Finite Group Databasebeta

← #28485860

Subgroups — #28485860

Order 15104.

500 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 93 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#21426991C211yesmaximal, characteristic
3#21424196C211yesmaximal, characteristic
4#21422616C211yesmaximal, characteristic
5#21422002C211yesmaximal, characteristic
6#21421982C211yesmaximal, characteristic
7C59×#1996C211yesmaximal, characteristic
8C2×#14273686C2211yescharacteristic
9#16837740—22no—
10#16837735—21no—
11#16837730—21no—
12#16837442C412yescomplemented
13#16837411C412yescomplemented
14#16837366C2211yescharacteristic
15#16837366C2211yescharacteristic
16#16837298—21no—
17#16837290—22no—
18#16837269—21no—
19C4⋊C4×Dic59—21no—
20#16837253—21no—
21C4×C22⋊Dic59C2211yescharacteristic
22#16837162—22no—
23#16837139—21no—
24#16837139—21no—
25C2×C22⋊Q8×C59C2211yescharacteristic
26C59×#380C2211yescharacteristic
27C4×C22⋊C4×C59C2211yescharacteristic
28C59×#342—22no—
29C2×#11745664—21no—
30C2×#11745664C2×C412yes—
31#14273686—21no—
32#14273686—21no—
33#14273686C2×C412yes—
34#14273686D412yes—
35C2×C4⋊Dic59D411yescharacteristic
36C2×C4⋊Dic59D411yescharacteristic
37C2×Dic59⋊C4—22no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C4×Dic59—21no—
41C2×C4×Dic59—21no—
42C2×C4×Dic59—22no—
43#14273660—22no—
44#14273658—22no—
45#14273651—22no—
46#14273651—22no—
47C22⋊(C59⋊C8)—21no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)—21no—
51C59⋊#95—22no—
52C59⋊#95—22no—
53C59⋊#95D411yescharacteristic
54C59⋊#95D411yescharacteristic
55C22×Q8×C59C2×C411yescharacteristic
56C23×C4×C59C2311yescharacteristic
57C22⋊Q8×C59—21no—
58C22⋊Q8×C59—21no—
59C22⋊Q8×C59—21no—
60C22⋊Q8×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59C2×C411yescharacteristic
64C2×C22⋊C4×C59—21no—
65C2×C22⋊C4×C59—21no—
66C2×C22⋊C4×C59D412yes—
67C2×C42×C59—21no—
68#95×C59—21no—
69#95×C59—21no—
70C24×C59C2×D411yescharacteristic
71C2×Q8×C59—21no—
72C2×Q8×C59—21no—
73C2×Q8×C59—42no—
74C22×C4×C59—22no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—42no—
78C22×C4×C59—41no—
79C22×C4×C59C22×C411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59#4111yescharacteristic
84C22×C4×C59#4111yescharacteristic
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C4⋊C4×C59—21no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—42no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C22⋊C4×C59—22no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—42no—
99C42×C59—42no—
100C22⋊Dic59—22no—
101C22⋊Dic59—22no—
102C22⋊Dic59—42no—
103C4⋊Dic59—21no—
104C4⋊Dic59—21no—
105C4⋊Dic59C22⋊C412yes—
106C4⋊Dic59C22⋊C412yes—
107Dic59⋊C4—44no—
108C4×Dic59—22no—
109C4×Dic59—22no—
110C4×Dic59—42no—
111C4×Dic59—82no—
112#11745664—42no—
113#11745664—42no—
114C2×C59⋊C8—22no—
115C2×C59⋊C8—41no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59C4×D411yescharacteristic
121C23×C59#12011yescharacteristic
122C23×C59#12111yescharacteristic
123Q8×C59—42no—
124Q8×C59—81no—
125C2×C4×C59—22no—
126C2×C4×C59—22no—
127C2×C4×C59—22no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—42no—
132C2×C4×C59—42no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—42no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59—81no—
140C2×C4×C59—81no—
141C2×C4×C59C2×C22⋊C411yescharacteristic
142C2×C4×C59C4×D411yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12311yescharacteristic
145C2×Dic59—22no—
146C2×Dic59—22no—
147C2×Dic59—22no—
148C2×Dic59—41no—
149C2×Dic59—42no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C59⋊C8—42no—
157C59⋊C8—81no—
158#9362—591nomaximal, Sylow
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59#38611yescharacteristic
170C22×C59#45511yescharacteristic
171C22×C59#45611yescharacteristic
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—82no—
176C4×C59—82no—
177C4×C59—81no—
178C4×C59—81no—
179C4×C59#41012yes—
180Dic59—42no—
181Dic59—42no—
182Dic59—42no—
183Dic59—81no—
184#3095—591no—
185#2347—591no—
186#1996D5911yescharacteristic
187#1826—591no—
188#1508—591no—
189#1305—591no—
190#1297—591no—
191C2×C59—21no—
192C2×C59—21no—
193C2×C59—41no—
194C2×C59—41no—
195C2×C59#188211yescharacteristic
196C2×C59#189911yescharacteristic
197C2×C59#190611yescharacteristic
198C2×C22⋊Q8C2×D5911yescharacteristic
199C2×#117—591no—
200#485—1182no—
201#484—1181no—
202#483—1181no—
203#428—592no—
204#419—592no—
205#407—591no—
206#407—591no—
207#389—1181no—
208#388—1182no—
209#383—1181no—
210#380C2×D5911yescharacteristic
211C4×C4⋊C4—1181no—
212C4×C4⋊C4—1181no—
213C4×C22⋊C4—591no—
214C4×C22⋊C4C2×D5911yescharacteristic
215#342—22no—
216#339—1182no—
217#328—1181no—
218#328—1181no—
219C59#936211yescharacteristic, complemented, Sylow
220C22×Q8C4×D5911yescharacteristic
221C23×C4C22×D5911yescharacteristic
222C2×M16—592no—
223C2×M16—1181no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228#117—592no—
229#117—592no—
230#117—1181no—
231#117—1181no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—591no—
235C2×C4⋊C4—591no—
236C2×C4⋊C4—1182no—
237C2×C4⋊C4C4×D5911yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1181no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4C22⋊D5912yes—
243C2×C42—21no—
244C2×C42—1181no—
245C2×C42—1182no—
246C2×C42—1181no—
247#107—1182no—
248#106—1182no—
249#103—1182no—
250#103—1182no—
251C22⋊C8—1181no—
252C22⋊C8—1181no—
253C22⋊C8—1181no—
254C22⋊C8—1181no—
255#95—21no—
256#95—21no—
257#95—591no—
258#95—591no—
259#95—1182no—
260#95—1182no—
261C24C2×C22⋊D5911yescharacteristic
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—1181no—
271C22×C4—1181no—
272C22×C4—1181no—
273C22×C4—1181no—
274C22×C4C2×C4×D5911yescharacteristic
275C22×C4#1174568511yescharacteristic
276C22×C4D4×D5911yescharacteristic
277C22×C4D4×D5911yescharacteristic
278C22×C4D4×D5911yescharacteristic
279C22×C4#1174568711yescharacteristic
280M16—2362no—
281M16—2362no—
282C2×C8—1182no—
283C2×C8—2361no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—592no—
290C4⋊C4—592no—
291C4⋊C4—1181no—
292C4⋊C4—1181no—
293C4⋊C4—2364no—
294C22⋊C4—22no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—1182no—
300C22⋊C4—1182no—
301C22⋊C4—2362no—
302C42—42no—
303C42—1182no—
304C42—1182no—
305C42—4722no—
306C42—2362no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23C4×C22⋊D5911yescharacteristic
312C23#1427369511yescharacteristic
313C23#1427369811yescharacteristic
314Q8—42no—
315Q8—81no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—1182no—
333C2×C4—1182no—
334C2×C4—1182no—
335C2×C4—2361no—
336C2×C4—2361no—
337C2×C4—2361no—
338C2×C4—2361no—
339C2×C4—2362no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4C22⋊C4×D5911yescharacteristic
344C2×C4(C22⋊D59)⋊C411yescharacteristic
345C2×C4#1427370011yescharacteristic
346C2×C4#1427370911yescharacteristic
347C8—4721no—
348C8—2362no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1683727811yescharacteristic
360C22#1683757911yescharacteristic
361C22#1683758411yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—82no—
367C4—81no—
368C4—81no—
369C4—4721no—
370C4—2362no—
371C4—2362no—
372C4—2362no—
373C4#1683737412yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2142446811yescharacteristic
379C2#2142455111yescharacteristic
380C2#2142459411yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1996C2
socleC22×C59#386
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#1996D59
hypocenterC59#9362
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
12. #16837442C4semidirect
13. #16837411C4semidirect
219. C59#9362semidirect

Maximal subgroups

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

subgroupindex
#214340402
#214269912
#214241962
#214226162
#214220022
#214219822
C59×#19962
#936259