Finite Group Databasebeta

← #28485768

Subgroups — #28485768

Order 15104.

474 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433110C211yesmaximal, characteristic, complemented
2#21422694C211yesmaximal, characteristic, complemented
3#21422530C211yesmaximal, characteristic, complemented
4#21422448C211yesmaximal, characteristic, complemented
5#21422007C211yesmaximal, characteristic, complemented
6#21421985C211yesmaximal, characteristic, complemented
7C59×#3040C211yesmaximal, characteristic
8#16837533—21no—
9#16837474—22no—
10D4×C59⋊C8—21no—
11#16837459—21no—
12#16837459C2211yescharacteristic
13#16837459C2211yescharacteristic
14#16837455C2211yescharacteristic
15C4×#11745664C2211yescharacteristic
16C2×#14273638—21no—
17#16837142—21no—
18#16837142—21no—
19Q8⋊(C59⋊C8)—21no—
20D4⋊(C59⋊C8)—21no—
21#16837126—22no—
22C2×#137×C59C411yescharacteristic
23C2×AGL(1,ℤ/8)×C59C411yescharacteristic
24C2×#124×C59C2211yescharacteristic
25C4×#41×C59C2211yescharacteristic
26C59×#464—21no—
27C59×#464—21no—
28C59×#463—21no—
29C59×#463—21no—
30C59×#423C2211yescharacteristic
31C59×#419C411yescharacteristic
32C59×#418C411yescharacteristic
33C2×#11745664—21no—
34C22×C59⋊C8—22no—
35C4⋊(C59⋊C8)—22no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—21no—
41C22⋊(C59⋊C8)—21no—
42C22⋊(C59⋊C8)D412yes—
43#14273638—22no—
44#14273638—21no—
45#14273638—21no—
46#14273638D411yescharacteristic
47#14273638D411yescharacteristic
48C4×C59⋊C8—21no—
49C4×C59⋊C8—21no—
50C2×#41×C59C2×C411yescharacteristic
51C22×Q8×C59C2×C411yescharacteristic
52C22×D4×C59C2×C411yescharacteristic
53#137×C59—42no—
54AGL(1,ℤ/8)×C59—42no—
55C2×Q16×C59—21no—
56C2×SD16×C59—21no—
57C2×SD16×C59—21no—
58C2×D8×C59—21no—
59C2×M16×C59C2×C411yescharacteristic
60C2×M16×C59C2×C411yescharacteristic
61#124×C59—21no—
62#124×C59—21no—
63#124×C59—21no—
64#124×C59—21no—
65C4×Q8×C59D411yescharacteristic
66C4×Q8×C59D411yescharacteristic
67C4×D4×C59—21no—
68C4×D4×C59—21no—
69C4×D4×C59D411yescharacteristic
70C4×D4×C59D411yescharacteristic
71#117×C59—21no—
72#117×C59C2×C411yescharacteristic
73C2×C22⋊C4×C59—21no—
74C2×C22⋊C4×C59—21no—
75C2×C42×C59—21no—
76C2×C42×C59C2311yescharacteristic
77C4⋊C8×C59—21no—
78C4⋊C8×C59—21no—
79#103×C59—21no—
80#103×C59—21no—
81#102×C59—21no—
82#102×C59—21no—
83C24×C59—21no—
84#41×C59—22no—
85#41×C59—41no—
86C2×Q8×C59—21no—
87C2×Q8×C59—21no—
88C2×Q8×C59—41no—
89C2×Q8×C59—41no—
90C2×Q8×C59C22⋊C411yescharacteristic
91C2×D4×C59—21no—
92C2×D4×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—41no—
95C2×D4×C59—41no—
96C2×D4×C59C22⋊C411yescharacteristic
97C22×C4×C59—21no—
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—21no—
101C22×C4×C59C22×C411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59#4111yescharacteristic
104Q16×C59—42no—
105SD16×C59—42no—
106SD16×C59—42no—
107D8×C59—42no—
108M16×C59—41no—
109M16×C59—41no—
110C2×C8×C59—21no—
111C2×C8×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59C22⋊C411yescharacteristic
115C4⋊C4×C59C22⋊C411yescharacteristic
116C22⋊C4×C59—21no—
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—41no—
120C22⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C22⋊C4×C59—41no—
123C42×C59—21no—
124C42×C59—21no—
125C42×C59—21no—
126C42×C59C2×D411yescharacteristic
127C42×C59C2×D411yescharacteristic
128C42×C59C2×D411yescharacteristic
129C42×C59#4111yescharacteristic
130#11745664—42no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—22no—
134C2×C59⋊C8—44no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—41no—
137C23×C59—21no—
138C23×C59—42no—
139C23×C59—41no—
140C23×C59—81no—
141C23×C59—81no—
142C23×C59C4×D411yescharacteristic
143Q8×C59—21no—
144Q8×C59—21no—
145Q8×C59—41no—
146Q8×C59—41no—
147Q8×C59—81no—
148D4×C59—21no—
149D4×C59—42no—
150D4×C59—41no—
151D4×C59—41no—
152D4×C59—81no—
153D4×C59#10012yes—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—42no—
162C2×C4×C59—42no—
163C2×C4×C59—41no—
164C2×C4×C59—81no—
165C2×C4×C59—81no—
166C2×C4×C59C2×C22⋊C411yescharacteristic
167C2×C4×C59C4×D411yescharacteristic
168C2×C4×C59#12011yescharacteristic
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12111yescharacteristic
171C2×C4×C59#12311yescharacteristic
172C8×C59—41no—
173C8×C59—41no—
174C59⋊C8—22no—
175C59⋊C8—42no—
176C59⋊C8—41no—
177C59⋊C8—81no—
178C59⋊C8—81no—
179#9349—591nomaximal, Sylow
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—22no—
183C22×C59—21no—
184C22×C59—41no—
185C22×C59—41no—
186C22×C59—81no—
187C22×C59—81no—
188C22×C59—81no—
189C22×C59—162no—
190C22×C59#38611yescharacteristic
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—81no—
200C4×C59C2×#10011yescharacteristic
201C4×C59#41311yescharacteristic
202#3040D5911yescharacteristic
203#2970—591no—
204#1527—591no—
205#1488—591no—
206#1469—591no—
207#1307—591no—
208#1299—591no—
209C2×C59—41no—
210C2×C59—42no—
211C2×C59—161no—
212C2×C59#188111yescharacteristic
213C2×C59#190011yescharacteristic
214C2×C59#190611yescharacteristic
215C2×#137Dic5911yescharacteristic
216C2×AGL(1,ℤ/8)Dic5911yescharacteristic
217C2×#124C2×D5911yescharacteristic
218C4×#41C2×D5911yescharacteristic
219#464—21no—
220#464—21no—
221#463—21no—
222#463—21no—
223#446—1181no—
224#435—1182no—
225C8×D4—1181no—
226#432—591no—
227#432—591no—
228#432—1181no—
229#431—591no—
230#423C2×D5911yescharacteristic
231#419Dic5911yescharacteristic
232#418Dic5911yescharacteristic
233C4×M16—591no—
234C2×#97—1181no—
235#329—1181no—
236#329—1181no—
237Q8⋊C8—1181no—
238D4⋊C8—1181no—
239#323—1182no—
240C59#934911yescharacteristic, complemented, Sylow
241C2×#41C2×Dic5911yescharacteristic
242C22×Q8C2×Dic5911yescharacteristic
243C22×D4C2×Dic5911yescharacteristic
244#137—42no—
245AGL(1,ℤ/8)—42no—
246C2×Q16—21no—
247C2×SD16—21no—
248C2×SD16—21no—
249C2×D8—21no—
250C2×M16—1181no—
251C2×M16C2×Dic5911yescharacteristic
252C2×M16C2×Dic5911yescharacteristic
253C22×C8—1182no—
254#124—21no—
255#124—21no—
256#124—21no—
257#124—21no—
258C4×Q8C22⋊D5911yescharacteristic
259C4×Q8C22⋊D5911yescharacteristic
260C4×D4—21no—
261C4×D4—21no—
262C4×D4C22⋊D5911yescharacteristic
263C4×D4C22⋊D5911yescharacteristic
264#117—21no—
265#117C2×Dic5911yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—21no—
269C2×C42C22×D5911yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—1182no—
273C4⋊C8—1181no—
274C4⋊C8—1181no—
275C4⋊C8—1181no—
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280C22⋊C8—592no—
281C22⋊C8—1182no—
282C22⋊C8—1181no—
283C22⋊C8—1181no—
284#97—591no—
285#97—591no—
286#97—1182no—
287#97—1181no—
288#97—1181no—
289C4×C8—1181no—
290C4×C8—1181no—
291C24—21no—
292#41—22no—
293#41—41no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8C22⋊Dic5911yescharacteristic
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C22⋊Dic5911yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4D4×D5911yescharacteristic
310C22×C4#1174568711yescharacteristic
311C22×C4C22×Dic5911yescharacteristic
312Q16—42no—
313SD16—42no—
314SD16—42no—
315D8—42no—
316M16—41no—
317M16—41no—
318M16—2362no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—1181no—
322C2×C8—1181no—
323C2×C8—1182no—
324C2×C8—2364no—
325C2×C8—2361no—
326C2×C8—2361no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4C22⋊Dic5911yescharacteristic
330C4⋊C4C22⋊Dic5911yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42D4×D5911yescharacteristic
342C42#1174568711yescharacteristic
343C42C2×C22⋊D5911yescharacteristic
344C42C2×C22⋊D5911yescharacteristic
345C23—21no—
346C23—42no—
347C23—41no—
348C23—81no—
349C23—81no—
350C23D4×Dic5911yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—41no—
354Q8—41no—
355Q8—81no—
356D4—21no—
357D4—42no—
358D4—41no—
359D4—41no—
360D4—81no—
361D4#1427364412yes—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4C2×C22⋊Dic5911yescharacteristic
375C2×C4D4×Dic5911yescharacteristic
376C2×C4#1427369511yescharacteristic
377C2×C4#1427369911yescharacteristic
378C2×C4#1427369911yescharacteristic
379C2×C4#1427370811yescharacteristic
380C8—41no—
381C8—41no—
382C8—1182no—
383C8—4721no—
384C8—4721no—
385C8—2362no—
386C8—2361no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—162no—
397C22#1683728211yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—81no—
407C4C2×#1427364411yescharacteristic
408C4#1683738611yescharacteristic
409C2—41no—
410C2—42no—
411C2—161no—
412C2#2142446011yescharacteristic
413C2#2142456011yescharacteristic
414C2#2142459911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3040C2
socleC22×C59#386
prime radicalC2×AGL(1,ℤ/8)×C59C4
cosocleC2×C42×C59C23
hypercenter#3040D59
hypocenterC59#9349
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3040

Complements

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

NH ≅ G/Ntype
1. #21433110C2semidirect
2. #21422694C2semidirect
3. #21422530C2semidirect
4. #21422448C2semidirect
5. #21422007C2semidirect
6. #21421985C2semidirect
240. C59#9349semidirect

Maximal subgroups

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

subgroupindex
#214331102
#214226942
#214225302
#214224482
#214220072
#214219852
C59×#30402
#934959