Finite Group Databasebeta

← #28486757

Subgroups — #28486757

Order 15104.

482 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434040C211yesmaximal, characteristic
2#21424196C211yesmaximal, characteristic
3#21424171C211yesmaximal, characteristic
4#21422629C211yesmaximal, characteristic
5#21422004C211yesmaximal, characteristic
6#21421949C211yesmaximal, characteristic
7C59×#2332C211yesmaximal, characteristic
8C2×#14273686C2211yescharacteristic
9#16837740—22no—
10#16837735C412yescomplemented
11#16837730C412yescomplemented
12#16837442—21no—
13#16837411—21no—
14#16837366C2211yescharacteristic
15#16837366C2211yescharacteristic
16#16837265C2211yescharacteristic
17#16837168—21no—
18#16837168—21no—
19#16837162—21no—
20#16837162—21no—
21#16837139—22no—
22C2×C22⋊Q8×C59C2211yescharacteristic
23C59×#391—21no—
24C59×#388—21no—
25C59×#388—21no—
26C59×#387—21no—
27C59×#387—21no—
28C59×#381C2211yescharacteristic
29C4×C4⋊C4×C59—21no—
30C4×C22⋊C4×C59C2211yescharacteristic
31C2×#11745664—21no—
32C2×#11745664—21no—
33#14273686—21no—
34#14273686—21no—
35#14273686—21no—
36#14273686—21no—
37C2×C4⋊Dic59C2×C411yescharacteristic
38C2×C4⋊Dic59C2×C411yescharacteristic
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—21no—
41C2×C4×Dic59—21no—
42#14273660—22no—
43#14273658—22no—
44#14273651—22no—
45#14273651—22no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)C2×C412yes—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C22×Q8×C59D411yescharacteristic
53C23×C4×C59C2311yescharacteristic
54C22⋊Q8×C59—21no—
55C22⋊Q8×C59—21no—
56C22⋊Q8×C59C2×C412yes—
57C22⋊Q8×C59D412yes—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59D411yescharacteristic
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C2×C22⋊C4×C59D411yescharacteristic
65C2×C22⋊C4×C59D411yescharacteristic
66C2×C42×C59—21no—
67C2×C42×C59—21no—
68C2×C42×C59D411yescharacteristic
69C2×C42×C59D411yescharacteristic
70#95×C59—21no—
71#95×C59—21no—
72#95×C59—21no—
73#95×C59—21no—
74#95×C59—21no—
75#95×C59—21no—
76#95×C59—21no—
77C24×C59C2×D411yescharacteristic
78C2×Q8×C59—21no—
79C2×Q8×C59—41no—
80C2×Q8×C59—41no—
81C2×Q8×C59C22⋊C412yes—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59C22×C411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59#4111yescharacteristic
95C22×C4×C59#4111yescharacteristic
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—41no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—42no—
103C4⋊C4×C59—81no—
104C4⋊C4×C59C22⋊C412yes—
105C22⋊C4×C59—22no—
106C22⋊C4×C59—22no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C42×C59—42no—
111C42×C59—41no—
112C42×C59—41no—
113C42×C59—81no—
114C22⋊Dic59—41no—
115C22⋊Dic59—41no—
116C22⋊Dic59—41no—
117C22⋊Dic59—41no—
118C4⋊Dic59—21no—
119C4⋊Dic59—21no—
120C4⋊Dic59—21no—
121C4⋊Dic59—21no—
122C4×Dic59—41no—
123C4×Dic59—41no—
124#11745664—41no—
125#11745664—41no—
126#11745664—41no—
127#11745664—41no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—41no—
130C23×C59—21no—
131C23×C59—21no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59C4×D411yescharacteristic
135C23×C59#12011yescharacteristic
136C23×C59#12111yescharacteristic
137Q8×C59—42no—
138Q8×C59—81no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—22no—
144C2×C4×C59—22no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59C2×C22⋊C411yescharacteristic
166C2×C4×C59C4×D411yescharacteristic
167C2×C4×C59#12111yescharacteristic
168C2×C4×C59#12311yescharacteristic
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—81no—
172C2×Dic59—81no—
173C2×Dic59—81no—
174C2×Dic59—81no—
175C2×Dic59—81no—
176C59⋊C8—42no—
177C59⋊C8—81no—
178#9526—591nomaximal, Sylow
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—81no—
188C22×C59—81no—
189C22×C59#38611yescharacteristic
190C22×C59#45811yescharacteristic
191C22×C59#45811yescharacteristic
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—42no—
197C4×C59—42no—
198C4×C59—42no—
199C4×C59—81no—
200C4×C59—81no—
201C4×C59—81no—
202Dic59—81no—
203Dic59—81no—
204Dic59—161no—
205#3095—591no—
206#2332D5911yescharacteristic
207#1826—591no—
208#1820—591no—
209#1511—591no—
210#1306—591no—
211#1284—591no—
212C2×C59—21no—
213C2×C59—21no—
214C2×C59—41no—
215C2×C59—41no—
216C2×C59#189011yescharacteristic
217C2×C59#190511yescharacteristic
218C2×C59#190611yescharacteristic
219C2×C22⋊Q8C2×D5911yescharacteristic
220C2×#117—591no—
221#485—1182no—
222#484—592no—
223#483—592no—
224#428—1181no—
225#419—1181no—
226#407—591no—
227#407—591no—
228#391—21no—
229#388—21no—
230#388—21no—
231#387—21no—
232#387—21no—
233#381—591no—
234#381C2×D5911yescharacteristic
235C4×C4⋊C4—21no—
236C4×C22⋊C4C2×D5911yescharacteristic
237#342—1181no—
238#342—1181no—
239#339—1181no—
240#339—1181no—
241#328—1182no—
242C59#952611yescharacteristic, complemented, Sylow
243C22×Q8C22⋊D5911yescharacteristic
244C23×C4C22×D5911yescharacteristic
245C2×M16—1181no—
246C2×M16—1181no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8C4×D5912yes—
250C22⋊Q8D23612yes—
251#117—1181no—
252#117—1181no—
253#117—1181no—
254#117—1181no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—591no—
259C2×C4⋊C4—591no—
260C2×C4⋊C4C22⋊D5911yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—1181no—
264C2×C22⋊C4—1181no—
265C2×C22⋊C4C22⋊D5911yescharacteristic
266C2×C22⋊C4C22⋊D5911yescharacteristic
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—1181no—
270C2×C42C22⋊D5911yescharacteristic
271C2×C42C22⋊D5911yescharacteristic
272#107—1182no—
273#106—1182no—
274#103—1182no—
275#103—1182no—
276C22⋊C8—592no—
277C22⋊C8—1181no—
278C22⋊C8—1181no—
279C22⋊C8—1181no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—1181no—
288#95—1181no—
289C24C2×C22⋊D5911yescharacteristic
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8(C2×D59)⋊C412yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—1181no—
303C22×C4—1181no—
304C22×C4—2361no—
305C22×C4C2×C4×D5911yescharacteristic
306C22×C4C2×D23611yescharacteristic
307C22×C4#1174568511yescharacteristic
308C22×C4#1174568511yescharacteristic
309C22×C4C2×C22⋊D5911yescharacteristic
310C22×C4C2×C22⋊D5911yescharacteristic
311M16—2361no—
312M16—2361no—
313M16—2361no—
314M16—2361no—
315C2×C8—1182no—
316C2×C8—2361no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—42no—
321C4⋊C4—81no—
322C4⋊C4—1181no—
323C4⋊C4—1181no—
324C4⋊C4—1181no—
325C4⋊C4—1181no—
326C4⋊C4(C2×D59)⋊C412yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—2361no—
333C22⋊C4—2361no—
334C22⋊C4—2361no—
335C22⋊C4—2361no—
336C42—41no—
337C42—41no—
338C42—42no—
339C42—81no—
340C42—2361no—
341C42—2361no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23C4×C22⋊D5911yescharacteristic
347C23#1427369611yescharacteristic
348C23#1427370111yescharacteristic
349Q8—42no—
350Q8—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—4721no—
378C2×C4—4721no—
379C2×C4—4721no—
380C2×C4—4721no—
381C2×C4—4721no—
382C2×C4—2361no—
383C2×C4—2361no—
384C2×C4C2×(C2×D59)⋊C411yescharacteristic
385C2×C4C4×C22⋊D5911yescharacteristic
386C2×C4#1427370111yescharacteristic
387C2×C4#1427371011yescharacteristic
388C8—4721no—
389C8—2362no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22#1683728111yescharacteristic
401C22#1683759211yescharacteristic
402C22#1683759211yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—9441no—
414C4—4721no—
415C4—4721no—
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#2142450911yescharacteristic
421C2#2142459011yescharacteristic
422C2#2142459711yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2332C2
socleC22×C59#386
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#2332D59
hypocenterC59#9526
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2332

Complements

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

NH ≅ G/Ntype
10. #16837735C4semidirect
11. #16837730C4semidirect
242. C59#9526semidirect

Maximal subgroups

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

subgroupindex
#214340402
#214241962
#214241712
#214226292
#214220042
#214219492
C59×#23322
#952659