Finite Group Databasebeta

← #28485352

Subgroups — #28485352

Order 15104.

422 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 109 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#21435046C211yesmaximal, characteristic
2#21433036C211yesmaximal, characteristic
3#21425069C211yesmaximal, characteristic
4#21422780C211yesmaximal, characteristic
5#21422661C211yesmaximal, characteristic
6#21421955C211yesmaximal, characteristic
7C59×#1275C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9C2×#14273727C2211yescharacteristic
10#16837782C412yescomplemented
11#16837753C412yescomplemented
12#16837613—22no—
13#16837560—22no—
14#16837530—22no—
15#16837530—22no—
16#16837428—21no—
17#16837426C2211yescharacteristic
18#16837357—21no—
19#16837268C2211yescharacteristic
20#16837165—21no—
21#16837165—21no—
22#16837149—22no—
23#16837140—22no—
24C59×#431C2211yescharacteristic
25C59×#431C2211yescharacteristic
26C2×#97×C59C2211yescharacteristic
27C22×Dic118—21no—
28C2×Dic236—24no—
29C2×#11745664—21no—
30#14273727—21no—
31#14273727—21no—
32#14273727—21no—
33#14273727C2×C412yes—
34#14273727C2×C411yescharacteristic
35#14273727C2×C411yescharacteristic
36#14273719—21no—
37#14273706—21no—
38#14273706D412yes—
39C4×Dic118—21no—
40C4×Dic118D412yes—
41#14273686—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—21no—
44C2×C4⋊Dic59—21no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)D412yes—
47Dic59⋊C8—24no—
48#14273652—24no—
49#14273637—22no—
50C8×Dic59—22no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C22×C8×C59D411yescharacteristic
54C22×C8×C59D411yescharacteristic
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—21no—
57C4⋊C8×C59—21no—
58C22⋊C8×C59—21no—
59C22⋊C8×C59—21no—
60#97×C59—21no—
61#97×C59—21no—
62C4×C8×C59—21no—
63C4×C8×C59C2×C412yes—
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59#4111yescharacteristic
67C22×Dic59—41no—
68C22×Dic59—41no—
69C2×Dic118—22no—
70C2×Dic118—21no—
71C2×Dic118—41no—
72C2×Dic118—82no—
73C2×Dic118C22⋊C412yes—
74C2×C8×C59—21no—
75C2×C8×C59—21no—
76C2×C8×C59—22no—
77C2×C8×C59—21no—
78C2×C8×C59—21no—
79C2×C8×C59—21no—
80C2×C8×C59—41no—
81C2×C8×C59Q1612yescomplemented
82C42×C59C22×C411yescharacteristic
83C42×C59C2×D411yescharacteristic
84C42×C59C2×D411yescharacteristic
85C42×C59#4111yescharacteristic
86C22⋊Dic59—22no—
87C4⋊Dic59—21no—
88C4⋊Dic59—21no—
89C4⋊Dic59—21no—
90C4⋊Dic59—22no—
91C4⋊Dic59—22no—
92C4⋊Dic59—41no—
93C4⋊Dic59—41no—
94C4⋊Dic59—41no—
95C4⋊Dic59—41no—
96C4⋊Dic59C22⋊C412yes—
97Dic59⋊C4—22no—
98Dic59⋊C4—22no—
99C4×Dic59—22no—
100#11745664—41no—
101#11745664—41no—
102C2×C59⋊C8—22no—
103Dic236—84no—
104C23×C59#12111yescharacteristic
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59C2×C22⋊C411yescharacteristic
108C2×C4×C59C4×D411yescharacteristic
109C2×C4×C59C4×D411yescharacteristic
110C2×C4×C59#12011yescharacteristic
111C2×C4×C59#12111yescharacteristic
112C2×C4×C59#12311yescharacteristic
113C2×C4×C59C2×Q1611yescharacteristic
114C2×C4×C59#13511yescharacteristic
115C2×C4×C59#13711yescharacteristic
116C2×C4×C59#13711yescharacteristic
117C2×Dic59—22no—
118C2×Dic59—22no—
119C2×Dic59—42no—
120C2×Dic59—82no—
121C2×Dic59—81no—
122C2×Dic59—81no—
123C2×Dic59—81no—
124Dic118—42no—
125Dic118—44no—
126Dic118—81no—
127Dic118—82no—
128C8×C59—22no—
129C8×C59—42no—
130C8×C59—41no—
131C8×C59—81no—
132C59⋊C8—42no—
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×C59C4×Q1611yescharacteristic
144C4×C59#44111yescharacteristic
145C4×C59#45111yescharacteristic
146C4×C59#45211yescharacteristic
147Dic59—42no—
148Dic59—42no—
149Dic59—82no—
150Dic59—161no—
151#3209—591no—
152#2961—591no—
153#1993—591no—
154#1549—591no—
155#1518—591no—
156#1286—591no—
157#1275D5911yescharacteristic
158C2×C59—21no—
159C2×C59—21no—
160C2×C59#187811yescharacteristic
161C2×C59#190211yescharacteristic
162C2×C59#191011yescharacteristic
163#533—591no—
164C2×C4⋊Q8—591no—
165#494—592no—
166#487—592no—
167#462—1182no—
168#451—1182no—
169#446—1182no—
170#446—1182no—
171#431C2×D5911yescharacteristic
172#431C2×D5911yescharacteristic
173#424—1181no—
174#423—591no—
175#405—1181no—
176C2×#97C2×D5911yescharacteristic
177#382—591no—
178#340—1181no—
179#340—1181no—
180#332—1182no—
181#328—1182no—
182C59#929011yescharacteristic, complemented, Sylow
183C22×Q8—1181no—
184C2×Q16—1184no—
185C2×M16—1181no—
186C22×C8D23611yescharacteristic
187C22×C8D23611yescharacteristic
188C4⋊Q8—591no—
189C4⋊Q8—591no—
190C4⋊Q8—592no—
191C4⋊Q8—1181no—
192C4⋊Q8—1181no—
193C4⋊Q8—1181no—
194#125—1181no—
195C22⋊Q8—592no—
196C22⋊Q8—1181no—
197C4×Q8—592no—
198C4×Q8—1181no—
199#117—1181no—
200C2×C4⋊C4—1181no—
201C2×C4⋊C4—1181no—
202C2×C4⋊C4—1181no—
203C2×C42C22×D5911yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—592no—
207C4⋊C8—1184no—
208C4⋊C8—1181no—
209#103—1184no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212#97—21no—
213#97—21no—
214#97—1182no—
215C4×C8—21no—
216C4×C8—1182no—
217C4×C8C4×D5912yes—
218#95—1181no—
219#95—1181no—
220C2×Q8—592no—
221C2×Q8—1181no—
222C2×Q8—1182no—
223C2×Q8—4722no—
224C2×Q8—2361no—
225C22×C4—2361no—
226C22×C4—2361no—
227C22×C4C2×D23611yescharacteristic
228C22×C4D4×D5911yescharacteristic
229C22×C4#1174568911yescharacteristic
230Q16—4724no—
231M16—2361no—
232M16—2361no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—22no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—41no—
240C2×C8—1182no—
241C2×C8Dic23612yescomplemented
242C4⋊C4—592no—
243C4⋊C4—1181no—
244C4⋊C4—1181no—
245C4⋊C4—1181no—
246C4⋊C4—1182no—
247C4⋊C4—1182no—
248C4⋊C4—1182no—
249C4⋊C4—1182no—
250C4⋊C4—2361no—
251C4⋊C4—2361no—
252C4⋊C4—2361no—
253C4⋊C4—2361no—
254C22⋊C4—1182no—
255C42—1182no—
256C42C2×C4×D5911yescharacteristic
257C42#1174568511yescharacteristic
258C42D4×D5911yescharacteristic
259C42D4×D5911yescharacteristic
260C23#1427370211yescharacteristic
261Q8—4722no—
262Q8—4721no—
263Q8—2364no—
264Q8—2362no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—1182no—
268C2×C4—1182no—
269C2×C4—4721no—
270C2×C4—4721no—
271C2×C4—4721no—
272C2×C4—4722no—
273C2×C4—2362no—
274C2×C4C22⋊C4×D5911yescharacteristic
275C2×C4#1427368911yescharacteristic
276C2×C4C4×D23611yescharacteristic
277C2×C4#1427369711yescharacteristic
278C2×C4#1427370011yescharacteristic
279C2×C4#1427371211yescharacteristic
280C2×C4C2×Dic23611yescharacteristic
281C2×C4#1427374911yescharacteristic
282C2×C4#1427376011yescharacteristic
283C2×C4#1427376011yescharacteristic
284C8—22no—
285C8—42no—
286C8—41no—
287C8—81no—
288C8—2362no—
289C22—21no—
290C22—21no—
291C22#1683728011yescharacteristic
292C22#1683761311yescharacteristic
293C22#1683762511yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—9441no—
299C4—4722no—
300C4—2362no—
301C4—2362no—
302C4C4×Dic23611yescharacteristic
303C4#1683751011yescharacteristic
304C4#1683756011yescharacteristic
305C4#1683756611yescharacteristic
306C2—21no—
307C2—21no—
308C2#2142443611yescharacteristic
309C2#2142457011yescharacteristic
310C2#2142462211yescharacteristic
subgroupG/H
centerC22#16837280
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1275C2
socleC22×C59#386
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#1275D59
hypocenterC59#9290
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
10. #16837782C4semidirect
11. #16837753C4semidirect
81. C2×C8×C59Q16semidirect
182. C59#9290semidirect
241. C2×C8Dic236semidirect

Maximal subgroups

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

subgroupindex
#214350462
#214330362
#214250692
#214227802
#214226612
#214219552
C59×#12752
#929059