Finite Group Databasebeta

← #28499009

Subgroups — #28499009

Order 15104.

432 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 107 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#21429917C211yesmaximal, characteristic
2#21423835C211yesmaximal, characteristic
3#21422127C214yesmaximal, complemented
4C59×#1458C211yesmaximal, characteristic
5C22×#11745664—21no—
6C2×#14273727C411yescharacteristic
7#16837341—21no—
8#16837338—21no—
9#16837322C2212yes—
10#16837314C412yes—
11#16837300—21no—
12#16837268C411yescharacteristic
13#16837262—21no—
14#16837171—21no—
15#16837171C2212yes—
16C2×#117×C59C2211yescharacteristic
17C59×#431—21no—
18C59×#324C2212yes—
19C59×#323—21no—
20C22×Dic118C2×C412yes—
21C2×#11745664—24no—
22C2×#11745664—22no—
23C2×#11745664—41no—
24C2×#11745664—42no—
25C22×C59⋊C8—41no—
26#14273727—41no—
27#14273727—41no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59—21no—
31C2×Dic59⋊C4—22no—
32C2×C22⋊Dic59—22no—
33C59⋊#95—21no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95C2×C412yes—
37C23×C4×C59D411yescharacteristic
38#117×C59—41no—
39#117×C59—41no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C4⋊C4×C59C2311yescharacteristic
42C2×C22⋊C4×C59C2×C411yescharacteristic
43C2×C22⋊C4×C59D411yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C4⋊C8×C59—41no—
47C22⋊C8×C59—22no—
48C4×C8×C59—41no—
49C24×C59C22⋊C411yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—22no—
53C22×C4×C59C22⋊C411yescharacteristic
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59SD1612yes—
57C22×C4×C59Q1612yes—
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D411yescharacteristic
61C22×Dic59—22no—
62C22×Dic59—22no—
63C2×Dic118—42no—
64C2×Dic118—82no—
65C2×C8×C59—42no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C22⋊C4×C59—41no—
69C22⋊C4×C59—41no—
70C42×C59—21no—
71C42×C59—21no—
72C42×C59—21no—
73C42×C59—21no—
74C22⋊Dic59—82no—
75C4⋊Dic59—42no—
76C4⋊Dic59—41no—
77C4⋊Dic59—41no—
78C4⋊Dic59—42no—
79Dic59⋊C4—82no—
80#11745664—24no—
81#11745664—44no—
82#11745664—42no—
83C2×C59⋊C8—42no—
84C2×C59⋊C8—44no—
85C23×C59—21no—
86C23×C59—21no—
87C23×C59—41no—
88C23×C59—41no—
89C23×C59#10312yes—
90C23×C59C2×C22⋊C411yescharacteristic
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59#9911yescharacteristic
106C2×C4×C59#9911yescharacteristic
107C2×C4×C59#10312yes—
108C2×C4×C59#10412yes—
109C2×C4×C59C2×SD1611yescharacteristic
110C2×C4×C59C2×Q1611yescharacteristic
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—42no—
114C2×Dic59—42no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117Dic118—84no—
118C8×C59—82no—
119C59⋊C8—44no—
120#11724—591nomaximal, Sylow
121C22×C59—21no—
122C22×C59—21no—
123C22×C59—22no—
124C22×C59—42no—
125C22×C59—41no—
126C22×C59—41no—
127C22×C59—81no—
128C22×C59C2×#9911yescharacteristic
129C22×C59C2×#10311yescharacteristic
130C22×C59C2×#10411yescharacteristic
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—41no—
134C4×C59—41no—
135C4×C59—41no—
136C4×C59—41no—
137C4×C59#32812yes—
138C4×C59#32812yes—
139Dic59—82no—
140Dic59—82no—
141#2628—591no—
142#1742—591no—
143#1458D5911yescharacteristic
144#1354—594no—
145C2×C59—21no—
146C2×C59—21no—
147C2×C59—42no—
148C2×C59C2×#32811yescharacteristic
149C2×C59#212211yescharacteristic
150C2×C59#213511yescharacteristic
151C22×M16—1181no—
152C2×C4⋊Q8—591no—
153C2×#117C2×D5911yescharacteristic
154#431—21no—
155#400—1181no—
156#399—1181no—
157#395—592no—
158#393—592no—
159#389—1181no—
160#382—591no—
161#380—1181no—
162#343—592no—
163#343—1181no—
164#324C2×D5912yes—
165#323—21no—
166C59#1172411yescharacteristic, complemented, Sylow
167C22×Q8—592no—
168C23×C4D23611yescharacteristic
169C2×M16—1184no—
170C2×M16—1182no—
171C2×M16—2361no—
172C2×M16—2362no—
173C22×C8—2361no—
174C4⋊Q8—2361no—
175C4⋊Q8—2361no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—1182no—
179C2×C4⋊C4—1181no—
180C2×C4⋊C4—1181no—
181C2×C4⋊C4—1181no—
182C2×C4⋊C4D23611yescharacteristic
183C2×C4⋊C4C22×D5911yescharacteristic
184C2×C22⋊C4—1182no—
185C2×C22⋊C4C4×D5911yescharacteristic
186C2×C22⋊C4C22⋊D5911yescharacteristic
187C2×C42C4×D5911yescharacteristic
188C2×C42C22⋊D5911yescharacteristic
189C4⋊C8—41no—
190C22⋊C8—22no—
191C4×C8—41no—
192#95—592no—
193#95—1181no—
194#95—1182no—
195#95—1182no—
196C24(C2×D59)⋊C411yescharacteristic
197C2×Q8—4722no—
198C2×Q8—2362no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—1182no—
203C22×C4—1182no—
204C22×C4#1174566012yes—
205C22×C4Dic23612yes—
206C22×C4(C2×D59)⋊C411yescharacteristic
207C22×C4(C2×D59)⋊C411yescharacteristic
208C22×C4(C2×D59)⋊C411yescharacteristic
209C22×C4C2×C4×D5911yescharacteristic
210C22×C4C2×D23611yescharacteristic
211C22×C4C2×C22⋊D5911yescharacteristic
212M16—1184no—
213M16—2364no—
214M16—2362no—
215C2×C8—42no—
216C2×C8—2362no—
217C2×C8—2364no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—4722no—
221C4⋊C4—2362no—
222C4⋊C4—2361no—
223C4⋊C4—2361no—
224C4⋊C4—2362no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—4722no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1427365212yes—
237C23C2×(C2×D59)⋊C411yescharacteristic
238Q8—4724no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—2362no—
254C2×C4—2362no—
255C2×C4—2362no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4—2362no—
259C2×C4#1427364111yescharacteristic
260C2×C4#1427364111yescharacteristic
261C2×C4#1427365212yes—
262C2×C4#1427365312yes—
263C2×C4C2×#1174566011yescharacteristic
264C2×C4C2×Dic23611yescharacteristic
265C8—82no—
266C8—2364no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1427364111yescharacteristic
275C22C2×#1427365211yescharacteristic
276C22C2×#1427365311yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—4722no—
284C4—4722no—
285C4#1683714012yes—
286C4#1683714012yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1683714011yescharacteristic
291C2#2142574511yescharacteristic
292C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1458C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1458D59
hypocenterC59#11724
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #21422127C2semidirect
166. C59#11724semidirect

Maximal subgroups

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

subgroupindex
#214299172
#214238352
#214221272
C59×#14582
#1172459