
279C22⋊C8—1182no—
280#97—1182no—
281#97—1181no—
282C4×C8—591no—
283C4×C8—591no—
284C4×C8—591no—
285C4×C8—591no—
286C4×C8—1182no—
287C4×C8C4×D5911yescharacteristic
288C4×C8C4×D5911yescharacteristic
289C4×C8C22×D5911yescharacteristic
290C4×C8C22×D5911yescharacteristic
291C2×Q8—592no—
292C2×D4—592no—
293C2×D4—1182no—
294C22×C4—1182no—
295C22×C4C2×C4×D5911yescharacteristic
296C22×C4D4×D5911yescharacteristic
297C22×C4#1174568911yescharacteristic
298Q16—1182no—
299SD16—2362no—
300D8—1182no—
301M16—1184no—
302M16—4722no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—82no—
308C2×C8—591no—
309C2×C8—591no—
310C2×C8—591no—
311C2×C8—591no—
312C2×C8—1181no—
313C2×C8—1181no—
314C2×C8—4722no—
315C2×C8C2×C4×D5911yescharacteristic
316C2×C8C2×C4×D5911yescharacteristic
317C2×C8C2×C4×D5911yescharacteristic
318C2×C8C2×C4×D5911yescharacteristic
319C4⋊C4—592no—
320C4⋊C4—592no—
321C4⋊C4—1182no—
322C4⋊C4—1182no—
323C4⋊C4—1182no—
324C4⋊C4—1182no—
325C22⋊C4—1182no—
326C22⋊C4—1182no—
327C22⋊C4—1182no—
328C22⋊C4—1182no—
329C22⋊C4—1182no—
330C42—1182no—
331C42C2×C4×D5911yescharacteristic
332C42D4×D5911yescharacteristic
333C42#1174568911yescharacteristic
334C42C23×D5911yescharacteristic
335C23—1182no—
336C23#1427368911yescharacteristic
337Q8—1182no—
338Q8—1182no—
339D4—1182no—
340D4—1182no—
341D4—2364no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—1182no—
348C2×C4—1182no—
349C2×C4—1182no—
350C2×C4—2362no—
351C2×C4—2362no—
352C2×C4#1427368911yescharacteristic
353C2×C4#1427368911yescharacteristic
354C2×C4#1427368911yescharacteristic
355C2×C4C22×C4×D5911yescharacteristic
356C2×C4C2×D4×D5911yescharacteristic
357C2×C4C2×#1174568911yescharacteristic
358C8—21no—
359C8—21no—
360C8—21no—
361C8—42no—
362C8—42no—
363C8—1182no—
364C8—1182no—
365C8—1181no—
366C8—1181no—
367C8—1181no—
368C8—1181no—
369C8#1427373312yes—
370C22—21no—
371C22—21no—
372C22—2362no—
373C22—2362no—
374C22—2362no—
375C22#1683751812yes—
376C22C2×#1427368911yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—2362no—
382C4—2362no—
383C4—2362no—
384C4C2×#1427373311yescharacteristic
385C4#1683816811yescharacteristic
386C4#1683820311yescharacteristic
387C4#1683821511yescharacteristic
388C2—22no—
389C2—4722no—
390C2#2143283911yescharacteristic
391C2#2143297011yescharacteristic
392C2C2×#1683751811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC2×C59×#335C2
socleC22×C59C2×C4×D4
prime radical#16837638C4
cosocleC42×C59C24
hypercenterC2×#335D59
hypocenterC59#18087
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #21433252C2semidirect
3. #21422992C2semidirect
4. #21422980C2semidirect
5. #21422973C2semidirect
6. #21422952C2semidirect
7. #21422902C2semidirect
8. #21422890C2semidirect
9. #21422507C2semidirect
10. C8×#11745664C2semidirect
11. C2×C59×#335C2semidirect
200. C2×#335D59semidirect
241. C59#18087semidirect

Maximal subgroups

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

subgroupindex
#214346662
#214332522
#214229922
#214229802
#214229732
#214229522
#214229022
#214228902
#214225072
C8×#117456642
C2×C59×#3352
#1808759