Finite Group Databasebeta

← #28489213

Subgroups — #28489213

Order 15104.

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

normal subgroups only

This group has 552 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#21434868C211yesmaximal, characteristic
2C2×#16837730C211yesmaximal, characteristic
3#21428581C211yesmaximal, characteristic
4#21424180C211yesmaximal, characteristic
5#21422667C211yesmaximal, characteristic
6#21421944C211yesmaximal, characteristic
7C59×#1996C211yesmaximal, characteristic
8C2×#14273706C2211yescharacteristic
9C22×C4⋊Dic59C2211yescharacteristic
10#16837737—21no—
11#16837735—21no—
12#16837730—22no—
13#16837730—22no—
14#16837641—22no—
15#16837558—21no—
16#16837557—21no—
17C2×#14273658—21no—
18C2×#14273651—21no—
19C2×C22⋊(C59⋊C8)C2211yescharacteristic
20C2×C22⋊(C59⋊C8)C2211yescharacteristic
21#16837342—22no—
22#16837308—21no—
23#16837268—21no—
24#16837263—22no—
25#16837164—22no—
26#16837138—22no—
27C2×C22⋊Q8×C59C2211yescharacteristic
28C59×#380C2211yescharacteristic
29C4×C22⋊C4×C59C2211yescharacteristic
30C59×#342—22no—
31C22×Dic118D411yescharacteristic
32C23×Dic59—21no—
33C2×#11745672—22no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8D412yes—
36#14273706—21no—
37#14273706—21no—
38#14273706—21no—
39#14273706—21no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—21no—
44C2×C4⋊Dic59—21no—
45C2×C4⋊Dic59D411yescharacteristic
46C2×C4⋊Dic59D412yes—
47C2×Dic59⋊C4—21no—
48C2×C22⋊Dic59—21no—
49#14273660—22no—
50#14273658—24no—
51#14273651—22no—
52#14273651—24no—
53#14273650—22no—
54C22⋊(C59⋊C8)—21no—
55C22⋊(C59⋊C8)—21no—
56C22⋊(C59⋊C8)—21no—
57C22⋊(C59⋊C8)D412yes—
58C59⋊#95—24no—
59C59⋊#95—24no—
60C22×Q8×C59D411yescharacteristic
61C23×C4×C59C2311yescharacteristic
62C22⋊Q8×C59—21no—
63C22⋊Q8×C59—21no—
64C22⋊Q8×C59—21no—
65C22⋊Q8×C59—21no—
66C2×C4⋊C4×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59D411yescharacteristic
69C2×C22⋊C4×C59—21no—
70C2×C22⋊C4×C59—21no—
71C2×C22⋊C4×C59D412yes—
72C2×C42×C59—21no—
73#95×C59—21no—
74#95×C59—21no—
75C24×C59C2×D411yescharacteristic
76C2×Q8×C59—21no—
77C2×Q8×C59—21no—
78C2×Q8×C59—42no—
79C22×C4×C59—22no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—42no—
83C22×C4×C59—41no—
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59#4111yescharacteristic
90C22×Dic59—21no—
91C22×Dic59—21no—
92C22×Dic59—22no—
93C22×Dic59—41no—
94C22×Dic59—41no—
95C22×Dic59—42no—
96C22×Dic59—42no—
97C2×Dic118—21no—
98C2×Dic118—21no—
99C2×Dic118—42no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—42no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C22⋊C4×C59—22no—
106C22⋊C4×C59—22no—
107C22⋊C4×C59—41no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—42no—
110C42×C59—42no—
111C22⋊Dic59—41no—
112C22⋊Dic59—41no—
113#11745672—82no—
114C4⋊Dic59—22no—
115C4⋊Dic59—21no—
116C4⋊Dic59—22no—
117C4⋊Dic59—21no—
118C4⋊Dic59—41no—
119C4⋊Dic59—41no—
120Dic59⋊C4—41no—
121Dic59⋊C4—41no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—42no—
125C2×C59⋊C8—42no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59#12011yescharacteristic
129C23×C59#12111yescharacteristic
130C23×C59#12711yescharacteristic
131C23×C59#13512yes—
132C23×C59#13712yes—
133Q8×C59—42no—
134Q8×C59—42no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—42no—
146C2×C4×C59—81no—
147C2×C4×C59—81no—
148C2×C4×C59—81no—
149C2×C4×C59—81no—
150C2×C4×C59—81no—
151C2×C4×C59#12011yescharacteristic
152C2×C4×C59#12011yescharacteristic
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59#12111yescharacteristic
155C2×Dic59—42no—
156C2×Dic59—42no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—42no—
160C2×Dic59—44no—
161C2×Dic59—82no—
162C2×Dic59—81no—
163C2×Dic59—81no—
164Dic118—42no—
165Dic118—81no—
166C59⋊C8—42no—
167C59⋊C8—42no—
168#9961—591nomaximal, Sylow
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—42no—
176C22×C59—42no—
177C22×C59#39211yescharacteristic
178C22×C59#45212yes—
179C22×C59#46412yes—
180C22×C59#49511yescharacteristic
181C22×C59#49711yescharacteristic
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—82no—
186C4×C59—81no—
187C4×C59—81no—
188C4×C59—161no—
189C4×C59#45812yes—
190Dic59—81no—
191Dic59—82no—
192Dic59—81no—
193#3189—591no—
194C2×#483—591no—
195#2503—591no—
196#1996D5911yescharacteristic
197#1823—591no—
198#1520—591no—
199#1282—591no—
200C2×C59—22no—
201C2×C59—21no—
202C2×C59—22no—
203C2×C59—21no—
204C2×C59#200911yescharacteristic
205C2×C59#201511yescharacteristic
206C2×C59#202211yescharacteristic
207C2×C22⋊Q8—591no—
208C2×C22⋊Q8C2×D5911yescharacteristic
209C22×C4⋊C4—591no—
210#484—1181no—
211#484—1181no—
212#483—1182no—
213#483—1182no—
214#467—1182no—
215#451—1181no—
216#451—1181no—
217C2×#106—1181no—
218C2×#103—1181no—
219C2×C22⋊C8—591no—
220C2×C22⋊C8—591no—
221#400—1182no—
222#391—1181no—
223#382—1181no—
224#381—1182no—
225#380C2×D5911yescharacteristic
226C4×C22⋊C4C2×D5911yescharacteristic
227#342—22no—
228#340—1182no—
229#328—1182no—
230C59#996111yescharacteristic, complemented, Sylow
231C22×Q8—591no—
232C22×Q8D23611yescharacteristic
233C23×C4—1181no—
234C23×C4C22×D5911yescharacteristic
235C2×Q16—1182no—
236C22×C8—592no—
237C22×C8—1181no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—1181no—
243C22⋊Q8—1181no—
244C22⋊Q8—1181no—
245C22⋊Q8—1181no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—592no—
249C2×C4⋊C4—591no—
250C2×C4⋊C4—1181no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—1181no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4—1181no—
255C2×C4⋊C4—1181no—
256C2×C4⋊C4D23611yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—1181no—
260C2×C22⋊C4C22⋊D5912yes—
261C2×C42—21no—
262#107—1182no—
263#106—1184no—
264#103—1182no—
265#103—1184no—
266#103—1182no—
267C22⋊C8—592no—
268C22⋊C8—1181no—
269C22⋊C8—1181no—
270C22⋊C8—1181no—
271#95—21no—
272#95—21no—
273#95—1184no—
274#95—1184no—
275C24C2×C22⋊D5911yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×Q8—1181no—
280C2×Q8—1181no—
281C2×Q8—2362no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—1181no—
288C22×C4—1181no—
289C22×C4—1182no—
290C22×C4—2361no—
291C22×C4—2361no—
292C22×C4—2362no—
293C22×C4—2362no—
294C22×C4C2×D23611yescharacteristic
295C22×C4#1174568511yescharacteristic
296C22×C4D4×D5911yescharacteristic
297C22×C4D4×D5911yescharacteristic
298C22×C4D4×D5911yescharacteristic
299C22×C4D4×D5911yescharacteristic
300Q16—4722no—
301C2×C8—1182no—
302C2×C8—1182no—
303C2×C8—2362no—
304C2×C8—2362no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—1182no—
311C4⋊C4—1181no—
312C4⋊C4—1182no—
313C4⋊C4—1181no—
314C4⋊C4—2361no—
315C4⋊C4—2361no—
316C4⋊C4—2361no—
317C4⋊C4—2361no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—2361no—
324C22⋊C4—2361no—
325C42—42no—
326C23—41no—
327C23—41no—
328C23#1427369511yescharacteristic
329C23#1427370111yescharacteristic
330C23#1427372311yescharacteristic
331C23#1427374712yes—
332C23#1427375812yes—
333Q8—42no—
334Q8—42no—
335Q8—4721no—
336Q8—2362no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—4721no—
354C2×C4—4721no—
355C2×C4—4722no—
356C2×C4—2362no—
357C2×C4—2362no—
358C2×C4—2364no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2362no—
362C2×C4#1427369711yescharacteristic
363C2×C4#1427369711yescharacteristic
364C2×C4#1427370011yescharacteristic
365C2×C4#1427370011yescharacteristic
366C8—2362no—
367C8—2362no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1683731111yescharacteristic
377C22#1683756212yes—
378C22#1683762412yes—
379C22#1683778411yescharacteristic
380C22#1683779411yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—81no—
386C4—81no—
387C4—161no—
388C4—4721no—
389C4—4722no—
390C4—4721no—
391C4#1683759312yes—
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2#2142515511yescharacteristic
397C2#2142518911yescharacteristic
398C2#2142522611yescharacteristic
subgroupG/H
centerC22#16837311
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1996C2
socleC22×C59#392
prime radicalC24×C59C2×D4
hypercenter#1996D59
hypocenterC59#9961
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
230. C59#9961semidirect

Maximal subgroups

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

subgroupindex
#214348682
C2×#168377302
#214285812
#214241802
#214226672
#214219442
C59×#19962
#996159