Finite Group Databasebeta

← #28484697

Subgroups — #28484697

Order 15104.

458 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 115 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#21429917C211yesmaximal, characteristic
2#21422778C212yesmaximal, complemented
3#21422410C211yesmaximal, characteristic
4#21421956C212yesmaximal, complemented
5C59×#2921C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7#16837455C2212yes—
8#16837341—21no—
9#16837338—21no—
10#16837322C412yes—
11#16837314C412yes—
12#16837300—21no—
13#16837268C2211yescharacteristic
14#16837262—21no—
15#16837150—24no—
16#16837128—21no—
17#16837126—21no—
18C2×#117×C59C2211yescharacteristic
19C59×#433—24no—
20C59×#423C2212yes—
21C59×#407—21no—
22C4×M16×C59—21no—
23C22×Dic118C2×C412yes—
24#14273727—22no—
25#14273727—22no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—21no—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—22no—
31C4⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33C4×C59⋊C8—22no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—21no—
37C59⋊#95C2×C412yes—
38C23×C4×C59D411yescharacteristic
39C2×M16×C59—22no—
40#117×C59—22no—
41#117×C59—22no—
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C42×C59D411yescharacteristic
47C2×C42×C59C2311yescharacteristic
48C4⋊C8×C59—22no—
49C4⋊C8×C59D414yes—
50C22⋊C8×C59—24no—
51#97×C59—22no—
52C4×C8×C59—22no—
53C24×C59C22⋊C411yescharacteristic
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×Dic59—22no—
65C22×Dic59—22no—
66C2×Dic118—24no—
67C2×Dic118—82no—
68M16×C59—42no—
69M16×C59—42no—
70C2×C8×C59—24no—
71C4⋊C4×C59—22no—
72C4⋊C4×C59—22no—
73C22⋊C4×C59—22no—
74C22⋊C4×C59—22no—
75C42×C59—21no—
76C42×C59—21no—
77C42×C59C2×D412yes—
78C42×C59#4112yes—
79C22⋊Dic59—82no—
80C4⋊Dic59—41no—
81C4⋊Dic59—42no—
82C4⋊Dic59—41no—
83C4⋊Dic59—42no—
84Dic59⋊C4—82no—
85C2×C59⋊C8—42no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59C2×C22⋊C411yescharacteristic
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59C4×D412yes—
104C2×C4×C59#12011yescharacteristic
105C2×C4×C59#12112yes—
106C2×C4×C59#12311yescharacteristic
107C2×C4×C59#13512yes—
108C2×C4×C59#13712yes—
109C2×Dic59—42no—
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—42no—
114C2×Dic59—42no—
115Dic118—84no—
116C8×C59—44no—
117C59⋊C8—82no—
118#9192—591nomaximal, Sylow
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—42no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—41no—
125C22×C59—81no—
126C22×C59#38611yescharacteristic
127C22×C59#41611yescharacteristic
128C22×C59#41911yescharacteristic
129C4×C59—22no—
130C4×C59—22no—
131C4×C59—41no—
132C4×C59—41no—
133C4×C59—41no—
134C4×C59#45212yes—
135C4×C59#48512yes—
136Dic59—82no—
137Dic59—82no—
138#2921D5911yescharacteristic
139#2628—591no—
140#1548—592no—
141#1458—591no—
142#1286—592no—
143C2×C59—21no—
144C2×C59—21no—
145C2×C59—81no—
146C2×C59#187311yescharacteristic
147C2×C59#190911yescharacteristic
148C2×C59#191411yescharacteristic
149C2×C4⋊Q8—591no—
150C2×#117C2×D5911yescharacteristic
151#433—24no—
152#431—592no—
153#423C2×D5912yes—
154#407—21no—
155C4×M16—21no—
156#400—1181no—
157#399—1181no—
158#395—592no—
159#393—592no—
160#389—1181no—
161#382—591no—
162#380—1181no—
163#333—1184no—
164#324—1181no—
165#323—1181no—
166C59#919211yescharacteristic, complemented, Sylow
167C22×Q8—592no—
168C23×C4C22⋊D5911yescharacteristic
169C2×M16—22no—
170C4⋊Q8—1182no—
171C4⋊Q8—1182no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—1182no—
175C2×C4⋊C4—1181no—
176C2×C4⋊C4—1181no—
177C2×C4⋊C4—1181no—
178C2×C4⋊C4C4×D5911yescharacteristic
179C2×C4⋊C4D23611yescharacteristic
180C2×C22⋊C4—1182no—
181C2×C22⋊C4C4×D5911yescharacteristic
182C2×C22⋊C4D23611yescharacteristic
183C2×C42C22⋊D5911yescharacteristic
184C2×C42C22×D5911yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—1182no—
187C4⋊C8C22⋊D5914yes—
188C22⋊C8—24no—
189C22⋊C8—1182no—
190#97—22no—
191C4×C8—22no—
192C4×C8—1182no—
193#95—592no—
194#95—1182no—
195#95—1182no—
196#95—1181no—
197C24(C2×D59)⋊C411yescharacteristic
198C2×Q8—1184no—
199C2×Q8—4722no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—1182no—
205C22×C4—1182no—
206C22×C4(C2×D59)⋊C411yescharacteristic
207C22×C4(C2×D59)⋊C411yescharacteristic
208C22×C4(C2×D59)⋊C411yescharacteristic
209C22×C4C2×C4×D5911yescharacteristic
210C22×C4C2×D23611yescharacteristic
211C22×C4C2×C22⋊D5911yescharacteristic
212M16—42no—
213M16—42no—
214C2×C8—24no—
215C2×C8—2362no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—4722no—
219C4⋊C4—2361no—
220C4⋊C4—2362no—
221C4⋊C4—2361no—
222C4⋊C4—2362no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—4722no—
226C42—21no—
227C42—21no—
228C42#1174568512yes—
229C42C2×C22⋊D5912yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×(C2×D59)⋊C411yescharacteristic
236Q8—4724no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—2362no—
249C2×C4—2362no—
250C2×C4—2362no—
251C2×C4—2362no—
252C2×C4—2362no—
253C2×C4—2362no—
254C2×C4C4×C22⋊D5912yes—
255C2×C4#1427369611yescharacteristic
256C2×C4#1427370112yes—
257C2×C4#1427371011yescharacteristic
258C2×C4#1427374712yes—
259C2×C4#1427375712yes—
260C8—44no—
261C8—4722no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1683728111yescharacteristic
270C22#1683739611yescharacteristic
271C22#1683741011yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—4722no—
278C4—4722no—
279C4#1683756412yes—
280C4#1683774312yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2142440811yescharacteristic
285C2#2142461811yescharacteristic
286C2#2142464311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2921C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2921D59
hypocenterC59#9192
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #21422778C2semidirect
4. #21421956C2semidirect
166. C59#9192semidirect

Maximal subgroups

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

subgroupindex
#214299172
#214227782
#214224102
#214219562
C59×#29212
#919259