Finite Group Databasebeta

← #28488411

Subgroups — #28488411

Order 15104.

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

normal subgroups only

This group has 484 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
1C2×#16837641C211yesmaximal, characteristic
2#21424132C211yesmaximal, characteristic
3#21423874C211yesmaximal, characteristic
4#21422615C211yesmaximal, characteristic
5#21422461C211yesmaximal, characteristic
6#21422460C211yesmaximal, characteristic
7C59×#1508C211yesmaximal, characteristic
8C22×#11745672C411yescharacteristic, complemented
9C23×C59⋊C8C2211yescharacteristic
10C2×#14273706C2211yescharacteristic
11#16837641—21no—
12#16837641—21no—
13#16837641—21no—
14#16837641—21no—
15C2×#14273662—21no—
16C2×#14273660C411yescharacteristic, complemented
17C2×#14273651C411yescharacteristic, complemented
18C2×#14273650C411yescharacteristic, complemented
19#16837366C2211yescharacteristic
20C2×C22⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273638—21no—
22#16837166—21no—
23#16837157—21no—
24#16837138—22no—
25#16837128—22no—
26#16837127—22no—
27C2×C22⋊Q8×C59C2211yescharacteristic
28C59×#407C2211yescharacteristic
29C4×C22⋊C4×C59C2211yescharacteristic
30C59×#328—22no—
31C22×Dic118C2×C411yescharacteristic
32C2×#11745672—21no—
33C2×#11745672—21no—
34C2×#11745672—42no—
35C2×#11745664—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—42no—
38C22×C59⋊C8C2×C411yescharacteristic
39C22×C59⋊C8C2×C411yescharacteristic
40C22×C59⋊C8D411yescharacteristic
41C22×C59⋊C8D411yescharacteristic
42C22×C59⋊C8D411yescharacteristic
43C22×C59⋊C8D411yescharacteristic
44#14273706—21no—
45#14273706—21no—
46#14273706—21no—
47#14273706—21no—
48C2×C4⋊Dic59C2×C411yescharacteristic
49C2×Dic59⋊C4—21no—
50C2×C22⋊Dic59—21no—
51#14273662—42no—
52#14273660—21no—
53#14273660—21no—
54#14273651—21no—
55#14273651—21no—
56#14273650—21no—
57#14273650—21no—
58C22⋊(C59⋊C8)—21no—
59C22⋊(C59⋊C8)—21no—
60C22⋊(C59⋊C8)—21no—
61C22⋊(C59⋊C8)D412yes—
62#14273638—42no—
63C22×Q8×C59C2×C411yescharacteristic
64C23×C4×C59C2311yescharacteristic
65C2×M16×C59—21no—
66C22⋊Q8×C59—21no—
67C22⋊Q8×C59—21no—
68C22⋊Q8×C59—21no—
69C22⋊Q8×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59C2×C411yescharacteristic
72C2×C22⋊C4×C59—21no—
73C2×C22⋊C4×C59—21no—
74C2×C42×C59—21no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59—21no—
77#95×C59D412yes—
78C24×C59#4111yescharacteristic
79C2×Q8×C59—21no—
80C2×Q8×C59—21no—
81C2×Q8×C59—41no—
82C2×Q8×C59—41no—
83C22×C4×C59—21no—
84C22×C4×C59—22no—
85C22×C4×C59—21no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59C22×C411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C22×Dic59—21no—
95C22×Dic59—21no—
96C2×Dic118—21no—
97C2×Dic118—21no—
98C2×Dic118—41no—
99C2×Dic118—41no—
100M16×C59—41no—
101M16×C59—41no—
102C2×C8×C59—41no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C42×C59—41no—
111C42×C59—41no—
112C22⋊Dic59—41no—
113C22⋊Dic59—41no—
114#11745672—42no—
115#11745672—42no—
116C4⋊Dic59—21no—
117C4⋊Dic59—21no—
118Dic59⋊C4—41no—
119Dic59⋊C4—41no—
120#11745664—41no—
121#11745664—41no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—22no—
125C2×C59⋊C8—22no—
126C2×C59⋊C8—41no—
127C2×C59⋊C8—41no—
128C2×C59⋊C8—41no—
129C2×C59⋊C8—41no—
130C2×C59⋊C8—41no—
131C2×C59⋊C8—41no—
132C2×C59⋊C8C22⋊C411yescharacteristic
133C2×C59⋊C8C22⋊C411yescharacteristic
134C2×C59⋊C8C22⋊C411yescharacteristic
135C2×C59⋊C8C22⋊C411yescharacteristic
136C23×C59—22no—
137C23×C59—22no—
138C23×C59—41no—
139C23×C59C4×D411yescharacteristic
140C23×C59#12111yescharacteristic
141C23×C59#12411yescharacteristic
142Q8×C59—41no—
143Q8×C59—41no—
144Q8×C59—81no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—42no—
155C2×C4×C59—42no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—42no—
159C2×C4×C59—42no—
160C2×C4×C59—81no—
161C2×C4×C59C2×C22⋊C411yescharacteristic
162C2×C4×C59C4×D411yescharacteristic
163C2×C4×C59#12111yescharacteristic
164C2×C4×C59#12711yescharacteristic
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—42no—
168C2×Dic59—81no—
169Dic118—41no—
170Dic118—41no—
171Dic118—81no—
172C8×C59—81no—
173C59⋊C8—21no—
174C59⋊C8—21no—
175C59⋊C8—21no—
176C59⋊C8—42no—
177C59⋊C8—81no—
178C59⋊C8#9912yes—
179#9817—591nomaximal, Sylow
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—44no—
188C22×C59#39011yescharacteristic
189C22×C59#44412yes—
190C22×C59#47312yes—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—82no—
195C4×C59—81no—
196C4×C59—81no—
197C4×C59C2×#9911yescharacteristic
198C4×C59#41011yescharacteristic
199Dic59—81no—
200Dic59—81no—
201C2×#467—591no—
202#1810—591no—
203#1752—591no—
204#1508—591no—
205#1508D5911yescharacteristic
206#1472—591no—
207#1472—591no—
208C2×C59—22no—
209C2×C59—22no—
210C2×C59—41no—
211C2×C59#196411yescharacteristic
212C2×C59#197511yescharacteristic
213C2×C59#197911yescharacteristic
214C22×Q16—591no—
215C23×C8—591no—
216C2×C22⋊Q8—591no—
217C2×C22⋊Q8C2×D5911yescharacteristic
218#467—1181no—
219#467—1181no—
220#467—1181no—
221#467—1181no—
222C2×#108—1181no—
223C2×#107—591no—
224C2×#103—591no—
225C2×#103—591no—
226#407—591no—
227#407C2×D5911yescharacteristic
228C2×C22⋊C8—591no—
229C2×#97—1181no—
230C4×C22⋊C4C2×D5911yescharacteristic
231#341—1181no—
232#336—1181no—
233#328—22no—
234#328—1182no—
235#324—1182no—
236#324—1182no—
237C59#981711yescharacteristic, complemented, Sylow
238C22×Q8—591no—
239C22×Q8C4×D5911yescharacteristic
240C23×C4C22×D5911yescharacteristic
241C2×Q16—1181no—
242C2×Q16—1181no—
243C2×Q16—2362no—
244C2×M16—21no—
245C2×M16—1181no—
246C22×C8—591no—
247C22×C8—591no—
248C22×C8—591no—
249C22×C8—591no—
250C22×C8—591no—
251C22×C8—591no—
252C22×C8—1181no—
253C22×C8—2362no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—1181no—
259C22⋊Q8—1181no—
260C22⋊Q8—1181no—
261C22⋊Q8—1181no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—591no—
264C2×C4⋊C4—1181no—
265C2×C4⋊C4C4×D5911yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1181no—
269C2×C42—21no—
270#108—2362no—
271#107—1181no—
272#107—1181no—
273#103—1181no—
274#103—1181no—
275#103—1181no—
276#103—1181no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—592no—
280C22⋊C8—1181no—
281C22⋊C8—1181no—
282C22⋊C8—1181no—
283#97—2362no—
284#95C22⋊D5912yes—
285C24#1174568711yescharacteristic
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—1181no—
291C2×Q8—1181no—
292C2×Q8—2361no—
293C2×Q8—2361no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—1181no—
300C22×C4—1181no—
301C22×C4C2×C4×D5911yescharacteristic
302C22×C4#1174568511yescharacteristic
303C22×C4D4×D5911yescharacteristic
304C22×C4D4×D5911yescharacteristic
305C22×C4D4×D5911yescharacteristic
306C22×C4C2×C22⋊D5911yescharacteristic
307Q16—2362no—
308Q16—2362no—
309M16—41no—
310M16—41no—
311M16—2361no—
312M16—2361no—
313C2×C8—41no—
314C2×C8—591no—
315C2×C8—591no—
316C2×C8—591no—
317C2×C8—591no—
318C2×C8—1182no—
319C2×C8—1182no—
320C2×C8—1182no—
321C2×C8—1182no—
322C2×C8—2361no—
323C2×C8—2361no—
324C2×C8—2361no—
325C2×C8—2361no—
326C2×C8—2361no—
327C2×C8—2361no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—1181no—
333C4⋊C4—1181no—
334C4⋊C4—2361no—
335C4⋊C4—2361no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—2361no—
340C22⋊C4—2361no—
341C42—41no—
342C42—41no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23(C22⋊D59)⋊C411yescharacteristic
347C23#1427369911yescharacteristic
348C23#1427371311yescharacteristic
349Q8—41no—
350Q8—41no—
351Q8—81no—
352Q8—4721no—
353Q8—2361no—
354Q8—2361no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—4721no—
372C2×C4—2362no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4C22⋊C4×D5911yescharacteristic
376C2×C4C4×C22⋊D5911yescharacteristic
377C2×C4#1427370011yescharacteristic
378C2×C4#1427372311yescharacteristic
379C8—81no—
380C8—592no—
381C8—1181no—
382C8—1181no—
383C8—1181no—
384C8—4721no—
385C8—2362no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22#1683730211yescharacteristic
395C22#1683752112yes—
396C22#1683767512yes—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—81no—
402C4—81no—
403C4—4721no—
404C4—4721no—
405C4#99×D5911yescharacteristic
406C4#1683737411yescharacteristic
407C2—22no—
408C2—22no—
409C2—41no—
410C2#2142493011yescharacteristic
411C2#2142498611yescharacteristic
412C2#2142501011yescharacteristic
subgroupG/H
centerC22#16837302
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1508C2
socleC22×C59#390
prime radicalC24×C59#41
cosocleC23×C4×C59C23
hypercenter#1508D59
hypocenterC59#9817
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
8. C22×#11745672C4semidirect
16. C2×#14273660C4semidirect
17. C2×#14273651C4semidirect
18. C2×#14273650C4semidirect
237. C59#9817semidirect

Maximal subgroups

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

subgroupindex
C2×#168376412
#214241322
#214238742
#214226152
#214224612
#214224602
C59×#15082
#981759