Finite Group Databasebeta

← #28484158

Subgroups — #28484158

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×C59⋊C8C211yesmaximal, characteristic, complemented
2#21422483C211yesmaximal, characteristic, complemented
3C2×D4⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
4#21422417C211yesmaximal, characteristic, complemented
5#21421947C211yesmaximal, characteristic, complemented
6#21421934C211yesmaximal, characteristic, complemented
7C4×AGL(1,ℤ/8)×C59C211yesmaximal, characteristic
8Q8×C59⋊C8—21no—
9D4×C59⋊C8—22no—
10D4×C59⋊C8—21no—
11#16837455—21no—
12#16837455C2211yescharacteristic
13#16837455C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×C4×C59⋊C8—21no—
16C2×C4×C59⋊C8C2211yescharacteristic
17Q8⋊(C59⋊C8)—21no—
18D4⋊(C59⋊C8)—21no—
19D4⋊(C59⋊C8)—21no—
20D4⋊(C59⋊C8)—21no—
21#16837126—22no—
22C2×AGL(1,ℤ/8)×C59C411yescharacteristic
23C4×#41×C59C2211yescharacteristic
24C2×C4×D4×C59C2211yescharacteristic
25C59×#442—21no—
26C59×#440—21no—
27C4×SD16×C59—21no—
28C4×D8×C59—21no—
29C59×#428C411yescharacteristic
30C59×#418C411yescharacteristic
31C59×#417C411yescharacteristic
32C4×M16×C59C2211yescharacteristic
33C22×C59⋊C8—22no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—21no—
36C4⋊(C59⋊C8)—22no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C22⋊(C59⋊C8)—22no—
42C22⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)D412yes—
44C4×C59⋊C8—22no—
45C4×C59⋊C8—21no—
46C4×C59⋊C8—21no—
47C4×C59⋊C8—21no—
48C4×C59⋊C8D411yescharacteristic
49C4×C59⋊C8D411yescharacteristic
50C2×#41×C59C2×C411yescharacteristic
51C22×D4×C59C2×C411yescharacteristic
52C23×C4×C59—21no—
53AGL(1,ℤ/8)×C59—22no—
54AGL(1,ℤ/8)×C59C814yescomplemented
55C2×SD16×C59—21no—
56C2×D8×C59—21no—
57C2×M16×C59C2×C411yescharacteristic
58C2×M16×C59C2×C411yescharacteristic
59C4×Q8×C59D411yescharacteristic
60C4×Q8×C59D411yescharacteristic
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59—21no—
66C4×D4×C59—21no—
67C4×D4×C59D411yescharacteristic
68C4×D4×C59D411yescharacteristic
69#117×C59—21no—
70#117×C59C2×C411yescharacteristic
71C2×C4⋊C4×C59C2×C411yescharacteristic
72C2×C22⋊C4×C59—21no—
73C2×C42×C59—21no—
74C2×C42×C59C2311yescharacteristic
75#107×C59—21no—
76#106×C59—21no—
77#103×C59—21no—
78#102×C59—21no—
79#102×C59—21no—
80#102×C59—21no—
81#97×C59—21no—
82C4×C8×C59—21no—
83C24×C59—21no—
84#41×C59—21no—
85#41×C59—21no—
86#41×C59C2×C812yes—
87#41×C59M1612yes—
88C2×Q8×C59C22⋊C411yescharacteristic
89C2×D4×C59—21no—
90C2×D4×C59—21no—
91C2×D4×C59—21no—
92C2×D4×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—21no—
95C2×D4×C59C22⋊C411yescharacteristic
96C2×D4×C59C2×C812yescomplemented
97C22×C4×C59—21no—
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—21no—
101C22×C4×C59—41no—
102C22×C4×C59—41no—
103C22×C4×C59—81no—
104C22×C4×C59—81no—
105C22×C4×C59C22×C411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59#4111yescharacteristic
108SD16×C59—24no—
109D8×C59—24no—
110M16×C59—21no—
111M16×C59—21no—
112M16×C59—21no—
113M16×C59C2×C812yescomplemented
114C2×C8×C59—21no—
115C2×C8×C59—21no—
116C4⋊C4×C59—21no—
117C4⋊C4×C59—21no—
118C4⋊C4×C59—21no—
119C4⋊C4×C59—21no—
120C4⋊C4×C59C22⋊C411yescharacteristic
121C4⋊C4×C59C22⋊C411yescharacteristic
122C22⋊C4×C59—21no—
123C22⋊C4×C59—21no—
124C22⋊C4×C59—21no—
125C22⋊C4×C59—41no—
126C22⋊C4×C59—41no—
127C42×C59—21no—
128C42×C59—21no—
129C42×C59—21no—
130C42×C59C2×D411yescharacteristic
131C42×C59C2×D411yescharacteristic
132C42×C59C2×D411yescharacteristic
133C42×C59#4111yescharacteristic
134C2×C59⋊C8—21no—
135C2×C59⋊C8—22no—
136C2×C59⋊C8—21no—
137C2×C59⋊C8—42no—
138C2×C59⋊C8—44no—
139C2×C59⋊C8—41no—
140C2×C59⋊C8—41no—
141C2×C59⋊C8—81no—
142C23×C59—21no—
143C23×C59—22no—
144C23×C59—41no—
145C23×C59—41no—
146C23×C59—81no—
147C23×C59—81no—
148C23×C59C4×D411yescharacteristic
149Q8×C59—21no—
150Q8×C59C22⋊C812yes—
151D4×C59—22no—
152D4×C59—22no—
153D4×C59—22no—
154D4×C59—21no—
155D4×C59—22no—
156D4×C59—41no—
157D4×C59—41no—
158D4×C59C22⋊C812yes—
159C2×C4×C59—21no—
160C2×C4×C59—22no—
161C2×C4×C59—22no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—21no—
166C2×C4×C59—41no—
167C2×C4×C59—42no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—82no—
172C2×C4×C59—81no—
173C2×C4×C59—81no—
174C2×C4×C59—81no—
175C2×C4×C59C2×C22⋊C411yescharacteristic
176C2×C4×C59C4×D411yescharacteristic
177C2×C4×C59#12011yescharacteristic
178C2×C4×C59#12111yescharacteristic
179C2×C4×C59#12111yescharacteristic
180C2×C4×C59#12311yescharacteristic
181C2×C4×C59C22×C811yescharacteristic
182C2×C4×C59C2×M1611yescharacteristic
183C2×C4×C59#13111yescharacteristic
184C2×C4×C59#13111yescharacteristic
185C8×C59—22no—
186C8×C59—41no—
187C59⋊C8—22no—
188C59⋊C8—42no—
189C59⋊C8—41no—
190C59⋊C8—81no—
191C59⋊C8—81no—
192#9095—591nomaximal, Sylow
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—22no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59—42no—
199C22×C59—82no—
200C22×C59—82no—
201C22×C59—81no—
202C22×C59—81no—
203C22×C59#38611yescharacteristic
204C22×C59C8×D411yescharacteristic
205C22×C59#43511yescharacteristic
206C4×C59—22no—
207C4×C59—21no—
208C4×C59—21no—
209C4×C59—21no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59—81no—
213C4×C59C2×C22⋊C811yescharacteristic
214C4×C59#40811yescharacteristic
215C4×C59C8×D411yescharacteristic
216C4×C59#43511yescharacteristic
217C4×C59#47111yescharacteristic
218C4×C59#47211yescharacteristic
219C8×#41—591no—
220C4×AGL(1,ℤ/8)D5911yescharacteristic
221#1477—591no—
222C2×D4⋊C8—591no—
223#1460—591no—
224#1283—591no—
225#1278—591no—
226C2×C59—21no—
227C2×C59—21no—
228C2×C59—42no—
229C2×C59—82no—
230C2×C59#186911yescharacteristic
231C2×C59#190611yescharacteristic
232C2×C59#190711yescharacteristic
233C2×AGL(1,ℤ/8)Dic5911yescharacteristic
234C4×#41C2×D5911yescharacteristic
235C2×C4×D4C2×D5911yescharacteristic
236C8×Q8—1181no—
237#442—21no—
238#440—21no—
239C4×SD16—21no—
240C4×D8—21no—
241C8×D4—1182no—
242C8×D4—1181no—
243#431—591no—
244#431—591no—
245#431—1181no—
246#428Dic5911yescharacteristic
247C2×C4⋊C8—591no—
248#418Dic5911yescharacteristic
249#417Dic5911yescharacteristic
250C4×M16C2×D5911yescharacteristic
251C2×C4×C8—591no—
252C2×C4×C8—1181no—
253Q8⋊C8—1181no—
254D4⋊C8—1181no—
255D4⋊C8—1181no—
256D4⋊C8—1181no—
257#323—1182no—
258C59#909511yescharacteristic, complemented, Sylow
259C2×#41C2×Dic5911yescharacteristic
260C22×D4C2×Dic5911yescharacteristic
261C23×C4—21no—
262AGL(1,ℤ/8)—22no—
263AGL(1,ℤ/8)C59⋊C814yescomplemented
264C2×SD16—21no—
265C2×D8—21no—
266C2×M16C2×Dic5911yescharacteristic
267C2×M16C2×Dic5911yescharacteristic
268C22×C8—1182no—
269C22×C8—1181no—
270C22×C8—1181no—
271C4×Q8C22⋊D5911yescharacteristic
272C4×Q8C22⋊D5911yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4C22⋊D5911yescharacteristic
280C4×D4C22⋊D5911yescharacteristic
281#117—21no—
282#117C2×Dic5911yescharacteristic
283C2×C4⋊C4C2×Dic5911yescharacteristic
284C2×C22⋊C4—21no—
285C2×C42—21no—
286C2×C42C22×D5911yescharacteristic
287#107—21no—
288#106—21no—
289C4⋊C8—1182no—
290C4⋊C8—1181no—
291C4⋊C8—1181no—
292C4⋊C8—1181no—
293C4⋊C8—1181no—
294#103—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298C22⋊C8—592no—
299C22⋊C8—1182no—
300C22⋊C8—1181no—
301#97—21no—
302C4×C8—21no—
303C4×C8—591no—
304C4×C8—591no—
305C4×C8—1182no—
306C4×C8—1181no—
307C4×C8—1181no—
308C4×C8—1181no—
309C24—21no—
310#41—21no—
311#41—21no—
312#41C2×C59⋊C812yes—
313#41#1174566412yes—
314C2×Q8C22⋊Dic5911yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4C2×C59⋊C812yescomplemented
322C2×D4C22⋊Dic5911yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4D4×D5911yescharacteristic
332C22×C4#1174568711yescharacteristic
333C22×C4C22×Dic5911yescharacteristic
334SD16—24no—
335D8—24no—
336M16—21no—
337M16—21no—
338M16—21no—
339M16C2×C59⋊C812yescomplemented
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—1181no—
343C2×C8—1181no—
344C2×C8—1182no—
345C2×C8—4721no—
346C2×C8—2362no—
347C2×C8—2364no—
348C2×C8—2361no—
349C2×C8—2361no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4C22⋊Dic5911yescharacteristic
355C4⋊C4C22⋊Dic5911yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42D4×D5911yescharacteristic
365C42#1174568711yescharacteristic
366C42C2×C22⋊D5911yescharacteristic
367C42C2×C22⋊D5911yescharacteristic
368C23—21no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—81no—
373C23—81no—
374C23D4×Dic5911yescharacteristic
375Q8—21no—
376Q8C22⋊(C59⋊C8)12yes—
377D4—22no—
378D4—22no—
379D4—22no—
380D4—22no—
381D4—21no—
382D4—41no—
383D4—41no—
384D4C22⋊(C59⋊C8)12yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4C2×C22⋊Dic5911yescharacteristic
402C2×C4D4×Dic5911yescharacteristic
403C2×C4#1427369511yescharacteristic
404C2×C4#1427369911yescharacteristic
405C2×C4#1427369911yescharacteristic
406C2×C4#1427370811yescharacteristic
407C2×C4C22×C59⋊C811yescharacteristic
408C2×C4C2×#1174566411yescharacteristic
409C2×C4#1427373411yescharacteristic
410C2×C4#1427373411yescharacteristic
411C8—22no—
412C8—41no—
413C8—1182no—
414C8—4721no—
415C8—4721no—
416C8—2361no—
417C8—2362no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22#1683728211yescharacteristic
429C22D4×C59⋊C811yescharacteristic
430C22#1683747411yescharacteristic
431C4—22no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4C2×C22⋊(C59⋊C8)11yescharacteristic
439C4#1683736911yescharacteristic
440C4D4×C59⋊C811yescharacteristic
441C4#1683747411yescharacteristic
442C4#1683766311yescharacteristic
443C4#1683767011yescharacteristic
444C2—21no—
445C2—21no—
446C2—42no—
447C2—82no—
448C2#2142438911yescharacteristic
449C2#2142459911yescharacteristic
450C2#2142460611yescharacteristic
subgroupG/H
centerC2×C4#14273699
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC4×AGL(1,ℤ/8)×C59C2
socleC22×C59#386
prime radicalC2×AGL(1,ℤ/8)×C59C4
cosocleC2×C42×C59C23
hypercenterC4×AGL(1,ℤ/8)D59
hypocenterC59#9095
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×C59⋊C8C2semidirect
2. #21422483C2semidirect
3. C2×D4⋊(C59⋊C8)C2semidirect
4. #21422417C2semidirect
5. #21421947C2semidirect
6. #21421934C2semidirect
54. AGL(1,ℤ/8)×C59C8semidirect
96. C2×D4×C59C2×C8semidirect
113. M16×C59C2×C8semidirect
258. C59#9095semidirect
263. AGL(1,ℤ/8)C59⋊C8semidirect
321. C2×D4C2×C59⋊C8semidirect
339. M16C2×C59⋊C8semidirect

Maximal subgroups

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

subgroupindex
#41×C59⋊C82
#214224832
C2×D4⋊(C59⋊C8)2
#214224172
#214219472
#214219342
C4×AGL(1,ℤ/8)×C592
#909559