Finite Group Databasebeta

← #28502245

Subgroups — #28502245

Order 15104.

312 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 65 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#21437082C211yesmaximal, characteristic
2#21426348C211yesmaximal, characteristic, complemented
3#21423463C211yesmaximal, characteristic, complemented
4#21422146C211yesmaximal, characteristic, complemented
5#21422074C211yesmaximal, characteristic, complemented
6#21422057C211yesmaximal, characteristic
7C59×#2208C211yesmaximal, characteristic
8#16838122—21no—
9C2×#14273673—22no—
10#16837785—21no—
11#16837782C2211yescharacteristic
12#16837684—21no—
13#16837623—22no—
14#16837607—21no—
15C8×Dic118C2211yescharacteristic
16#16837496—21no—
17C4×C59⋊C16C2211yescharacteristic
18#16837155C2211yescharacteristic
19#16837148—21no—
20C2×SD32×C59—21no—
21C59×#494C2211yescharacteristic
22C4×D8×C59C2211yescharacteristic
23C59×C4⋊C16C2211yescharacteristic
24C59×#358—21no—
25C2×Dic236—21no—
26C2×Dic236D412yes—
27C2×#11745670—22no—
28#14273727—21no—
29#14273720—42no—
30#14273714—22no—
31#14273706—41no—
32C4×Dic118D411yescharacteristic
33#14273692—22no—
34#14273686—41no—
35#14273673—44no—
36C2×C59⋊C16—21no—
37#14273658—41no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)D411yescharacteristic
40Dic59⋊C8—21no—
41#14273650—22no—
42#14273650—41no—
43C8×Dic59—21no—
44C2×Q16×C59—21no—
45C2×Q16×C59—21no—
46C2×D8×C59D411yescharacteristic
47C4⋊Q8×C59—21no—
48C4×D4×C59—21no—
49SD32×C59—42no—
50C2×C16×C59—21no—
51#107×C59D411yescharacteristic
52#102×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×Q8×C59—41no—
55C2×D4×C59—21no—
56C22×C4×C59—41no—
57C2×Dic118—22no—
58C2×Dic118D811yescharacteristic
59Q16×C59—42no—
60Q16×C59—41no—
61Q16×C59—41no—
62D8×C59—21no—
63D8×C59—21no—
64C2×C8×C59C2×D411yescharacteristic
65C2×C8×C59C2×D411yescharacteristic
66C4⋊C4×C59—21no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—41no—
69C22⋊C4×C59—41no—
70C42×C59C2×D411yescharacteristic
71C22⋊Dic59—42no—
72C22⋊Dic59—42no—
73#11745670—82no—
74C4⋊Dic59—41no—
75C4⋊Dic59—41no—
76C4⋊Dic59D811yescharacteristic
77Dic59⋊C4—21no—
78Dic59⋊C4—42no—
79C4×Dic59—21no—
80C4×Dic59—42no—
81C2×C59⋊C8—21no—
82C2×C59⋊C8—41no—
83Dic236—42no—
84Dic236—41no—
85Dic236—41no—
86C16×C59—41no—
87C59⋊C16—22no—
88C23×C59—41no—
89Q8×C59—81no—
90Q8×C59—81no—
91D4×C59—41no—
92D4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—81no—
95C2×C4×C59—81no—
96C2×C4×C59—81no—
97C2×C4×C59#12011yescharacteristic
98C2×C4×C59C2×D811yescharacteristic
99C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
100C2×Dic59—21no—
101C2×Dic59—21no—
102C2×Dic59—42no—
103Dic118—21no—
104Dic118—21no—
105Dic118—42no—
106Dic118—42no—
107C8×C59—21no—
108C8×C59#13511yescharacteristic
109C8×C59AGL(1,ℤ/8)11yescharacteristic
110C59⋊C8—41no—
111C59⋊C8—81no—
112#12335—591nomaximal, Sylow
113C22×C59—81no—
114C22×C59—81no—
115C22×C59—81no—
116C22×C59#44711yescharacteristic
117C4×C59—21no—
118C4×C59—81no—
119C4×C59—161no—
120C4×C59#44911yescharacteristic
121C4×C59#45311yescharacteristic
122C4×C59#50811yescharacteristic
123C4×C59#51011yescharacteristic
124Dic59—41no—
125Dic59—41no—
126Dic59—82no—
127#3393—591no—
128#2247—591no—
129#2208D5911yescharacteristic
130#1670—591no—
131#1362—591no—
132Q8⋊C16—591no—
133#1331—591no—
134C2×C59—161no—
135C2×C59#162111yescharacteristic
136C2×C59#218911yescharacteristic
137C2×C59#219511yescharacteristic
138#556—1181no—
139C2×Q32—1182no—
140C2×SD32—21no—
141#495—1181no—
142#494—591no—
143#494C2×D5911yescharacteristic
144#475—1181no—
145#464—1182no—
146#461—1181no—
147C8×Q8—591no—
148C4×Q16—1181no—
149C4×D8C2×D5911yescharacteristic
150C4⋊C16C2×D5911yescharacteristic
151#358—21no—
152C4×C16—591no—
153#335—591no—
154#332—1181no—
155C59#1233511yescharacteristic, complemented, Sylow
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—592no—
159C2×Q16—1181no—
160C2×SD16—1182no—
161C2×D8C22⋊D5911yescharacteristic
162C4⋊Q8—21no—
163C4⋊Q8—1181no—
164#126—2362no—
165#124—1182no—
166C22⋊Q8—2361no—
167C4×Q8—591no—
168C4×Q8—1182no—
169C4×D4—21no—
170#117—2361no—
171Q32—2364no—
172SD32—42no—
173C2×C16—21no—
174C2×C16—1181no—
175#107C22⋊D5911yescharacteristic
176#106—2361no—
177C4⋊C8—591no—
178C4⋊C8—1181no—
179C4⋊C8—1181no—
180#103—1182no—
181#103—2361no—
182#102—21no—
183C4×C8—1181no—
184C4×C8C22×D5911yescharacteristic
185C2×Q8—41no—
186C2×Q8—591no—
187C2×Q8—1182no—
188C2×D4—21no—
189C22×C4—41no—
190Q16—42no—
191Q16—41no—
192Q16—41no—
193Q16—2362no—
194Q16—2361no—
195Q16—2361no—
196SD16—4722no—
197D8—21no—
198D8—21no—
199C2×C8—1181no—
200C2×C8—2361no—
201C2×C8D4×D5911yescharacteristic
202C2×C8C2×C22⋊D5911yescharacteristic
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—591no—
207C4⋊C4—1181no—
208C4⋊C4—2362no—
209C4⋊C4—2361no—
210C4⋊C4—2361no—
211C22⋊C4—41no—
212C22⋊C4—2362no—
213C22⋊C4—2362no—
214C42—1181no—
215C42—2362no—
216C42D4×D5911yescharacteristic
217C16—41no—
218C16—1182no—
219C23—41no—
220Q8—81no—
221Q8—81no—
222Q8—1181no—
223Q8—1181no—
224Q8—2362no—
225Q8—2362no—
226D4—41no—
227D4—41no—
228C2×C4—41no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—1181no—
233C2×C4—1181no—
234C2×C4—2362no—
235C2×C4#1427369511yescharacteristic
236C2×C4D8×D5911yescharacteristic
237C2×C4#1427375111yescharacteristic
238C8—21no—
239C8—4721no—
240C8—2361no—
241C8#1427374711yescharacteristic
242C8#1427375311yescharacteristic
243C22—81no—
244C22—81no—
245C22—81no—
246C22#1683753411yescharacteristic
247C4—21no—
248C4—81no—
249C4—161no—
250C4—4722no—
251C4—2361no—
252C4—2361no—
253C4#1683754511yescharacteristic
254C4#1683756811yescharacteristic
255C4#1683783311yescharacteristic
256C4#1683784111yescharacteristic
257C2—161no—
258C2#2142316011yescharacteristic
259C2#2142603311yescharacteristic
260C2#2142607111yescharacteristic
subgroupG/H
centerC22#16837534
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2208C2
socleC22×C59#447
prime radicalC2×D8×C59D4
hypercenter#2208D59
hypocenterC59#12335
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
2. #21426348C2semidirect
3. #21423463C2semidirect
4. #21422146C2semidirect
5. #21422074C2semidirect
155. C59#12335semidirect

Maximal subgroups

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

subgroupindex
#214370822
#214263482
#214234632
#214221462
#214220742
#214220572
C59×#22082
#1233559