Finite Group Databasebeta

← #28501911

Subgroups — #28501911

Order 15104.

344 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 67 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#21434871C211yesmaximal, characteristic
2#21426298C212yesmaximal, complemented
3#21422107C212yesmaximal, complemented
4#21422020C211yesmaximal, characteristic
5C59×#2076C211yesmaximal, characteristic
6C2×#14273706—21no—
7#16837738—22no—
8#16837643—21no—
9#16837643C2212yes—
10#16837560—22no—
11#16837227—22no—
12#16837205—22no—
13C22⋊(C59⋊C16)C2212yes—
14C2×#107×C59C2211yescharacteristic
15C2×C22⋊C8×C59C2211yescharacteristic
16C59×#380—21no—
17C59×#340—21no—
18C59×#340C2211yescharacteristic
19C22×Dic118—21no—
20C2×Dic236—21no—
21C2×Dic236D412yes—
22#14273706—22no—
23#14273706—22no—
24#14273706—41no—
25#14273706—41no—
26C2×C4⋊Dic59—21no—
27C2×Dic59⋊C4—41no—
28C2×C22⋊Dic59—41no—
29C2×C59⋊C16—22no—
30#14273661—21no—
31#14273661D412yes—
32#14273652—22no—
33#14273652—22no—
34C23×C4×C59D411yescharacteristic
35C22×C8×C59D411yescharacteristic
36C22×C8×C59C2311yescharacteristic
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C22⋊C4×C59—41no—
40#107×C59—22no—
41C22⋊C8×C59—22no—
42#95×C59—41no—
43C24×C59D811yescharacteristic
44C22×C4×C59—21no—
45C22×C4×C59—21no—
46C22×C4×C59—41no—
47C22×C4×C59—41no—
48C22×C4×C59—41no—
49C22×C4×C59—41no—
50C22×C4×C59D811yescharacteristic
51C22×C4×C59C2×D411yescharacteristic
52C22×Dic59—41no—
53C22×Dic59—41no—
54C2×Dic118—22no—
55C2×Dic118—41no—
56C2×Dic118—81no—
57C2×Dic118—81no—
58C2×C8×C59—21no—
59C2×C8×C59—41no—
60C2×C8×C59—41no—
61C2×C8×C59C2×D412yes—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—41no—
65C4⋊C4×C59—41no—
66C22⋊C4×C59—82no—
67C22⋊Dic59—42no—
68C22⋊Dic59—42no—
69C4⋊Dic59—22no—
70C4⋊Dic59—41no—
71Dic59⋊C4—42no—
72Dic59⋊C4—42no—
73Dic236—42no—
74Dic236—81no—
75C59⋊C16—42no—
76C23×C59—21no—
77C23×C59—21no—
78C23×C59—41no—
79C23×C59—41no—
80C23×C59Q3212yes—
81C23×C59C2×D811yescharacteristic
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—81no—
89C2×C4×C59—81no—
90C2×C4×C59—81no—
91C2×C4×C59—81no—
92C2×C4×C59—81no—
93C2×C4×C59—81no—
94C2×C4×C59—81no—
95C2×C4×C59#12011yescharacteristic
96C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
97C2×Dic59—42no—
98C2×Dic59—42no—
99C2×Dic59—82no—
100C2×Dic59—82no—
101Dic118—42no—
102Dic118—42no—
103Dic118—82no—
104C8×C59—22no—
105C8×C59—41no—
106#12282—591nomaximal, Sylow
107C22×C59—21no—
108C22×C59—21no—
109C22×C59—22no—
110C22×C59—42no—
111C22×C59—41no—
112C22×C59—41no—
113C22×C59—81no—
114C22×C59#44711yescharacteristic
115C22×C59C2×Q3211yescharacteristic
116C22×C59#51011yescharacteristic
117C4×C59—21no—
118C4×C59—41no—
119C4×C59—161no—
120C4×C59—161no—
121C4×C59#45511yescharacteristic
122C4×C59#45811yescharacteristic
123Dic59—82no—
124Dic59—82no—
125#3189—591no—
126#2236—592no—
127#2076D5911yescharacteristic
128#1348—592no—
129#1314—591no—
130C2×C59—21no—
131C2×C59—21no—
132C2×C59—42no—
133C2×C59#160711yescharacteristic
134C2×C59#218811yescharacteristic
135C2×C59#219011yescharacteristic
136C2×C22⋊Q8—1181no—
137#484—1182no—
138#467—592no—
139#467—1181no—
140#451—1182no—
141C2×#107C2×D5911yescharacteristic
142C2×C22⋊C8C2×D5911yescharacteristic
143#380—21no—
144#366—1182no—
145#358—1182no—
146C22⋊C16—592no—
147#340—21no—
148#340C2×D5911yescharacteristic
149C59#1228211yescharacteristic, complemented, Sylow
150C22×Q8—1181no—
151C23×C4C22⋊D5911yescharacteristic
152C2×Q16—592no—
153C2×Q16—1181no—
154C22×C8C22⋊D5911yescharacteristic
155C22×C8C22×D5911yescharacteristic
156C22⋊Q8—1182no—
157C22⋊Q8—1182no—
158C22⋊Q8—2361no—
159C22⋊Q8—2361no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—1181no—
163C2×C4⋊C4—2361no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—2361no—
166C2×C16—1182no—
167#107—22no—
168#107—592no—
169#107—1181no—
170#103—1182no—
171#103—1182no—
172C22⋊C8—22no—
173#95—41no—
174C24#1174566911yescharacteristic
175C2×Q8—1182no—
176C2×Q8—4721no—
177C2×Q8—4721no—
178C2×Q8—2361no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—2361no—
186C22×C4—2361no—
187C22×C4#1174566911yescharacteristic
188C22×C4C2×C22⋊D5911yescharacteristic
189Q16—4721no—
190Q16—2362no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8D4×D5912yes—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—1182no—
200C4⋊C4—2361no—
201C4⋊C4—2362no—
202C4⋊C4—2362no—
203C22⋊C4—82no—
204C22⋊C4—2362no—
205C22⋊C4—2362no—
206C16—2362no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23#1427367312yes—
212C23C2×#1174566911yescharacteristic
213Q8—4722no—
214Q8—2362no—
215Q8—2362no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—4722no—
230C2×C4—4722no—
231C2×C4—2362no—
232C2×C4—2362no—
233C2×C4#1427369511yescharacteristic
234C2×C4#1427375211yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#1683753511yescharacteristic
245C22C2×#1427367311yescharacteristic
246C22#1683784211yescharacteristic
247C4—21no—
248C4—41no—
249C4—161no—
250C4—161no—
251C4—4722no—
252C4—4722no—
253C4#1683757911yescharacteristic
254C4#1683759111yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2#2142308111yescharacteristic
259C2#2142602911yescharacteristic
260C2#2142604111yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#2076C2
socleC22×C59#447
prime radicalC24×C59D8
hypercenter#2076D59
hypocenterC59#12282
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2076

Complements

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

NH ≅ G/Ntype
2. #21426298C2semidirect
3. #21422107C2semidirect
149. C59#12282semidirect

Maximal subgroups

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

subgroupindex
#214348712
#214262982
#214221072
#214220202
C59×#20762
#1228259