Finite Group Databasebeta

← #28486034

Subgroups — #28486034

Order 15104.

496 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 99 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#21429783C211yesmaximal, characteristic
2#21422693C212yesmaximal, complemented
3#21422430C211yesmaximal, characteristic, complemented
4#21421990C212yesmaximal, complemented
5C59×#2921C211yesmaximal, characteristic, complemented
6C2×#14273715C2211yescharacteristic
7#16837459C2212yes—
8#16837322C411yescharacteristic
9#16837319C411yescharacteristic
10#16837314C411yescharacteristic
11#16837311C411yescharacteristic
12#16837302—21no—
13#16837293—21no—
14#16837292—21no—
15#16837281—21no—
16#16837248C2211yescharacteristic
17#16837141—24no—
18#16837128—21no—
19#16837126—21no—
20C2×#117×C59C2211yescharacteristic
21C59×#433—24no—
22C59×#423—21no—
23C59×#407—21no—
24C4×M16×C59C2212yes—
25C22×Dic118C2×C411yescharacteristic
26C22×D236C2×C411yescharacteristic
27C22×C22⋊D59—21no—
28#14273715—22no—
29#14273715—22no—
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32C2×(C2×D59)⋊C4—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)—22no—
39C22⋊(C59⋊C8)—22no—
40#14273638—22no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48#117×C59—22no—
49#117×C59—22no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—24no—
57C22⋊C8×C59—24no—
58#97×C59D414yescomplemented
59C4×C8×C59—22no—
60C24×C59C22⋊C411yescharacteristic
61C23×D59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—41no—
65C22×C4×C59—41no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C2×C22⋊D59—42no—
73C2×C22⋊D59—42no—
74C2×C22⋊D59—81no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C2×D236—22no—
79C2×D236—81no—
80C2×Dic118—22no—
81C2×Dic118—81no—
82M16×C59—42no—
83M16×C59—42no—
84C2×C8×C59—24no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—22no—
87C22⋊C4×C59—22no—
88C22⋊C4×C59—22no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59C2×D412yes—
92C42×C59#4112yes—
93C22⋊Dic59—81no—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—42no—
96(C2×D59)⋊C4—41no—
97(C2×D59)⋊C4—42no—
98(C2×D59)⋊C4—41no—
99(C2×D59)⋊C4—81no—
100Dic59⋊C4—81no—
101C4×Dic59—42no—
102C2×C59⋊C8—42no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59C2×C22⋊C411yescharacteristic
109C22×D59—41no—
110C22×D59—41no—
111C22×D59—41no—
112C22×D59—41no—
113C22×D59—41no—
114C22×D59—41no—
115C22×D59—41no—
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—41no—
124C2×C4×C59—41no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59C4×D412yes—
130C2×C4×C59#12011yescharacteristic
131C2×C4×C59#12112yes—
132C2×C4×C59#12311yescharacteristic
133C22⋊D59—84no—
134C2×Dic59—22no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143D236—82no—
144Dic118—82no—
145C8×C59—44no—
146C59⋊C8—82no—
147#9389—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—42no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—81no—
155C22×C59#38611yescharacteristic
156C22×C59#42111yescharacteristic
157C22×C59#42111yescharacteristic
158C2×D59—81no—
159C2×D59—81no—
160C2×D59—81no—
161C2×D59—81no—
162C2×D59—81no—
163C2×D59—81no—
164C2×D59—81no—
165C4×C59—22no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—22no—
169C4×C59—41no—
170C4×C59—41no—
171C4×C59—41no—
172Dic59—42no—
173Dic59—81no—
174Dic59—81no—
175#2921D5911yescharacteristic, complemented
176#2610—591no—
177#1527—592no—
178#1464—591no—
179#1300—592no—
180C2×C59—21no—
181C2×C59—21no—
182C2×C59—81no—
183C2×C59#188411yescharacteristic
184C2×C59#189711yescharacteristic
185C2×C59#191411yescharacteristic
186D59—161no—
187C2×#124—591no—
188C2×#117C2×D5911yescharacteristic
189#433—24no—
190#432—592no—
191#423—21no—
192#407—21no—
193C4×M16C2×D5912yes—
194#395—591no—
195#394—591no—
196#393—591no—
197#392—591no—
198#390—1181no—
199#388—1181no—
200#388—1181no—
201#386—1181no—
202#376—591no—
203#329—1184no—
204#324—1181no—
205#323—1181no—
206C59#938911yescharacteristic, complemented, Sylow
207C22×Q8—591no—
208C22×D4—591no—
209C22×D4—1181no—
210C23×C4C22⋊D5911yescharacteristic
211C2×M16—22no—
212#124—1182no—
213#124—1182no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—1181no—
217C2×C4⋊C4C4×D5911yescharacteristic
218C2×C4⋊C4D23611yescharacteristic
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1181no—
221C2×C22⋊C4—1181no—
222C2×C22⋊C4—1181no—
223C2×C22⋊C4—1181no—
224C2×C22⋊C4—1181no—
225C2×C22⋊C4C4×D5911yescharacteristic
226C2×C22⋊C4D23611yescharacteristic
227C2×C42—1181no—
228C2×C42C22⋊D5911yescharacteristic
229C2×C42C22×D5911yescharacteristic
230C4⋊C8—24no—
231C4⋊C8—1182no—
232C22⋊C8—24no—
233C22⋊C8—1182no—
234#97—1182no—
235#97C22⋊D5914yescomplemented
236C4×C8—22no—
237#95—591no—
238#95—591no—
239#95—1181no—
240#95—1181no—
241#95—1181no—
242C24—1181no—
243C24(C2×D59)⋊C411yescharacteristic
244C2×Q8—1182no—
245C2×Q8—4721no—
246C2×D4—1182no—
247C2×D4—4721no—
248C2×D4—4721no—
249C2×D4—2362no—
250C2×D4—2362no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4—1181no—
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—24no—
267C2×C8—2362no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—4721no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—4721no—
274C22⋊C4—4721no—
275C22⋊C4—2362no—
276C22⋊C4—2362no—
277C22⋊C4—2361no—
278C22⋊C4—2362no—
279C22⋊C4—2361no—
280C42—21no—
281C42—21no—
282C42—2362no—
283C42#1174568512yes—
284C42C2×C22⋊D5912yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—2361no—
291C23—2361no—
292C23—2361no—
293C23—2361no—
294C23—2361no—
295C23—2361no—
296C23—2361no—
297C23C2×(C2×D59)⋊C411yescharacteristic
298Q8—4722no—
299D4—4722no—
300D4—4724no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—1182no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4C4×C22⋊D5912yes—
324C2×C4#1427369611yescharacteristic
325C2×C4#1427370112yes—
326C2×C4#1427371011yescharacteristic
327C8—44no—
328C8—4722no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—4721no—
337C22—4721no—
338C22—4721no—
339C22—4721no—
340C22—4721no—
341C22—4721no—
342C22—4721no—
343C22#1683728111yescharacteristic
344C22#1683742011yescharacteristic
345C22#1683742011yescharacteristic
346C4—22no—
347C4—22no—
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—4721no—
354C4—4721no—
355C4—2362no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—9441no—
360C2#2142447911yescharacteristic
361C2#2142454411yescharacteristic
362C2#2142464311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2921C2
socleC22×C59#386
prime radical#16837311C4
cosocleC2×C42×C59C23
hypercenter#2921D59
hypocenterC59#9389
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #21422693C2semidirect
3. #21422430C2semidirect
4. #21421990C2semidirect
5. C59×#2921C2semidirect
58. #97×C59D4semidirect
175. #2921D59semidirect
206. C59#9389semidirect
235. #97C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214297832
#214226932
#214224302
#214219902
C59×#29212
#938959