Finite Group Databasebeta

← #28485617

Subgroups — #28485617

Order 15104.

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

normal subgroups only

This group has 444 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#21434226C211yesmaximal, characteristic
2#21426747C211yesmaximal, characteristic
3#21425040C211yesmaximal, characteristic
4#21422609C211yesmaximal, characteristic
5#21421990C211yesmaximal, characteristic
6#21421951C211yesmaximal, characteristic
7C59×#1491C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837732—21no—
10#16837687C412yescomplemented
11#16837612C412yescomplemented
12#16837558—21no—
13#16837507—22no—
14#16837459C2211yescharacteristic
15#16837309—21no—
16#16837301—21no—
17#16837288—22no—
18#16837270—21no—
19#16837268—21no—
20C4×C4⋊Dic59C2211yescharacteristic
21#16837248C2211yescharacteristic
22#16837163—22no—
23#16837140—21no—
24#16837140—21no—
25C2×C4×Q8×C59C2211yescharacteristic
26C59×#432C2211yescharacteristic
27C4×M16×C59C2211yescharacteristic
28C59×Q8⋊C8—22no—
29C2×#11745672—22no—
30#14273727C2×C411yescharacteristic
31#14273727C2×C411yescharacteristic
32#14273719—21no—
33#14273706—21no—
34#14273706—21no—
35C4×Dic118—21no—
36C4×Dic118—21no—
37#14273686—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59—22no—
40C2×C4×Dic59—21no—
41C2×C4×Dic59D412yes—
42#14273658—22no—
43C4⋊(C59⋊C8)C2×C412yes—
44#14273651—22no—
45#14273650—22no—
46C22⋊(C59⋊C8)—21no—
47#14273638—21no—
48C59⋊#95—22no—
49C59⋊#95—22no—
50C59⋊#95—22no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C59⋊#95D411yescharacteristic
54C59⋊#95D411yescharacteristic
55C22×Q8×C59D411yescharacteristic
56C2×M16×C59—21no—
57C4×Q8×C59—21no—
58C4×Q8×C59—21no—
59C4×Q8×C59C2×C412yes—
60C4×Q8×C59D412yes—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C42×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C22⋊C8×C59—21no—
67#97×C59—21no—
68#97×C59—21no—
69C4×C8×C59—21no—
70C2×Q8×C59—21no—
71C2×Q8×C59—42no—
72C2×Q8×C59C22⋊C412yes—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×Dic59—21no—
79C22×Dic59—22no—
80C22×Dic59—21no—
81C2×Dic118—21no—
82C2×Dic118—21no—
83M16×C59—22no—
84M16×C59—41no—
85C2×C8×C59—41no—
86C2×C8×C59—41no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—22no—
89C4⋊C4×C59—21no—
90C4⋊C4×C59C22⋊C412yes—
91C42×C59—22no—
92C42×C59—22no—
93C42×C59C22×C411yescharacteristic
94C42×C59C2×D411yescharacteristic
95C42×C59#4111yescharacteristic
96C42×C59#4111yescharacteristic
97C22⋊Dic59—41no—
98#11745672—44no—
99C4⋊Dic59—21no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102C4⋊Dic59—21no—
103C4⋊Dic59—42no—
104C4⋊Dic59—82no—
105Dic59⋊C4—41no—
106Dic59⋊C4—41no—
107C4×Dic59—42no—
108C4×Dic59—41no—
109C4×Dic59—42no—
110C2×C59⋊C8—22no—
111C23×C59#12011yescharacteristic
112Q8×C59—22no—
113Q8×C59—22no—
114Q8×C59—42no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59C2×C22⋊C411yescharacteristic
125C2×C4×C59C4×D411yescharacteristic
126C2×C4×C59C4×D411yescharacteristic
127C2×C4×C59#12111yescharacteristic
128C2×C4×C59#12111yescharacteristic
129C2×C4×C59#12311yescharacteristic
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—42no—
133C2×Dic59—41no—
134C2×Dic59—42no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—42no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141Dic118—42no—
142Dic118—81no—
143C8×C59—42no—
144C8×C59—81no—
145C59⋊C8—42no—
146#9328—591nomaximal, Sylow
147C22×C59—21no—
148C22×C59—21no—
149C22×C59#38611yescharacteristic
150C22×C59#45511yescharacteristic
151C22×C59#45611yescharacteristic
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—42no—
158C4×C59—42no—
159C4×C59#42112yes—
160Dic59—81no—
161Dic59—82no—
162Dic59—81no—
163Dic59—81no—
164Dic59—81no—
165#3113—591no—
166#2320—591no—
167#1986—591no—
168#1506—591no—
169#1491D5911yescharacteristic
170#1300—591no—
171#1285—591no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59#188211yescharacteristic
175C2×C59#189711yescharacteristic
176C2×C59#190511yescharacteristic
177#533—591no—
178C2×C4×Q8C2×D5911yescharacteristic
179#483—1181no—
180#475—592no—
181#462—592no—
182#451—1181no—
183#441—1182no—
184#432—591no—
185#432C2×D5911yescharacteristic
186C4×M16C2×D5911yescharacteristic
187#391—1181no—
188#389—1181no—
189#387—1182no—
190#383—1181no—
191#382—1181no—
192C4×C4⋊C4—591no—
193#376—591no—
194#339—1182no—
195#328—1181no—
196#328—1181no—
197Q8⋊C8—22no—
198C59#932811yescharacteristic, complemented, Sylow
199C22×Q8D23611yescharacteristic
200C2×Q16—1182no—
201C2×M16—21no—
202C4⋊Q8—591no—
203C4⋊Q8—591no—
204#125—1181no—
205C22⋊Q8—1181no—
206C22⋊Q8—1181no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—1181no—
210C4×Q8—1181no—
211C4×Q8C4×D5912yes—
212C4×Q8C22⋊D5912yes—
213#117—1181no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1182no—
216C2×C4⋊C4—1181no—
217C2×C4⋊C4D23611yescharacteristic
218C2×C42—21no—
219C2×C42—592no—
220C2×C42—1181no—
221C2×C42C22×D5911yescharacteristic
222#106—1182no—
223C4⋊C8—21no—
224C4⋊C8—592no—
225#103—1182no—
226#103—1182no—
227C22⋊C8—21no—
228C22⋊C8—1181no—
229#97—21no—
230#97—21no—
231#97—1181no—
232C4×C8—21no—
233#95—591no—
234#95—591no—
235#95—1182no—
236#95—1182no—
237#95—1182no—
238#95—1181no—
239#95—1181no—
240C2×Q8—21no—
241C2×Q8—42no—
242C2×Q8—1181no—
243C2×Q8—1181no—
244C2×Q8(C2×D59)⋊C412yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4—1182no—
250C22×C4C2×D23611yescharacteristic
251C22×C4D4×D5911yescharacteristic
252C22×C4D4×D5911yescharacteristic
253Q16—2364no—
254M16—22no—
255M16—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—1182no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—1181no—
263C4⋊C4—1181no—
264C4⋊C4—1181no—
265C4⋊C4—1181no—
266C4⋊C4—4722no—
267C4⋊C4—2362no—
268C4⋊C4—2361no—
269C4⋊C4—2361no—
270C4⋊C4(C2×D59)⋊C412yes—
271C22⋊C4—2361no—
272C42—22no—
273C42—22no—
274C42—2362no—
275C42—2362no—
276C42—2361no—
277C42C2×C4×D5911yescharacteristic
278C42#1174568711yescharacteristic
279C42#1174568711yescharacteristic
280C42C2×C22⋊D5911yescharacteristic
281C23#1427369711yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285Q8—4721no—
286Q8—2362no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—2361no—
297C2×C4—2361no—
298C2×C4—2362no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2362no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2362no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4C2×(C2×D59)⋊C411yescharacteristic
308C2×C4(C22⋊D59)⋊C411yescharacteristic
309C2×C4(C22⋊D59)⋊C411yescharacteristic
310C2×C4#1427369911yescharacteristic
311C2×C4#1427369911yescharacteristic
312C2×C4#1427371111yescharacteristic
313C8—42no—
314C8—81no—
315C8—2362no—
316C22—21no—
317C22—21no—
318C22#1683727911yescharacteristic
319C22#1683758111yescharacteristic
320C22#1683758611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—4721no—
329C4—4722no—
330C4—4721no—
331C4—4721no—
332C4—4721no—
333C4#1683741812yes—
334C2—21no—
335C2—21no—
336C2#2142446311yescharacteristic
337C2#2142454211yescharacteristic
338C2#2142458811yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1491C2
socleC22×C59#386
prime radicalC23×C59#120
cosocleC2×C42×C59C23
hypercenter#1491D59
hypocenterC59#9328
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
10. #16837687C4semidirect
11. #16837612C4semidirect
198. C59#9328semidirect

Maximal subgroups

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

subgroupindex
#214342262
#214267472
#214250402
#214226092
#214219902
#214219512
C59×#14912
#932859