Finite Group Databasebeta

← #28485350

Subgroups — #28485350

Order 15104.

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

normal subgroups only

This group has 422 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#21435044C211yesmaximal, characteristic
2#21433033C211yesmaximal, characteristic
3#21425070C211yesmaximal, characteristic
4#21422661C211yesmaximal, characteristic
5#21421956C211yesmaximal, characteristic
6#21421926C211yesmaximal, characteristic
7C59×#1549C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837781C412yescomplemented
10#16837752C412yescomplemented
11#16837609—22no—
12#16837558—22no—
13#16837531—22no—
14#16837531—22no—
15#16837455C2211yescharacteristic
16#16837430—21no—
17#16837360—21no—
18C2×#14273638C2211yescharacteristic
19#16837268C2211yescharacteristic
20#16837164—21no—
21#16837164—21no—
22#16837140—22no—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C59×#332—22no—
27C2×#11745672—24no—
28C22×C59⋊C8D411yescharacteristic
29C22×C59⋊C8D411yescharacteristic
30#14273727C2×C411yescharacteristic
31#14273727C2×C411yescharacteristic
32#14273719—21no—
33#14273706—21no—
34#14273706D412yes—
35C4×Dic118—21no—
36C4×Dic118D412yes—
37#14273686—21no—
38C2×C4⋊Dic59—21no—
39C4⋊(C59⋊C8)—21no—
40Dic59⋊C8—24no—
41#14273651—24no—
42C22⋊(C59⋊C8)—21no—
43#14273638—21no—
44#14273638—21no—
45#14273637—22no—
46C4×C59⋊C8C2×C412yes—
47C8×Dic59—22no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C22×Q8×C59—21no—
51C2×M16×C59—21no—
52C4⋊Q8×C59—21no—
53C4⋊Q8×C59—21no—
54C4⋊Q8×C59—21no—
55C4⋊Q8×C59C2×C412yes—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—21no—
60C4⋊C8×C59—21no—
61C4⋊C8×C59D412yes—
62C22⋊C8×C59—21no—
63C4×C8×C59—21no—
64C2×Q8×C59—22no—
65C2×Q8×C59—41no—
66C2×Q8×C59—82no—
67C22×C4×C59—41no—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59#4111yescharacteristic
71C22×Dic59—41no—
72C2×Dic118—21no—
73C2×Dic118C22⋊C412yes—
74M16×C59—41no—
75M16×C59—41no—
76C2×C8×C59—22no—
77C2×C8×C59—41no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C42×C59C22×C411yescharacteristic
83C42×C59C2×D411yescharacteristic
84C42×C59C2×D411yescharacteristic
85C42×C59#4111yescharacteristic
86C22⋊Dic59—22no—
87#11745672—84no—
88C4⋊Dic59—21no—
89C4⋊Dic59—21no—
90C4⋊Dic59—21no—
91C4⋊Dic59—41no—
92C4⋊Dic59—41no—
93C4⋊Dic59C22⋊C412yes—
94Dic59⋊C4—22no—
95Dic59⋊C4—22no—
96C4×Dic59—22no—
97C2×C59⋊C8—21no—
98C2×C59⋊C8—21no—
99C2×C59⋊C8—22no—
100C2×C59⋊C8—21no—
101C2×C59⋊C8—21no—
102C2×C59⋊C8—21no—
103C2×C59⋊C8Q1612yescomplemented
104C23×C59#12111yescharacteristic
105Q8×C59—44no—
106Q8×C59—82no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59—82no—
111C2×C4×C59C2×C22⋊C411yescharacteristic
112C2×C4×C59C4×D411yescharacteristic
113C2×C4×C59C4×D411yescharacteristic
114C2×C4×C59#12011yescharacteristic
115C2×C4×C59#12111yescharacteristic
116C2×C4×C59#12311yescharacteristic
117C2×C4×C59C2×Q1611yescharacteristic
118C2×C4×C59#13511yescharacteristic
119C2×C4×C59#13711yescharacteristic
120C2×C4×C59#13711yescharacteristic
121C2×Dic59—22no—
122C2×Dic59—22no—
123C2×Dic59—81no—
124C2×Dic59—81no—
125C2×Dic59—81no—
126Dic118—42no—
127Dic118—81no—
128C8×C59—42no—
129C8×C59—81no—
130C59⋊C8—22no—
131C59⋊C8—42no—
132C59⋊C8—41no—
133#9290—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59#38611yescharacteristic
137C22×C59#46211yescharacteristic
138C22×C59#46411yescharacteristic
139C4×C59—21no—
140C4×C59—21no—
141C4×C59—21no—
142C4×C59—21no—
143C4×C59—82no—
144C4×C59C4×Q1611yescharacteristic
145C4×C59#44111yescharacteristic
146C4×C59#45111yescharacteristic
147C4×C59#45211yescharacteristic
148Dic59—42no—
149Dic59—42no—
150Dic59—161no—
151#3209—591no—
152#2961—591no—
153#1993—591no—
154#1549D5911yescharacteristic
155#1518—591no—
156#1286—591no—
157#1275—591no—
158C2×C59—21no—
159C2×C59—21no—
160C2×C59#187811yescharacteristic
161C2×C59#190211yescharacteristic
162C2×C59#191011yescharacteristic
163#533—591no—
164C2×C4⋊Q8C2×D5911yescharacteristic
165#494—592no—
166#487—592no—
167#462—1182no—
168#451—1182no—
169#446—1182no—
170#446—1182no—
171#431—591no—
172#431C2×D5911yescharacteristic
173#424—1181no—
174#423C2×D5911yescharacteristic
175#405—1181no—
176C2×#97—591no—
177#382—591no—
178#340—1181no—
179#340—1181no—
180#332—22no—
181#328—1182no—
182C59#929011yescharacteristic, complemented, Sylow
183C22×Q8—21no—
184C2×Q16—1184no—
185C2×M16—21no—
186C22×C8—591no—
187C22×C8—591no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—591no—
192C4⋊Q8—591no—
193C4⋊Q8C4×D5912yes—
194#125—1181no—
195C22⋊Q8—592no—
196C22⋊Q8—1181no—
197C4×Q8—592no—
198C4×Q8—1181no—
199#117—1181no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—1181no—
203C2×C42C22×D5911yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—1184no—
207C4⋊C8—1181no—
208C4⋊C8C22⋊D5912yes—
209#103—1184no—
210C22⋊C8—21no—
211C22⋊C8—1181no—
212#97—1182no—
213#97—1181no—
214#97—1181no—
215C4×C8—21no—
216C4×C8—592no—
217C4×C8—1182no—
218#95—1181no—
219#95—1181no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—592no—
224C2×Q8—1181no—
225C22×C4—41no—
226C22×C4—2361no—
227C22×C4#1174568511yescharacteristic
228C22×C4D4×D5911yescharacteristic
229C22×C4D4×D5911yescharacteristic
230Q16—4724no—
231M16—41no—
232M16—41no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—592no—
236C2×C8—1181no—
237C2×C8—1181no—
238C2×C8—1182no—
239C2×C8—1181no—
240C2×C8—1181no—
241C2×C8—1181no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—592no—
247C4⋊C4—1181no—
248C4⋊C4—1181no—
249C4⋊C4—1181no—
250C4⋊C4—1182no—
251C4⋊C4—1182no—
252C4⋊C4—2361no—
253C4⋊C4—2361no—
254C22⋊C4—1182no—
255C42—1182no—
256C42C2×C4×D5911yescharacteristic
257C42D4×D5911yescharacteristic
258C42#1174568711yescharacteristic
259C42C2×C22⋊D5911yescharacteristic
260C23#1427370011yescharacteristic
261Q8—44no—
262Q8—82no—
263Q8—4721no—
264Q8—2362no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—1182no—
270C2×C4—1182no—
271C2×C4—4721no—
272C2×C4—4721no—
273C2×C4—4721no—
274C2×C4C22⋊C4×D5911yescharacteristic
275C2×C4(C22⋊D59)⋊C411yescharacteristic
276C2×C4C4×C22⋊D5911yescharacteristic
277C2×C4#1427369511yescharacteristic
278C2×C4#1427369811yescharacteristic
279C2×C4#1427370911yescharacteristic
280C2×C4Q16×D5911yescharacteristic
281C2×C4#1427374811yescharacteristic
282C2×C4#1427375611yescharacteristic
283C2×C4#1427375911yescharacteristic
284C8—42no—
285C8—81no—
286C8—1182no—
287C8—2362no—
288C8—2361no—
289C22—21no—
290C22—21no—
291C22#1683727811yescharacteristic
292C22#1683761111yescharacteristic
293C22#1683762711yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—9441no—
300C4—2362no—
301C4—2362no—
302C4#1683749411yescharacteristic
303C4#1683750611yescharacteristic
304C4#1683755611yescharacteristic
305C4#1683756111yescharacteristic
306C2—21no—
307C2—21no—
308C2#2142443411yescharacteristic
309C2#2142457211yescharacteristic
310C2#2142462011yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1549C2
socleC22×C59#386
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#1549D59
hypocenterC59#9290
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
9. #16837781C4semidirect
10. #16837752C4semidirect
103. C2×C59⋊C8Q16semidirect
182. C59#9290semidirect

Maximal subgroups

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

subgroupindex
#214350442
#214330332
#214250702
#214226612
#214219562
#214219262
C59×#15492
#929059