Finite Group Databasebeta

← #28499378

Subgroups — #28499378

Order 15104.

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

normal subgroups only

This group has 472 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#21422101C212yesmaximal, complemented
4#21422098C212yesmaximal, complemented
5C59×#1822C211yesmaximal, characteristic, complemented
6C2×C4×D236C411yescharacteristic
7C2×C4×C22⋊D59—21no—
8#16837459—21no—
9#16837332C2211yescharacteristic
10#16837327C411yescharacteristic
11#16837324—21no—
12#16837319C2211yescharacteristic
13#16837318—21no—
14#16837311C411yescharacteristic
15C4×C4⋊Dic59C411yescharacteristic
16C4×(C2×D59)⋊C4—21no—
17#16837128C2212yes—
18#16837126—21no—
19C2×#117×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#343—21no—
22C59×#343C2212yes—
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
46C2×M16×C59—22no—
47#117×C59—41no—
48#117×C59—41no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C42×C59C2×C411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C22⋊C8×C59—41no—
56C22⋊C8×C59—41no—
57C24×C59C22⋊C411yescharacteristic
58C23×D59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59—41no—
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C2×C22⋊D59—42no—
70C2×C22⋊D59—42no—
71C2×C22⋊D59—42no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C2×D236—41no—
76C2×D236—42no—
77C2×C4×D59—42no—
78C2×C4×D59—42no—
79C2×C4×D59—41no—
80C2×C4×D59—41no—
81M16×C59—42no—
82M16×C59—42no—
83C2×C8×C59—42no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
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—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59C2×C22⋊C411yescharacteristic
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—42no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59#9911yescharacteristic
130C2×C4×C59#9911yescharacteristic
131C2×C4×C59AGL(1,ℤ/8)12yes—
132C22⋊D59—48no—
133C2×Dic59—22no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142D236—42no—
143D236—42no—
144C4×D59—82no—
145C4×D59—82no—
146C8×C59—82no—
147C59⋊C8—82no—
148#11804—591nomaximal, Sylow
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—42no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—81no—
156C22×C59C2×#9911yescharacteristic
157C22×C59#41811yescharacteristic
158C22×C59#42111yescharacteristic
159C2×D59—42no—
160C2×D59—42no—
161C2×D59—42no—
162C2×D59—81no—
163C2×D59—81no—
164C2×D59—81no—
165C2×D59—81no—
166C4×C59—22no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—41no—
170C4×C59—41no—
171C4×C59—41no—
172C4×C59—41no—
173Dic59—42no—
174Dic59—81no—
175Dic59—81no—
176#2392—591no—
177#1822D5911yescharacteristic, complemented
178#1464—591no—
179#1346—592no—
180#1345—592no—
181C2×C59—21no—
182C2×C59—21no—
183C2×C59—81no—
184C2×C59#151011yescharacteristic
185C2×C59#212011yescharacteristic
186C2×C59#213211yescharacteristic
187D59—82no—
188C2×C4×D4—591no—
189C2×C4×D4—1181no—
190C2×#117C2×D5911yescharacteristic
191#432—1181no—
192#407—21no—
193#398—591no—
194#397—591no—
195#396—1181no—
196#394—591no—
197#394—1181no—
198#392—591no—
199C4×C4⋊C4—591no—
200C4×C22⋊C4—1181no—
201#343—21no—
202#343C2×D5912yes—
203#324—592no—
204#323—1181no—
205C59#1180411yescharacteristic, complemented, Sylow
206C22×D4—591no—
207C22×D4—1181no—
208C23×C4—1181no—
209C23×C4C22⋊D5911yescharacteristic
210C2×M16—22no—
211C4×D4—2361no—
212C4×D4—2361no—
213C4×D4—2364no—
214#117—41no—
215#117—41no—
216C2×C4⋊C4—591no—
217C2×C4⋊C4—1181no—
218C2×C4⋊C4—1181no—
219C2×C4⋊C4C22⋊D5911yescharacteristic
220C2×C4⋊C4C22×D5911yescharacteristic
221C2×C22⋊C4—1181no—
222C2×C22⋊C4—1181no—
223C2×C22⋊C4—1181no—
224C2×C22⋊C4—1181no—
225C2×C22⋊C4—1181no—
226C2×C22⋊C4C4×D5911yescharacteristic
227C2×C22⋊C4D23611yescharacteristic
228C2×C42—1181no—
229C2×C42C4×D5911yescharacteristic
230C2×C42D23611yescharacteristic
231C4⋊C8—2361no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234C22⋊C8—1182no—
235#97—2361no—
236#95—591no—
237#95—591no—
238#95—1181no—
239#95—1181no—
240C24—1181no—
241C24(C2×D59)⋊C411yescharacteristic
242C2×D4—2361no—
243C2×D4—2362no—
244C2×D4—2362no—
245C2×D4—2362no—
246C2×D4—2362no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4—2362no—
255C22×C4—2362no—
256C22×C4—2361no—
257C22×C4—2361no—
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4(C2×D59)⋊C411yescharacteristic
260C22×C4(C2×D59)⋊C411yescharacteristic
261C22×C4C2×C4×D5911yescharacteristic
262C22×C4C2×D23611yescharacteristic
263C22×C4C2×C22⋊D5911yescharacteristic
264M16—42no—
265M16—42no—
266C2×C8—42no—
267C2×C8—2362no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—2361no—
271C4⋊C4—2362no—
272C4⋊C4—2362no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—2362no—
276C22⋊C4—2362no—
277C22⋊C4—2362no—
278C22⋊C4—2362no—
279C22⋊C4—2362no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—2362no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—1182no—
291C23—2361no—
292C23—2361no—
293C23—2361no—
294C23—2361no—
295C23—2361no—
296C23—2361no—
297C23C2×(C2×D59)⋊C411yescharacteristic
298D4—2362no—
299D4—2362no—
300D4—2368no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—1182no—
316C2×C4—4722no—
317C2×C4—4722no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4#1427364111yescharacteristic
327C2×C4#1427364111yescharacteristic
328C2×C4#1427375212yes—
329C8—82no—
330C8—4722no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—4721no—
339C22—4721no—
340C22—4721no—
341C22—4721no—
342C22—2362no—
343C22—2362no—
344C22—2362no—
345C22C2×#1427364111yescharacteristic
346C22#1683740611yescharacteristic
347C22#1683742011yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—4721no—
356C4—4721no—
357C4—2362no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—4722no—
362C2#2142262411yescharacteristic
363C2#2142573811yescharacteristic
364C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1822C2
socleC22×C59C2×#99
prime radical#16837311C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1822D59
hypocenterC59#11804
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #21422430C2semidirect
3. #21422101C2semidirect
4. #21422098C2semidirect
5. C59×#1822C2semidirect
177. #1822D59semidirect
205. C59#11804semidirect

Maximal subgroups

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

subgroupindex
#214273322
#214224302
#214221012
#214220982
C59×#18222
#1180459