Finite Group Databasebeta

← #28499503

Subgroups — #28499503

Order 15104.

384 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 83 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#21427387C211yesmaximal, characteristic
2#21422440C211yesmaximal, characteristic
3#21422125C212yesmaximal
4#21422101C212yesmaximal
5C59×#1801C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic
7#16837459—21no—
8#16837333—21no—
9#16837332C411yescharacteristic
10#16837332C2211yescharacteristic
11#16837322C411yescharacteristic
12#16837322C2211yescharacteristic
13#16837321—21no—
14C4×Dic59⋊C4—21no—
15C4×Dic59⋊C4—21no—
16#16837248C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#423—21no—
21C2×#101×C59—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C4×Dic118—41no—
25C4×Dic118—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×C4×Dic59—21no—
33C2×C4×Dic59—21no—
34C4⋊(C59⋊C8)—41no—
35C22⋊(C59⋊C8)—22no—
36#14273638—41no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C22×Q8×C59D411yescharacteristic
45C2×M16×C59—22no—
46C4×Q8×C59—41no—
47C4×Q8×C59—41no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C4⋊C8×C59—41no—
55C4⋊C8×C59—41no—
56#101×C59—42no—
57C2×Q8×C59—21no—
58C2×Q8×C59—21no—
59C2×Q8×C59—41no—
60C2×Q8×C59—41no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×Dic118—41no—
73C2×Dic118—42no—
74M16×C59—42no—
75M16×C59—42no—
76C2×C8×C59—42no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C4⋊Dic59—41no—
87Dic59⋊C4—42no—
88Dic59⋊C4—42no—
89Dic59⋊C4—42no—
90Dic59⋊C4—42no—
91Dic59⋊C4—41no—
92Dic59⋊C4—41no—
93C4×Dic59—24no—
94C4×Dic59—41no—
95C4×Dic59—41no—
96C2×C59⋊C8—42no—
97C23×C59C2×C22⋊C411yescharacteristic
98Q8×C59—42no—
99Q8×C59—42no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
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—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59#9911yescharacteristic
116C2×C4×C59#9911yescharacteristic
117C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C59#13711yescharacteristic
119C2×Dic59—22no—
120C2×Dic59—22no—
121C2×Dic59—22no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131Dic118—42no—
132Dic118—42no—
133C8×C59—82no—
134C59⋊C8—82no—
135#11831—591nomaximal, Sylow
136C22×C59—21no—
137C22×C59—21no—
138C22×C59C2×#9911yescharacteristic
139C22×C59#41711yescharacteristic
140C22×C59#42111yescharacteristic
141C4×C59—21no—
142C4×C59—21no—
143C4×C59—21no—
144C4×C59—41no—
145C4×C59—41no—
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—41no—
149C4×C59#35612yes—
150Dic59—42no—
151Dic59—42no—
152Dic59—42no—
153Dic59—81no—
154#2399—591no—
155#1801D5911yescharacteristic
156#1467—591no—
157#1354—592no—
158#1346—592no—
159C2×C59—21no—
160C2×C59—21no—
161C2×C59#151311yescharacteristic
162C2×C59#212011yescharacteristic
163C2×C59C2×#35611yescharacteristic
164C2×C4×Q8—591no—
165C2×C4×Q8C2×D5911yescharacteristic
166#432—1181no—
167#423—21no—
168C2×#101—21no—
169#398—591no—
170#398—591no—
171#398—1181no—
172#395—591no—
173#395—591no—
174#395—1181no—
175C4×C4⋊C4—1181no—
176C4×C4⋊C4—1181no—
177#376—591no—
178#343C2×D5912yes—
179#324—592no—
180#324—1181no—
181C59#1183111yescharacteristic, complemented, Sylow
182C22×Q8—591no—
183C22×Q8C22⋊D5911yescharacteristic
184C2×M16—22no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—2361no—
188C4×Q8—2361no—
189C2×C4⋊C4—591no—
190C2×C4⋊C4—1181no—
191C2×C4⋊C4—1181no—
192C2×C4⋊C4—1181no—
193C2×C4⋊C4—1181no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4C4×D5911yescharacteristic
196C2×C4⋊C4D23611yescharacteristic
197C2×C4⋊C4C22×D5911yescharacteristic
198C2×C42—1181no—
199C2×C42—1181no—
200C2×C42C4×D5911yescharacteristic
201C2×C42D23611yescharacteristic
202C2×C42C22⋊D5911yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—2361no—
206#101—42no—
207C22⋊C8—1182no—
208#97—2361no—
209#95—591no—
210#95—591no—
211#95—1181no—
212#95—1181no—
213#95—1181no—
214#95—1181no—
215#95—1181no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—2361no—
221C2×Q8—2362no—
222C22×C4—1181no—
223C22×C4—1181no—
224C22×C4—1181no—
225C22×C4—1181no—
226C22×C4(C2×D59)⋊C411yescharacteristic
227C22×C4(C2×D59)⋊C411yescharacteristic
228C22×C4(C2×D59)⋊C411yescharacteristic
229C22×C4(C2×D59)⋊C411yescharacteristic
230C22×C4C2×C4×D5911yescharacteristic
231C22×C4C2×D23611yescharacteristic
232C22×C4C2×C22⋊D5911yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—2362no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—2361no—
241C4⋊C4—2362no—
242C4⋊C4—2362no—
243C4⋊C4—2362no—
244C4⋊C4—2362no—
245C4⋊C4—2361no—
246C4⋊C4—2361no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—1184no—
254C42—2361no—
255C42—2361no—
256C23C2×(C2×D59)⋊C411yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—2362no—
260Q8—2362no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
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—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—1182no—
277C2×C4—1182no—
278C2×C4—1182no—
279C2×C4—2361no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4—2361no—
286C2×C4—2361no—
287C2×C4—2361no—
288C2×C4#1427364111yescharacteristic
289C2×C4#1427364111yescharacteristic
290C2×C4#1427375311yescharacteristic
291C2×C4#1427375811yescharacteristic
292C8—82no—
293C8—4722no—
294C22—21no—
295C22—21no—
296C22C2×#1427364111yescharacteristic
297C22#1683740111yescharacteristic
298C22#1683742011yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—4721no—
308C4—2362no—
309C4—2362no—
310C4—2362no—
311C4#1683720012yes—
312C2—21no—
313C2—21no—
314C2#2142263711yescharacteristic
315C2#2142573811yescharacteristic
316C2C2×#1683720011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1801C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1801D59
hypocenterC59#11831
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
181. C59#11831semidirect

Maximal subgroups

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

subgroupindex
#214273872
#214224402
#214221252
#214221012
C59×#18012
#1183159