Finite Group Databasebeta

← #28476803

Subgroups — #28476803

Order 15104.

492 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 111 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#21426639C211yesmaximal, characteristic
2#21424211C212yesmaximal, complemented
3#21421922C211yesmaximal, characteristic, complemented
4#21421916C212yesmaximal
5C59×#1272C211yesmaximal, characteristic, complemented
6C2×#14273716C411yescharacteristic
7C2×Q8×Dic59C411yescharacteristic
8C2×#14273689C411yescharacteristic
9C2×Dic59⋊C8—22no—
10#16837304C2211yescharacteristic
11#16837294—22no—
12#16837267C411yescharacteristic
13#16837249—22no—
14C42×Dic59C2211yescharacteristic
15#16837157C2212yes—
16#16837156—22no—
17C59×#391C2211yescharacteristic
18C59×#336C2212yes—
19C22×D236C2×C411yescharacteristic
20C22×C4×D59—21no—
21C22×C59⋊C8—22no—
22#14273716—22no—
23#14273716C814yescomplemented
24Q8×Dic59—42no—
25#14273689—42no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×(C2×D59)⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—21no—
31C2×C4×Dic59—21no—
32C2×C4×Dic59C2×C411yescharacteristic
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59D412yes—
35C2×C4×Dic59D412yes—
36Dic59⋊C8—44no—
37C59⋊#95—22no—
38C22×Q8×C59C2×C411yescharacteristic
39C22×C8×C59—22no—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C42×C59C2311yescharacteristic
42#95×C59—22no—
43C23×D59—21no—
44C2×Q8×C59—21no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59C2×C812yes—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59C22×C411yescharacteristic
51C22×C4×C59C2×D412yes—
52C22×Dic59—21no—
53C22×Dic59C22⋊C411yescharacteristic
54C22×Dic59C22⋊C412yes—
55C22×Dic59C22⋊C411yescharacteristic
56C2×D236—21no—
57C2×D236—41no—
58C2×D236—41no—
59C2×D236C2×C812yes—
60C2×C4×D59—44no—
61C2×C4×D59—42no—
62C2×C8×C59—42no—
63C2×C8×C59—42no—
64C2×C8×C59—42no—
65C4⋊C4×C59—41no—
66C42×C59—41no—
67(C2×D59)⋊C4—28no—
68(C2×D59)⋊C4—42no—
69(C2×D59)⋊C4—42no—
70(C2×D59)⋊C4—42no—
71C4⋊Dic59—42no—
72C4⋊Dic59—41no—
73C4×Dic59—22no—
74C4×Dic59—22no—
75C4×Dic59—22no—
76C4×Dic59—42no—
77C4×Dic59—42no—
78C4×Dic59C2×C812yes—
79C4×Dic59M1612yes—
80C2×C59⋊C8—42no—
81C2×C59⋊C8—42no—
82C2×C59⋊C8—42no—
83C23×C59C2×C22⋊C411yescharacteristic
84C22×D59—22no—
85C22×D59—41no—
86C22×D59—41no—
87C22×D59—42no—
88C22×D59—42no—
89Q8×C59—42no—
90Q8×C59—42no—
91C2×C4×C59—22no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—42no—
100C2×C4×C59C22×C811yescharacteristic
101C2×C4×C59C2×M1611yescharacteristic
102C2×C4×C59AGL(1,ℤ/8)12yes—
103C2×C4×C59#13712yes—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—21no—
108C2×Dic59—22no—
109C2×Dic59—41no—
110C2×Dic59—42no—
111C2×Dic59C22⋊C814yes—
112C2×Dic59#10012yes—
113D236—42no—
114D236—42no—
115C4×D59—84no—
116C8×C59—82no—
117C59⋊C8—82no—
118#7601—591nomaximal, Sylow
119C22×C59C2×C22⋊C811yescharacteristic
120C22×C59#41011yescharacteristic
121C22×C59C2×#10011yescharacteristic
122C22×C59#41712yes—
123C22×C59#42112yes—
124C2×D59—42no—
125C2×D59—44no—
126C2×D59—81no—
127C2×D59—82no—
128C2×D59—81no—
129C4×C59—21no—
130C4×C59—21no—
131C4×C59—22no—
132C4×C59—22no—
133C4×C59—42no—
134C4×C59—81no—
135Dic59—24no—
136Dic59—22no—
137Dic59—41no—
138Dic59—81no—
139C4×#124—591no—
140#1831—592no—
141#1272—591no—
142#1272D5911yescharacteristic, complemented
143#1270—592no—
144C2×C59#146011yescharacteristic
145C2×C59#147712yes—
146C2×C59#151212yes—
147C2×C59#153311yescharacteristic
148C2×C59#156011yescharacteristic
149D59—82no—
150C2×#124—591no—
151C2×C4×Q8—591no—
152C2×C4×D4—591no—
153C2×C4⋊C8—1182no—
154#391C2×D5911yescharacteristic
155#390—591no—
156#388—1182no—
157#382—591no—
158C4×C22⋊C4—1182no—
159C43—591no—
160#336—592no—
161#336—1182no—
162#336C2×D5912yes—
163C59#760111yescharacteristic, complemented, Sylow
164C22×Q8C4×D5911yescharacteristic
165C22×D4—591no—
166C23×C4—1181no—
167C22×C8—22no—
168C22×C8—1182no—
169#124—594no—
170#124—1182no—
171C4×Q8—2362no—
172C4×D4—2362no—
173C2×C4⋊C4—591no—
174C2×C4⋊C4—1181no—
175C2×C4⋊C4C4×D5911yescharacteristic
176C2×C22⋊C4—1182no—
177C2×C22⋊C4—1181no—
178C2×C22⋊C4—1182no—
179C2×C42—591no—
180C2×C42—591no—
181C2×C42—592no—
182C2×C42—592no—
183C2×C42—1181no—
184C2×C42C22×D5911yescharacteristic
185C4⋊C8—2364no—
186#95—22no—
187#95—1182no—
188C24—1181no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8C8×D5912yes—
193C2×D4—592no—
194C2×D4—1181no—
195C2×D4—2361no—
196C2×D4—2361no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—592no—
200C22×C4—591no—
201C22×C4—591no—
202C22×C4—1181no—
203C22×C4—2362no—
204C22×C4—2364no—
205C22×C4C2×C4×D5911yescharacteristic
206C22×C4D4×D5912yes—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—2362no—
211C2×C8—2362no—
212C2×C8—2362no—
213C4⋊C4—41no—
214C4⋊C4—2361no—
215C4⋊C4—2362no—
216C22⋊C4—1188no—
217C22⋊C4—2362no—
218C22⋊C4—2362no—
219C22⋊C4—2362no—
220C42—41no—
221C42—592no—
222C42—592no—
223C42—1182no—
224C42—1182no—
225C42—1182no—
226C42—2362no—
227C42—2362no—
228C23—1182no—
229C23—2361no—
230C23—2362no—
231C23—2362no—
232C23—2361no—
233C23C22⋊C4×D5911yescharacteristic
234Q8—42no—
235Q8—42no—
236D4—2362no—
237D4—2362no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—592no—
248C2×C4—594no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1181no—
253C2×C4—1182no—
254C2×C4—4724no—
255C2×C4—2361no—
256C2×C4—2362no—
257C2×C4C2×C8×D5911yescharacteristic
258C2×C4M16×D5911yescharacteristic
259C2×C4#1427375412yes—
260C2×C4#1427375912yes—
261C8—82no—
262C8—4722no—
263C22—4722no—
264C22—4721no—
265C22—4721no—
266C22—2362no—
267C22—2364no—
268C22C22⋊C8×D5911yescharacteristic
269C22#1683737411yescharacteristic
270C22#100×D5911yescharacteristic
271C22#1683740212yes—
272C22#1683741712yes—
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—1182no—
280C4—1184no—
281C4—4721no—
282C4—2361no—
283C2—4722no—
284C2#2142241511yescharacteristic
285C2#2142248212yes—
286C2#2142263312yes—
287C2#2142271811yescharacteristic
288C2#2142283011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1272D59
hypocenterC59#7601
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #21424211C2semidirect
3. #21421922C2semidirect
5. C59×#1272C2semidirect
23. #14273716C8semidirect
142. #1272D59semidirect
163. C59#7601semidirect

Maximal subgroups

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

subgroupindex
#214266392
#214242112
#214219222
#214219162
C59×#12722
#760159