Finite Group Databasebeta

← #28498179

Subgroups — #28498179

Order 15104.

440 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 83 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#21426808C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422041C212yesmaximal, complemented
4#21422029C212yesmaximal, complemented
5C59×#1465C211yesmaximal, characteristic
6C2×#14273714C411yescharacteristic
7C2×C4×Dic118C411yescharacteristic
8#16837366—21no—
9#16837307C2211yescharacteristic
10#16837305C411yescharacteristic
11#16837290—21no—
12#16837290—21no—
13#16837275—21no—
14#16837275—21no—
15#16837271C2211yescharacteristic
16#16837248C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C22×Dic118C2×C411yescharacteristic
24C2×#11745664—41no—
25#14273714—22no—
26#14273714—22no—
27C4×Dic118—41no—
28C4×Dic118—41no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C4×Dic59—21no—
38C2×C4×Dic59C2×C411yescharacteristic
39C2×C4×Dic59C2×C411yescharacteristic
40C22⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)—41no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C22×D4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C2×M16×C59—41no—
50C4×D4×C59—41no—
51C4×D4×C59—41no—
52C2×C4⋊C4×C59C2311yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59D411yescharacteristic
55C2×C42×C59C2×C411yescharacteristic
56C22⋊C8×C59—22no—
57C22⋊C8×C59—41no—
58C24×C59C22⋊C411yescharacteristic
59C24×C59C22⋊C411yescharacteristic
60C2×D4×C59—21no—
61C2×D4×C59—21no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—81no—
69C22×C4×C59—81no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C2×Dic118—21no—
80C2×Dic118—41no—
81C2×Dic118—41no—
82C2×Dic118M1612yescomplemented
83M16×C59—82no—
84C2×C8×C59—42no—
85C4⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C42×C59—21no—
89C42×C59—21no—
90C22⋊Dic59—24no—
91C22⋊Dic59—24no—
92C22⋊Dic59—81no—
93C22⋊Dic59—81no—
94C4⋊Dic59—41no—
95Dic59⋊C4—41no—
96Dic59⋊C4—41no—
97Dic59⋊C4—81no—
98Dic59⋊C4—81no—
99C4×Dic59—21no—
100C4×Dic59—42no—
101C4×Dic59—41no—
102C4×Dic59M1612yescomplemented
103#11745664—82no—
104C2×C59⋊C8—42no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59C2×C22⋊C411yescharacteristic
116D4×C59—42no—
117D4×C59—42no—
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—42no—
125C2×C4×C59—42no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—81no—
130C2×C4×C59—81no—
131C2×C4×C59#9911yescharacteristic
132C2×C4×C59#9911yescharacteristic
133C2×C4×C59C2×M1611yescharacteristic
134C2×C4×C59C2×M1611yescharacteristic
135C2×Dic59—22no—
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—
145C2×Dic59—41no—
146C2×Dic59—41no—
147Dic118—42no—
148Dic118—42no—
149C8×C59—82no—
150C59⋊C8—82no—
151#11535—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—42no—
155C22×C59—42no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—81no—
163C22×C59—81no—
164C22×C59#40711yescharacteristic
165C22×C59C2×#9911yescharacteristic
166C22×C59#41311yescharacteristic
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—21no—
171C4×C59—41no—
172C4×C59—41no—
173C4×C59—41no—
174Dic59—42no—
175Dic59—42no—
176Dic59—81no—
177Dic59—81no—
178#2326—591no—
179#1465—591no—
180#1465D5911yescharacteristic
181#1325—592no—
182#1319—592no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—81no—
186C2×C59—81no—
187C2×C59#146511yescharacteristic
188C2×C59#212711yescharacteristic
189C2×C59#213511yescharacteristic
190C2×#124—591no—
191C2×C4×Q8—591no—
192C2×C4×D4C2×D5911yescharacteristic
193#407—21no—
194#407—1181no—
195#391—591no—
196#390—591no—
197#388—1181no—
198#388—1181no—
199#385—1181no—
200#385—1181no—
201#384—591no—
202#376—591no—
203#324—592no—
204#324C2×D5912yes—
205#323—21no—
206#323—1181no—
207C59#1153511yescharacteristic, complemented, Sylow
208C22×Q8—591no—
209C22×D4C4×D5911yescharacteristic
210C23×C4D23611yescharacteristic
211C23×C4C22⋊D5911yescharacteristic
212C2×M16—41no—
213C2×M16—2361no—
214#124—1182no—
215#124—1182no—
216C4×Q8—2361no—
217C4×Q8—2361no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—591no—
221C2×C4⋊C4—1181no—
222C2×C4⋊C4—1181no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4C22×D5911yescharacteristic
225C2×C22⋊C4—1181no—
226C2×C22⋊C4—1181no—
227C2×C22⋊C4—1181no—
228C2×C22⋊C4—1181no—
229C2×C22⋊C4D23611yescharacteristic
230C2×C22⋊C4C22⋊D5911yescharacteristic
231C2×C42—591no—
232C2×C42—591no—
233C2×C42—1181no—
234C2×C42C4×D5911yescharacteristic
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—1182no—
238C22⋊C8—2361no—
239#95—1181no—
240#95—1181no—
241#95—1181no—
242#95—1181no—
243C24(C2×D59)⋊C411yescharacteristic
244C24(C2×D59)⋊C411yescharacteristic
245C2×Q8—592no—
246C2×Q8—1181no—
247C2×Q8—2361no—
248C2×Q8—2361no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—1181no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4(C2×D59)⋊C411yescharacteristic
264C22×C4(C2×D59)⋊C411yescharacteristic
265C22×C4C2×C4×D5911yescharacteristic
266C22×C4C2×D23611yescharacteristic
267C22×C4C2×C22⋊D5911yescharacteristic
268M16—82no—
269M16—4722no—
270C2×C8—42no—
271C2×C8—2362no—
272C4⋊C4—41no—
273C4⋊C4—4721no—
274C4⋊C4—4721no—
275C4⋊C4—2361no—
276C4⋊C4—2361no—
277C4⋊C4—2361no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—1184no—
281C22⋊C4—1184no—
282C22⋊C4—4721no—
283C22⋊C4—4721no—
284C42—21no—
285C42—21no—
286C42—592no—
287C42—1181no—
288C42—2361no—
289C42—2362no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×(C2×D59)⋊C411yescharacteristic
301Q8—2362no—
302Q8—2362no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—1182no—
319C2×C4—1182no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4#1427364111yescharacteristic
331C2×C4#1427364111yescharacteristic
332C2×C4M16×D5911yescharacteristic
333C2×C4M16×D5911yescharacteristic
334C8—82no—
335C8—4722no—
336C22—21no—
337C22—21no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22#1683736511yescharacteristic
349C22C2×#1427364111yescharacteristic
350C22#1683738411yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—4721no—
359C4—4721no—
360C4—2362no—
361C4—2362no—
362C2—21no—
363C2—21no—
364C2—81no—
365C2—81no—
366C2#2142243211yescharacteristic
367C2#2142576311yescharacteristic
368C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1465C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1465D59
hypocenterC59#11535
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #21422041C2semidirect
4. #21422029C2semidirect
82. C2×Dic118M16semidirect
102. C4×Dic59M16semidirect
207. C59#11535semidirect

Maximal subgroups

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

subgroupindex
#214268082
#214224342
#214220412
#214220292
C59×#14652
#1153559