Finite Group Databasebeta

← #28500200

Subgroups — #28500200

Order 15104.

308 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 85 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#21434683C211yesmaximal, characteristic
2#21425850C211yesmaximal, characteristic
3#21422853C211yesmaximal, characteristic
4#21422151C211yesmaximal, characteristic, complemented
5#21422146C211yesmaximal, characteristic, complemented
6#21422144C211yesmaximal, characteristic, complemented
7C59×#1364C211yesmaximal, characteristic, complemented
8#16837957—21no—
9C2×#14273666—21no—
10#16837782C2211yescharacteristic
11#16837754—21no—
12#16837644—21no—
13#16837644C412yes—
14#16837496—21no—
15#16837496C412yes—
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×Dic236—21no—
26C2×Dic236C2×C412yes—
27C2×#11745659—41no—
28C2×C8×D59C2×C411yescharacteristic
29C2×C8×D59D411yescharacteristic
30#14273727—21no—
31#14273718—41no—
32#14273707—22no—
33#14273703—41no—
34Q8×Dic59—41no—
35#14273692—22no—
36#14273685—41no—
37C42×D59D411yescharacteristic
38#14273666—42no—
39C2×C59⋊C16—21no—
40C4⋊(C59⋊C8)—21no—
41Dic59⋊C8—41no—
42#14273650—22no—
43#14273650—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—
54C2×C4×D59C22⋊C411yescharacteristic
55C2×C4×D59D811yescharacteristic
56C2×C4×D59SD1611yescharacteristic
57C2×Dic118—22no—
58C2×C8×C59—41no—
59C2×C8×C59C22×C411yescharacteristic
60C2×C8×C59C2×D411yescharacteristic
61C4⋊C4×C59—22no—
62C4⋊C4×C59—41no—
63C42×C59C2×D411yescharacteristic
64(C2×D59)⋊C4—42no—
65C4⋊Dic59—41no—
66C4⋊Dic59—41no—
67C4⋊Dic59—42no—
68Dic59⋊C4—42no—
69C4×Dic59—42no—
70C4×Dic59C22⋊C411yescharacteristic
71C4×Dic59D811yescharacteristic
72C4×Dic59SD1611yescharacteristic
73C2×C59⋊C8—41no—
74C2×C59⋊C8C22⋊C411yescharacteristic
75C2×C59⋊C8C22⋊C411yescharacteristic
76Dic236—42no—
77Dic236—41no—
78Dic236—41no—
79#11745659—82no—
80C8×D59—21no—
81C8×D59—21no—
82C8×D59—41no—
83C16×C59—41no—
84C59⋊C16—41no—
85C22×D59#10211yescharacteristic
86Q8×C59—81no—
87Q8×C59—81no—
88C2×C4×C59—42no—
89C2×C4×C59C2×C22⋊C411yescharacteristic
90C2×C4×C59C2×D811yescharacteristic
91C2×C4×C59C2×SD1611yescharacteristic
92C2×Dic59—42no—
93C2×Dic59#10211yescharacteristic
94C2×Dic59#10211yescharacteristic
95C2×Dic59#10211yescharacteristic
96C4×D59—21no—
97C4×D59—21no—
98C4×D59—21no—
99C4×D59—21no—
100C4×D59—22no—
101Dic118—42no—
102Dic118—42no—
103C8×C59—21no—
104C8×C59—81no—
105C8×C59AGL(1,ℤ/8)11yescharacteristic
106C8×C59#13711yescharacteristic
107C59⋊C8—21no—
108C59⋊C8—21no—
109C59⋊C8—81no—
110#11988—591nomaximal, Sylow
111C22×C59C2×#10211yescharacteristic
112C2×D59—21no—
113C2×D59—21no—
114C2×D59—21no—
115C4×C59—21no—
116C4×C59—82no—
117C4×C59#41311yescharacteristic
118C4×C59#41711yescharacteristic
119C4×C59#50911yescharacteristic
120C4×C59#51011yescharacteristic
121Dic59—21no—
122Dic59—21no—
123Dic59—82no—
124Dic59#36212yes—
125#3170—591no—
126#2150—591no—
127#1567—591no—
128#1364—591no—
129#1364D5911yescharacteristic, complemented
130#1362—591no—
131#1362—591no—
132C2×C59#154711yescharacteristic
133C2×C59#213911yescharacteristic
134C2×C59C2×#36211yescharacteristic
135D59—41no—
136#533—1181no—
137C2×M32—1181no—
138#500C2×D5911yescharacteristic
139#494—591no—
140#487—1181no—
141#467—592no—
142#467—1181no—
143C4×Q16—592no—
144C4×Q16—1181no—
145#423—1181no—
146C2×C4×C8—591no—
147C4⋊C16—591no—
148C4⋊C16—1181no—
149C4⋊C16C2×D5911yescharacteristic
150#335—591no—
151#335—1181no—
152#335C2×D5911yescharacteristic
153C59#1198811yescharacteristic, complemented, Sylow
154C2×Q16—592no—
155C2×Q16—1181no—
156C2×M16—2361no—
157C22×C8—591no—
158C22×C8—591no—
159C4⋊Q8—21no—
160C4⋊Q8—1181no—
161#125—2361no—
162C22⋊Q8—1182no—
163C22⋊Q8—2361no—
164C4×Q8—1182no—
165C4×Q8—2361no—
166#117—2361no—
167C2×C42—591no—
168M32—2362no—
169C2×C16—21no—
170C2×C16—1181no—
171#107—21no—
172#107C4×D5912yes—
173C4⋊C8—21no—
174C4⋊C8—1181no—
175C4⋊C8—2361no—
176#103—1182no—
177#103—1182no—
178C4×C8—591no—
179C4×C8—591no—
180C4×C8—591no—
181C4×C8C22×D5911yescharacteristic
182C2×Q8—41no—
183C2×Q8—1182no—
184C22×C4—591no—
185C22×C4—591no—
186C22×C4—591no—
187Q16—2362no—
188Q16—2361no—
189Q16—2361no—
190M16—4722no—
191C2×C8—41no—
192C2×C8—591no—
193C2×C8—591no—
194C2×C8—1181no—
195C2×C8—1181no—
196C2×C8—2361no—
197C2×C8—2361no—
198C2×C8C2×C4×D5911yescharacteristic
199C2×C8D4×D5911yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—41no—
202C4⋊C4—2362no—
203C4⋊C4—2362no—
204C4⋊C4—2361no—
205C4⋊C4—2361no—
206C22⋊C4—2362no—
207C42—591no—
208C42—591no—
209C42—591no—
210C42—2362no—
211C42D4×D5911yescharacteristic
212C16—41no—
213C16—2361no—
214C23—591no—
215Q8—81no—
216Q8—81no—
217Q8—2362no—
218Q8—2362no—
219C2×C4—42no—
220C2×C4—591no—
221C2×C4—591no—
222C2×C4—591no—
223C2×C4—1182no—
224C2×C4—1181no—
225C2×C4—1181no—
226C2×C4—1181no—
227C2×C4—1181no—
228C2×C4—2362no—
229C2×C4C22⋊C4×D5911yescharacteristic
230C2×C4D8×D5911yescharacteristic
231C2×C4SD16×D5911yescharacteristic
232C8—21no—
233C8—81no—
234C8—1181no—
235C8—1181no—
236C8—4721no—
237C8#1427375411yescharacteristic
238C8#1427375911yescharacteristic
239C22—1181no—
240C22—1181no—
241C22—1181no—
242C22#102×D5911yescharacteristic
243C4—21no—
244C4—82no—
245C4—592no—
246C4—1181no—
247C4—1181no—
248C4—4722no—
249C4#1683738511yescharacteristic
250C4#1683740211yescharacteristic
251C4#1683783611yescharacteristic
252C4#1683784111yescharacteristic
253C2—2361no—
254C2#2142277211yescharacteristic
255C2#2142580511yescharacteristic
256C2#362×D5911yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1364C2
socleC22×C59C2×#102
prime radicalC22×D59#102
cosocleC4×C8×C59C23
hypercenter#1364D59
hypocenterC59#11988
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
4. #21422151C2semidirect
5. #21422146C2semidirect
6. #21422144C2semidirect
7. C59×#1364C2semidirect
129. #1364D59semidirect
153. C59#11988semidirect

Maximal subgroups

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

subgroupindex
#214346832
#214258502
#214228532
#214221512
#214221462
#214221442
C59×#13642
#1198859