Finite Group Databasebeta

← #28498140

Subgroups — #28498140

Order 15104.

470 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C4×#14273708C211yesmaximal, characteristic
2#21422437C211yesmaximal, characteristic
3#21422045C212yesmaximal, complemented
4#21422025C212yesmaximal, complemented
5C59×#1465C211yesmaximal, characteristic
6C2×#14273708C411yescharacteristic
7C22×C4×Dic59C411yescharacteristic
8#16837459—21no—
9#16837290—21no—
10#16837283—21no—
11#16837282C411yescharacteristic
12#16837274C2211yescharacteristic
13#16837263C2211yescharacteristic
14C4×Dic59⋊C4—21no—
15C4×C22⋊Dic59—21no—
16C4×C22⋊Dic59C411yescharacteristic
17#16837126—21no—
18#16837126C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C23×Dic59C2×C411yescharacteristic
24C23×Dic59C2×C411yescharacteristic
25#14273708—22no—
26#14273708—22no—
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59C2×C411yescharacteristic
32C2×C22⋊Dic59C2×C411yescharacteristic
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—21no—
36C2×C4×Dic59—41no—
37C2×C4×Dic59—41no—
38C4⋊(C59⋊C8)—41no—
39C22⋊(C59⋊C8)—22no—
40#14273638—41no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C22×D4×C59C2×C411yescharacteristic
47C23×C4×C59C2×C411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C2×M16×C59—41no—
50C4×D4×C59—41no—
51C4×D4×C59—41no—
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2311yescharacteristic
55C2×C42×C59D411yescharacteristic
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—42no—
69C22×C4×C59—42no—
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—
79C22×Dic59—21no—
80C22×Dic59—42no—
81C22×Dic59—41no—
82C22×Dic59—41no—
83C22×Dic59—41no—
84C22×Dic59M1612yescomplemented
85M16×C59—82no—
86C2×C8×C59—42no—
87C4⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C42×C59—21no—
91C42×C59—21no—
92C22⋊Dic59—24no—
93C22⋊Dic59—21no—
94C22⋊Dic59—42no—
95C22⋊Dic59—41no—
96C22⋊Dic59M1612yescomplemented
97Dic59⋊C4—24no—
98Dic59⋊C4—42no—
99C4×Dic59—42no—
100C4×Dic59—42no—
101C4×Dic59—41no—
102C4×Dic59—41no—
103C2×C59⋊C8—42no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59C2×C22⋊C411yescharacteristic
114C23×C59C2×M1611yescharacteristic
115C23×C59C2×M1611yescharacteristic
116D4×C59—42no—
117D4×C59—42no—
118C2×C4×C59—22no—
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—42no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—81no—
132C2×C4×C59#9911yescharacteristic
133C2×C4×C59#9911yescharacteristic
134C2×Dic59—22no—
135C2×Dic59—22no—
136C2×Dic59—22no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—42no—
141C2×Dic59—42no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—42no—
146C2×Dic59—42no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C8×C59—82no—
151C59⋊C8—82no—
152#11526—591nomaximal, Sylow
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59—42no—
160C22×C59—42no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—81no—
167C22×C59#40711yescharacteristic
168C22×C59C2×#9911yescharacteristic
169C22×C59#41311yescharacteristic
170C4×C59—21no—
171C4×C59—41no—
172C4×C59—41no—
173C4×C59—41no—
174C4×C59—41no—
175C4×C59#35212yes—
176Dic59—42no—
177Dic59—42no—
178Dic59—42no—
179Dic59—81no—
180C4×#123—591no—
181#1466—591no—
182#1465D5911yescharacteristic
183#1326—592no—
184#1317—592no—
185C2×C59—21no—
186C2×C59—21no—
187C2×C59—41no—
188C2×C59—41no—
189C2×C59—81no—
190C2×C59#146511yescharacteristic
191C2×C59C2×#35211yescharacteristic
192C2×C59#212011yescharacteristic
193C2×#123—591no—
194C2×C4×D4C2×D5911yescharacteristic
195C22×C42—591no—
196#432—1181no—
197#407—21no—
198#388—1181no—
199#387—1181no—
200#386—591no—
201#385—591no—
202#381—591no—
203C4×C4⋊C4—1181no—
204C4×C22⋊C4—591no—
205C4×C22⋊C4—1181no—
206#324—21no—
207#323—592no—
208#323—1181no—
209#323C2×D5912yes—
210C59#1152611yescharacteristic, complemented, Sylow
211C22×D4C4×D5911yescharacteristic
212C23×C4—591no—
213C23×C4—591no—
214C23×C4C4×D5911yescharacteristic
215C23×C4C22⋊D5911yescharacteristic
216C2×M16—41no—
217#123—1182no—
218#123—1182no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—1181no—
222C2×C4⋊C4—1181no—
223C2×C4⋊C4C22⋊D5911yescharacteristic
224C2×C22⋊C4—591no—
225C2×C22⋊C4—591no—
226C2×C22⋊C4—1181no—
227C2×C22⋊C4—1181no—
228C2×C22⋊C4D23611yescharacteristic
229C2×C22⋊C4C22×D5911yescharacteristic
230C2×C42—1181no—
231C2×C42—1181no—
232C2×C42—1181no—
233C2×C42—2361no—
234C2×C42—2361no—
235C2×C42D23611yescharacteristic
236C4⋊C8—2361no—
237C22⋊C8—22no—
238C22⋊C8—41no—
239C22⋊C8—1182no—
240#97—2361no—
241#95—1181no—
242#95—1181no—
243#95—1181no—
244#95—1181no—
245#95—1181no—
246C24(C2×D59)⋊C411yescharacteristic
247C24C2×C4×D5911yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—592no—
259C22×C4—1181no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4—1181no—
264C22×C4—2362no—
265C22×C4—2361no—
266C22×C4—2361no—
267C22×C4—2361no—
268C22×C4(C2×D59)⋊C411yescharacteristic
269C22×C4(C2×D59)⋊C411yescharacteristic
270C22×C4(C2×D59)⋊C411yescharacteristic
271C22×C4C2×D23611yescharacteristic
272C22×C4C2×C22⋊D5911yescharacteristic
273M16—82no—
274C2×C8—42no—
275C2×C8—2362no—
276C4⋊C4—41no—
277C4⋊C4—1184no—
278C4⋊C4—2362no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—592no—
282C22⋊C4—1184no—
283C22⋊C4—1181no—
284C22⋊C4—2362no—
285C22⋊C4—2361no—
286C42—21no—
287C42—21no—
288C42—2362no—
289C42—2362no—
290C42—2361no—
291C42—2361no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23C2×(C2×D59)⋊C411yescharacteristic
302C23M16×D5911yescharacteristic
303C23M16×D5911yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—1182no—
321C2×C4—1182no—
322C2×C4—1182no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2362no—
330C2×C4—2362no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4—2362no—
335C2×C4—2362no—
336C2×C4#1427364111yescharacteristic
337C2×C4#1427364111yescharacteristic
338C8—82no—
339C8—4722no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22#1683736511yescharacteristic
355C22C2×#1427364111yescharacteristic
356C22#1683738411yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—4721no—
363C4—2362no—
364C4—2362no—
365C4—2362no—
366C4#1683719112yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—81no—
372C2#2142243211yescharacteristic
373C2C2×#1683719111yescharacteristic
374C2#2142573811yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1465C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C22⋊C4×C59C23
hypercenter#1465D59
hypocenterC59#11526
LCSG ▷ C24×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. #21422045C2semidirect
4. #21422025C2semidirect
84. C22×Dic59M16semidirect
96. C22⋊Dic59M16semidirect
210. C59#11526semidirect

Maximal subgroups

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

subgroupindex
C4×#142737082
#214224372
#214220452
#214220252
C59×#14652
#1152659