Finite Group Databasebeta

← #28499430

Subgroups — #28499430

Order 15104.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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#21427364C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic
3#21422125C212yesmaximal, complemented
4#21422122C212yesmaximal, complemented
5C59×#1822C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic
7#16837459—21no—
8#16837341C2211yescharacteristic
9#16837338C411yescharacteristic
10#16837335—21no—
11#16837325—21no—
12#16837322C2211yescharacteristic
13#16837314C411yescharacteristic
14C4×Dic59⋊C4—21no—
15C4×C4⋊Dic59C411yescharacteristic
16C4×C22⋊Dic59—21no—
17#16837128C2212yes—
18#16837126—21no—
19C2×#117×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#343—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C4×Dic118—41no—
25C4×Dic118—41no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35C4⋊(C59⋊C8)—41no—
36C22⋊(C59⋊C8)—22no—
37#14273638—41no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C2×M16×C59—22no—
46#117×C59—41no—
47#117×C59—41no—
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59C2311yescharacteristic
50C2×C22⋊C4×C59C2×C411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C22⋊C8×C59—41no—
55C22⋊C8×C59—41no—
56C24×C59C22⋊C411yescharacteristic
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—22no—
60C22×C4×C59—41no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—21no—
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C2×Dic118—41no—
72C2×Dic118—42no—
73M16×C59—42no—
74M16×C59—42no—
75C2×C8×C59—42no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C22⋊Dic59—42no—
85C22⋊Dic59—42no—
86C4⋊Dic59—42no—
87C4⋊Dic59—41no—
88Dic59⋊C4—42no—
89Dic59⋊C4—42no—
90Dic59⋊C4—42no—
91C4×Dic59—42no—
92C4×Dic59—41no—
93C4×Dic59—41no—
94C2×C59⋊C8—42no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59C2×C22⋊C411yescharacteristic
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59#9911yescharacteristic
116C2×C4×C59#9911yescharacteristic
117C2×C4×C59#13712yes—
118C2×Dic59—22no—
119C2×Dic59—22no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130Dic118—42no—
131Dic118—42no—
132C8×C59—82no—
133C59⋊C8—82no—
134#11816—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—42no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—81no—
142C22×C59C2×#9911yescharacteristic
143C22×C59#41911yescharacteristic
144C22×C59#42111yescharacteristic
145C4×C59—22no—
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—41no—
152Dic59—42no—
153Dic59—42no—
154Dic59—81no—
155Dic59—81no—
156#2396—591no—
157#1822D5911yescharacteristic
158#1464—591no—
159#1354—592no—
160#1353—592no—
161C2×C59—21no—
162C2×C59—21no—
163C2×C59—81no—
164C2×C59#151111yescharacteristic
165C2×C59#212011yescharacteristic
166C2×C59#213211yescharacteristic
167C2×C4×Q8—591no—
168C2×#117C2×D5911yescharacteristic
169#432—1181no—
170#407—21no—
171#400—591no—
172#399—591no—
173#398—1181no—
174#396—1181no—
175#395—591no—
176#393—591no—
177C4×C4⋊C4—591no—
178C4×C4⋊C4—1181no—
179C4×C22⋊C4—1181no—
180#343—21no—
181#343C2×D5912yes—
182#324—592no—
183#323—1181no—
184C59#1181611yescharacteristic, complemented, Sylow
185C22×Q8—591no—
186C23×C4C22⋊D5911yescharacteristic
187C2×M16—22no—
188C4×Q8—2361no—
189C4×Q8—2361no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—591no—
193C2×C4⋊C4—1181no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4C22⋊D5911yescharacteristic
198C2×C4⋊C4C22×D5911yescharacteristic
199C2×C22⋊C4—1181no—
200C2×C22⋊C4—1181no—
201C2×C22⋊C4C4×D5911yescharacteristic
202C2×C22⋊C4D23611yescharacteristic
203C2×C42—1181no—
204C2×C42—1181no—
205C2×C42C4×D5911yescharacteristic
206C2×C42D23611yescharacteristic
207C4⋊C8—2361no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—1182no—
211#97—2361no—
212#95—591no—
213#95—591no—
214#95—1181no—
215#95—1181no—
216#95—1181no—
217#95—1181no—
218C24(C2×D59)⋊C411yescharacteristic
219C2×Q8—2361no—
220C2×Q8—2362no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—1181no—
226C22×C4—1181no—
227C22×C4—1181no—
228C22×C4—1181no—
229C22×C4(C2×D59)⋊C411yescharacteristic
230C22×C4(C2×D59)⋊C411yescharacteristic
231C22×C4(C2×D59)⋊C411yescharacteristic
232C22×C4C2×C4×D5911yescharacteristic
233C22×C4C2×D23611yescharacteristic
234C22×C4C2×C22⋊D5911yescharacteristic
235M16—42no—
236M16—42no—
237C2×C8—42no—
238C2×C8—2362no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—2361no—
242C4⋊C4—2362no—
243C4⋊C4—2362no—
244C4⋊C4—2362no—
245C4⋊C4—2362no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—2362no—
249C22⋊C4—2362no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—2362no—
255C42—2361no—
256C42—2361no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×(C2×D59)⋊C411yescharacteristic
263Q8—2362no—
264Q8—2362no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—1182no—
280C2×C4—1182no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4—2361no—
286C2×C4—2361no—
287C2×C4—2361no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4—2361no—
291C2×C4#1427364111yescharacteristic
292C2×C4#1427364111yescharacteristic
293C2×C4#1427375712yes—
294C8—82no—
295C8—4722no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1427364111yescharacteristic
304C22#1683741011yescharacteristic
305C22#1683742011yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—4721no—
314C4—4721no—
315C4—2362no—
316C4—2362no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2142262811yescharacteristic
321C2#2142573811yescharacteristic
322C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1822C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1822D59
hypocenterC59#11816
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #21422125C2semidirect
4. #21422122C2semidirect
184. C59#11816semidirect

Maximal subgroups

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

subgroupindex
#214273642
#214224302
#214221252
#214221222
C59×#18222
#1181659