Finite Group Databasebeta

← #28490747

Subgroups — #28490747

Order 15104.

496 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 103 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#21435249C211yesmaximal, characteristic, complemented
2#21433893C211yesmaximal, characteristic
3C2×#16837682C211yesmaximal, characteristic
4#21422776C211yesmaximal, characteristic, complemented
5#21422500C211yesmaximal, characteristic, complemented
6#21421992C211yesmaximal, characteristic, complemented
7C59×#1460C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9C2×#14273727C2211yescharacteristic
10#16837807—21no—
11#16837753—21no—
12#16837738—22no—
13#16837702—22no—
14#16837698—22no—
15#16837692—21no—
16#16837682—21no—
17#16837682—22no—
18#16837682—22no—
19#16837442—21no—
20C2×#14273660—21no—
21#16837426C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23#16837407—21no—
24C2×#14273648—21no—
25#16837150—22no—
26#16837134—22no—
27C2×C4×D4×C59C2211yescharacteristic
28C59×#431C2211yescharacteristic
29C59×#431C2211yescharacteristic
30C59×#323—22no—
31C22×Dic118—21no—
32C2×#11745664—21no—
33C22×C59⋊C8—21no—
34#14273727—21no—
35#14273727—21no—
36#14273727—21no—
37#14273727D412yes—
38#14273727D411yescharacteristic
39#14273727D411yescharacteristic
40#14273719—21no—
41#14273706—21no—
42#14273706—21no—
43C4×Dic118—21no—
44C4×Dic118—21no—
45#14273686—21no—
46C2×C4⋊Dic59—21no—
47C2×C4⋊Dic59—21no—
48#14273661—24no—
49#14273660—24no—
50#14273660—22no—
51#14273658—22no—
52C4⋊(C59⋊C8)—21no—
53C4⋊(C59⋊C8)—21no—
54C4⋊(C59⋊C8)D412yes—
55C4⋊(C59⋊C8)D412yes—
56#14273652—24no—
57#14273648—24no—
58#14273648—22no—
59#14273648—22no—
60C22×D4×C59D411yescharacteristic
61C23×C4×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59—21no—
66C2×C4⋊C4×C59D411yescharacteristic
67C2×C22⋊C4×C59—21no—
68C2×C42×C59C2311yescharacteristic
69C4⋊C8×C59—21no—
70C4⋊C8×C59—21no—
71C22⋊C8×C59—21no—
72C22⋊C8×C59D412yes—
73C4×C8×C59—21no—
74C4×C8×C59—21no—
75C24×C59—21no—
76C2×D4×C59—21no—
77C2×D4×C59—21no—
78C2×D4×C59—21no—
79C2×D4×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59Q1612yes—
82C22×C4×C59—21no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59—82no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×Dic59—41no—
90C2×Dic118—21no—
91C2×Dic118—21no—
92C2×Dic118—22no—
93C2×Dic118—41no—
94C2×Dic118—82no—
95C2×C8×C59—22no—
96C2×C8×C59—41no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C42×C59C2×D411yescharacteristic
102C42×C59C2×D411yescharacteristic
103C42×C59C2×D411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C22⋊Dic59—41no—
106C4⋊Dic59—21no—
107C4⋊Dic59—22no—
108C4⋊Dic59—21no—
109C4⋊Dic59—22no—
110C4⋊Dic59—21no—
111C4⋊Dic59—21no—
112C4⋊Dic59—41no—
113C4⋊Dic59—41no—
114Dic59⋊C4—41no—
115Dic59⋊C4—41no—
116C4×Dic59—41no—
117#11745664—41no—
118#11745664—41no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—22no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—41no—
123C23×C59—22no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—81no—
127C23×C59—81no—
128C23×C59#12111yescharacteristic
129D4×C59—22no—
130D4×C59—22no—
131D4×C59—41no—
132D4×C59—41no—
133C2×C4×C59—22no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—82no—
137C2×C4×C59—82no—
138C2×C4×C59—81no—
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59#12011yescharacteristic
141C2×C4×C59#12011yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12711yescharacteristic
145C2×C4×C59C2×Q1611yescharacteristic
146C2×C4×C59#13511yescharacteristic
147C2×C4×C59#13711yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—42no—
152C2×Dic59—82no—
153Dic118—42no—
154Dic118—42no—
155Dic118—82no—
156Dic118—82no—
157C8×C59—42no—
158C8×C59—81no—
159C59⋊C8—42no—
160C59⋊C8—42no—
161#10190—591nomaximal, Sylow
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—42no—
165C22×C59—82no—
166C22×C59—82no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59#39211yescharacteristic
170C22×C59#46711yescharacteristic
171C22×C59#47011yescharacteristic
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—81no—
176C4×C59#45111yescharacteristic
177C4×C59#45111yescharacteristic
178C4×C59#45211yescharacteristic
179C4×C59#45211yescharacteristic
180C4×C59#47212yes—
181Dic59—81no—
182Dic59—81no—
183Dic59—82no—
184#3230—591no—
185#3077—591no—
186C2×#474—591no—
187#1548—591no—
188#1481—591no—
189#1460D5911yescharacteristic
190#1301—591no—
191C2×C59—21no—
192C2×C59—21no—
193C2×C59—82no—
194C2×C59#200111yescharacteristic
195C2×C59#200911yescharacteristic
196C2×C59#201811yescharacteristic
197#533—591no—
198C2×C4⋊Q8—591no—
199C2×C4×D4C2×D5911yescharacteristic
200#500—1181no—
201#487—1181no—
202#484—1182no—
203#478—1182no—
204#477—1182no—
205#476—1181no—
206#474—1182no—
207#474—1181no—
208#474—1182no—
209#431C2×D5911yescharacteristic
210#431C2×D5911yescharacteristic
211#428—1181no—
212C2×#107—1181no—
213#423—591no—
214C2×C4⋊C8—591no—
215#418—1181no—
216C2×#102—1181no—
217#333—1182no—
218Q8⋊C8—1182no—
219#323—22no—
220C59#1019011yescharacteristic, complemented, Sylow
221C22×Q8—1181no—
222C22×D4D23611yescharacteristic
223C23×C4—21no—
224C2×M16—1181no—
225C22×C8—1181no—
226C4⋊Q8—592no—
227C4⋊Q8—591no—
228C4⋊Q8—591no—
229C4⋊Q8—1181no—
230C4⋊Q8—1181no—
231C4⋊Q8—1181no—
232#125—1181no—
233C22⋊Q8—1181no—
234C22⋊Q8—1181no—
235C4×Q8—1181no—
236C4×Q8—1181no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241#117—1181no—
242C2×C4⋊C4—1181no—
243C2×C4⋊C4—1181no—
244C2×C4⋊C4D23611yescharacteristic
245C2×C22⋊C4—21no—
246C2×C42C22×D5911yescharacteristic
247#107—1184no—
248#107—1182no—
249#107—1184no—
250#106—1182no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—592no—
254C4⋊C8—592no—
255C4⋊C8—1181no—
256C4⋊C8—1181no—
257#103—1184no—
258#102—1184no—
259#102—1182no—
260#102—1182no—
261C22⋊C8—21no—
262C22⋊C8C22⋊D5912yes—
263C4×C8—21no—
264C4×C8—21no—
265C24—21no—
266C2×Q8—1181no—
267C2×Q8—1182no—
268C2×Q8—1181no—
269C2×Q8—4722no—
270C2×Q8—2361no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4Dic23612yes—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—2361no—
282C22×C4C2×D23611yescharacteristic
283C22×C4#1174568511yescharacteristic
284C22×C4C2×C22⋊D5911yescharacteristic
285M16—2361no—
286M16—2361no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—1182no—
290C2×C8—1182no—
291C2×C8—2361no—
292C2×C8—2361no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—1182no—
296C4⋊C4—1181no—
297C4⋊C4—1181no—
298C4⋊C4—1182no—
299C4⋊C4—1181no—
300C4⋊C4—1181no—
301C4⋊C4—2361no—
302C4⋊C4—2361no—
303C4⋊C4—2361no—
304C4⋊C4—2361no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—2361no—
308C42—2361no—
309C42D4×D5911yescharacteristic
310C42D4×D5911yescharacteristic
311C42D4×D5911yescharacteristic
312C42D4×D5911yescharacteristic
313C23—22no—
314C23—41no—
315C23—41no—
316C23—81no—
317C23—81no—
318C23#1427370111yescharacteristic
319Q8—4722no—
320Q8—4722no—
321Q8—2362no—
322Q8—2362no—
323D4—22no—
324D4—22no—
325D4—41no—
326D4—41no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—4722no—
334C2×C4—2361no—
335C2×C4—2361no—
336C2×C4—2362no—
337C2×C4#1427369511yescharacteristic
338C2×C4#1427369711yescharacteristic
339C2×C4#1427369711yescharacteristic
340C2×C4#1427370011yescharacteristic
341C2×C4#1427370011yescharacteristic
342C2×C4#1427372311yescharacteristic
343C2×C4C2×Dic23611yescharacteristic
344C2×C4#1427374911yescharacteristic
345C2×C4#1427376011yescharacteristic
346C2×C4#1427376011yescharacteristic
347C8—42no—
348C8—81no—
349C8—2362no—
350C8—2362no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—82no—
355C22—82no—
356C22—81no—
357C22—81no—
358C22#1683731111yescharacteristic
359C22#1683764311yescharacteristic
360C22#1683766011yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—4721no—
366C4—4722no—
367C4—4721no—
368C4#1683756011yescharacteristic
369C4#1683756011yescharacteristic
370C4#1683756611yescharacteristic
371C4#1683756611yescharacteristic
372C4#1683767112yes—
373C2—21no—
374C2—21no—
375C2—82no—
376C2#2142510611yescharacteristic
377C2#2142515511yescharacteristic
378C2#2142520411yescharacteristic
subgroupG/H
centerC22#16837311
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1460C2
socleC22×C59#392
prime radicalC22×D4×C59D4
hypercenter#1460D59
hypocenterC59#10190
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
1. #21435249C2semidirect
4. #21422776C2semidirect
5. #21422500C2semidirect
6. #21421992C2semidirect
220. C59#10190semidirect

Maximal subgroups

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

subgroupindex
#214352492
#214338932
C2×#168376822
#214227762
#214225002
#214219922
C59×#14602
#1019059