Finite Group Databasebeta

← #28498271

Subgroups — #28498271

Order 15104.

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

normal subgroups only

This group has 408 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#21428554C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic, complemented
3#21422041C212yesmaximal, complemented
4#21422027C212yesmaximal, complemented
5C59×#1464C211yesmaximal, characteristic, complemented
6C2×#14273701C411yescharacteristic
7#16837455—21no—
8#16837317—21no—
9#16837317—21no—
10#16837303C411yescharacteristic
11#16837303C2211yescharacteristic
12#16837293—21no—
13#16837293—21no—
14#16837271C2211yescharacteristic
15#16837267C411yescharacteristic
16C4×C22⋊Dic59C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C2×#117×C59C2211yescharacteristic
20C59×#432—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25#14273701—41no—
26#14273701—41no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—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—21no—
36C2×C4×Dic59C2×C411yescharacteristic
37C2×C4×Dic59C2×C411yescharacteristic
38C4⋊(C59⋊C8)—41no—
39C22⋊(C59⋊C8)—22no—
40C4×C59⋊C8—41no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C23×C4×C59C2×C411yescharacteristic
45#117×C59—41no—
46#117×C59—41no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C4⋊C4×C59C2311yescharacteristic
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C4⋊C8×C59—41no—
54C22⋊C8×C59—22no—
55#97×C59—41no—
56C24×C59C22⋊C411yescharacteristic
57C23×D59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—41no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C2×C22⋊D59—42no—
69C2×C22⋊D59—81no—
70C2×C22⋊D59—81no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C2×D236—22no—
75C2×D236—41no—
76C2×D236—41no—
77C2×C8×C59—42no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C22⋊Dic59—42no—
87(C2×D59)⋊C4—42no—
88(C2×D59)⋊C4—42no—
89(C2×D59)⋊C4—41no—
90(C2×D59)⋊C4—41no—
91(C2×D59)⋊C4—81no—
92(C2×D59)⋊C4—81no—
93C4⋊Dic59—41no—
94Dic59⋊C4—81no—
95Dic59⋊C4—81no—
96C4×Dic59—22no—
97C4×Dic59—41no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59C2×C22⋊C411yescharacteristic
105C22×D59—41no—
106C22×D59—41no—
107C22×D59—41no—
108C22×D59—41no—
109C22×D59—41no—
110C22×D59—41no—
111C22×D59—41no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59#13112yes—
129C22⋊D59—84no—
130C2×Dic59—22no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139D236—82no—
140C8×C59—82no—
141C59⋊C8—82no—
142#11554—591nomaximal, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—81no—
149C22×C59—81no—
150C22×C59#40811yescharacteristic
151C22×C59C2×#9911yescharacteristic
152C22×C59#41311yescharacteristic
153C2×D59—81no—
154C2×D59—81no—
155C2×D59—81no—
156C2×D59—81no—
157C2×D59—81no—
158C2×D59—81no—
159C2×D59—81no—
160C4×C59—22no—
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59—41no—
167Dic59—42no—
168Dic59—81no—
169Dic59—81no—
170#2500—591no—
171#1464D5911yescharacteristic, complemented
172#1458—591no—
173#1325—592no—
174#1318—592no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59—81no—
178C2×C59#146811yescharacteristic
179C2×C59#212811yescharacteristic
180C2×C59#213211yescharacteristic
181D59—161no—
182C2×#121—591no—
183C2×#117C2×D5911yescharacteristic
184#432—21no—
185#431—1181no—
186#394—1181no—
187#394—1181no—
188#390—591no—
189#390—591no—
190#388—1181no—
191#388—1181no—
192#384—591no—
193#382—591no—
194C4×C22⋊C4—591no—
195#324—592no—
196#324C2×D5912yes—
197#323—21no—
198#323—1181no—
199C59#1155411yescharacteristic, complemented, Sylow
200C22×D4—591no—
201C22×D4—1181no—
202C23×C4C4×D5911yescharacteristic
203#121—2361no—
204#121—2361no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—591no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4C4×D5911yescharacteristic
211C2×C4⋊C4C22×D5911yescharacteristic
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C22⋊C4—1181no—
215C2×C22⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4D23611yescharacteristic
219C2×C22⋊C4C22⋊D5911yescharacteristic
220C2×C42—591no—
221C2×C42—591no—
222C2×C42D23611yescharacteristic
223C2×C42C22⋊D5911yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—2361no—
226C22⋊C8—22no—
227C22⋊C8—1182no—
228#97—41no—
229C4×C8—2361no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233C24—1181no—
234C24(C2×D59)⋊C411yescharacteristic
235C2×D4—1182no—
236C2×D4—4721no—
237C2×D4—4721no—
238C2×D4—2361no—
239C2×D4—2361no—
240C2×D4—2362no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—1181no—
246C22×C4—1181no—
247C22×C4—1181no—
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4C2×C4×D5911yescharacteristic
252C22×C4C2×D23611yescharacteristic
253C22×C4C2×C22⋊D5911yescharacteristic
254C2×C8—42no—
255C2×C8—2362no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—4721no—
259C4⋊C4—4721no—
260C4⋊C4—2361no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—4721no—
264C22⋊C4—4721no—
265C22⋊C4—2362no—
266C22⋊C4—2362no—
267C22⋊C4—2362no—
268C22⋊C4—2361no—
269C22⋊C4—2361no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—1182no—
275C42—2361no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—2361no—
282C23—2361no—
283C23—2361no—
284C23—2361no—
285C23—2361no—
286C23—2361no—
287C23—2361no—
288C23C2×(C2×D59)⋊C411yescharacteristic
289D4—4722no—
290D4—4724no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—1182no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4#1427364111yescharacteristic
315C2×C4#1427364111yescharacteristic
316C2×C4#1427373512yes—
317C8—82no—
318C8—4722no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—4721no—
327C22—4721no—
328C22—4721no—
329C22—4721no—
330C22—4721no—
331C22—4721no—
332C22—4721no—
333C22#1683737011yescharacteristic
334C22C2×#1427364111yescharacteristic
335C22#1683738411yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—4721no—
344C4—4721no—
345C4—2362no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—9441no—
350C2#2142244511yescharacteristic
351C2#2142576711yescharacteristic
352C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1464C2
socleC22×C59C2×#99
prime radicalC2×#14273701C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1464D59
hypocenterC59#11554
LCSG ▷ C22×C4×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. #21422041C2semidirect
4. #21422027C2semidirect
5. C59×#1464C2semidirect
171. #1464D59semidirect
199. C59#11554semidirect

Maximal subgroups

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

subgroupindex
#214285542
#214224102
#214220412
#214220272
C59×#14642
#1155459