Finite Group Databasebeta

← #28500165

Subgroups — #28500165

Order 15104.

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

normal subgroups only

This group has 358 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
2C4⋊C16×D59C211yesmaximal, characteristic, complemented
3#21422853C211yesmaximal, characteristic, complemented
4#21422151C211yesmaximal, characteristic, complemented
5#21422143C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic, complemented
7C59×#1364C211yesmaximal, characteristic, complemented
8#16837953—21no—
9C2×C16×D59—21no—
10#16837779C2211yescharacteristic
11#16837750—21no—
12#16837639—21no—
13#16837639C412yes—
14#16837484—21no—
15#16837484C412yes—
16#16837424—21no—
17C4×C8×D59C2211yescharacteristic
18C4⋊(C59⋊C16)C2211yescharacteristic
19Dic59⋊C16—21no—
20#16837155C2211yescharacteristic
21#16837154—21no—
22C59×#500C2211yescharacteristic
23C59×C4⋊C16C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×#11745689—41no—
26C2×D472—21no—
27C2×D472C2×C412yes—
28C2×#11745659—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—
39C16×D59—42no—
40C4⋊(C59⋊C8)—21no—
41Dic59⋊C8—41no—
42#14273647—22no—
43#14273647—22no—
44C4×C59⋊C8D411yescharacteristic
45C8×Dic59C2×C411yescharacteristic
46C8×Dic59D411yescharacteristic
47C4⋊Q8×C59—21no—
48C2×C16×C59—21no—
49#107×C59—21no—
50#107×C59C2×C412yes—
51C4⋊C8×C59—21no—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59—41no—
54#11745689—82no—
55#11745689—82no—
56C2×D236—22no—
57C2×D236—42no—
58C2×D236—41no—
59C2×D236—41no—
60C2×C4×D59—42no—
61C2×C4×D59C22⋊C411yescharacteristic
62C2×C4×D59D811yescharacteristic
63C2×C4×D59SD1611yescharacteristic
64C2×C8×C59—41no—
65C2×C8×C59C22×C411yescharacteristic
66C2×C8×C59C2×D411yescharacteristic
67C4⋊C4×C59—22no—
68C4⋊C4×C59—41no—
69C42×C59C2×D411yescharacteristic
70(C2×D59)⋊C4—42no—
71(C2×D59)⋊C4—42no—
72C4×Dic59C22⋊C411yescharacteristic
73C4×Dic59D811yescharacteristic
74C4×Dic59SD1611yescharacteristic
75C2×C59⋊C8—41no—
76C2×C59⋊C8C22⋊C411yescharacteristic
77C2×C59⋊C8C22⋊C411yescharacteristic
78D472—42no—
79D472—41no—
80D472—41no—
81#11745659—82no—
82C8×D59—21no—
83C8×D59—21no—
84C8×D59—41no—
85C16×C59—41no—
86C59⋊C16—41no—
87C22×D59—42no—
88C22×D59#10211yescharacteristic
89Q8×C59—81no—
90Q8×C59—81no—
91C2×C4×C59—42no—
92C2×C4×C59C2×C22⋊C411yescharacteristic
93C2×C4×C59C2×D811yescharacteristic
94C2×C4×C59C2×SD1611yescharacteristic
95C2×Dic59#10211yescharacteristic
96C2×Dic59#10211yescharacteristic
97C2×Dic59#10211yescharacteristic
98D236—42no—
99D236—42no—
100D236—82no—
101D236—84no—
102D236—81no—
103D236—81no—
104C4×D59—21no—
105C4×D59—21no—
106C4×D59—41no—
107C4×D59—84no—
108C4×D59D1612yes—
109C4×D59SD3212yes—
110C8×C59—21no—
111C8×C59—81no—
112C8×C59AGL(1,ℤ/8)11yescharacteristic
113C8×C59#13711yescharacteristic
114C59⋊C8—21no—
115C59⋊C8—21no—
116C59⋊C8—81no—
117#11981—591nomaximal, Sylow
118C22×C59C2×#10211yescharacteristic
119C2×D59—21no—
120C2×D59—21no—
121C2×D59—82no—
122C2×D59—82no—
123C2×D59—82no—
124C2×D59#35712yes—
125C4×C59—21no—
126C4×C59—82no—
127C4×C59#41311yescharacteristic
128C4×C59#41711yescharacteristic
129C4×C59C2×D1611yescharacteristic
130C4×C59C2×SD3211yescharacteristic
131Dic59—21no—
132Dic59—21no—
133Dic59#35712yes—
134#3169—591no—
135C2×C4⋊C16—591no—
136#1567—591no—
137#1364—591no—
138#1364D5911yescharacteristic, complemented
139#1361—591no—
140#1361—591no—
141C2×C59#154711yescharacteristic
142C2×C59C2×#35711yescharacteristic
143C2×C59#214811yescharacteristic
144D59—22no—
145D59—162no—
146#532—1181no—
147C22×C16—1181no—
148#500C2×D5911yescharacteristic
149#493—591no—
150#486—1181no—
151#466—592no—
152#466—1181no—
153C4×D8—592no—
154C4×D8—1181no—
155#423—1181no—
156C2×C4×C8—591no—
157C4⋊C16—591no—
158C4⋊C16—1181no—
159C4⋊C16C2×D5911yescharacteristic
160#335—591no—
161#335—1181no—
162#335C2×D5911yescharacteristic
163C59#1198111yescharacteristic, complemented, Sylow
164C2×#41—2361no—
165C2×D8—592no—
166C2×D8—1181no—
167C2×M16—2361no—
168C22×C8—591no—
169C22×C8—591no—
170C4⋊Q8—21no—
171#127—1181no—
172#124—2361no—
173#121—1182no—
174#121—2361no—
175C4×D4—1182no—
176C4×D4—2361no—
177C2×C42—591no—
178C2×C16—21no—
179C2×C16—1181no—
180C2×C16—2362no—
181#107—21no—
182#107C4×D5912yes—
183C4⋊C8—21no—
184C4⋊C8—1181no—
185C4⋊C8—2361no—
186#102—1182no—
187#102—1182no—
188C4×C8—591no—
189C4×C8—591no—
190C4×C8—591no—
191C4×C8C22×D5911yescharacteristic
192#41—4722no—
193#41—4722no—
194C2×Q8—41no—
195C2×D4—1182no—
196C2×D4—2362no—
197C2×D4—2361no—
198C2×D4—2361no—
199C22×C4—591no—
200C22×C4—591no—
201C22×C4—591no—
202C22×C4—2362no—
203D8—2362no—
204D8—2361no—
205D8—2361no—
206M16—4722no—
207C2×C8—41no—
208C2×C8—591no—
209C2×C8—591no—
210C2×C8—1181no—
211C2×C8—1181no—
212C2×C8—2361no—
213C2×C8—2361no—
214C2×C8C2×C4×D5911yescharacteristic
215C2×C8D4×D5911yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—41no—
218C22⋊C4—2362no—
219C22⋊C4—2362no—
220C42—591no—
221C42—591no—
222C42—591no—
223C42D4×D5911yescharacteristic
224C16—41no—
225C16—2361no—
226C23—591no—
227C23—2362no—
228Q8—81no—
229Q8—81no—
230D4—4722no—
231D4—4724no—
232D4—4721no—
233D4—4721no—
234D4—2362no—
235D4—2362no—
236C2×C4—42no—
237C2×C4—591no—
238C2×C4—592no—
239C2×C4—592no—
240C2×C4—591no—
241C2×C4—591no—
242C2×C4—1181no—
243C2×C4—1181no—
244C2×C4—4724no—
245C2×C4—2361no—
246C2×C4C22⋊C4×D5911yescharacteristic
247C2×C4D8×D5911yescharacteristic
248C2×C4SD16×D5911yescharacteristic
249C8—21no—
250C8—81no—
251C8—1181no—
252C8—1181no—
253C8—4721no—
254C8#1427375411yescharacteristic
255C8#1427375911yescharacteristic
256C22—592no—
257C22—1181no—
258C22—1181no—
259C22—4722no—
260C22—4722no—
261C22—4722no—
262C22#102×D5911yescharacteristic
263C4—21no—
264C4—82no—
265C4—592no—
266C4—1181no—
267C4—1181no—
268C4#1683738511yescharacteristic
269C4#1683740211yescharacteristic
270C4D16×D5911yescharacteristic
271C4SD32×D5911yescharacteristic
272C2—1182no—
273C2—9442no—
274C2#2142277211yescharacteristic
275C2#357×D5911yescharacteristic
276C2#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#11981
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. C4⋊C16×D59C2semidirect
3. #21422853C2semidirect
4. #21422151C2semidirect
5. #21422143C2semidirect
6. #21422141C2semidirect
7. C59×#1364C2semidirect
138. #1364D59semidirect
163. C59#11981semidirect

Maximal subgroups

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

subgroupindex
#214346762
C4⋊C16×D592
#214228532
#214221512
#214221432
#214221412
C59×#13642
#1198159