Finite Group Databasebeta

← #28499709

Subgroups — #28499709

Order 15104.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#21428404C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422097C214yesmaximal, complemented
4C59×#1460C211yesmaximal, characteristic
5C2×#14273708—21no—
6C22×C4⋊Dic59C411yescharacteristic, complemented
7C2×#14273642—21no—
8#16837332C412yescomplemented
9#16837327C2212yes—
10#16837300—21no—
11#16837263—21no—
12#16837262—21no—
13#16837262—21no—
14#16837262C411yescharacteristic, complemented
15#16837169C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#431—21no—
18C59×#323—21no—
19C59×#323C2212yes—
20C23×Dic59—21no—
21#14273708—42no—
22#14273708—42no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—41no—
26C2×C4⋊Dic59—41no—
27C2×C4⋊Dic59—41no—
28C2×C4⋊Dic59C2×C412yes—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33#14273642—42no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—41no—
37C59⋊#95C2×C412yes—
38C22×D4×C59D411yescharacteristic
39C23×C4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2311yescharacteristic
46C2×C42×C59D411yescharacteristic
47C4⋊C8×C59—41no—
48C22⋊C8×C59—22no—
49C4×C8×C59—41no—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59D812yes—
55C2×D4×C59SD1612yes—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—22no—
68C22×Dic59—22no—
69C22×Dic59—42no—
70C22×Dic59—42no—
71C22×Dic59—42no—
72C22×Dic59—42no—
73C2×C8×C59—42no—
74C4⋊C4×C59—41no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C42×C59—21no—
78C42×C59—21no—
79C22⋊Dic59—42no—
80C22⋊Dic59—42no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C22⋊Dic59—42no—
84C4⋊Dic59—42no—
85C4⋊Dic59—41no—
86C4⋊Dic59—41no—
87C4⋊Dic59—42no—
88C4⋊Dic59—82no—
89Dic59⋊C4—44no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59#10212yes—
99C23×C59C2×C22⋊C411yescharacteristic
100C23×C59C2×D811yescharacteristic
101C23×C59C2×SD1611yescharacteristic
102D4×C59—42no—
103D4×C59—42no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—81no—
117C2×C4×C59#9911yescharacteristic
118C2×C4×C59#9911yescharacteristic
119C2×C4×C59#10212yes—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—84no—
127C2×Dic59—82no—
128C2×Dic59—82no—
129C2×Dic59—82no—
130C8×C59—82no—
131#11874—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—22no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—42no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—81no—
146C22×C59C2×#9911yescharacteristic
147C22×C59C2×#10211yescharacteristic
148C22×C59#42111yescharacteristic
149C4×C59—21no—
150C4×C59—22no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59#35212yes—
155Dic59—82no—
156Dic59—82no—
157Dic59—162no—
158#2486—591no—
159#1799—591no—
160#1460D5911yescharacteristic
161#1344—594no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—41no—
165C2×C59—42no—
166C2×C59—41no—
167C2×C59#151911yescharacteristic
168C2×C59C2×#35211yescharacteristic
169C2×C59#212911yescharacteristic
170C2×#123—1181no—
171C2×C4×D4C2×D5911yescharacteristic
172C22×C4⋊C4—591no—
173#431—21no—
174C2×#99—1181no—
175#398—592no—
176#397—592no—
177#389—1181no—
178#381—1181no—
179#380—591no—
180#380—1181no—
181#380—1181no—
182#342—592no—
183#323—21no—
184#323C2×D5912yes—
185C59#1187411yescharacteristic, complemented, Sylow
186C22×D4D23611yescharacteristic
187C23×C4—1181no—
188C23×C4C4×D5911yescharacteristic
189C23×C4D23611yescharacteristic
190#123—2362no—
191#123—2362no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—592no—
195C2×C4⋊C4—1182no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4—1181no—
198C2×C4⋊C4—2361no—
199C2×C4⋊C4—2361no—
200C2×C4⋊C4—2361no—
201C2×C4⋊C4C4×D5911yescharacteristic
202C2×C22⋊C4—1181no—
203C2×C22⋊C4—1182no—
204C2×C22⋊C4—1182no—
205C2×C22⋊C4C22⋊D5911yescharacteristic
206C2×C22⋊C4C22×D5911yescharacteristic
207C2×C42C22⋊D5911yescharacteristic
208C4⋊C8—41no—
209#99—2362no—
210C22⋊C8—22no—
211C4×C8—41no—
212#95—592no—
213#95—1182no—
214#95—1182no—
215#95—2361no—
216C24(C2×D59)⋊C411yescharacteristic
217C24C2×D23611yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#1174566012yes—
221C2×D4D47212yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—1182no—
229C22×C4—1182no—
230C22×C4—2362no—
231C22×C4—2362no—
232C22×C4—2362no—
233C22×C4—2362no—
234C22×C4(C2×D59)⋊C411yescharacteristic
235C22×C4(C2×D59)⋊C411yescharacteristic
236C22×C4(C2×D59)⋊C411yescharacteristic
237C22×C4C2×C4×D5911yescharacteristic
238C22×C4C2×C22⋊D5911yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—4722no—
242C4⋊C4—2362no—
243C4⋊C4—2362no—
244C4⋊C4—2361no—
245C4⋊C4—2361no—
246C4⋊C4—2364no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—2362no—
250C22⋊C4—2362no—
251C22⋊C4—2362no—
252C22⋊C4—2362no—
253C22⋊C4—2362no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#1427364912yes—
265C23C2×(C2×D59)⋊C411yescharacteristic
266C23C2×D47211yescharacteristic
267C23C2×#1174566011yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—4722no—
284C2×C4—4722no—
285C2×C4—4722no—
286C2×C4—4724no—
287C2×C4—2362no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4—2362no—
292C2×C4—2362no—
293C2×C4#1427364111yescharacteristic
294C2×C4#1427364111yescharacteristic
295C2×C4#1427364912yes—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1427364111yescharacteristic
312C22C2×#1427364911yescharacteristic
313C22#1683741811yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—9442no—
320C4—4722no—
321C4—4722no—
322C4#1683719312yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2142266511yescharacteristic
329C2C2×#1683719311yescharacteristic
330C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1460C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1460D59
hypocenterC59#11874
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #21422097C2semidirect
6. C22×C4⋊Dic59C4semidirect
8. #16837332C4semidirect
14. #16837262C4semidirect
185. C59#11874semidirect

Maximal subgroups

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

subgroupindex
#214284042
#214240842
#214220972
C59×#14602
#1187459