Finite Group Databasebeta

← #28498231

Subgroups — #28498231

Order 15104.

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

normal subgroups only

This group has 460 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#21428366C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic, complemented
3#21422045C212yesmaximal, complemented
4#21422023C212yesmaximal
5C59×#1464C211yesmaximal, characteristic, complemented
6C22×(C2×D59)⋊C4C411yescharacteristic
7#16837455—21no—
8#16837337—21no—
9#16837328—21no—
10#16837293—21no—
11#16837293—21no—
12#16837274C411yescharacteristic
13#16837274C2211yescharacteristic
14#16837265C411yescharacteristic
15#16837257C411yescharacteristic
16#16837257C2211yescharacteristic
17#16837128—21no—
18#16837126C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#432—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C24×D59C2×C411yescharacteristic
24C23×Dic59C2×C411yescharacteristic
25C2×Dic59⋊C4—21no—
26C2×Dic59⋊C4—21no—
27C2×C22⋊Dic59C2×C411yescharacteristic
28C2×C22⋊Dic59C2×C411yescharacteristic
29C2×(C2×D59)⋊C4—21no—
30C2×(C2×D59)⋊C4—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—41no—
36C2×(C2×D59)⋊C4—41no—
37C4⋊(C59⋊C8)—41no—
38C22⋊(C59⋊C8)—22no—
39C4×C59⋊C8—41no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C23×C4×C59C2×C411yescharacteristic
45#117×C59—41no—
46#117×C59—41no—
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59C2311yescharacteristic
51C2×C42×C59D411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C4⋊C8×C59—41no—
54C22⋊C8×C59—22no—
55#97×C59—41no—
56C24×C59C22×C411yescharacteristic
57C23×D59—21no—
58C23×D59—22no—
59C23×D59—42no—
60C23×D59—41no—
61C23×D59—41no—
62C23×D59—81no—
63C22×C4×C59—22no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—41no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—41no—
77C22×Dic59—81no—
78C2×C8×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C22⋊Dic59—22no—
88C22⋊Dic59—41no—
89(C2×D59)⋊C4—42no—
90(C2×D59)⋊C4—42no—
91(C2×D59)⋊C4—42no—
92(C2×D59)⋊C4—41no—
93(C2×D59)⋊C4—41no—
94(C2×D59)⋊C4—81no—
95(C2×D59)⋊C4—81no—
96Dic59⋊C4—81no—
97Dic59⋊C4—81no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104C23×C59#13112yes—
105C22×D59—22no—
106C22×D59—41no—
107C22×D59—41no—
108C22×D59—41no—
109C22×D59—42no—
110C22×D59—41no—
111C22×D59—41no—
112C22×D59—41no—
113C22×D59—42no—
114C22×D59—84no—
115C22×D59—84no—
116C22×D59—82no—
117C22×D59—82no—
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—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—42no—
128C2×C4×C59—42no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59#9911yescharacteristic
134C2×C4×C59#9911yescharacteristic
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—82no—
145C8×C59—82no—
146C59⋊C8—82no—
147#11546—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—22no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59#40811yescharacteristic
156C22×C59C2×#9911yescharacteristic
157C22×C59#41311yescharacteristic
158C2×D59—42no—
159C2×D59—42no—
160C2×D59—42no—
161C2×D59—81no—
162C2×D59—81no—
163C2×D59—84no—
164C2×D59—81no—
165C2×D59—81no—
166C2×D59—84no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—41no—
170C4×C59—41no—
171C4×C59—41no—
172C4×C59—41no—
173C4×C59—41no—
174Dic59—81no—
175Dic59—81no—
176Dic59—81no—
177#2482—591no—
178#1464D5911yescharacteristic, complemented
179#1458—591no—
180#1326—592no—
181#1316—592no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—42no—
185C2×C59#146811yescharacteristic
186C2×C59#212011yescharacteristic
187C2×C59#212211yescharacteristic
188D59—82no—
189C2×#117C2×D5911yescharacteristic
190C22×C22⋊C4—591no—
191#432—21no—
192#431—1181no—
193#399—1181no—
194#397—1181no—
195#388—1181no—
196#388—1181no—
197#385—591no—
198#385—591no—
199#381—591no—
200#379—591no—
201#379—591no—
202#324—21no—
203#324—1181no—
204#323—592no—
205#323C2×D5912yes—
206C59#1154611yescharacteristic, complemented, Sylow
207C25—591no—
208C23×C4—591no—
209C23×C4C4×D5911yescharacteristic
210#117—41no—
211#117—41no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4D23611yescharacteristic
215C2×C4⋊C4C22⋊D5911yescharacteristic
216C2×C22⋊C4—591no—
217C2×C22⋊C4—591no—
218C2×C22⋊C4—1181no—
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1181no—
221C2×C22⋊C4—1181no—
222C2×C22⋊C4—1181no—
223C2×C22⋊C4—1181no—
224C2×C22⋊C4—2361no—
225C2×C22⋊C4—2361no—
226C2×C22⋊C4C4×D5911yescharacteristic
227C2×C22⋊C4C22×D5911yescharacteristic
228C2×C42D23611yescharacteristic
229C2×C42C22⋊D5911yescharacteristic
230C4⋊C8—41no—
231C4⋊C8—2361no—
232C22⋊C8—22no—
233C22⋊C8—1182no—
234#97—41no—
235C4×C8—2361no—
236#95—1181no—
237#95—1181no—
238#95—1181no—
239#95—1181no—
240C24—1181no—
241C24—1182no—
242C24—4721no—
243C24—2361no—
244C24—2361no—
245C24—2362no—
246C24C2×C4×D5911yescharacteristic
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4—4721no—
255C22×C4—2361no—
256C22×C4(C2×D59)⋊C411yescharacteristic
257C22×C4(C2×D59)⋊C411yescharacteristic
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4(C2×D59)⋊C411yescharacteristic
260C22×C4C2×D23611yescharacteristic
261C22×C4C2×C22⋊D5911yescharacteristic
262C2×C8—42no—
263C2×C8—2362no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—4721no—
267C4⋊C4—4721no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—1182no—
271C22⋊C4—4721no—
272C22⋊C4—4721no—
273C22⋊C4—2362no—
274C22⋊C4—2362no—
275C22⋊C4—2362no—
276C22⋊C4—2361no—
277C22⋊C4—2361no—
278C22⋊C4—2361no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23—1182no—
288C23—4722no—
289C23—4722no—
290C23—4724no—
291C23—4724no—
292C23—2361no—
293C23—2361no—
294C23—2361no—
295C23—2362no—
296C23—2362no—
297C23—2361no—
298C23—2361no—
299C23—2361no—
300C23C2×(C2×D59)⋊C411yescharacteristic
301C23#1427373512yes—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
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—2361no—
327C2×C4#1427364111yescharacteristic
328C2×C4#1427364111yescharacteristic
329C8—82no—
330C8—4722no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—4721no—
339C22—4721no—
340C22—4724no—
341C22—4724no—
342C22—4721no—
343C22—4721no—
344C22—2362no—
345C22—2362no—
346C22—2362no—
347C22#1683737011yescharacteristic
348C22C2×#1427364111yescharacteristic
349C22#1683738411yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4721no—
358C4—4721no—
359C4—4721no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—4722no—
364C2#2142244511yescharacteristic
365C2#2142573511yescharacteristic
366C2#2142574711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1464C2
socleC22×C59C2×#99
prime radicalC24×D59C2×C4
cosocleC2×C22⋊C4×C59C23
hypercenter#1464D59
hypocenterC59#11546
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #21422410C2semidirect
3. #21422045C2semidirect
5. C59×#1464C2semidirect
178. #1464D59semidirect
206. C59#11546semidirect

Maximal subgroups

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

subgroupindex
#214283662
#214224102
#214220452
#214220232
C59×#14642
#1154659