Finite Group Databasebeta

← #28501892

Subgroups — #28501892

Order 15104.

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

normal subgroups only

This group has 396 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#21434847C211yesmaximal, characteristic
2#21426277C212yesmaximal, complemented
3#21425519C211yesmaximal, characteristic, complemented
4#21422106C212yesmaximal, complemented
5C59×#1314C211yesmaximal, characteristic, complemented
6C2×#14273701—21no—
7#16837716—22no—
8#16837638—21no—
9#16837638C2212yes—
10#16837538—22no—
11C2×#14273661C2211yescharacteristic, complemented
12#16837262—21no—
13#16837228—22no—
14#16837204—22no—
15#16837165—21no—
16#16837165C2211yescharacteristic, complemented
17C2×C22⋊C8×C59C2211yescharacteristic
18C59×C22⋊C16C2212yescomplemented
19C22×D236—21no—
20C22×C22⋊D59—41no—
21C2×D472—21no—
22C2×D472D412yes—
23#14273701—22no—
24#14273701—22no—
25#14273701—41no—
26#14273701—41no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—21no—
30C2×C22⋊Dic59—41no—
31C2×(C2×D59)⋊C4—41no—
32#14273661—21no—
33#14273661—22no—
34#14273661D412yes—
35#14273649—22no—
36#14273649—22no—
37C59⋊#95—41no—
38C23×C4×C59D411yescharacteristic
39C22×C8×C59D411yescharacteristic
40C22×C8×C59C2311yescharacteristic
41C2×C16×C59—22no—
42C22⋊C8×C59—22no—
43C24×C59D811yescharacteristic
44C23×D59—41no—
45C22×C4×C59—21no—
46C22×C4×C59—21no—
47C22×C4×C59—41no—
48C22×C4×C59—41no—
49C22×C4×C59D811yescharacteristic
50C22×C4×C59C2×D411yescharacteristic
51C2×C22⋊D59—42no—
52C2×C22⋊D59—42no—
53C2×C22⋊D59—82no—
54C2×C22⋊D59—82no—
55C22×Dic59—41no—
56C22×Dic59—41no—
57C22×Dic59—41no—
58C2×D236—22no—
59C2×D236—41no—
60C2×D236—81no—
61C2×D236—81no—
62C2×C8×C59—21no—
63C2×C8×C59—41no—
64C2×C8×C59—41no—
65C2×C8×C59C2×D412yes—
66C22⋊Dic59—82no—
67(C2×D59)⋊C4—42no—
68(C2×D59)⋊C4—42no—
69C4⋊Dic59—22no—
70C4⋊Dic59—41no—
71C4⋊Dic59—41no—
72C4⋊Dic59—41no—
73C4⋊Dic59—41no—
74C4⋊Dic59—41no—
75D472—42no—
76D472—81no—
77C16×C59—42no—
78C23×C59—21no—
79C23×C59—21no—
80C23×C59—41no—
81C23×C59—41no—
82C23×C59D1612yes—
83C23×C59C2×D811yescharacteristic
84C22×D59—42no—
85C22×D59—82no—
86C22×D59—82no—
87C22×D59—82no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—81no—
95C2×C4×C59#12011yescharacteristic
96C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
97C22⋊D59—84no—
98C22⋊D59—82no—
99C22⋊D59—82no—
100C2×Dic59—42no—
101C2×Dic59—82no—
102C2×Dic59—81no—
103C2×Dic59—81no—
104C2×Dic59—81no—
105C2×Dic59—81no—
106C2×Dic59—81no—
107C2×Dic59—81no—
108D236—42no—
109D236—42no—
110D236—82no—
111C8×C59—22no—
112C8×C59—41no—
113#12278—591nomaximal, Sylow
114C22×C59—21no—
115C22×C59—21no—
116C22×C59—22no—
117C22×C59—42no—
118C22×C59—41no—
119C22×C59—41no—
120C22×C59—81no—
121C22×C59#44711yescharacteristic
122C22×C59C2×D1611yescharacteristic
123C22×C59#50911yescharacteristic
124C2×D59—82no—
125C2×D59—82no—
126C2×D59—82no—
127C2×D59—164no—
128C4×C59—21no—
129C4×C59—41no—
130C4×C59#45511yescharacteristic
131C4×C59#45811yescharacteristic
132Dic59—82no—
133Dic59—161no—
134Dic59—161no—
135#3186—591no—
136#2232—592no—
137#2075—591no—
138#1347—592no—
139#1314D5911yescharacteristic, complemented
140C2×C59—21no—
141C2×C59—21no—
142C2×C59—42no—
143C2×C59#160711yescharacteristic
144C2×C59#218411yescharacteristic
145C2×C59#219111yescharacteristic
146D59—162no—
147C2×#121—1181no—
148#480—1182no—
149#466—592no—
150#466—1181no—
151#447—1182no—
152C2×#107—591no—
153C2×C22⋊C8C2×D5911yescharacteristic
154#380—1181no—
155#366—1182no—
156#357—1182no—
157C22⋊C16C2×D5912yescomplemented
158#340—591no—
159#340—1181no—
160C59#1227811yescharacteristic, complemented, Sylow
161C22×D4—1181no—
162C22×D4—2361no—
163C23×C4D23611yescharacteristic
164C2×D8—592no—
165C2×D8—1181no—
166C22×C8D23611yescharacteristic
167C22×C8C22×D5911yescharacteristic
168#121—1182no—
169#121—1182no—
170#121—2361no—
171#121—2361no—
172C2×C4⋊C4—1181no—
173C2×C4⋊C4—1181no—
174C2×C4⋊C4—1181no—
175C2×C22⋊C4—2361no—
176C2×C22⋊C4—2361no—
177C2×C16—22no—
178#107—592no—
179#107—1181no—
180#107—1182no—
181#102—1182no—
182#102—1182no—
183C22⋊C8—22no—
184#95—2361no—
185C24—2361no—
186C24D47211yescharacteristic
187C2×D4—1182no—
188C2×D4—4721no—
189C2×D4—4721no—
190C2×D4—4722no—
191C2×D4—4722no—
192C2×D4—2361no—
193C2×D4—2362no—
194C2×D4—2362no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—2361no—
200C22×C4—2361no—
201C22×C4—2361no—
202C22×C4D47211yescharacteristic
203C22×C4C2×D23611yescharacteristic
204D8—4721no—
205D8—2362no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D5912yes—
210C4⋊C4—1182no—
211C4⋊C4—2361no—
212C4⋊C4—2361no—
213C4⋊C4—2361no—
214C4⋊C4—2361no—
215C4⋊C4—2361no—
216C22⋊C4—4722no—
217C22⋊C4—2362no—
218C22⋊C4—2362no—
219C16—42no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—4722no—
225C23—4722no—
226C23—4722no—
227C23—2362no—
228C23D94412yes—
229C23C2×D47211yescharacteristic
230D4—4722no—
231D4—4724no—
232D4—4722no—
233D4—4722no—
234D4—2362no—
235D4—2362no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—4721no—
244C2×C4—4721no—
245C2×C4—4721no—
246C2×C4—4721no—
247C2×C4—4721no—
248C2×C4—4721no—
249C2×C4—4722no—
250C2×C4—2362no—
251C2×C4#1427369711yescharacteristic
252C2×C4#1427375511yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—9444no—
263C22—4722no—
264C22—4722no—
265C22—4722no—
266C22#1683753811yescharacteristic
267C22C2×D94411yescharacteristic
268C22#1683784011yescharacteristic
269C4—21no—
270C4—41no—
271C4—9441no—
272C4—9441no—
273C4—4722no—
274C4#1683758211yescharacteristic
275C4#1683759311yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—9442no—
280C2#2142308411yescharacteristic
281C2#2142601011yescharacteristic
282C2#2142604911yescharacteristic
subgroupG/H
centerC22#16837538
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#1314C2
socleC22×C59#447
prime radical#21434847C2
hypercenter#1314D59
hypocenterC59#12278
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #21426277C2semidirect
3. #21425519C2semidirect
4. #21422106C2semidirect
5. C59×#1314C2semidirect
11. C2×#14273661C22semidirect
16. #16837165C22semidirect
18. C59×C22⋊C16C22semidirect
139. #1314D59semidirect
157. C22⋊C16C2×D59semidirect
160. C59#12278semidirect

Maximal subgroups

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

subgroupindex
#214348472
#214262772
#214255192
#214221062
C59×#13142
#1227859