Finite Group Databasebeta

← #28506568

Subgroups — #28506568

Order 15104.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16837224C211yesmaximal, characteristic, complemented
2C2×#16837216C211yesmaximal, characteristic
3#21425811C211yesmaximal, characteristic
4#21424970C211yesmaximal, characteristic
5#21423982C211yesmaximal, characteristic, complemented
6#21422175C211yesmaximal, characteristic, complemented
7C59×#1386C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×#14273667C2211yescharacteristic
10C2×#14273663C2211yescharacteristic
11C2×#14273647—21no—
12#16837358—21no—
13C2×#14273637C2211yescharacteristic
14#16837303—21no—
15#16837224—22no—
16#16837216—22no—
17#16837202C414yescomplemented
18#16837161—21no—
19C2×M32×C59C2211yescharacteristic
20C59×#427C2211yescharacteristic
21C2×#106×C59C2211yescharacteristic
22C22×D236—21no—
23C2×D472—21no—
24C2×D472—21no—
25C2×D472—21no—
26C2×D472—21no—
27C2×D472—21no—
28C2×D472C2×C412yes—
29C2×#11745664—21no—
30C22×C59⋊C8D411yescharacteristic
31C2×(C2×D59)⋊C4—42no—
32C2×C4×Dic59D411yescharacteristic
33#14273667—21no—
34#14273667—21no—
35C2×C59⋊C16C2×C412yes—
36#14273663—21no—
37#14273663D412yes—
38#14273647—42no—
39#14273638—41no—
40#14273637—21no—
41#14273637—21no—
42C4×C59⋊C8—41no—
43C22×C8×C59C2311yescharacteristic
44#117×C59—21no—
45C2×C4⋊C4×C59—21no—
46M32×C59—21no—
47M32×C59—21no—
48C2×C16×C59—21no—
49#107×C59C2×C412yes—
50#106×C59—21no—
51#106×C59—21no—
52#106×C59—21no—
53C23×D59—41no—
54C22×C4×C59—41no—
55C22×C4×C59C2×D411yescharacteristic
56C22×Dic59—21no—
57C2×D236—22no—
58C2×D236—41no—
59C2×D236—41no—
60C2×D236—41no—
61C2×D236—41no—
62C2×D236—41no—
63C2×C8×C59C22×C411yescharacteristic
64C2×C8×C59C2×D411yescharacteristic
65C2×C8×C59#4111yescharacteristic
66C2×C8×C59#4111yescharacteristic
67C2×C8×C59#4111yescharacteristic
68C2×C8×C59#4111yescharacteristic
69C4⋊C4×C59—22no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—41no—
72C22⋊C4×C59—41no—
73C42×C59—41no—
74(C2×D59)⋊C4—84no—
75C4×Dic59—21no—
76C4×Dic59—21no—
77#11745664—22no—
78#11745664—41no—
79C2×C59⋊C8—21no—
80C2×C59⋊C8—21no—
81C2×C59⋊C8—41no—
82C2×C59⋊C8SD1612yes—
83D472—22no—
84D472—22no—
85D472—41no—
86D472—41no—
87C16×C59—41no—
88C59⋊C16—22no—
89C23×C59AGL(1,ℤ/8)11yescharacteristic
90C22×D59—42no—
91C22×D59—82no—
92C22×D59—82no—
93C22×D59—81no—
94C22×D59—81no—
95C2×C4×C59—42no—
96C2×C4×C59—82no—
97C2×C4×C59—81no—
98C2×C4×C59C4×D411yescharacteristic
99C2×C4×C59#12111yescharacteristic
100C2×C4×C59#12311yescharacteristic
101C2×C4×C59C2×SD1611yescharacteristic
102C2×C4×C59#13511yescharacteristic
103C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
104C2×Dic59—42no—
105C2×Dic59—41no—
106D236—42no—
107D236—42no—
108D236—81no—
109D236—81no—
110C8×C59#11711yescharacteristic
111C8×C59C4×D411yescharacteristic
112C8×C59#12411yescharacteristic
113C8×C59#12611yescharacteristic
114C59⋊C8—22no—
115C59⋊C8—42no—
116#13142—591nomaximal, Sylow
117C22×C59—22no—
118C22×C59#44211yescharacteristic
119C22×C59#46011yescharacteristic
120C22×C59#48211yescharacteristic
121C2×D59—84no—
122C2×D59—82no—
123C2×D59—162no—
124C2×D59—161no—
125C2×D59—161no—
126C4×C59—82no—
127C4×C59—161no—
128C4×C59#38811yescharacteristic
129C4×C59C4×SD1611yescharacteristic
130C4×C59#46311yescharacteristic
131C4×C59#48111yescharacteristic
132Dic59—81no—
133C2×#365—591no—
134C2×#361—591no—
135#2140—591no—
136#1972—591no—
137#1773—591no—
138#1386D5911yescharacteristic, complemented
139#1375—591no—
140C2×C59—21no—
141C2×C59—21no—
142C2×C59#192611yescharacteristic
143C2×C59AGL(1,ℤ/16)12yes—
144D59—162no—
145C22×D8—591no—
146C2×M32—591no—
147C2×M32C2×D5911yescharacteristic
148C2×#108—591no—
149#427C2×D5911yescharacteristic
150C2×#106C2×D5911yescharacteristic
151C2×#102—1181no—
152#405—1181no—
153C2×#97—591no—
154#390—1181no—
155#365—1182no—
156#361—1182no—
157#357—594no—
158#339—1181no—
159C59#1314211yescharacteristic, complemented, Sylow
160C22×D4—1181no—
161C2×D8—592no—
162C2×D8—1181no—
163C2×D8—1181no—
164C2×D8—1181no—
165C2×D8—1181no—
166C2×D8—1181no—
167C2×M16—1181no—
168C22×C8—591no—
169C22×C8C22×D5911yescharacteristic
170#117—21no—
171C2×C4⋊C4—21no—
172C2×C22⋊C4—2362no—
173C2×C42—591no—
174M32—21no—
175M32—21no—
176M32—1181no—
177M32—1181no—
178C2×C16—21no—
179C2×C16—592no—
180#108—592no—
181#108—1181no—
182#107C4×D5912yes—
183#106—21no—
184#106—21no—
185#106—21no—
186#102—2362no—
187#97—1181no—
188#97—1181no—
189#97—2361no—
190C4×C8—2361no—
191C24—2361no—
192C2×D4—1182no—
193C2×D4—2361no—
194C2×D4—2361no—
195C2×D4—2361no—
196C2×D4—2361no—
197C2×D4—2361no—
198C22×C4—41no—
199C22×C4—1181no—
200C22×C4D4×D5911yescharacteristic
201D8—1182no—
202D8—1182no—
203D8—2361no—
204D8—2361no—
205M16—1182no—
206M16—2361no—
207C2×C8—592no—
208C2×C8—1181no—
209C2×C8—1181no—
210C2×C8—2361no—
211C2×C8C2×C4×D5911yescharacteristic
212C2×C8#1174568511yescharacteristic
213C2×C8#1174568511yescharacteristic
214C2×C8D4×D5911yescharacteristic
215C2×C8#1174568711yescharacteristic
216C2×C8#1174568911yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—4724no—
222C42—41no—
223C42—1181no—
224C42—1181no—
225C16—41no—
226C16—1182no—
227C23—4722no—
228C23—4722no—
229C23—4721no—
230C23—4721no—
231C23—2362no—
232C23#1427375111yescharacteristic
233D4—4721no—
234D4—4721no—
235D4—2362no—
236D4—2362no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—2362no—
241C2×C4—2361no—
242C2×C4(C22⋊D59)⋊C411yescharacteristic
243C2×C4#1427370011yescharacteristic
244C2×C4#1427371211yescharacteristic
245C2×C4SD16×D5911yescharacteristic
246C2×C4#1427375011yescharacteristic
247C2×C4#1427375411yescharacteristic
248C8—1182no—
249C8—2362no—
250C8#1427368411yescharacteristic
251C8#1427368911yescharacteristic
252C8#1427371311yescharacteristic
253C8#1427372211yescharacteristic
254C22—22no—
255C22—9441no—
256C22—9441no—
257C22—9442no—
258C22—4724no—
259C22—4722no—
260C22#1683751111yescharacteristic
261C22#1683760111yescharacteristic
262C22#1683772911yescharacteristic
263C4—82no—
264C4—161no—
265C4—4721no—
266C4#1683729111yescharacteristic
267C4#1683748711yescharacteristic
268C4#1683762011yescharacteristic
269C4#1683772211yescharacteristic
270C2—21no—
271C2—21no—
272C2—9442no—
273C2#2142472111yescharacteristic
274C2#2142599112yes—
subgroupG/H
centerC22#16837511
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1386C2
socleC22×C59#442
prime radicalC22×D472C22
cosocleC22×C8×C59C23
hypercenter#1386D59
hypocenterC59#13142
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
1. C2×#16837224C2semidirect
5. #21423982C2semidirect
6. #21422175C2semidirect
7. C59×#1386C2semidirect
17. #16837202C4semidirect
138. #1386D59semidirect
159. C59#13142semidirect

Maximal subgroups

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

subgroupindex
C2×#168372242
C2×#168372162
#214258112
#214249702
#214239822
#214221752
C59×#13862
#1314259