Finite Group Databasebeta

← #28498068

Subgroups — #28498068

Order 15104.

598 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#16837307C211yesmaximal, characteristic
2#21422426C211yesmaximal, characteristic
3#21422029C214yesmaximal, complemented
4C59×#1463C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic
6C22×C4×Dic59C411yescharacteristic
7#16837366—21no—
8#16837307—21no—
9#16837307—21no—
10#16837307C412yes—
11#16837307C2212yes—
12C2×C59⋊#95—21no—
13C2×C59⋊#95—21no—
14#16837128—21no—
15#16837128C2212yes—
16C22×C4⋊C4×C59C2211yescharacteristic
17C59×#407—21no—
18C59×#324—21no—
19C59×#324C2212yes—
20C22×Dic118—21no—
21C22×Dic118—24no—
22C22×Dic118—41no—
23C22×Dic118—41no—
24C22×Dic118—42no—
25C22×Dic118C2×C412yes—
26C23×Dic59—21no—
27C23×Dic59—21no—
28C2×#11745664—41no—
29C2×C4×Dic59—21no—
30C2×C4×Dic59—42no—
31C2×C4×Dic59C2×C412yes—
32C22⋊(C59⋊C8)—22no—
33C22⋊(C59⋊C8)—41no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C23×C4×C59C2×C411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C2×M16×C59—41no—
42C2×C4⋊C4×C59—41no—
43C2×C4⋊C4×C59—41no—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59C2311yescharacteristic
48C22⋊C8×C59—22no—
49C22⋊C8×C59—41no—
50C24×C59C22⋊C411yescharacteristic
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—22no—
54C22×C4×C59—22no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—24no—
68C22×Dic59—24no—
69C22×Dic59—22no—
70C22×Dic59—22no—
71C22×Dic59—42no—
72C22×Dic59—42no—
73C22×Dic59—42no—
74C22×Dic59—41no—
75C22×Dic59—41no—
76C2×Dic118—24no—
77C2×Dic118—22no—
78C2×Dic118—42no—
79C2×Dic118—48no—
80C2×Dic118—48no—
81C2×Dic118—42no—
82C2×Dic118—42no—
83C2×Dic118—42no—
84C2×Dic118M1614yescomplemented
85M16×C59—82no—
86C2×C8×C59—42no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4×Dic59—42no—
92C4×Dic59—42no—
93#11745664—82no—
94C2×C59⋊C8—42no—
95C23×C59—22no—
96C23×C59—22no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59#9911yescharacteristic
100C23×C59#9911yescharacteristic
101C23×C59C2×C22⋊C411yescharacteristic
102C2×C4×C59—24no—
103C2×C4×C59—24no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—81no—
119C2×C4×C59—81no—
120C2×C4×C59#10012yes—
121C2×C4×C59C2×M1612yes—
122C2×Dic59—24no—
123C2×Dic59—24no—
124C2×Dic59—44no—
125C2×Dic59—44no—
126C2×Dic59—48no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131Dic118—48no—
132Dic118—44no—
133Dic118—44no—
134C8×C59—82no—
135C59⋊C8—82no—
136#11506—591nomaximal, Sylow
137C22×C59—24no—
138C22×C59—24no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59#35412yes—
146C22×C59#35612yes—
147C22×C59#40711yescharacteristic
148C22×C59C2×#9911yescharacteristic
149C22×C59C2×#10011yescharacteristic
150C4×C59—22no—
151C4×C59—22no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59—41no—
155C4×C59—41no—
156Dic59—44no—
157Dic59—44no—
158C2×#391—591no—
159#1463—591no—
160#1463D5911yescharacteristic
161#1319—594no—
162C2×C59—21no—
163C2×C59—22no—
164C2×C59—22no—
165C2×C59—21no—
166C2×C59#146211yescharacteristic
167C2×C59C2×#35411yescharacteristic
168C2×C59C2×#35611yescharacteristic
169C23×Q8—591no—
170C22×C4⋊C4C2×D5911yescharacteristic
171C22×C42—591no—
172#407—21no—
173#407—1181no—
174#391—592no—
175#391—592no—
176#391—1181no—
177#391—1181no—
178C2×#95—1181no—
179C2×#95—1181no—
180#324—21no—
181#324—592no—
182#324—1181no—
183#324C2×D5912yes—
184C59#1150611yescharacteristic, complemented, Sylow
185C22×Q8—592no—
186C22×Q8—1184no—
187C22×Q8—1181no—
188C22×Q8—2362no—
189C22×Q8—2361no—
190C22×Q8—2361no—
191C23×C4—1181no—
192C23×C4—1181no—
193C23×C4C4×D5911yescharacteristic
194C23×C4D23611yescharacteristic
195C23×C4C22⋊D5911yescharacteristic
196C2×M16—41no—
197C2×M16—2361no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C4×D5911yescharacteristic
201C2×C4⋊C4D23611yescharacteristic
202C2×C4⋊C4C22⋊D5911yescharacteristic
203C2×C4⋊C4C22×D5911yescharacteristic
204C2×C42—592no—
205C2×C42—1181no—
206C2×C42—2362no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8—1182no—
210C22⋊C8—2361no—
211#95—1182no—
212#95—1182no—
213#95—1182no—
214#95—1182no—
215C24(C2×D59)⋊C411yescharacteristic
216C2×Q8—594no—
217C2×Q8—1184no—
218C2×Q8—1182no—
219C2×Q8—2368no—
220C2×Q8—2368no—
221C2×Q8—2362no—
222C2×Q8—2362no—
223C2×Q8—2362no—
224C2×Q8—2362no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1184no—
236C22×C4—1184no—
237C22×C4—1182no—
238C22×C4—1182no—
239C22×C4—2362no—
240C22×C4—2362no—
241C22×C4—2362no—
242C22×C4—2361no—
243C22×C4—2361no—
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4(C2×D59)⋊C411yescharacteristic
246C22×C4(C2×D59)⋊C411yescharacteristic
247C22×C4C2×C4×D5911yescharacteristic
248C22×C4C2×D23611yescharacteristic
249C22×C4C2×C22⋊D5911yescharacteristic
250M16—82no—
251M16—4722no—
252C2×C8—42no—
253C2×C8—2362no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C42—2362no—
259C42—2362no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—21no—
264C23#1427364111yescharacteristic
265C23#1427364111yescharacteristic
266C23C2×(C2×D59)⋊C411yescharacteristic
267Q8—2364no—
268Q8—2368no—
269Q8—2364no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—1184no—
289C2×C4—1184no—
290C2×C4—2368no—
291C2×C4—2362no—
292C2×C4—2364no—
293C2×C4—2364no—
294C2×C4—2362no—
295C2×C4—2362no—
296C2×C4—2362no—
297C2×C4#1427364312yes—
298C2×C4M16×D5912yes—
299C8—82no—
300C8—4722no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22#1683719612yes—
310C22#1683720012yes—
311C22#1683736511yescharacteristic
312C22C2×#1427364111yescharacteristic
313C22C2×#1427364311yescharacteristic
314C4—22no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—2364no—
321C4—2364no—
322C2—21no—
323C2—22no—
324C2—22no—
325C2—21no—
326C2#2142242211yescharacteristic
327C2C2×#1683719611yescharacteristic
328C2C2×#1683720011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1463C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1463D59
hypocenterC59#11506
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #21422029C2semidirect
84. C2×Dic118M16semidirect
184. C59#11506semidirect

Maximal subgroups

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

subgroupindex
C2×#168373072
#214224262
#214220292
C59×#14632
#1150659