Finite Group Databasebeta

← #28497990

Subgroups — #28497990

Order 15104.

382 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 83 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#21432789C211yesmaximal, characteristic
2#21425715C212yesmaximal, complemented
3#21422049C211yesmaximal, characteristic, complemented
4#21422031C212yesmaximal, complemented
5C59×#1328C211yesmaximal, characteristic, complemented
6C2×#14273735C2211yescharacteristic
7C22×C8×D59C411yescharacteristic
8#16837914C411yescharacteristic
9M32×D59—22no—
10#16837472—22no—
11#16837467—22no—
12#16837368C411yescharacteristic
13#16837365C2211yescharacteristic
14#16837358C411yescharacteristic
15#16837184C2212yes—
16#16837183—22no—
17C59×#424C2211yescharacteristic
18C59×#349C2212yes—
19C2×#11745685C2×C411yescharacteristic
20C22×C4×D59C2×C411yescharacteristic
21#14273735—41no—
22#14273735—41no—
23C2×#11745664C2×C411yescharacteristic
24C2×#11745659D412yes—
25C22×C59⋊C8C2×C411yescharacteristic
26C2×C8×D59—21no—
27C2×C8×D59—21no—
28C2×C8×D59—41no—
29C2×C8×D59—41no—
30C2×C8×D59D412yes—
31#14273711C812yes—
32#14273707—21no—
33#14273702—21no—
34#14273697C812yes—
35C4×D236—22no—
36#14273667—22no—
37#14273666—42no—
38C16×D59—42no—
39(C2×D59)⋊C8—22no—
40(C2×D59)⋊C8—22no—
41#14273638—21no—
42C4×C59⋊C8—21no—
43C2×M16×C59C2311yescharacteristic
44C22×C8×C59C2×C411yescharacteristic
45#117×C59C2×C411yescharacteristic
46M32×C59—22no—
47C4⋊C8×C59—22no—
48C23×D59C2×C811yescharacteristic
49C22×C4×C59C22×C411yescharacteristic
50C2×C22⋊D59C2×C812yes—
51C22×Dic59C2×C811yescharacteristic
52#11745685—41no—
53#11745685—41no—
54C2×D236—22no—
55C2×D236M1611yescharacteristic
56C2×C4×D59—21no—
57C2×C4×D59—21no—
58C2×C4×D59—41no—
59C2×C4×D59—41no—
60C2×C4×D59C22⋊C412yes—
61C2×Dic118M1611yescharacteristic
62M16×C59—41no—
63C2×C8×C59—21no—
64C2×C8×C59—41no—
65C2×C8×C59C2×D412yes—
66C4⋊C4×C59—21no—
67C22⋊C4×C59C2×C812yes—
68C42×C59—21no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—22no—
71C4⋊Dic59—22no—
72#11745664—41no—
73C2×C59⋊C8—21no—
74C2×C59⋊C8—21no—
75C2×C59⋊C8—21no—
76C2×C59⋊C8C22⋊C411yescharacteristic
77C2×C59⋊C8C22⋊C411yescharacteristic
78#11745659—42no—
79C8×D59—22no—
80C8×D59—22no—
81C8×D59—42no—
82C8×D59—41no—
83C8×D59—41no—
84C16×C59—42no—
85C59⋊C16—42no—
86C23×C59C22×C811yescharacteristic
87C22×D59—21no—
88C22×D59—42no—
89C22×D59—41no—
90C22×D59C22⋊C812yes—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—22no—
94C2×C4×C59C2×C22⋊C411yescharacteristic
95C2×C4×C59C2×M1611yescharacteristic
96C22⋊D59—42no—
97C2×Dic59—21no—
98C2×Dic59—41no—
99C2×Dic59C22⋊C812yes—
100D236—41no—
101D236—44no—
102C4×D59—22no—
103C4×D59—22no—
104C4×D59—42no—
105C4×D59—41no—
106C4×D59—41no—
107Dic118—41no—
108C8×C59—22no—
109C8×C59—41no—
110C59⋊C8—21no—
111C59⋊C8—41no—
112C59⋊C8#9911yescharacteristic
113C59⋊C8#9911yescharacteristic
114#11483—591nomaximal, Sylow
115C22×C59—21no—
116C22×C59—41no—
117C22×C59C2×C22⋊C811yescharacteristic
118C2×D59—22no—
119C2×D59—22no—
120C2×D59—41no—
121C2×D59—42no—
122C2×D59—41no—
123C2×D59—42no—
124C2×D59—42no—
125C4×C59—21no—
126C4×C59—42no—
127C4×C59C2×#9911yescharacteristic
128C4×C59#41311yescharacteristic
129Dic59—22no—
130Dic59—41no—
131#2929—591no—
132#2115—592no—
133#1328—591no—
134#1328D5911yescharacteristic, complemented
135#1320—592no—
136C2×C59—21no—
137C2×C59—41no—
138C2×C59#145911yescharacteristic
139D59—42no—
140D59—42no—
141C2×#131—591no—
142C23×C8—591no—
143#525—591no—
144C2×M32—1182no—
145#435—1182no—
146C8×D4—1182no—
147#424C2×D5911yescharacteristic
148#408—591no—
149#407—591no—
150#405—591no—
151#349—592no—
152#349—1182no—
153#349C2×D5912yes—
154C59#1148311yescharacteristic, complemented, Sylow
155C2×#41—591no—
156C23×C4—591no—
157#131—2361no—
158#131—2361no—
159C2×M16—592no—
160C2×M16—591no—
161C2×M16C22×D5911yescharacteristic
162C22×C8—591no—
163C22×C8—592no—
164C22×C8—1181no—
165C22×C8—1181no—
166C22×C8—2361no—
167C22×C8—2361no—
168C22×C8C4×D5911yescharacteristic
169#123—592no—
170C22⋊Q8—1181no—
171#121—1181no—
172#120—592no—
173C4×D4—1182no—
174#117C4×D5911yescharacteristic
175M32—22no—
176M32—1182no—
177M32—2362no—
178C2×C16—2362no—
179C4⋊C8—22no—
180C22⋊C8—1182no—
181C22⋊C8—1182no—
182#97—1181no—
183C4×C8—1181no—
184C24—591no—
185#41—2361no—
186#41—2361no—
187C2×Q8—591no—
188C2×D4—592no—
189C2×D4—591no—
190C2×D4—1182no—
191C22×C4—592no—
192C22×C4—591no—
193C22×C4—1181no—
194C22×C4—1181no—
195C22×C4—2361no—
196C22×C4—2361no—
197C22×C4C2×C4×D5911yescharacteristic
198M16—41no—
199M16—2362no—
200M16—2361no—
201C2×C8—21no—
202C2×C8—41no—
203C2×C8—591no—
204C2×C8—591no—
205C2×C8—1182no—
206C2×C8—1182no—
207C2×C8—1181no—
208C2×C8—1181no—
209C2×C8—1181no—
210C2×C8—2361no—
211C2×C8—2361no—
212C2×C8—2362no—
213C2×C8D4×D5912yes—
214C4⋊C4—21no—
215C4⋊C4—1182no—
216C22⋊C4—1182no—
217C22⋊C4—1182no—
218C22⋊C4C8×D5912yes—
219C42—21no—
220C16—42no—
221C16—2362no—
222C23—592no—
223C23—1181no—
224C23—2361no—
225C23—2362no—
226C23C2×C8×D5911yescharacteristic
227Q8—2361no—
228D4—2361no—
229D4—2364no—
230D4—2362no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—592no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—1181no—
238C2×C4—2362no—
239C2×C4—2361no—
240C2×C4—2361no—
241C2×C4—2361no—
242C2×C4C22⋊C4×D5911yescharacteristic
243C2×C4M16×D5911yescharacteristic
244C8—22no—
245C8—41no—
246C8—591no—
247C8—591no—
248C8—1181no—
249C8—2361no—
250C22—21no—
251C22—41no—
252C22—1182no—
253C22—1182no—
254C22—2362no—
255C22—2361no—
256C22—2362no—
257C22—2362no—
258C22—2361no—
259C22C22⋊C8×D5911yescharacteristic
260C4—21no—
261C4—42no—
262C4—1182no—
263C4—2361no—
264C4#99×D5911yescharacteristic
265C4#1683738511yescharacteristic
266C2—21no—
267C2—41no—
268C2—2362no—
269C2—2362no—
270C2#2142241311yescharacteristic
subgroupG/H
centerC4#99×D59
commutatorC23×C59C22×C8
FrattiniC2×M16C22×D59
FittingC59×#1328C2
socleC2×C59#1459
prime radical#16837914C4
cosocleC2×M16×C59C23
hypercenter#1328D59
hypocenterC59#11483
LCSG ▷ C23×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1328

Complements

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

NH ≅ G/Ntype
2. #21425715C2semidirect
3. #21422049C2semidirect
4. #21422031C2semidirect
5. C59×#1328C2semidirect
134. #1328D59semidirect
154. C59#11483semidirect

Maximal subgroups

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

subgroupindex
#214327892
#214257152
#214220492
#214220312
C59×#13282
#1148359