Finite Group Databasebeta

← #28499352

Subgroups — #28499352

Order 15104.

486 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 79 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#21427332C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic, complemented
3#21422095C212yesmaximal, complemented
4#21422092C212yesmaximal
5C59×#1820C211yesmaximal, characteristic, complemented
6C2×C4×D236C411yescharacteristic, complemented
7C2×C4×C22⋊D59—21no—
8#16837459—21no—
9#16837332C411yescharacteristic, complemented
10#16837327C2211yescharacteristic
11#16837324—21no—
12#16837319C411yescharacteristic, complemented
13#16837318—21no—
14#16837311C2211yescharacteristic
15C4×C4⋊Dic59C411yescharacteristic, complemented
16C4×(C2×D59)⋊C4—21no—
17#16837128—21no—
18#16837126C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#381—21no—
21C59×#342—21no—
22C59×#342C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C4×D59—21no—
26C4×D236—41no—
27C4×D236—41no—
28C4×C22⋊D59—44no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×C4×Dic59—21no—
38C4⋊(C59⋊C8)—41no—
39C22⋊(C59⋊C8)—22no—
40#14273638—41no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46#117×C59—41no—
47#117×C59—41no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59—22no—
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59C2×C411yescharacteristic
54C2×C42×C59D411yescharacteristic
55#95×C59—41no—
56#95×C59—41no—
57C24×C59C2×D411yescharacteristic
58C23×D59—21no—
59C22×C4×C59—22no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—41no—
63C22×C4×C59—42no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C2×C22⋊D59—42no—
71C2×C22⋊D59—42no—
72C2×C22⋊D59—42no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C2×D236—41no—
77C2×D236—42no—
78C2×C4×D59—42no—
79C2×C4×D59—42no—
80C2×C4×D59—41no—
81C2×C4×D59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—42no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59—21no—
92C22⋊Dic59—42no—
93(C2×D59)⋊C4—42no—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—42no—
96(C2×D59)⋊C4—42no—
97C4⋊Dic59—42no—
98C4⋊Dic59—41no—
99Dic59⋊C4—42no—
100C4×Dic59—42no—
101C2×C59⋊C8—42no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59C2×C22⋊C411yescharacteristic
107C23×C59AGL(1,ℤ/8)12yes—
108C22×D59—22no—
109C22×D59—41no—
110C22×D59—41no—
111C22×D59—41no—
112C22×D59—41no—
113C22×D59—41no—
114C22×D59—41no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—82no—
131C2×C4×C59—82no—
132C2×C4×C59—82no—
133C2×C4×C59#9911yescharacteristic
134C2×C4×C59#9911yescharacteristic
135C22⋊D59—48no—
136C2×Dic59—22no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145D236—42no—
146D236—42no—
147C4×D59—82no—
148C4×D59—82no—
149C59⋊C8—82no—
150#11798—591nomaximal, Sylow
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—22no—
154C22×C59—42no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—81no—
158C22×C59C2×#9911yescharacteristic
159C22×C59#41811yescharacteristic
160C22×C59#42111yescharacteristic
161C2×D59—42no—
162C2×D59—42no—
163C2×D59—42no—
164C2×D59—81no—
165C2×D59—81no—
166C2×D59—81no—
167C2×D59—81no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—41no—
171C4×C59—41no—
172C4×C59—41no—
173C4×C59—41no—
174C4×C59—41no—
175C4×C59—162no—
176Dic59—42no—
177Dic59—81no—
178Dic59—81no—
179#2392—591no—
180#1820D5911yescharacteristic, complemented
181#1464—591no—
182#1344—592no—
183#1343—592no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59—42no—
187C2×C59#151011yescharacteristic
188C2×C59#212011yescharacteristic
189C2×C59#212611yescharacteristic
190D59—82no—
191C2×C4×D4—591no—
192C2×C4×D4—1181no—
193C2×#117C2×D5911yescharacteristic
194#432—1181no—
195#398—591no—
196#397—591no—
197#396—1181no—
198#394—591no—
199#394—1181no—
200#392—591no—
201#381—21no—
202C4×C4⋊C4—591no—
203C4×C22⋊C4—1181no—
204#342—21no—
205#342C2×D5912yes—
206#324—1181no—
207#323—592no—
208C59#1179811yescharacteristic, complemented, Sylow
209C22×D4—591no—
210C22×D4—1181no—
211C23×C4—1181no—
212C23×C4C22⋊D5911yescharacteristic
213C4×D4—2361no—
214C4×D4—2361no—
215C4×D4—2364no—
216#117—41no—
217#117—41no—
218C2×C4⋊C4—591no—
219C2×C4⋊C4—1181no—
220C2×C4⋊C4—1181no—
221C2×C4⋊C4C4×D5911yescharacteristic
222C2×C4⋊C4D23611yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—1181no—
225C2×C22⋊C4—1181no—
226C2×C22⋊C4—1181no—
227C2×C22⋊C4—1181no—
228C2×C22⋊C4—1181no—
229C2×C22⋊C4C22⋊D5911yescharacteristic
230C2×C22⋊C4C22×D5911yescharacteristic
231C2×C42—1181no—
232C2×C42C4×D5911yescharacteristic
233C2×C42D23611yescharacteristic
234C4⋊C8—2361no—
235C22⋊C8—1182no—
236#97—2361no—
237#95—41no—
238#95—41no—
239#95—591no—
240#95—591no—
241#95—1181no—
242#95—1181no—
243C24—1181no—
244C24C2×C22⋊D5911yescharacteristic
245C2×D4—2361no—
246C2×D4—2362no—
247C2×D4—2362no—
248C2×D4—2362no—
249C2×D4—2362no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4—1181no—
258C22×C4—2362no—
259C22×C4—2362no—
260C22×C4—2361no—
261C22×C4—2361no—
262C22×C4(C2×D59)⋊C411yescharacteristic
263C22×C4(C2×D59)⋊C411yescharacteristic
264C22×C4(C2×D59)⋊C411yescharacteristic
265C22×C4(C2×D59)⋊C411yescharacteristic
266C22×C4C2×C4×D5911yescharacteristic
267C22×C4C2×D23611yescharacteristic
268C2×C8—2362no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—2361no—
272C4⋊C4—2362no—
273C4⋊C4—2362no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—2362no—
279C22⋊C4—2362no—
280C22⋊C4—2362no—
281C22⋊C4—2362no—
282C22⋊C4—2362no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—2362no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—1182no—
293C23—2361no—
294C23—2361no—
295C23—2361no—
296C23—2361no—
297C23—2361no—
298C23—2361no—
299C23C2×(C2×D59)⋊C411yescharacteristic
300C23#1427375212yes—
301D4—2362no—
302D4—2362no—
303D4—2368no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—1182no—
323C2×C4—4722no—
324C2×C4—4722no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4#1427364111yescharacteristic
334C2×C4#1427364111yescharacteristic
335C8—4722no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—4721no—
344C22—4721no—
345C22—4721no—
346C22—4721no—
347C22—2362no—
348C22—2362no—
349C22—2362no—
350C22C2×#1427364111yescharacteristic
351C22#1683740611yescharacteristic
352C22#1683742011yescharacteristic
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—162no—
361C4—4721no—
362C4—4721no—
363C4—2362no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—4722no—
368C2#2142262411yescharacteristic
369C2#2142573511yescharacteristic
370C2#2142575911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1820C2
socleC22×C59C2×#99
prime radical#16837311C22
cosocleC2×C22⋊C4×C59C23
hypercenter#1820D59
hypocenterC59#11798
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #21422430C2semidirect
3. #21422095C2semidirect
5. C59×#1820C2semidirect
6. C2×C4×D236C4semidirect
9. #16837332C4semidirect
12. #16837319C4semidirect
15. C4×C4⋊Dic59C4semidirect
180. #1820D59semidirect
208. C59#11798semidirect

Maximal subgroups

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

subgroupindex
#214273322
#214224302
#214220952
#214220922
C59×#18202
#1179859