Finite Group Databasebeta

← #28509088

Subgroups — #28509088

Order 15104.

308 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 67 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#21435179C211yesmaximal, characteristic
2#21425785C211yesmaximal, characteristic
3#21422270C212yesmaximal, complemented
4#21422255C212yesmaximal, complemented
5C59×#2129C211yesmaximal, characteristic
6#16838014C411yescharacteristic, complemented
7#16837797C2211yescharacteristic
8#16837781C411yescharacteristic, complemented
9#16837770C411yescharacteristic, complemented
10#16837756C2211yescharacteristic
11#16837621—21no—
12#16837609—21no—
13#16837564—21no—
14#16837558—21no—
15#16837463C411yescharacteristic, complemented
16C2×#14273646—21no—
17#16837201—21no—
18#16837199C2212yes—
19C59×#536C2211yescharacteristic
20C59×#410—21no—
21C59×#354—21no—
22C59×#354C2212yes—
23C2×#11745672—21no—
24C2×#11745672—21no—
25C2×#11745672—21no—
26C2×#11745670—21no—
27C2×#11745664—41no—
28#14273727C2×C411yescharacteristic
29#14273720—41no—
30#14273719C2×C411yescharacteristic
31#14273717—41no—
32#14273706—21no—
33C4×Dic118—21no—
34C4⋊(C59⋊C8)—21no—
35#14273651—21no—
36#14273651—21no—
37#14273651—21no—
38#14273648—21no—
39#14273646—22no—
40#14273646—41no—
41C22⋊(C59⋊C8)—21no—
42#14273638C2×C411yescharacteristic
43C4×C59⋊C8C2×C411yescharacteristic
44C2×#41×C59D411yescharacteristic
45C22×Q8×C59D411yescharacteristic
46C4⋊Q8×C59C2×C411yescharacteristic
47C4⋊Q8×C59D411yescharacteristic
48#124×C59D411yescharacteristic
49#124×C59C2311yescharacteristic
50#123×C59—41no—
51C22⋊Q8×C59—41no—
52#117×C59—41no—
53#117×C59C2×C411yescharacteristic
54#99×C59—22no—
55#99×C59—41no—
56#41×C59—81no—
57C2×Q8×C59—21no—
58C2×Q8×C59—81no—
59C2×Q8×C59C22⋊C411yescharacteristic
60C2×Q8×C59C22⋊C411yescharacteristic
61C2×Q8×C59C2×D411yescharacteristic
62C2×D4×C59—21no—
63C2×D4×C59C2×D411yescharacteristic
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59C22⋊C411yescharacteristic
67C2×Dic118—21no—
68C4⋊C4×C59—21no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—81no—
72C22⋊C4×C59—21no—
73C22⋊C4×C59—41no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—42no—
76C42×C59—81no—
77C42×C59C22⋊C411yescharacteristic
78C42×C59C22×C411yescharacteristic
79C22⋊Dic59—41no—
80#11745672—81no—
81#11745672—81no—
82#11745672—81no—
83#11745670—81no—
84C4⋊Dic59—21no—
85C4⋊Dic59—41no—
86Dic59⋊C4—41no—
87Dic59⋊C4—41no—
88Dic59⋊C4—41no—
89C4×Dic59—41no—
90#11745664—42no—
91#11745664—81no—
92C2×C59⋊C8—21no—
93C2×C59⋊C8—21no—
94C2×C59⋊C8—81no—
95C23×C59—21no—
96C23×C59#9911yescharacteristic
97Q8×C59—41no—
98Q8×C59—41no—
99Q8×C59—41no—
100Q8×C59—81no—
101D4×C59—41no—
102D4×C59—81no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—82no—
114C2×C4×C59#9911yescharacteristic
115C2×C4×C59C2×C22⋊C411yescharacteristic
116C2×Dic59—41no—
117C2×Dic59—41no—
118Dic118—41no—
119Dic118—41no—
120C59⋊C8—41no—
121C59⋊C8—41no—
122C59⋊C8—82no—
123#13608—591nomaximal, Sylow
124C22×C59—21no—
125C22×C59—41no—
126C22×C59—41no—
127C22×C59—81no—
128C22×C59C2×#9911yescharacteristic
129C4×C59—21no—
130C4×C59—21no—
131C4×C59—41no—
132C4×C59—41no—
133C4×C59—41no—
134C4×C59—41no—
135C4×C59—41no—
136C4×C59—162no—
137Dic59—81no—
138Dic59—81no—
139#3223—591no—
140#2133—591no—
141#2129D5911yescharacteristic
142#1414—592no—
143#1407—592no—
144C2×C59—21no—
145C2×C59—41no—
146C2×C59—81no—
147C2×C59#212711yescharacteristic
148#541—591no—
149#536C2×D5911yescharacteristic
150#497—591no—
151#494—591no—
152#491—591no—
153#488—591no—
154#464—1181no—
155#462—1181no—
156#452—1181no—
157#451—1181no—
158#433—591no—
159C2×#101—1181no—
160#410—21no—
161#356—1181no—
162#355—592no—
163#354—21no—
164#354C2×D5912yes—
165C59#1360811yescharacteristic, complemented, Sylow
166C2×#41C22⋊D5911yescharacteristic
167C22×Q8D23611yescharacteristic
168C2×Q16—1181no—
169C2×Q16—1181no—
170C2×Q16—1181no—
171C2×SD16—1181no—
172C2×M16—2361no—
173C4⋊Q8—591no—
174C4⋊Q8C4×D5911yescharacteristic
175C4⋊Q8D23611yescharacteristic
176#126—2361no—
177#125—591no—
178#125—2361no—
179#124C22⋊D5911yescharacteristic
180#124C22×D5911yescharacteristic
181#123—41no—
182C22⋊Q8—41no—
183C22⋊Q8—1181no—
184C4×Q8—1181no—
185#117—41no—
186#117C4×D5911yescharacteristic
187C4⋊C8—1181no—
188#103—1181no—
189#103—1181no—
190#103—1181no—
191#102—1181no—
192#101—1182no—
193#101—2361no—
194#99—22no—
195#99—41no—
196C22⋊C8—1181no—
197#97—591no—
198C4×C8—591no—
199#41—81no—
200C2×Q8—21no—
201C2×Q8—81no—
202C2×Q8—1181no—
203C2×Q8(C2×D59)⋊C411yescharacteristic
204C2×Q8(C2×D59)⋊C411yescharacteristic
205C2×Q8C2×D23611yescharacteristic
206C2×D4—21no—
207C2×D4C2×C22⋊D5911yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4(C2×D59)⋊C411yescharacteristic
211Q16—4721no—
212Q16—4721no—
213Q16—4721no—
214SD16—4721no—
215M16—4721no—
216M16—2362no—
217C2×C8—1181no—
218C2×C8—1181no—
219C2×C8—4721no—
220C4⋊C4—21no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—81no—
224C4⋊C4—1181no—
225C4⋊C4—2361no—
226C4⋊C4—2361no—
227C4⋊C4—2361no—
228C4⋊C4—2361no—
229C22⋊C4—21no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—42no—
233C22⋊C4—2361no—
234C42—81no—
235C42—2361no—
236C42(C2×D59)⋊C411yescharacteristic
237C42C2×C4×D5911yescharacteristic
238C23—21no—
239C23#1427364111yescharacteristic
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—81no—
244Q8—2361no—
245Q8—2361no—
246D4—41no—
247D4—81no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—82no—
259C2×C4—2361no—
260C2×C4—2361no—
261C2×C4#1427364111yescharacteristic
262C2×C4C2×(C2×D59)⋊C411yescharacteristic
263C8—4722no—
264C8—2361no—
265C8—2361no—
266C22—21no—
267C22—41no—
268C22—41no—
269C22—81no—
270C22C2×#1427364111yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—162no—
279C4—4721no—
280C4—4721no—
281C2—21no—
282C2—41no—
283C2—81no—
284C2#2142576311yescharacteristic
subgroupG/H
centerC2#21425763
commutatorC42×C59C22×C4
Frattini#124C22×D59
FittingC59×#2129C2
socleC2×C59#2127
prime radicalC2×#41×C59D4
cosocle#124×C59C23
hypercenter#2129D59
hypocenterC59#13608
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #2129

Complements

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

NH ≅ G/Ntype
3. #21422270C2semidirect
4. #21422255C2semidirect
6. #16838014C4semidirect
8. #16837781C4semidirect
9. #16837770C4semidirect
15. #16837463C4semidirect
165. C59#13608semidirect

Maximal subgroups

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

subgroupindex
#214351792
#214257852
#214222702
#214222552
C59×#21292
#1360859