Finite Group Databasebeta

← #28486529

Subgroups — #28486529

Order 15104.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#21433120C211yesmaximal, characteristic
2#21424088C211yesmaximal, characteristic
3#21422525C212yesmaximal, complemented
4#21421954C212yesmaximal, complemented
5C59×#2580C211yesmaximal, characteristic
6D4×C59⋊C8—24no—
7#16837455C2212yes—
8#16837426C2212yes—
9C2×#14273644—21no—
10C2×C22⋊(C59⋊C8)—22no—
11#16837171—21no—
12Q8⋊(C59⋊C8)—24no—
13C2×C4×Q8×C59C2211yescharacteristic
14C2×C4×D4×C59C2211yescharacteristic
15C59×#399C412yes—
16C59×#393C412yes—
17C59×#390—21no—
18C59×#389—21no—
19C59×#385—22no—
20C59×#382C2211yescharacteristic
21C2×#11745664—22no—
22C22×C59⋊C8—24no—
23C4⋊(C59⋊C8)—22no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26#14273644—42no—
27C22⋊(C59⋊C8)—24no—
28C22⋊(C59⋊C8)—22no—
29C4×C59⋊C8D414yes—
30C22×Q8×C59C2×C411yescharacteristic
31C22×D4×C59D411yescharacteristic
32C23×C4×C59D412yes—
33C4×Q8×C59—22no—
34C4×Q8×C59—22no—
35C4×D4×C59—22no—
36C4×D4×C59—22no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59C2×C412yes—
45C2×C42×C59—21no—
46C2×C42×C59C2311yescharacteristic
47#95×C59—22no—
48#95×C59—21no—
49#95×C59C2×C412yes—
50C24×C59C22⋊C412yes—
51C2×Q8×C59—22no—
52C2×Q8×C59—42no—
53C2×D4×C59—22no—
54C2×D4×C59—22no—
55C2×D4×C59SD1612yes—
56C2×D4×C59Q1612yes—
57C22×C4×C59—21no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—82no—
63C22×C4×C59C22⋊C412yes—
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C4⋊C4×C59—22no—
68C4⋊C4×C59—22no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—82no—
73C22⋊C4×C59—24no—
74C22⋊C4×C59—44no—
75C22⋊C4×C59—82no—
76C42×C59—42no—
77C42×C59C2×D412yes—
78C42×C59#4112yes—
79#11745664—42no—
80#11745664—42no—
81C2×C59⋊C8—24no—
82C2×C59⋊C8—42no—
83C2×C59⋊C8—84no—
84C23×C59—22no—
85C23×C59—22no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59#10314yes—
89C23×C59C2×C22⋊C411yescharacteristic
90Q8×C59—24no—
91Q8×C59—42no—
92D4×C59—24no—
93D4×C59—24no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—44no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—41no—
104C2×C4×C59—42no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—82no—
109C2×C4×C59C4×D412yes—
110C2×C4×C59#12011yescharacteristic
111C2×C4×C59#12112yes—
112C2×C4×C59#12311yescharacteristic
113C2×C4×C59C2×SD1611yescharacteristic
114C2×C4×C59C2×Q1611yescharacteristic
115C2×C4×C59#13512yes—
116C59⋊C8—44no—
117C59⋊C8—82no—
118#9483—591nomaximal, Sylow
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—24no—
122C22×C59—44no—
123C22×C59—42no—
124C22×C59—42no—
125C22×C59—82no—
126C22×C59#38611yescharacteristic
127C22×C59C2×#10311yescharacteristic
128C22×C59#41611yescharacteristic
129C4×C59—22no—
130C4×C59—21no—
131C4×C59—21no—
132C4×C59—42no—
133C4×C59—42no—
134C4×C59—82no—
135C4×C59—81no—
136C4×C59#46512yes—
137C4×C59#46712yes—
138#2972—591no—
139#2580D5911yescharacteristic
140#1800—591no—
141#1487—592no—
142#1286—592no—
143C2×C59—21no—
144C2×C59—21no—
145C2×C59—44no—
146C2×C59#189111yescharacteristic
147C2×C59#189211yescharacteristic
148C2×C59#191111yescharacteristic
149C2×C4×Q8C2×D5911yescharacteristic
150C2×C4×D4C2×D5911yescharacteristic
151C8×D4—1184no—
152#431—592no—
153#423—592no—
154C2×#100—1181no—
155C2×C22⋊C8—1182no—
156#399Dic5912yes—
157#393Dic5912yes—
158#390—21no—
159#389—21no—
160#385—22no—
161#382C2×D5911yescharacteristic
162#343—1181no—
163Q8⋊C8—1184no—
164C59#948311yescharacteristic, complemented, Sylow
165C22×Q8C2×Dic5911yescharacteristic
166C22×D4C22⋊D5911yescharacteristic
167C23×C4C22⋊D5912yes—
168C2×M16—1182no—
169C22×C8—1184no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×Dic5911yescharacteristic
178C2×C4⋊C4C22⋊D5911yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×Dic5912yes—
182C2×C42—21no—
183C2×C42C22×D5911yescharacteristic
184C4⋊C8—1182no—
185C4⋊C8—1182no—
186C4⋊C8—1182no—
187#100—2362no—
188C22⋊C8—1184no—
189C22⋊C8—1182no—
190C4×C8—594no—
191#95—22no—
192#95—21no—
193#95C2×Dic5912yes—
194C24C22⋊Dic5912yes—
195C2×Q8—22no—
196C2×Q8—42no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4#1174567112yes—
200C2×D4#1174567212yes—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—82no—
207C22×C4C22⋊Dic5912yes—
208C22×C4C22×Dic5911yescharacteristic
209C22×C4C2×C22⋊D5911yescharacteristic
210C22×C4C2×C22⋊D5911yescharacteristic
211M16—2362no—
212M16—2362no—
213C2×C8—1184no—
214C2×C8—4724no—
215C2×C8—2362no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C22⋊C4—24no—
223C22⋊C4—44no—
224C22⋊C4—82no—
225C42—42no—
226C42D4×D5912yes—
227C42#1174568712yes—
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23#1427365114yes—
233C23C2×C22⋊Dic5911yescharacteristic
234Q8—24no—
235Q8—42no—
236D4—24no—
237D4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4D4×Dic5912yes—
254C2×C4#1427369511yescharacteristic
255C2×C4#1427369912yes—
256C2×C4#1427370811yescharacteristic
257C2×C4C2×#1174567111yescharacteristic
258C2×C4C2×#1174567211yescharacteristic
259C2×C4#1427374712yes—
260C8—4722no—
261C8—2364no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#1683728211yescharacteristic
270C22C2×#1427365111yescharacteristic
271C22#1683739811yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—42no—
277C4—82no—
278C4—81no—
279C4#1683763312yes—
280C4#1683764112yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#2142451511yescharacteristic
285C2#2142452011yescharacteristic
286C2#2142462911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2580C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#2580D59
hypocenterC59#9483
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #21422525C2semidirect
4. #21421954C2semidirect
164. C59#9483semidirect

Maximal subgroups

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

subgroupindex
#214331202
#214240882
#214225252
#214219542
C59×#25802
#948359