Finite Group Databasebeta

← #28608388

Subgroups — #28608388

Order 15104.

722 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273749C211yesmaximal, characteristic
2#21432764C212yesmaximal, complemented
3#21424982C212yesmaximal
4#21424743C214yesmaximal, complemented
5#21424113C212yesmaximal, complemented
6#21423978C212yesmaximal, complemented
7C8×C4⋊Dic59C211yesmaximal, characteristic, complemented
8C59×#1848C211yesmaximal, characteristic, complemented
9C2×#14273749C2211yescharacteristic
10C4×#11745685C2212yes—
11C4×Dic236C411yescharacteristic, complemented
12C4×Dic236C411yescharacteristic, complemented
13C4×#11745660C412yescomplemented
14C4×#11745660C412yescomplemented
15C4×D472C411yescharacteristic, complemented
16C4×D472C411yescharacteristic, complemented
17#16837462—24no—
18#16837458—22no—
19#16837454—22no—
20#16837440C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273653C2214yes—
23#16837399C2212yes—
24#16837370C2214yes—
25#16837360C2214yes—
26C2×C8×Dic59C2212yes—
27C4×C4⋊Dic59C2211yescharacteristic
28#16837163C2214yes—
29#16837156C2212yes—
30C2×#108×C59C2212yes—
31C2×#108×C59C2212yes—
32C59×#423C2212yes—
33C2×C4×C8×C59C2211yescharacteristic
34C2×#11745685C2312yes—
35#14273749—22no—
36#14273749—22no—
37C2×Dic236C2×C411yescharacteristic
38C2×#11745660C2×C412yes—
39C2×D472C2×C411yescharacteristic
40C22×C59⋊C8C2312yes—
41C4×Dic118C2×C412yes—
42C4×Dic118C2×C412yes—
43C4×D236C2×C412yes—
44C4×D236C2×C412yes—
45C4×C22⋊D59—22no—
46C4×C22⋊D59—22no—
47#14273686C2312yes—
48#14273684—22no—
49C2×C4⋊Dic59—21no—
50C2×C4×Dic59C2312yes—
51C42×D59—22no—
52#14273661C2×C411yescharacteristic
53#14273661C2×C411yescharacteristic
54#14273659C2×C412yes—
55C4⋊(C59⋊C8)D411yescharacteristic
56C4⋊(C59⋊C8)D411yescharacteristic
57C4⋊(C59⋊C8)D412yes—
58#14273653—44no—
59#14273652C2×C412yes—
60#14273652C2×C412yes—
61#14273649C2×C412yes—
62#14273649C2×C412yes—
63(C2×D59)⋊C8—24no—
64(C2×D59)⋊C8—24no—
65#14273637—24no—
66C8×Dic59—24no—
67C8×Dic59—24no—
68C59⋊#95—21no—
69C2×M16×C59C2314yes—
70C22×C8×C59C2311yescharacteristic
71C22×C8×C59C2311yescharacteristic
72C2×C42×C59C2311yescharacteristic
73#108×C59—24no—
74#108×C59—42no—
75C4⋊C8×C59—22no—
76C4⋊C8×C59—22no—
77C4×C8×C59C2×C411yescharacteristic
78C4×C8×C59C2×C411yescharacteristic
79C4×C8×C59C2×C411yescharacteristic
80C4×C8×C59C2×C411yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C22×C4×C59#4111yescharacteristic
83C22×C4×C59C2411yescharacteristic
84C2×C22⋊D59—22no—
85C22×Dic59—22no—
86#11745685—22no—
87#11745685—22no—
88#11745685—22no—
89#11745685—22no—
90C2×D236C22×C412yes—
91C2×C4×D59—22no—
92C2×C4×D59—22no—
93C2×C4×D59—22no—
94C2×Dic118C22×C412yes—
95M16×C59—24no—
96M16×C59—24no—
97C2×C8×C59—21no—
98C2×C8×C59—21no—
99C2×C8×C59—24no—
100C2×C8×C59C22×C411yescharacteristic
101C2×C8×C59C22×C411yescharacteristic
102C2×C8×C59C22×C411yescharacteristic
103C2×C8×C59C22×C411yescharacteristic
104C2×C8×C59#4112yes—
105C2×C8×C59#4112yes—
106C42×C59C22×C411yescharacteristic
107C42×C59C22×C411yescharacteristic
108C42×C59C2×D411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C22⋊Dic59—22no—
111(C2×D59)⋊C4—22no—
112(C2×D59)⋊C4—22no—
113C4⋊Dic59—24no—
114C4⋊Dic59C22×C412yes—
115C4⋊Dic59C22×C412yes—
116Dic59⋊C4—22no—
117Dic59⋊C4—22no—
118C4×Dic59—22no—
119C4×Dic59—22no—
120C4×Dic59—22no—
121C4×Dic59—22no—
122C4×Dic59—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—22no—
125C2×C59⋊C8C2×D412yes—
126C2×C59⋊C8C2×D412yes—
127Dic236—22no—
128D472—22no—
129#11745660—24no—
130C23×C59C2×#4111yescharacteristic
131C22×D59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—41no—
134C2×C4×C59C4×D411yescharacteristic
135C2×C4×C59C4×D411yescharacteristic
136C2×C4×C59C4×D411yescharacteristic
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59C23×C411yescharacteristic
139C2×C4×C59C22×D411yescharacteristic
140C2×C4×C59C2×#4111yescharacteristic
141C2×C4×C59C2×#4111yescharacteristic
142C2×C4×C59H(5,𝔽2)11yescharacteristic
143C2×C4×C592−1+411yescharacteristic
144C22⋊D59—42no—
145C22⋊D59—42no—
146C2×Dic59—24no—
147C2×Dic59—22no—
148C2×Dic59—22no—
149C2×Dic59—22no—
150C2×Dic59—42no—
151C2×Dic59—42no—
152D236—22no—
153D236—22no—
154C4×D59—42no—
155C4×D59—42no—
156C4×D59—44no—
157C4×D59—82no—
158Dic118—22no—
159Dic118—22no—
160C8×C59—21no—
161C8×C59—21no—
162C8×C59—44no—
163C8×C59#11712yes—
164C8×C59#11712yes—
165C59⋊C8—22no—
166C59⋊C8—22no—
167#22802—591nomaximal, Sylow
168C22×C59—21no—
169C22×C59—21no—
170C22×C59C2×C4×D411yescharacteristic
171C22×C59#55511yescharacteristic
172C22×C59#55611yescharacteristic
173C2×D59—42no—
174C2×D59—42no—
175C2×D59—42no—
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59#44312yes—
180C4×C59C2×#11711yescharacteristic
181C4×C59#52011yescharacteristic
182C4×C59#54711yescharacteristic
183C4×C59#55011yescharacteristic
184Dic59—44no—
185Dic59—42no—
186Dic59—42no—
187Dic59—42no—
188C4×#135—591no—
189#2925—592no—
190#1974—592no—
191#1929—594no—
192#1848D5911yescharacteristic, complemented
193#1806—592no—
194#1772—592no—
195C8×C4⋊C4—591no—
196C2×C59—21no—
197C2×C59—21no—
198C2×C59#235411yescharacteristic
199C2×C59C2×#44311yescharacteristic
200C2×C59#295711yescharacteristic
201D59—82no—
202C2×#135—591no—
203C4×#41—592no—
204C4×Q16—591no—
205C4×Q16—591no—
206C4×SD16—592no—
207C4×SD16—592no—
208C4×D8—591no—
209C4×D8—591no—
210#433—1184no—
211#432—1182no—
212#431—1182no—
213C2×#108C2×D5912yes—
214C2×#108C2×D5912yes—
215#427—591no—
216#423C2×D5912yes—
217C2×C4⋊C8—591no—
218C2×#104—594no—
219#416—592no—
220#408—594no—
221#405—594no—
222C2×C4×C8—592no—
223C2×C4×C8C2×D5911yescharacteristic
224C4×C4⋊C4—591no—
225#339—594no—
226#336—592no—
227C59#2280211yescharacteristic, complemented, Sylow
228C2×#41—592no—
229#135—1182no—
230#135—1182no—
231C2×Q16—591no—
232C2×SD16—592no—
233C2×D8—591no—
234C2×M16C22×D5914yes—
235C22×C8—592no—
236C22×C8C22×D5911yescharacteristic
237C22×C8C22×D5911yescharacteristic
238C4×Q8—592no—
239C4×Q8—592no—
240C4×D4—592no—
241C4×D4—592no—
242C4×D4—1182no—
243C4×D4—1182no—
244#117—592no—
245#117—1182no—
246C2×C4⋊C4—1181no—
247C2×C42—592no—
248C2×C42—1182no—
249C2×C42C22×D5911yescharacteristic
250#108—24no—
251#108—42no—
252#107—591no—
253#107—591no—
254#106—592no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—591no—
258C4⋊C8—591no—
259C4⋊C8—592no—
260#104—2364no—
261#103—592no—
262#103—592no—
263#102—592no—
264#102—592no—
265C22⋊C8—1184no—
266C22⋊C8—1184no—
267#97—1184no—
268C4×C8—1184no—
269C4×C8—1184no—
270C4×C8C4×D5911yescharacteristic
271C4×C8C4×D5911yescharacteristic
272C4×C8C4×D5911yescharacteristic
273C4×C8C4×D5911yescharacteristic
274#95—1181no—
275#41—1182no—
276#41—1182no—
277#41—1182no—
278#41—1182no—
279C2×Q8—592no—
280C2×D4—592no—
281C2×D4—1182no—
282C22×C4—1182no—
283C22×C4—1182no—
284C22×C4—1182no—
285C22×C4—1182no—
286C22×C4#1174568911yescharacteristic
287C22×C4#1174568911yescharacteristic
288C22×C4C23×D5911yescharacteristic
289Q16—1182no—
290SD16—1184no—
291D8—1182no—
292M16—24no—
293M16—24no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—24no—
297C2×C8—592no—
298C2×C8—592no—
299C2×C8—1182no—
300C2×C8—1182no—
301C2×C8C2×C4×D5911yescharacteristic
302C2×C8C2×C4×D5911yescharacteristic
303C2×C8C2×C4×D5911yescharacteristic
304C2×C8C2×C4×D5911yescharacteristic
305C2×C8#1174568712yes—
306C2×C8#1174568912yes—
307C4⋊C4—592no—
308C4⋊C4—592no—
309C4⋊C4—1184no—
310C4⋊C4—1182no—
311C4⋊C4—1182no—
312C22⋊C4—1182no—
313C22⋊C4—1182no—
314C22⋊C4—1182no—
315C42—1182no—
316C42—1182no—
317C42—1182no—
318C42—1182no—
319C42—1182no—
320C42C2×C4×D5911yescharacteristic
321C42C2×C4×D5911yescharacteristic
322C42D4×D5911yescharacteristic
323C42D4×D5911yescharacteristic
324C23—1182no—
325C23C2×#1174568911yescharacteristic
326Q8—1182no—
327Q8—1182no—
328D4—1182no—
329D4—1182no—
330D4—2362no—
331D4—2362no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—1184no—
335C2×C4—1182no—
336C2×C4—1182no—
337C2×C4—1182no—
338C2×C4—4722no—
339C2×C4—2362no—
340C2×C4—2362no—
341C2×C4—2364no—
342C2×C4—2362no—
343C2×C4—2362no—
344C2×C4#1427368911yescharacteristic
345C2×C4#1427368911yescharacteristic
346C2×C4#1427368911yescharacteristic
347C2×C4#1427368911yescharacteristic
348C2×C4C22×C4×D5911yescharacteristic
349C2×C4C2×D4×D5911yescharacteristic
350C2×C4C2×#1174568711yescharacteristic
351C2×C4C2×#1174568911yescharacteristic
352C2×C4#1427377211yescharacteristic
353C2×C4#1427377511yescharacteristic
354C8—21no—
355C8—21no—
356C8—44no—
357C8—1182no—
358C8—1182no—
359C8#1427368512yes—
360C8#1427368512yes—
361C22—21no—
362C22—21no—
363C22—2362no—
364C22—2362no—
365C22—2362no—
366C22C2×#1427368911yescharacteristic
367C22#1683811811yescharacteristic
368C22#1683812611yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—2364no—
373C4—2362no—
374C4—2362no—
375C4—2362no—
376C4#1683751812yes—
377C4C2×#1427368511yescharacteristic
378C4#1683788311yescharacteristic
379C4#1683805511yescharacteristic
380C4#1683807211yescharacteristic
381C2—21no—
382C2—21no—
383C2—4722no—
384C2#2142704711yescharacteristic
385C2C2×#1683751811yescharacteristic
386C2#2143301211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1848C2
socleC22×C59C2×C4×D4
prime radicalC2×#14273749C22
cosocleC22×C4×C59C24
hypercenter#1848D59
hypocenterC59#22802
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1848

Complements

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

NH ≅ G/Ntype
2. #21432764C2semidirect
4. #21424743C2semidirect
5. #21424113C2semidirect
6. #21423978C2semidirect
7. C8×C4⋊Dic59C2semidirect
8. C59×#1848C2semidirect
11. C4×Dic236C4semidirect
12. C4×Dic236C4semidirect
13. C4×#11745660C4semidirect
14. C4×#11745660C4semidirect
15. C4×D472C4semidirect
16. C4×D472C4semidirect
192. #1848D59semidirect
227. C59#22802semidirect

Maximal subgroups

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

subgroupindex
C4×#142737492
#214327642
#214249822
#214247432
#214241132
#214239782
C8×C4⋊Dic592
C59×#18482
#2280259