Finite Group Databasebeta

← #28501906

Subgroups — #28501906

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#21426281C212yesmaximal, complemented
3#21422104C212yesmaximal, complemented
4#21422020C211yesmaximal, characteristic, complemented
5C59×#2076C211yesmaximal, characteristic, complemented
6C2×#14273701—21no—
7#16837716—22no—
8#16837638—21no—
9#16837638C2212yes—
10#16837538—22no—
11#16837229—22no—
12#16837202—22no—
13C22⋊(C59⋊C16)C2212yescomplemented
14C2×#107×C59C2211yescharacteristic, complemented
15C2×C22⋊C8×C59C2211yescharacteristic
16C59×#380—21no—
17C59×#340—21no—
18C59×#340C2211yescharacteristic, complemented
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×(C2×D59)⋊C4—41no—
29C2×C59⋊C16—22no—
30#14273661—21no—
31#14273661D412yes—
32#14273649—22no—
33#14273649—22no—
34C23×C4×C59D411yescharacteristic
35C22×C8×C59D411yescharacteristic
36C22×C8×C59C2311yescharacteristic
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C22⋊C4×C59—41no—
40#107×C59—22no—
41C22⋊C8×C59—22no—
42#95×C59—41no—
43C24×C59D811yescharacteristic
44C23×D59—41no—
45C22×C4×C59—21no—
46C22×C4×C59—21no—
47C22×C4×C59—41no—
48C22×C4×C59—41no—
49C22×C4×C59—41no—
50C22×C4×C59—41no—
51C22×C4×C59D811yescharacteristic
52C22×C4×C59C2×D411yescharacteristic
53C2×C22⋊D59—42no—
54C2×C22⋊D59—42no—
55C2×C22⋊D59—82no—
56C2×C22⋊D59—82no—
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—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—41no—
69C4⋊C4×C59—41no—
70C22⋊C4×C59—82no—
71(C2×D59)⋊C4—42no—
72(C2×D59)⋊C4—42no—
73C4⋊Dic59—22no—
74C4⋊Dic59—41no—
75D472—42no—
76D472—81no—
77C59⋊C16—42no—
78C23×C59—21no—
79C23×C59—21no—
80C23×C59—41no—
81C23×C59—41no—
82C23×C59SD3212yes—
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—81no—
96C2×C4×C59—81no—
97C2×C4×C59—81no—
98C2×C4×C59—81no—
99C2×C4×C59—81no—
100C2×C4×C59—81no—
101C2×C4×C59#12011yescharacteristic
102C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
103C22⋊D59—84no—
104C22⋊D59—82no—
105C22⋊D59—82no—
106C2×Dic59—42no—
107C2×Dic59—82no—
108D236—42no—
109D236—42no—
110D236—82no—
111C8×C59—22no—
112C8×C59—41no—
113#12281—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×SD3211yescharacteristic
123C22×C59#50911yescharacteristic
124C2×D59—82no—
125C2×D59—82no—
126C2×D59—82no—
127C2×D59—164no—
128C4×C59—21no—
129C4×C59—41no—
130C4×C59—161no—
131C4×C59—161no—
132C4×C59#45511yescharacteristic
133C4×C59#45811yescharacteristic
134Dic59—82no—
135#3186—591no—
136#2233—592no—
137#2076D5911yescharacteristic, complemented
138#1347—592no—
139#1314—591no—
140C2×C59—21no—
141C2×C59—21no—
142C2×C59—42no—
143C2×C59#160711yescharacteristic
144C2×C59#218711yescharacteristic
145C2×C59#219011yescharacteristic
146D59—162no—
147C2×#121—1181no—
148#480—1182no—
149#466—592no—
150#466—1181no—
151#447—1182no—
152C2×#107C2×D5911yescharacteristic, complemented
153C2×C22⋊C8C2×D5911yescharacteristic
154#380—21no—
155#367—1182no—
156#357—1182no—
157C22⋊C16—592no—
158#340—21no—
159#340C2×D5911yescharacteristic, complemented
160C59#1228111yescharacteristic, complemented, Sylow
161C22×D4—1181no—
162C22×D4—2361no—
163C23×C4C22⋊D5911yescharacteristic
164C2×D8—592no—
165C2×D8—1181no—
166C22×C8C22⋊D5911yescharacteristic
167C22×C8C22×D5911yescharacteristic
168#121—1182no—
169#121—1182no—
170#121—2361no—
171#121—2361no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—1181no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—2361no—
177C2×C16—1182no—
178#107—22no—
179#107—592no—
180#107—1181no—
181#102—1182no—
182#102—1182no—
183C22⋊C8—22no—
184#95—41no—
185C24—2361no—
186C24#1174566911yescharacteristic
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—41no—
200C22×C4—41no—
201C22×C4—2361no—
202C22×C4#1174566911yescharacteristic
203C22×C4C2×C22⋊D5911yescharacteristic
204D8—4721no—
205D8—2362no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D5912yes—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—1182no—
215C4⋊C4—2361no—
216C22⋊C4—82no—
217C22⋊C4—2362no—
218C22⋊C4—2362no—
219C16—2362no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—4722no—
225C23—4722no—
226C23—4722no—
227C23—2362no—
228C23#1427367012yes—
229C23C2×#1174566911yescharacteristic
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—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—4722no—
250C2×C4—2362no—
251C2×C4#1427369511yescharacteristic
252C2×C4#1427375211yescharacteristic
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#1683753511yescharacteristic
267C22C2×#1427367011yescharacteristic
268C22#1683783711yescharacteristic
269C4—21no—
270C4—41no—
271C4—161no—
272C4—161no—
273C4—4722no—
274C4#1683757911yescharacteristic
275C4#1683759111yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—9442no—
280C2#2142308111yescharacteristic
281C2#2142602411yescharacteristic
282C2#2142604111yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#2076C2
socleC22×C59#447
prime radical#21434847C2
hypercenter#2076D59
hypocenterC59#12281
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2076

Complements

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

NH ≅ G/Ntype
2. #21426281C2semidirect
3. #21422104C2semidirect
4. #21422020C2semidirect
5. C59×#2076C2semidirect
13. C22⋊(C59⋊C16)C22semidirect
14. C2×#107×C59C22semidirect
18. C59×#340C22semidirect
137. #2076D59semidirect
152. C2×#107C2×D59semidirect
159. #340C2×D59semidirect
160. C59#12281semidirect

Maximal subgroups

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

subgroupindex
#214348472
#214262812
#214221042
#214220202
C59×#20762
#1228159