Finite Group Databasebeta

← #28493088

Subgroups — #28493088

Order 15104.

490 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 95 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#21425080C212yesmaximal
3#21424517C212yesmaximal
4#21424273C211yesmaximal, characteristic
5C59×#1289C211yesmaximal, characteristic
6C23×C59⋊C8C2211yescharacteristic
7C2×#14273706C2212yes—
8#16837738—22no—
9#16837643—21no—
10#16837643—21no—
11#16837560—22no—
12C2×Dic59⋊C8—21no—
13C2×#14273650—22no—
14C4×C22⋊Dic59C2211yescharacteristic
15#16837168—22no—
16#16837168—22no—
17#16837164—22no—
18#16837156—21no—
19C2×C22⋊C8×C59C2211yescharacteristic
20C59×#380C2212yes—
21C22×Dic118D412yes—
22C2×Dic236—21no—
23C2×Dic236—21no—
24C22×C59⋊C8—22no—
25C22×C59⋊C8—21no—
26C22×C59⋊C8—42no—
27#14273706—22no—
28#14273706—22no—
29#14273706—22no—
30#14273706—22no—
31C2×C4⋊Dic59D412yes—
32C2×Dic59⋊C4—22no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59D411yescharacteristic
35C2×C22⋊Dic59D411yescharacteristic
36C2×C4×Dic59—21no—
37#14273661—21no—
38#14273661—21no—
39Dic59⋊C8—42no—
40#14273652—22no—
41#14273652—22no—
42#14273650—44no—
43C59⋊#95—21no—
44C23×C4×C59C2311yescharacteristic
45C22×C8×C59D411yescharacteristic
46C22×C8×C59D411yescharacteristic
47C2×C4⋊C4×C59—22no—
48C2×C22⋊C4×C59—22no—
49C22⋊C8×C59—22no—
50#95×C59—22no—
51C24×C59C2×D411yescharacteristic
52C22×C4×C59—41no—
53C22×C4×C59—41no—
54C22×C4×C59—42no—
55C22×C4×C59C2×D412yes—
56C22×C4×C59C2×D411yescharacteristic
57C22×C4×C59#4111yescharacteristic
58C22×C4×C59#4112yes—
59C22×Dic59—21no—
60C22×Dic59—21no—
61C22×Dic59—22no—
62C22×Dic59—22no—
63C2×Dic118—22no—
64C2×Dic118—22no—
65C2×Dic118—42no—
66C2×Dic118—42no—
67C2×C8×C59—21no—
68C2×C8×C59—21no—
69C2×C8×C59—41no—
70C2×C8×C59—41no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—42no—
73C22⋊C4×C59—42no—
74C22⋊C4×C59—42no—
75C22⋊Dic59—21no—
76C22⋊Dic59—22no—
77C22⋊Dic59—42no—
78C22⋊Dic59—42no—
79C22⋊Dic59Q1612yescomplemented
80C4⋊Dic59—22no—
81C4⋊Dic59—22no—
82Dic59⋊C4—42no—
83Dic59⋊C4—42no—
84C4×Dic59—41no—
85C4×Dic59—41no—
86C2×C59⋊C8—24no—
87C2×C59⋊C8—22no—
88C2×C59⋊C8—41no—
89C2×C59⋊C8—42no—
90C2×C59⋊C8—41no—
91Dic236—81no—
92Dic236—81no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59#12112yes—
96C23×C59#12111yescharacteristic
97C23×C59C2×Q1611yescharacteristic
98C23×C59#13511yescharacteristic
99C23×C59#13712yes—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—22no—
104C2×C4×C59—42no—
105C2×C4×C59—41no—
106C2×C4×C59—82no—
107C2×C4×C59—82no—
108C2×C4×C59—82no—
109C2×C4×C59#12011yescharacteristic
110C2×C4×C59#12312yes—
111C2×C4×C59#12411yescharacteristic
112C2×Dic59—22no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—41no—
116C2×Dic59—41no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—41no—
120C2×Dic59—82no—
121C2×Dic59—82no—
122Dic118—44no—
123Dic118—42no—
124Dic118—42no—
125C8×C59—41no—
126C8×C59—41no—
127C59⋊C8—42no—
128#10566—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—22no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—42no—
136C22×C59#39411yescharacteristic
137C22×C59#46212yes—
138C22×C59#46412yes—
139C22×C59#46711yescharacteristic
140C22×C59#47011yescharacteristic
141C4×C59—22no—
142C4×C59—21no—
143C4×C59—162no—
144C4×C59B(4,𝔽2)11yescharacteristic
145C4×C59#45811yescharacteristic
146Dic59—42no—
147Dic59—81no—
148Dic59—82no—
149Dic59—82no—
150#3189—591no—
151#1996—592no—
152#1892—592no—
153#1850—591no—
154#1289D5911yescharacteristic
155C2×C59—21no—
156C2×C59—21no—
157C2×C59—22no—
158C2×C59—22no—
159C2×C59#204411yescharacteristic
160C2×C59#205311yescharacteristic
161C2×C59#205611yescharacteristic
162C23×C8—591no—
163C2×C22⋊Q8—592no—
164#484—1182no—
165#467—1181no—
166#467—1181no—
167#451—1182no—
168C2×C4⋊C8—1181no—
169C2×#103—1182no—
170C2×C22⋊C8C2×D5911yescharacteristic
171#380C2×D5912yes—
172C4×C22⋊C4—591no—
173#342—1182no—
174#342—1182no—
175#340—1182no—
176#336—1181no—
177C59#1056611yescharacteristic, complemented, Sylow
178C22×Q8—592no—
179C23×C4C22×D5911yescharacteristic
180C2×Q16—1181no—
181C2×Q16—1181no—
182C22×C8—1181no—
183C22×C8—1182no—
184C22×C8—2362no—
185C22×C8D23611yescharacteristic
186C22×C8D23611yescharacteristic
187C22⋊Q8—1182no—
188C22⋊Q8—1182no—
189C22⋊Q8—1182no—
190C22⋊Q8—1182no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—592no—
193C2×C4⋊C4—1182no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—591no—
196C2×C22⋊C4—591no—
197C2×C22⋊C4—1182no—
198C2×C42—1181no—
199#107—1181no—
200#107—1181no—
201C4⋊C8—2362no—
202#103—1182no—
203#103—1182no—
204#103—2364no—
205C22⋊C8—22no—
206#95—22no—
207#95—1181no—
208C24D4×D5911yescharacteristic
209C2×Q8—1182no—
210C2×Q8—1182no—
211C2×Q8—2362no—
212C2×Q8—2362no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—1181no—
217C22×C4—1182no—
218C22×C4—1182no—
219C22×C4—1181no—
220C22×C4C2×D23611yescharacteristic
221C22×C4#1174568512yes—
222C22×C4D4×D5912yes—
223C22×C4#1174568911yescharacteristic
224Q16—4721no—
225Q16—4721no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—1184no—
231C2×C8—1182no—
232C2×C8—2361no—
233C2×C8—2362no—
234C2×C8—2361no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—1182no—
238C4⋊C4—1182no—
239C4⋊C4—2362no—
240C4⋊C4—2362no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—592no—
244C22⋊C4—1181no—
245C22⋊C4—1182no—
246C22⋊C4—2362no—
247C22⋊C4—2362no—
248C42—2361no—
249C42—2361no—
250C23—41no—
251C23—41no—
252C23#1427370012yes—
253C23#1427370211yescharacteristic
254C23Q16×D5911yescharacteristic
255C23#1427374611yescharacteristic
256C23#1427375612yes—
257Q8—2364no—
258Q8—2362no—
259Q8—2362no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—1182no—
270C2×C4—4722no—
271C2×C4—4722no—
272C2×C4—2361no—
273C2×C4—2361no—
274C2×C4—2361no—
275C2×C4—2362no—
276C2×C4—2362no—
277C2×C4—2361no—
278C2×C4—2361no—
279C2×C4#1427369711yescharacteristic
280C2×C4#1427371212yes—
281C2×C4#1427371511yescharacteristic
282C8—41no—
283C8—41no—
284C8—2362no—
285C22—22no—
286C22—21no—
287C22—22no—
288C22—21no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#1683732011yescharacteristic
293C22#1683761112yes—
294C22#1683762312yes—
295C22#1683764411yescharacteristic
296C22#1683766111yescharacteristic
297C4—22no—
298C4—21no—
299C4—162no—
300C4—4721no—
301C4—4722no—
302C4—4722no—
303C4—2362no—
304C4#1683759011yescharacteristic
305C4#1683759311yescharacteristic
306C2—22no—
307C2—21no—
308C2—21no—
309C2—22no—
310C2#2142534811yescharacteristic
311C2#2142540311yescharacteristic
312C2#2142542011yescharacteristic
subgroupG/H
centerC22#16837320
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1289C2
socleC22×C59#394
prime radicalC24×C59C2×D4
hypercenter#1289D59
hypocenterC59#10566
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
79. C22⋊Dic59Q16semidirect
177. C59#10566semidirect

Maximal subgroups

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

subgroupindex
#214348712
#214250802
#214245172
#214242732
C59×#12892
#1056659