Finite Group Databasebeta

← #28499004

Subgroups — #28499004

Order 15104.

432 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 115 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#21423835C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic
3#21422123C214yesmaximal, complemented
4C59×#2628C211yesmaximal, characteristic
5C22×#11745664—21no—
6#16837455—21no—
7#16837171—21no—
8#16837171C2212yes—
9#16837128C2212yes—
10#16837126—21no—
11C2×C4⋊Q8×C59C411yescharacteristic
12C2×#117×C59C2211yescharacteristic
13C59×#400C2212yes—
14C59×#399C412yes—
15C59×#395—21no—
16C59×#393—21no—
17C59×#389—21no—
18C59×#382C411yescharacteristic
19C59×#380—21no—
20C2×#11745664—24no—
21C2×#11745664—22no—
22C2×#11745664—41no—
23C2×#11745664—42no—
24C22×C59⋊C8—41no—
25C4⋊(C59⋊C8)—41no—
26C22⋊(C59⋊C8)—22no—
27C4×C59⋊C8—41no—
28C22×Q8×C59—21no—
29C23×C4×C59D411yescharacteristic
30C4⋊Q8×C59—41no—
31C4⋊Q8×C59—41no—
32#117×C59—41no—
33#117×C59—41no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59C2×C412yes—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C4⋊C4×C59C2311yescharacteristic
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59C2×C411yescharacteristic
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59D411yescharacteristic
45#95×C59—21no—
46#95×C59—22no—
47#95×C59—22no—
48#95×C59C2×C412yes—
49C24×C59C22⋊C411yescharacteristic
50C2×Q8×C59—42no—
51C2×Q8×C59—82no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59SD1612yes—
61C22×C4×C59Q1612yes—
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C4⋊C4×C59—41no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—42no—
68C4⋊C4×C59—41no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—82no—
72C22⋊C4×C59—41no—
73C22⋊C4×C59—41no—
74C22⋊C4×C59—82no—
75C42×C59—21no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59—21no—
79#11745664—24no—
80#11745664—44no—
81#11745664—42no—
82C2×C59⋊C8—42no—
83C2×C59⋊C8—44no—
84C2×C59⋊C8—42no—
85C23×C59—21no—
86C23×C59—21no—
87C23×C59—41no—
88C23×C59—41no—
89C23×C59#10312yes—
90C23×C59C2×C22⋊C411yescharacteristic
91Q8×C59—84no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59#9911yescharacteristic
113C2×C4×C59#9911yescharacteristic
114C2×C4×C59#10312yes—
115C2×C4×C59#10412yes—
116C2×C4×C59C2×SD1611yescharacteristic
117C2×C4×C59C2×Q1611yescharacteristic
118C59⋊C8—44no—
119C59⋊C8—82no—
120#11723—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—82no—
138C4×C59—82no—
139C4×C59#32812yes—
140C4×C59#32812yes—
141#2628D5911yescharacteristic
142#1742—591no—
143#1458—591no—
144#1353—594no—
145C2×C59—21no—
146C2×C59—21no—
147C2×C59—42no—
148C2×C59C2×#32811yescharacteristic
149C2×C59#212311yescharacteristic
150C2×C59#213411yescharacteristic
151C22×M16—1181no—
152C2×C4⋊Q8Dic5911yescharacteristic
153C2×#117C2×D5911yescharacteristic
154#431—1181no—
155#400C2×D5912yes—
156#399Dic5912yes—
157#395—21no—
158#393—21no—
159#389—21no—
160#382Dic5911yescharacteristic
161#380—21no—
162#343—592no—
163#343—1181no—
164#324—592no—
165#323—1181no—
166C59#1172311yescharacteristic, complemented, Sylow
167C22×Q8—21no—
168C23×C4C22⋊D5911yescharacteristic
169C2×M16—1184no—
170C2×M16—1182no—
171C2×M16—2361no—
172C2×M16—2362no—
173C22×C8—2361no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2×Dic5912yes—
182C2×C4⋊C4C22⋊D5911yescharacteristic
183C2×C4⋊C4C22×D5911yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×Dic5911yescharacteristic
186C2×C22⋊C4C22⋊D5911yescharacteristic
187C2×C42C2×Dic5911yescharacteristic
188C2×C42C22⋊D5911yescharacteristic
189C4⋊C8—2361no—
190C22⋊C8—1182no—
191C4×C8—2361no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95C2×Dic5912yes—
196C24C22⋊Dic5911yescharacteristic
197C2×Q8—42no—
198C2×Q8—82no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4#1174567112yes—
205C22×C4#1174567212yes—
206C22×C4C22⋊Dic5911yescharacteristic
207C22×C4C22⋊Dic5911yescharacteristic
208C22×C4C22⋊Dic5911yescharacteristic
209C22×C4C22×Dic5911yescharacteristic
210C22×C4C2×C22⋊D5911yescharacteristic
211C22×C4C2×C22⋊D5911yescharacteristic
212M16—1184no—
213M16—2364no—
214M16—2362no—
215C2×C8—2362no—
216C2×C8—2362no—
217C2×C8—2364no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—82no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1427365112yes—
237C23C2×C22⋊Dic5911yescharacteristic
238Q8—84no—
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—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4#1427364211yescharacteristic
260C2×C4#1427364211yescharacteristic
261C2×C4#1427365112yes—
262C2×C4#1427365412yes—
263C2×C4C2×#1174567111yescharacteristic
264C2×C4C2×#1174567211yescharacteristic
265C8—4722no—
266C8—2364no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1427364211yescharacteristic
275C22C2×#1427365111yescharacteristic
276C22C2×#1427365411yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4#1683713912yes—
286C4#1683713912yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1683713911yescharacteristic
291C2#2142575011yescharacteristic
292C2#2142578811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2628C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2628D59
hypocenterC59#11723
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #21422123C2semidirect
166. C59#11723semidirect

Maximal subgroups

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

subgroupindex
#214238352
#214224102
#214221232
C59×#26282
#1172359