Finite Group Databasebeta

← #28508114

Subgroups — #28508114

Order 15104.

496 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 77 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×#16837192C212yesmaximal
2#21424510C212yesmaximal
3#21424305C211yesmaximal, characteristic
4#21422221C211yesmaximal, characteristic
5C59×#2724C211yesmaximal, characteristic
6#16837877C2211yescharacteristic
7#16837442—21no—
8C2×#14273644C2211yescharacteristic
9C2×#14273642C2212yes—
10#16837265—22no—
11#16837192—24no—
12#16837168—22no—
13#16837162—21no—
14C2×#123×C59C2212yes—
15C59×#398—21no—
16C59×#397—22no—
17C59×#390C2211yescharacteristic
18C59×#389—21no—
19C59×#385—22no—
20C2×#11745664—21no—
21D4×Dic59—41no—
22D4×Dic59—41no—
23#14273686—21no—
24C2×C22⋊Dic59—22no—
25C2×C22⋊Dic59—22no—
26#14273660—41no—
27#14273658—41no—
28#14273644—21no—
29#14273644D412yes—
30#14273642—22no—
31#14273642—22no—
32C59⋊#95—42no—
33C59⋊#95—42no—
34C22×D4×C59C2311yescharacteristic
35C23×C4×C59D412yes—
36#123×C59—22no—
37#123×C59—22no—
38#123×C59—22no—
39#123×C59—22no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—22no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59D412yes—
48C2×C42×C59D411yescharacteristic
49#95×C59—22no—
50#95×C59—22no—
51#95×C59—21no—
52C24×C59C2×D412yes—
53C2×D4×C59—41no—
54C2×D4×C59—41no—
55C2×D4×C59C2×D411yescharacteristic
56C2×D4×C59#4111yescharacteristic
57C2×D4×C59#4112yes—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—22no—
62C22×C4×C59—22no—
63C22×C4×C59—22no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—22no—
68C22×Dic59—22no—
69C22×Dic59—42no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—41no—
72C4⋊C4×C59—42no—
73C4⋊C4×C59—42no—
74C4⋊C4×C59—81no—
75C4⋊C4×C59—81no—
76C22⋊C4×C59—22no—
77C22⋊C4×C59—22no—
78C22⋊C4×C59—44no—
79C22⋊C4×C59—42no—
80C22⋊C4×C59—42no—
81C42×C59—41no—
82C22⋊Dic59—42no—
83C22⋊Dic59—41no—
84C22⋊Dic59—42no—
85C22⋊Dic59—42no—
86C4⋊Dic59—21no—
87C4⋊Dic59—21no—
88C4×Dic59—41no—
89#11745664—22no—
90#11745664—41no—
91C2×C59⋊C8—41no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—42no—
95C23×C59—42no—
96C23×C59#12011yescharacteristic
97C23×C59#12112yes—
98C23×C59#12312yes—
99D4×C59—41no—
100D4×C59—41no—
101D4×C59—42no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—22no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—44no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—41no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—82no—
117C2×C4×C59#12111yescharacteristic
118C2×C4×C59#12411yescharacteristic
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—82no—
122C2×Dic59—82no—
123C2×Dic59—82no—
124C2×Dic59—82no—
125C2×Dic59—82no—
126C59⋊C8—42no—
127#13421—591nomaximal, Sylow
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—22no—
131C22×C59—22no—
132C22×C59—42no—
133C22×C59—42no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—82no—
137C22×C59#39411yescharacteristic
138C22×C59B(4,𝔽2)11yescharacteristic
139C22×C59#45811yescharacteristic
140C4×C59—21no—
141C4×C59—21no—
142C4×C59—42no—
143C4×C59—82no—
144C4×C59—81no—
145C4×C59—82no—
146Dic59—81no—
147Dic59—81no—
148Dic59—162no—
149#2724D5911yescharacteristic
150C2×#352—592no—
151#1890—592no—
152#1856—591no—
153#1393—591no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59—42no—
157C2×C59—42no—
158C2×C59#204411yescharacteristic
159C2×C59#219811yescharacteristic
160C2×C59#220111yescharacteristic
161C2×#123C2×D5912yes—
162#518—591no—
163#428—1181no—
164C2×#100—591no—
165C2×#99—592no—
166#398—21no—
167#397—22no—
168#390C2×D5911yescharacteristic
169#389—21no—
170#385—22no—
171#381—1182no—
172#352—1184no—
173#342—1182no—
174#339—1181no—
175C59#1342111yescharacteristic, complemented, Sylow
176C22×D4C22×D5911yescharacteristic
177C23×C4C22⋊D5912yes—
178C2×M16—1181no—
179#123—22no—
180#123—22no—
181#123—22no—
182#123—22no—
183C4×D4—2361no—
184C4×D4—2361no—
185#117—1181no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C22⋊D5911yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—1182no—
194C2×C22⋊C4—1182no—
195C2×C22⋊C4C22⋊D5912yes—
196C2×C42C22⋊D5911yescharacteristic
197#107—2361no—
198#106—2361no—
199#100—592no—
200#100—1181no—
201#99—1182no—
202#99—1182no—
203#95—22no—
204#95—22no—
205#95—21no—
206#95—2362no—
207#95—2362no—
208C24C2×C22⋊D5912yes—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4D4×D5911yescharacteristic
212C2×D4#1174568711yescharacteristic
213C2×D4#1174568712yes—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—1182no—
223C22×C4—1182no—
224C22×C4—2362no—
225C22×C4C2×C22⋊D5911yescharacteristic
226M16—1182no—
227M16—2361no—
228C2×C8—2361no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—1181no—
236C4⋊C4—1181no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—2362no—
243C22⋊C4—2362no—
244C22⋊C4—2362no—
245C22⋊C4—2361no—
246C42—41no—
247C42—2361no—
248C23—22no—
249C23—22no—
250C23—42no—
251C23—42no—
252C23#1427369611yescharacteristic
253C23#1427369912yes—
254C23#1427370812yes—
255D4—41no—
256D4—41no—
257D4—42no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—4722no—
274C2×C4—4722no—
275C2×C4—4722no—
276C2×C4—4722no—
277C2×C4—4722no—
278C2×C4—2361no—
279C2×C4—2361no—
280C2×C4#1427369811yescharacteristic
281C2×C4#1427371411yescharacteristic
282C8—2362no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—22no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—82no—
292C22#1683731611yescharacteristic
293C22#1683758911yescharacteristic
294C22#1683759211yescharacteristic
295C4—21no—
296C4—21no—
297C4—42no—
298C4—82no—
299C4—81no—
300C4—82no—
301C4—9442no—
302C4—4721no—
303C4—4721no—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2142534411yescharacteristic
309C2#2142609211yescharacteristic
310C2#2142610911yescharacteristic
subgroupG/H
centerC22#16837589
commutator, prime radical, cosocleC22×D4×C59C23
FrattiniC22×D4C22×D59
FittingC59×#2724C2
socleC22×C59B(4,𝔽2)
hypercenter#2724D59
hypocenterC59#13421
LCSG ▷ C22×D4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2724

Complements

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

NH ≅ G/Ntype
175. C59#13421semidirect

Maximal subgroups

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

subgroupindex
C2×#168371922
#214245102
#214243052
#214222212
C59×#27242
#1342159