Finite Group Databasebeta

← #28498873

Subgroups — #28498873

Order 15104.

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

normal subgroups only

This group has 568 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#21429846C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic, complemented
3#21422092C214yesmaximal, complemented
4C59×#1740C211yesmaximal, characteristic, complemented
5C2×#14273724C411yescharacteristic, complemented
6C2×#14273701—21no—
7#16837455—21no—
8#16837319—21no—
9#16837319C412yescomplemented
10#16837311—21no—
11#16837311C2212yes—
12#16837304—21no—
13#16837268C411yescharacteristic, complemented
14#16837128—21no—
15#16837126C2212yes—
16C2×#117×C59—21no—
17C2×#117×C59C2211yescharacteristic
18C59×#342—21no—
19C59×#342C2212yes—
20C22×D236—21no—
21C22×D236—21no—
22C22×D236C2×C412yes—
23C22×C22⋊D59—22no—
24#14273724—41no—
25#14273724—41no—
26#14273701—42no—
27#14273701—42no—
28C2×C4⋊Dic59—21no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C4⋊(C59⋊C8)—41no—
33C22⋊(C59⋊C8)—22no—
34C4×C59⋊C8—41no—
35C59⋊#95—21no—
36C59⋊#95C2×C412yes—
37C23×C4×C59D411yescharacteristic
38#117×C59—24no—
39#117×C59—41no—
40#117×C59—41no—
41#117×C59—42no—
42C2×C4⋊C4×C59—41no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59C2311yescharacteristic
48C2×C42×C59—41no—
49C2×C42×C59C2×C411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C23×D59—22no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—22no—
56C22×C4×C59—42no—
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59D812yes—
62C22×C4×C59SD1612yes—
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C2×C22⋊D59—42no—
66C2×C22⋊D59—42no—
67C2×C22⋊D59—82no—
68C2×C22⋊D59—82no—
69C22×Dic59—22no—
70C2×D236—42no—
71C2×D236—42no—
72C2×D236—41no—
73C2×D236—41no—
74C2×D236—42no—
75C2×D236—42no—
76C2×D236—82no—
77C2×D236—82no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—44no—
81C22⋊C4×C59—24no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—42no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—21no—
89C42×C59—44no—
90(C2×D59)⋊C4—44no—
91(C2×D59)⋊C4—82no—
92(C2×D59)⋊C4—82no—
93C4⋊Dic59—42no—
94C2×C59⋊C8—42no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59C2×C22⋊C411yescharacteristic
100C23×C59C2×D811yescharacteristic
101C23×C59C2×SD1611yescharacteristic
102C22×D59—42no—
103C22×D59—42no—
104C22×D59—42no—
105C22×D59—42no—
106C22×D59—42no—
107C22×D59—42no—
108C22×D59—42no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—44no—
124C2×C4×C59—82no—
125C2×C4×C59—82no—
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59#10212yes—
129C2×C4×C59#10212yes—
130C2×C4×C59#10412yes—
131C22⋊D59—84no—
132C22⋊D59—84no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136D236—84no—
137D236—84no—
138D236—82no—
139D236—82no—
140C59⋊C8—82no—
141#11693—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—81no—
151C22×C59C2×#9911yescharacteristic
152C22×C59C2×#10211yescharacteristic
153C22×C59C2×#10411yescharacteristic
154C2×D59—82no—
155C2×D59—82no—
156C2×D59—82no—
157C2×D59—82no—
158C2×D59—82no—
159C2×D59—82no—
160C2×D59—82no—
161C4×C59—22no—
162C4×C59—41no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59—84no—
167C4×C59#32712yes—
168C4×C59#32712yes—
169Dic59—82no—
170#2619—591no—
171#1740D5911yescharacteristic, complemented
172#1458—591no—
173#1343—594no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—41no—
177C2×C59—41no—
178C2×C59C2×#32711yescharacteristic
179C2×C59#212211yescharacteristic
180C2×C59#212811yescharacteristic
181D59—162no—
182C2×#127—591no—
183C2×#121—1181no—
184C2×#117—21no—
185C2×#117C2×D5911yescharacteristic
186#431—1181no—
187#394—592no—
188#394—1181no—
189#392—592no—
190#392—1181no—
191#390—1181no—
192#382—591no—
193#342—21no—
194#342C2×D5912yes—
195#324—1181no—
196#323—592no—
197C59#1169311yescharacteristic, complemented, Sylow
198C22×D4—592no—
199C22×D4—1182no—
200C22×D4—1181no—
201C22×D4—1181no—
202C23×C4C22⋊D5911yescharacteristic
203#127—2361no—
204#127—2361no—
205#121—2362no—
206#121—2362no—
207#117—24no—
208#117—41no—
209#117—41no—
210#117—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4C4×D5911yescharacteristic
214C2×C4⋊C4D23611yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—1182no—
217C2×C22⋊C4—1182no—
218C2×C22⋊C4—1182no—
219C2×C22⋊C4C22⋊D5911yescharacteristic
220C2×C22⋊C4C22×D5911yescharacteristic
221C2×C42—41no—
222C2×C42C4×D5911yescharacteristic
223C2×C42D23611yescharacteristic
224C4⋊C8—2361no—
225C22⋊C8—1182no—
226C4×C8—2361no—
227#95—592no—
228#95—1181no—
229C24—1182no—
230C24C2×C22⋊D5911yescharacteristic
231C2×D4—4722no—
232C2×D4—4722no—
233C2×D4—4722no—
234C2×D4—4722no—
235C2×D4—2362no—
236C2×D4—2362no—
237C2×D4—2362no—
238C2×D4—2362no—
239C2×D4—2361no—
240C2×D4—2361no—
241C2×D4—2362no—
242C2×D4—2362no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—1182no—
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4#1174566912yes—
253C22×C4#1174567112yes—
254C22×C4C2×C4×D5911yescharacteristic
255C22×C4C2×D23611yescharacteristic
256C2×C8—2362no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—44no—
260C4⋊C4—2362no—
261C22⋊C4—24no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—4722no—
266C22⋊C4—4722no—
267C22⋊C4—2364no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—44no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—2362no—
278C23—2362no—
279C23—2362no—
280C23—2362no—
281C23—2362no—
282C23—2362no—
283C23—2362no—
284C23C2×(C2×D59)⋊C411yescharacteristic
285C23C2×#1174566911yescharacteristic
286C23C2×#1174567111yescharacteristic
287D4—4724no—
288D4—4724no—
289D4—4724no—
290D4—4724no—
291D4—4722no—
292D4—4722no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—2362no—
311C2×C4—2362no—
312C2×C4—2362no—
313C2×C4#1427364111yescharacteristic
314C2×C4#1427364111yescharacteristic
315C2×C4#1427364712yes—
316C2×C4#1427364712yes—
317C2×C4#1427365512yes—
318C8—4722no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—4722no—
329C22—4722no—
330C22—4722no—
331C22—4722no—
332C22—4722no—
333C22—4722no—
334C22—4722no—
335C22C2×#1427364111yescharacteristic
336C22C2×#1427364711yescharacteristic
337C22C2×#1427365511yescharacteristic
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—84no—
344C4—4722no—
345C4#1683713512yes—
346C4#1683713512yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—41no—
351C2—9442no—
352C2C2×#1683713511yescharacteristic
353C2#2142574511yescharacteristic
354C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1740C2
socleC22×C59C2×#99
prime radical#21429846C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1740D59
hypocenterC59#11693
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. #21422410C2semidirect
3. #21422092C2semidirect
4. C59×#1740C2semidirect
5. C2×#14273724C4semidirect
9. #16837319C4semidirect
13. #16837268C4semidirect
171. #1740D59semidirect
197. C59#11693semidirect

Maximal subgroups

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

subgroupindex
#214298462
#214224102
#214220922
C59×#17402
#1169359