Finite Group Databasebeta

← #28499841

Subgroups — #28499841

Order 15104.

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

normal subgroups only

This group has 372 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#21429458C211yesmaximal, characteristic
2#21422420C211yesmaximal, characteristic
3#21422125C212yesmaximal
4#21422122C212yesmaximal
5C59×#1801C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic
7#16837455—21no—
8#16837341C411yescharacteristic
9#16837341C2211yescharacteristic
10#16837322C411yescharacteristic
11#16837322C2211yescharacteristic
12#16837309—21no—
13#16837298—21no—
14#16837286—21no—
15#16837286—21no—
16#16837268C411yescharacteristic
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⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×C4×Dic59—21no—
32C4⋊(C59⋊C8)—41no—
33C22⋊(C59⋊C8)—22no—
34C4×C59⋊C8—41no—
35C59⋊#95—21no—
36C59⋊#95—21no—
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—41no—
58C2×Q8×C59—41no—
59C2×Q8×C59SD1612yes—
60C2×Q8×C59Q1612yes—
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—42no—
87C4⋊Dic59—41no—
88Dic59⋊C4—42no—
89Dic59⋊C4—42no—
90Dic59⋊C4—41no—
91Dic59⋊C4—41no—
92C4×Dic59—42no—
93C2×C59⋊C8—42no—
94C23×C59C2×C22⋊C411yescharacteristic
95Q8×C59—42no—
96Q8×C59—42no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59#9911yescharacteristic
111C2×C4×C59#9911yescharacteristic
112C2×C4×C59#10312yes—
113C2×C4×C59#10312yes—
114C2×C4×C59C2×SD1611yescharacteristic
115C2×C4×C59C2×Q1611yescharacteristic
116C2×Dic59—22no—
117C2×Dic59—41no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—42no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127Dic118—42no—
128Dic118—42no—
129C8×C59—82no—
130C59⋊C8—82no—
131#11905—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59C2×#9911yescharacteristic
135C22×C59C2×#10311yescharacteristic
136C22×C59#42111yescharacteristic
137C4×C59—21no—
138C4×C59—21no—
139C4×C59—21no—
140C4×C59—22no—
141C4×C59—22no—
142C4×C59—21no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59—41no—
146Dic59—42no—
147Dic59—81no—
148Dic59—81no—
149Dic59—81no—
150#2581—591no—
151#1801D5911yescharacteristic
152#1461—591no—
153#1354—592no—
154#1353—592no—
155C2×C59—21no—
156C2×C59—21no—
157C2×C59#152011yescharacteristic
158C2×C59#212011yescharacteristic
159C2×C59#213311yescharacteristic
160C2×C4×Q8—591no—
161C2×C4×Q8C2×D5911yescharacteristic
162#431—1181no—
163#423—21no—
164C2×#101—21no—
165#400—591no—
166#400—591no—
167#395—591no—
168#395—591no—
169#391—1181no—
170#389—1181no—
171#387—1181no—
172#387—1181no—
173#382—591no—
174#343C2×D5912yes—
175#324—592no—
176#324—1181no—
177C59#1190511yescharacteristic, complemented, Sylow
178C22×Q8—591no—
179C22×Q8C22⋊D5911yescharacteristic
180C2×M16—22no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—2361no—
184C4×Q8—2361no—
185C2×C4⋊C4—591no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4—1181no—
189C2×C4⋊C4—1181no—
190C2×C4⋊C4C4×D5911yescharacteristic
191C2×C4⋊C4D23611yescharacteristic
192C2×C4⋊C4C22×D5911yescharacteristic
193C2×C42—1181no—
194C2×C42C4×D5911yescharacteristic
195C2×C42D23611yescharacteristic
196C2×C42C22⋊D5911yescharacteristic
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—2361no—
200#101—42no—
201C22⋊C8—1182no—
202C4×C8—2361no—
203#95—591no—
204#95—591no—
205#95—1181no—
206#95—1181no—
207#95—1181no—
208#95—1181no—
209#95—1181no—
210#95—1181no—
211#95—1181no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—2361no—
215C2×Q8—2362no—
216C2×Q8#1174567012yes—
217C2×Q8#1174567212yes—
218C22×C4—1181no—
219C22×C4—1181no—
220C22×C4—1181no—
221C22×C4—1181no—
222C22×C4(C2×D59)⋊C411yescharacteristic
223C22×C4(C2×D59)⋊C411yescharacteristic
224C22×C4(C2×D59)⋊C411yescharacteristic
225C22×C4(C2×D59)⋊C411yescharacteristic
226C22×C4C2×C4×D5911yescharacteristic
227C22×C4C2×D23611yescharacteristic
228C22×C4C2×C22⋊D5911yescharacteristic
229M16—42no—
230M16—42no—
231C2×C8—42no—
232C2×C8—2362no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—2361no—
237C4⋊C4—2362no—
238C4⋊C4—2362no—
239C4⋊C4—2362no—
240C4⋊C4—2361no—
241C4⋊C4—2361no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—2362no—
249C23C2×(C2×D59)⋊C411yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—2362no—
253Q8—2362no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—1182no—
268C2×C4—2361no—
269C2×C4—2361no—
270C2×C4—2361no—
271C2×C4—2361no—
272C2×C4—2362no—
273C2×C4—2361no—
274C2×C4—2361no—
275C2×C4—2361no—
276C2×C4—2361no—
277C2×C4—2361no—
278C2×C4#1427364111yescharacteristic
279C2×C4#1427364111yescharacteristic
280C2×C4#1427365012yes—
281C2×C4#1427365012yes—
282C2×C4C2×#1174567011yescharacteristic
283C2×C4C2×#1174567211yescharacteristic
284C8—82no—
285C8—4722no—
286C22—21no—
287C22—21no—
288C22C2×#1427364111yescharacteristic
289C22C2×#1427365011yescharacteristic
290C22#1683742011yescharacteristic
291C4—21no—
292C4—22no—
293C4—22no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—4721no—
301C4—4721no—
302C4—4721no—
303C4—2362no—
304C2—21no—
305C2—21no—
306C2#2142266711yescharacteristic
307C2#2142573811yescharacteristic
308C2#2142578311yescharacteristic
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#11905
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
177. C59#11905semidirect

Maximal subgroups

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

subgroupindex
#214294582
#214224202
#214221252
#214221222
C59×#18012
#1190559