Finite Group Databasebeta

← #28484690

Subgroups — #28484690

Order 15104.

504 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 127 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#21432660C211yesmaximal, characteristic
2#21429917C211yesmaximal, characteristic
3#21422776C212yesmaximal, complemented
4#21421955C212yesmaximal, complemented
5C59×#1458C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273727C2211yescharacteristic
8#16837429—24no—
9#16837426—21no—
10#16837426C2212yes—
11#16837341—21no—
12#16837338—21no—
13#16837322C412yes—
14#16837314C412yes—
15#16837300—21no—
16#16837268C2211yescharacteristic
17#16837262—21no—
18#16837150—24no—
19C2×#117×C59C2211yescharacteristic
20C59×#431C2212yes—
21C59×#324—21no—
22C59×#323—21no—
23C22×Dic118C2×C412yes—
24C2×#11745664—22no—
25C2×#11745664—22no—
26C2×#11745664—42no—
27C2×#11745664—42no—
28C22×C59⋊C8—22no—
29#14273727—22no—
30#14273727—22no—
31C2×C4⋊Dic59—21no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×Dic59⋊C4—22no—
35C2×C22⋊Dic59—22no—
36C4⋊(C59⋊C8)—24no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)D414yes—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C59⋊#95—21no—
42C59⋊#95C2×C412yes—
43C23×C4×C59D411yescharacteristic
44#117×C59—22no—
45#117×C59—22no—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C22⋊C4×C59C2×C411yescharacteristic
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C2×C42×C59C2311yescharacteristic
52C4⋊C8×C59—22no—
53C22⋊C8×C59—22no—
54C4×C8×C59—22no—
55C24×C59C22⋊C411yescharacteristic
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59SD1612yes—
64C22×C4×C59Q1612yes—
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×Dic59—22no—
69C22×Dic59—22no—
70C2×Dic118—24no—
71C2×Dic118—82no—
72C2×C8×C59—42no—
73C4⋊C4×C59—22no—
74C4⋊C4×C59—22no—
75C22⋊C4×C59—22no—
76C22⋊C4×C59—22no—
77C42×C59—21no—
78C42×C59—21no—
79C42×C59C2×D412yes—
80C42×C59#4112yes—
81C22⋊Dic59—82no—
82C4⋊Dic59—41no—
83C4⋊Dic59—42no—
84C4⋊Dic59—41no—
85C4⋊Dic59—41no—
86C4⋊Dic59—41no—
87Dic59⋊C4—82no—
88#11745664—44no—
89#11745664—42no—
90#11745664—42no—
91C2×C59⋊C8—24no—
92C2×C59⋊C8—44no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59#10312yes—
98C23×C59C2×C22⋊C411yescharacteristic
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59#10312yes—
112C2×C4×C59C4×D412yes—
113C2×C4×C59#12011yescharacteristic
114C2×C4×C59#12112yes—
115C2×C4×C59#12311yescharacteristic
116C2×C4×C59C2×SD1611yescharacteristic
117C2×C4×C59C2×Q1611yescharacteristic
118C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C59#13711yescharacteristic
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126Dic118—84no—
127C8×C59—82no—
128C59⋊C8—44no—
129#9191—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—42no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—81no—
137C22×C59#38611yescharacteristic
138C22×C59C2×#10311yescharacteristic
139C22×C59#41711yescharacteristic
140C4×C59—22no—
141C4×C59—22no—
142C4×C59—22no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59#45011yescharacteristic
146C4×C59#45111yescharacteristic
147C4×C59#48311yescharacteristic
148C4×C59#48411yescharacteristic
149Dic59—82no—
150Dic59—82no—
151#2905—591no—
152#2628—591no—
153#1548—592no—
154#1458D5911yescharacteristic
155#1286—592no—
156C2×C59—21no—
157C2×C59—21no—
158C2×C59—42no—
159C2×C59#187211yescharacteristic
160C2×C59#190911yescharacteristic
161C2×C59#191411yescharacteristic
162C22×M16—1181no—
163C2×C4⋊Q8—591no—
164C2×#117C2×D5911yescharacteristic
165#431C2×D5912yes—
166#424—1184no—
167#423—592no—
168#423—1181no—
169#400—1181no—
170#399—1181no—
171#395—592no—
172#393—592no—
173#389—1181no—
174#382—591no—
175#380—1181no—
176#333—1184no—
177#324—21no—
178#323—21no—
179C59#919111yescharacteristic, complemented, Sylow
180C22×Q8—592no—
181C23×C4D23611yescharacteristic
182C2×M16—1182no—
183C2×M16—1182no—
184C2×M16—2362no—
185C2×M16—2362no—
186C22×C8—1182no—
187C4⋊Q8—1182no—
188C4⋊Q8—1182no—
189#117—22no—
190#117—22no—
191C2×C4⋊C4—1182no—
192C2×C4⋊C4—1181no—
193C2×C4⋊C4—1181no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4C4×D5911yescharacteristic
196C2×C4⋊C4C22⋊D5911yescharacteristic
197C2×C22⋊C4—1182no—
198C2×C22⋊C4C4×D5911yescharacteristic
199C2×C22⋊C4C22⋊D5911yescharacteristic
200C2×C42D23611yescharacteristic
201C2×C42C22×D5911yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—594no—
204C4⋊C8—1184no—
205C4⋊C8—1182no—
206C22⋊C8—22no—
207C4×C8—22no—
208#95—592no—
209#95—1182no—
210#95—1182no—
211#95—1181no—
212C24(C2×D59)⋊C411yescharacteristic
213C2×Q8—1184no—
214C2×Q8—4722no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—1182no—
220C22×C4—1182no—
221C22×C4#1174566012yes—
222C22×C4Dic23612yes—
223C22×C4(C2×D59)⋊C411yescharacteristic
224C22×C4(C2×D59)⋊C411yescharacteristic
225C22×C4(C2×D59)⋊C411yescharacteristic
226C22×C4C2×C4×D5911yescharacteristic
227C22×C4C2×D23611yescharacteristic
228C22×C4C2×C22⋊D5911yescharacteristic
229M16—2364no—
230M16—2362no—
231M16—2362no—
232C2×C8—42no—
233C2×C8—1184no—
234C2×C8—2364no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—4722no—
238C4⋊C4—2361no—
239C4⋊C4—2362no—
240C4⋊C4—2361no—
241C4⋊C4—2361no—
242C4⋊C4—2361no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—4722no—
246C42—21no—
247C42—21no—
248C42D4×D5912yes—
249C42#1174568712yes—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23#1427365212yes—
255C23C2×(C2×D59)⋊C411yescharacteristic
256Q8—4724no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—2362no—
270C2×C4—2362no—
271C2×C4—2362no—
272C2×C4—2362no—
273C2×C4—2362no—
274C2×C4—2362no—
275C2×C4#1427365212yes—
276C2×C4(C22⋊D59)⋊C412yes—
277C2×C4#1427369711yescharacteristic
278C2×C4#1427369912yes—
279C2×C4#1427371111yescharacteristic
280C2×C4C2×#1174566011yescharacteristic
281C2×C4C2×Dic23611yescharacteristic
282C2×C4#1427375511yescharacteristic
283C2×C4#1427376011yescharacteristic
284C8—82no—
285C8—2364no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22#1683727911yescharacteristic
294C22C2×#1427365211yescharacteristic
295C22#1683740411yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—4722no—
302C4—4722no—
303C4#1683755511yescharacteristic
304C4#1683756011yescharacteristic
305C4#1683773311yescharacteristic
306C4#1683773811yescharacteristic
307C2—21no—
308C2—21no—
309C2—42no—
310C2#2142440111yescharacteristic
311C2#2142461611yescharacteristic
312C2#2142464111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1458C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1458D59
hypocenterC59#9191
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #21422776C2semidirect
4. #21421955C2semidirect
179. C59#9191semidirect

Maximal subgroups

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

subgroupindex
#214326602
#214299172
#214227762
#214219552
C59×#14582
#919159