Finite Group Databasebeta

← #28483886

Subgroups — #28483886

Order 15104.

476 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 107 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#21434104C211yesmaximal, characteristic
2#21433033C211yesmaximal, characteristic, complemented
3#21424895C211yesmaximal, characteristic
4#21422751C211yesmaximal, characteristic, complemented
5#21421991C211yesmaximal, characteristic, complemented
6#21421940C211yesmaximal, characteristic, complemented
7C59×#1486C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837732C412yes—
10#16837692—21no—
11#16837607—21no—
12#16837553C412yes—
13#16837531—22no—
14#16837531—22no—
15C4×#11745670—22no—
16#16837455C2211yescharacteristic
17#16837430—21no—
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×C22⋊(C59⋊C8)—21no—
20C2×C22⋊(C59⋊C8)—21no—
21#16837360—21no—
22C2×#14273638C2211yescharacteristic
23#16837150—21no—
24#16837149—21no—
25C2×C4×D4×C59C2211yescharacteristic
26C59×#431C2211yescharacteristic
27C59×#423C2211yescharacteristic
28C59×D4⋊C8—22no—
29C2×#11745670—22no—
30C22×C59⋊C8—21no—
31C22×C59⋊C8D411yescharacteristic
32C22×C59⋊C8D411yescharacteristic
33#14273727D411yescharacteristic
34#14273727D411yescharacteristic
35#14273719—21no—
36#14273706—21no—
37#14273706C2×C412yes—
38C4×Dic118—21no—
39C4×Dic118D412yes—
40#14273686—21no—
41#14273658—22no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45Dic59⋊C8—24no—
46#14273650—22no—
47#14273648—22no—
48C22⋊(C59⋊C8)—42no—
49C22⋊(C59⋊C8)—42no—
50C22⋊(C59⋊C8)C2×C412yes—
51#14273638—21no—
52#14273638—21no—
53#14273637—22no—
54C4×C59⋊C8—21no—
55C8×Dic59—22no—
56C22×D4×C59C2×C411yescharacteristic
57C23×C4×C59—21no—
58C2×M16×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63C2×C4⋊C4×C59C2×C411yescharacteristic
64C2×C22⋊C4×C59—21no—
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59D412yes—
69C22⋊C8×C59—21no—
70C4×C8×C59—21no—
71C24×C59—21no—
72C2×D4×C59—21no—
73C2×D4×C59—21no—
74C2×D4×C59—42no—
75C22×C4×C59—21no—
76C22×C4×C59—41no—
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59#4111yescharacteristic
85C2×Dic118—21no—
86C2×Dic118C22⋊C412yes—
87M16×C59—41no—
88M16×C59—41no—
89C2×C8×C59—22no—
90C2×C8×C59—41no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—21no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C42×C59C2×D411yescharacteristic
96C42×C59C2×D411yescharacteristic
97C42×C59C2×D411yescharacteristic
98C42×C59#4111yescharacteristic
99C22⋊Dic59—22no—
100#11745670—44no—
101C4⋊Dic59—21no—
102C4⋊Dic59—21no—
103C4⋊Dic59—21no—
104C4⋊Dic59C22⋊C412yes—
105Dic59⋊C4—22no—
106Dic59⋊C4—22no—
107C4×Dic59—22no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—22no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—21no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8SD1612yescomplemented
118C23×C59—22no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59C4×D411yescharacteristic
126D4×C59—22no—
127D4×C59—22no—
128D4×C59—41no—
129D4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59#12011yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12311yescharacteristic
145C2×C4×C59C2×SD1611yescharacteristic
146C2×C4×C59#13511yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C2×Dic59—22no—
150C2×Dic59—22no—
151Dic118—42no—
152Dic118—81no—
153C8×C59—42no—
154C8×C59—81no—
155C59⋊C8—22no—
156C59⋊C8—42no—
157C59⋊C8—41no—
158C59⋊C8—81no—
159#9050—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—42no—
163C22×C59—42no—
164C22×C59—42no—
165C22×C59—81no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59#38611yescharacteristic
170C22×C59C4×SD1611yescharacteristic
171C22×C59#44011yescharacteristic
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—81no—
177C4×C59#44811yescharacteristic
178C4×C59#44911yescharacteristic
179C4×C59#46011yescharacteristic
180C4×C59#46411yescharacteristic
181Dic59—42no—
182Dic59—42no—
183#3101—591no—
184#2961—591no—
185#1955—591no—
186#1542—591no—
187#1486D5911yescharacteristic
188#1301—591no—
189#1281—591no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—82no—
193C2×C59#188011yescharacteristic
194C2×C59#188111yescharacteristic
195C2×C59#189111yescharacteristic
196#533—591no—
197C2×C4×D4C2×D5911yescharacteristic
198#483—592no—
199#476—1181no—
200#461—1181no—
201#450—592no—
202#446—1182no—
203#446—1182no—
204C4×SD16—1182no—
205#431—591no—
206#431C2×D5911yescharacteristic
207#424—1181no—
208#423C2×D5911yescharacteristic
209C2×C4⋊C8—591no—
210C2×C22⋊C8—1181no—
211C2×C22⋊C8—1181no—
212#405—1181no—
213C2×#97—591no—
214#333—1181no—
215#332—1181no—
216D4⋊C8—22no—
217C59#905011yescharacteristic, complemented, Sylow
218C22×D4C4×D5911yescharacteristic
219C23×C4—21no—
220C2×SD16—1182no—
221C2×M16—21no—
222C22×C8—591no—
223C22×C8—591no—
224C22×C8—1181no—
225C4⋊Q8—591no—
226C4⋊Q8—591no—
227#125—1181no—
228C22⋊Q8—592no—
229C22⋊Q8—1181no—
230C4×Q8—592no—
231C4×Q8—1181no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236#117—1181no—
237C2×C4⋊C4C4×D5911yescharacteristic
238C2×C22⋊C4—21no—
239C2×C42C22×D5911yescharacteristic
240#106—1182no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—1184no—
244C4⋊C8—1181no—
245C4⋊C8—1181no—
246C4⋊C8—1181no—
247C4⋊C8C22⋊D5912yes—
248#103—1182no—
249#102—1182no—
250C22⋊C8—21no—
251C22⋊C8—592no—
252C22⋊C8—2362no—
253C22⋊C8—2362no—
254#97—1182no—
255#97—1181no—
256#97—1181no—
257C4×C8—21no—
258C4×C8—1182no—
259C4×C8—1181no—
260C24—21no—
261C2×Q8—592no—
262C2×Q8—1181no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—42no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×C4×D5911yescharacteristic
274C22×C4D4×D5911yescharacteristic
275C22×C4#1174568711yescharacteristic
276SD16—2364no—
277M16—41no—
278M16—41no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—592no—
282C2×C8—1181no—
283C2×C8—1181no—
284C2×C8—1182no—
285C2×C8—1181no—
286C2×C8—1181no—
287C2×C8—1181no—
288C2×C8—2361no—
289C2×C8—2361no—
290C2×C8—2361no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—592no—
294C4⋊C4—1182no—
295C4⋊C4—1181no—
296C4⋊C4—1181no—
297C4⋊C4—1181no—
298C4⋊C4—1182no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—1182no—
302C42—1182no—
303C42#1174568511yescharacteristic
304C42D4×D5911yescharacteristic
305C42D4×D5911yescharacteristic
306C42C2×C22⋊D5911yescharacteristic
307C23—22no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23(C22⋊D59)⋊C411yescharacteristic
315Q8—4721no—
316Q8—2362no—
317D4—22no—
318D4—22no—
319D4—41no—
320D4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—1182no—
331C2×C4—1182no—
332C2×C4C22⋊C4×D5911yescharacteristic
333C2×C4C4×C22⋊D5911yescharacteristic
334C2×C4#1427369511yescharacteristic
335C2×C4#1427369811yescharacteristic
336C2×C4#1427370011yescharacteristic
337C2×C4#1427370911yescharacteristic
338C2×C4SD16×D5911yescharacteristic
339C2×C4#1427374611yescharacteristic
340C2×C4#1427375111yescharacteristic
341C2×C4#1427375611yescharacteristic
342C8—42no—
343C8—81no—
344C8—1182no—
345C8—4721no—
346C8—2362no—
347C8—2361no—
348C22—21no—
349C22—21no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#1683727811yescharacteristic
358C22#1683749311yescharacteristic
359C22#1683749911yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—2362no—
366C4—2362no—
367C4#1683753911yescharacteristic
368C4#1683754411yescharacteristic
369C4#1683760111yescharacteristic
370C4#1683762311yescharacteristic
371C2—21no—
372C2—21no—
373C2—82no—
374C2#2142444811yescharacteristic
375C2#2142445511yescharacteristic
376C2#2142451111yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1486C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#1486D59
hypocenterC59#9050
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #21433033C2semidirect
4. #21422751C2semidirect
5. #21421991C2semidirect
6. #21421940C2semidirect
117. C2×C59⋊C8SD16semidirect
217. C59#9050semidirect

Maximal subgroups

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

subgroupindex
#214341042
#214330332
#214248952
#214227512
#214219912
#214219402
C59×#14862
#905059