Finite Group Databasebeta

← #28500013

Subgroups — #28500013

Order 15104.

354 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 91 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#21434676C211yesmaximal, characteristic
2#21425894C211yesmaximal, characteristic, complemented
3#335×D59C211yesmaximal, characteristic, complemented
4#21422151C211yesmaximal, characteristic, complemented
5#21422143C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic, complemented
7C59×#1364C211yesmaximal, characteristic, complemented
8#16837953—21no—
9#16837779C2211yescharacteristic
10#16837750—21no—
11#16837639—21no—
12#16837639C412yes—
13#16837484—21no—
14#16837484C412yes—
15C4⋊C8×D59—21no—
16C4×C8×D59C2211yescharacteristic
17C4⋊(C59⋊C16)C2211yescharacteristic
18(C2×D59)⋊C16—21no—
19C16×Dic59—21no—
20#16837155C2211yescharacteristic
21#16837154—21no—
22C59×#500C2211yescharacteristic
23C59×C4⋊C16C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×#11745689—41no—
26C2×D472—21no—
27C2×D472C2×C412yes—
28C2×C8×D59—41no—
29C2×C8×D59C2×C411yescharacteristic
30C2×C8×D59D411yescharacteristic
31#14273724—21no—
32#14273716—41no—
33#14273702—22no—
34#14273702—41no—
35#14273689—22no—
36#14273689—41no—
37C42×D59D411yescharacteristic
38C2×C59⋊C16—21no—
39C4⋊(C59⋊C8)—21no—
40Dic59⋊C8—41no—
41#14273647—22no—
42#14273647—22no—
43C4×C59⋊C8D411yescharacteristic
44C8×Dic59C2×C411yescharacteristic
45C8×Dic59D411yescharacteristic
46C4⋊Q8×C59—21no—
47C2×C16×C59—21no—
48#107×C59—21no—
49#107×C59C2×C412yes—
50C4⋊C8×C59—21no—
51C4×C8×C59C2311yescharacteristic
52C2×Q8×C59—41no—
53#11745689—82no—
54#11745689—82no—
55C2×D236—22no—
56C2×D236—42no—
57C2×D236—41no—
58C2×D236—41no—
59C2×C4×D59—42no—
60C2×C4×D59C22⋊C411yescharacteristic
61C2×C4×D59D811yescharacteristic
62C2×C4×D59SD1611yescharacteristic
63C2×C8×C59—41no—
64C2×C8×C59C22×C411yescharacteristic
65C2×C8×C59C2×D411yescharacteristic
66C4⋊C4×C59—22no—
67C4⋊C4×C59—41no—
68C42×C59C2×D411yescharacteristic
69(C2×D59)⋊C4—42no—
70(C2×D59)⋊C4—42no—
71C4×Dic59C22⋊C411yescharacteristic
72C4×Dic59D811yescharacteristic
73C4×Dic59SD1611yescharacteristic
74C2×C59⋊C8—41no—
75C2×C59⋊C8C22⋊C411yescharacteristic
76C2×C59⋊C8C22⋊C411yescharacteristic
77D472—42no—
78D472—41no—
79D472—41no—
80C8×D59—21no—
81C8×D59—21no—
82C8×D59—41no—
83C8×D59—82no—
84C16×C59—41no—
85C59⋊C16—41no—
86C22×D59—42no—
87C22×D59#10211yescharacteristic
88Q8×C59—81no—
89Q8×C59—81no—
90C2×C4×C59—42no—
91C2×C4×C59C2×C22⋊C411yescharacteristic
92C2×C4×C59C2×D811yescharacteristic
93C2×C4×C59C2×SD1611yescharacteristic
94C2×Dic59#10211yescharacteristic
95C2×Dic59#10211yescharacteristic
96C2×Dic59#10211yescharacteristic
97D236—42no—
98D236—42no—
99D236—82no—
100D236—84no—
101D236—81no—
102D236—81no—
103C4×D59—21no—
104C4×D59—21no—
105C4×D59—41no—
106C4×D59—84no—
107C4×D59D1612yes—
108C4×D59SD3212yes—
109C8×C59—21no—
110C8×C59—81no—
111C8×C59#13511yescharacteristic
112C8×C59#13511yescharacteristic
113C59⋊C8—21no—
114C59⋊C8—21no—
115C59⋊C8—81no—
116#11947—591nomaximal, Sylow
117C22×C59C2×#10211yescharacteristic
118C2×D59—21no—
119C2×D59—21no—
120C2×D59—82no—
121C2×D59—82no—
122C2×D59—82no—
123C2×D59#35712yes—
124C4×C59—21no—
125C4×C59—82no—
126C4×C59#41311yescharacteristic
127C4×C59#41611yescharacteristic
128C4×C59C2×D1611yescharacteristic
129C4×C59C2×SD3211yescharacteristic
130Dic59—21no—
131Dic59—21no—
132Dic59#35712yes—
133#3169—591no—
134#2162—591no—
135C2×#335—591no—
136#1364—591no—
137#1364D5911yescharacteristic, complemented
138#1361—591no—
139#1361—591no—
140C2×C59#154011yescharacteristic
141C2×C59C2×#35711yescharacteristic
142C2×C59#214811yescharacteristic
143D59—22no—
144D59—162no—
145#532—1181no—
146#500C2×D5911yescharacteristic
147#493—591no—
148#486—1181no—
149#466—592no—
150#466—1181no—
151C4×D8—592no—
152C4×D8—1181no—
153C2×C4⋊C8—1181no—
154C2×C4×C8—591no—
155C4⋊C16—591no—
156C4⋊C16C2×D5911yescharacteristic
157C22⋊C16—1181no—
158C4×C16—1181no—
159#335—591no—
160#335—1181no—
161#335C2×D5911yescharacteristic
162C59#1194711yescharacteristic, complemented, Sylow
163C2×#41—2361no—
164C2×D8—592no—
165C2×D8—1181no—
166C22×C8—591no—
167C22×C8—591no—
168C22×C8—2361no—
169C4⋊Q8—21no—
170#127—1181no—
171#124—2361no—
172#121—1182no—
173#121—2361no—
174C4×D4—1182no—
175C4×D4—2361no—
176C2×C42—591no—
177C2×C16—21no—
178C2×C16—1181no—
179#107—21no—
180#107C4×D5912yes—
181C4⋊C8—21no—
182C4⋊C8—1181no—
183C4⋊C8—2361no—
184#102—1182no—
185#102—1182no—
186C4×C8—591no—
187C4×C8—591no—
188C4×C8—591no—
189C4×C8C22×D5911yescharacteristic
190#41—4722no—
191#41—4722no—
192C2×Q8—41no—
193C2×D4—1182no—
194C2×D4—2362no—
195C2×D4—2361no—
196C2×D4—2361no—
197C22×C4—591no—
198C22×C4—591no—
199C22×C4—591no—
200C22×C4—2362no—
201D8—2362no—
202D8—2361no—
203D8—2361no—
204C2×C8—41no—
205C2×C8—591no—
206C2×C8—591no—
207C2×C8—1181no—
208C2×C8—1181no—
209C2×C8—4722no—
210C2×C8—2361no—
211C2×C8—2361no—
212C2×C8C2×C4×D5911yescharacteristic
213C2×C8D4×D5911yescharacteristic
214C4⋊C4—22no—
215C4⋊C4—41no—
216C22⋊C4—2362no—
217C22⋊C4—2362no—
218C42—591no—
219C42—591no—
220C42—591no—
221C42D4×D5911yescharacteristic
222C16—41no—
223C16—2361no—
224C23—591no—
225C23—2362no—
226Q8—81no—
227Q8—81no—
228D4—4722no—
229D4—4724no—
230D4—4721no—
231D4—4721no—
232D4—2362no—
233D4—2362no—
234C2×C4—42no—
235C2×C4—591no—
236C2×C4—592no—
237C2×C4—592no—
238C2×C4—591no—
239C2×C4—591no—
240C2×C4—1181no—
241C2×C4—1181no—
242C2×C4—4724no—
243C2×C4—2361no—
244C2×C4C22⋊C4×D5911yescharacteristic
245C2×C4D8×D5911yescharacteristic
246C2×C4SD16×D5911yescharacteristic
247C8—21no—
248C8—81no—
249C8—1181no—
250C8—1181no—
251C8—4721no—
252C8#1427374811yescharacteristic
253C8#1427375011yescharacteristic
254C22—592no—
255C22—1181no—
256C22—1181no—
257C22—4722no—
258C22—4722no—
259C22—4722no—
260C22#102×D5911yescharacteristic
261C4—21no—
262C4—82no—
263C4—592no—
264C4—1181no—
265C4—1181no—
266C4#1683738511yescharacteristic
267C4#1683739711yescharacteristic
268C4D16×D5911yescharacteristic
269C4SD32×D5911yescharacteristic
270C2—1182no—
271C2—9442no—
272C2#2142274411yescharacteristic
273C2#357×D5911yescharacteristic
274C2#2142584411yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1364C2
socleC22×C59C2×#102
prime radical#21434676C2
cosocleC4×C8×C59C23
hypercenter#1364D59
hypocenterC59#11947
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
2. #21425894C2semidirect
3. #335×D59C2semidirect
4. #21422151C2semidirect
5. #21422143C2semidirect
6. #21422141C2semidirect
7. C59×#1364C2semidirect
137. #1364D59semidirect
162. C59#11947semidirect

Maximal subgroups

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

subgroupindex
#214346762
#214258942
#335×D592
#214221512
#214221432
#214221412
C59×#13642
#1194759