Finite Group Databasebeta

← #28484718

Subgroups — #28484718

Order 15104.

478 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 119 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#21432734C211yesmaximal, characteristic
2#21422768C212yesmaximal, complemented
3#21422410C211yesmaximal, characteristic
4#21421954C212yesmaximal, complemented
5C59×#2621C211yesmaximal, characteristic
6#16837463—24no—
7#16837455C2212yes—
8#16837426C2212yes—
9#16837366—21no—
10C4×#11745664—21no—
11#16837148—24no—
12#16837128—21no—
13#16837126—21no—
14C2×C4⋊Q8×C59C2211yescharacteristic
15C2×C22⋊Q8×C59—21no—
16C2×#117×C59C2211yescharacteristic
17C59×#398—21no—
18C59×#398C412yes—
19C59×#397—21no—
20C59×#397C412yes—
21C59×#391—21no—
22C59×#382C2211yescharacteristic
23C2×#11745664—22no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26C4⋊(C59⋊C8)D414yes—
27C22⋊(C59⋊C8)—24no—
28C22⋊(C59⋊C8)—22no—
29#14273638—22no—
30C4×C59⋊C8—22no—
31C4×C59⋊C8—22no—
32C22×Q8×C59—21no—
33C23×C4×C59D411yescharacteristic
34C4⋊Q8×C59—22no—
35C4⋊Q8×C59—22no—
36C22⋊Q8×C59—44no—
37#117×C59—22no—
38#117×C59—22no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59C2×C412yes—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59D411yescharacteristic
48C2×C42×C59D411yescharacteristic
49C2×C42×C59C2311yescharacteristic
50#95×C59—22no—
51#95×C59—22no—
52#95×C59—21no—
53#95×C59C2×C412yes—
54C24×C59C22⋊C411yescharacteristic
55C2×Q8×C59—41no—
56C2×Q8×C59—41no—
57C2×Q8×C59—42no—
58C2×Q8×C59—42no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59—22no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C4⋊C4×C59—22no—
72C4⋊C4×C59—22no—
73C4⋊C4×C59—24no—
74C4⋊C4×C59—44no—
75C4⋊C4×C59—42no—
76C4⋊C4×C59—42no—
77C22⋊C4×C59—22no—
78C22⋊C4×C59—22no—
79C22⋊C4×C59—44no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59C2×D412yes—
83C42×C59#4112yes—
84#11745664—42no—
85#11745664—42no—
86C2×C59⋊C8—24no—
87C2×C59⋊C8—42no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59C2×C22⋊C411yescharacteristic
94Q8×C59—82no—
95Q8×C59—82no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59C4×D412yes—
114C2×C4×C59#12011yescharacteristic
115C2×C4×C59#12112yes—
116C2×C4×C59#12311yescharacteristic
117C2×C4×C59#13512yes—
118C2×C4×C59AGL(1,ℤ/8)12yes—
119C59⋊C8—44no—
120C59⋊C8—82no—
121#9196—591nomaximal, Sylow
122C22×C59—21no—
123C22×C59—21no—
124C22×C59—42no—
125C22×C59—41no—
126C22×C59—41no—
127C22×C59—41no—
128C22×C59—81no—
129C22×C59#38611yescharacteristic
130C22×C59#41611yescharacteristic
131C22×C59#41811yescharacteristic
132C4×C59—22no—
133C4×C59—22no—
134C4×C59—41no—
135C4×C59—41no—
136C4×C59—41no—
137C4×C59—82no—
138C4×C59—82no—
139C4×C59#44912yes—
140C4×C59#48212yes—
141#2921—591no—
142#2621D5911yescharacteristic
143#1546—592no—
144#1458—591no—
145#1286—592no—
146C2×C59—21no—
147C2×C59—21no—
148C2×C59—81no—
149C2×C59#187511yescharacteristic
150C2×C59#190911yescharacteristic
151C2×C59#191511yescharacteristic
152C2×C4⋊Q8C2×D5911yescharacteristic
153C2×C22⋊Q8—21no—
154C2×#117C2×D5911yescharacteristic
155#433—1184no—
156#431—592no—
157#423—592no—
158#407—1181no—
159C4×M16—1181no—
160#398—21no—
161#398Dic5912yes—
162#397—21no—
163#397Dic5912yes—
164#391—21no—
165#382C2×D5911yescharacteristic
166#332—1184no—
167#324—1181no—
168#323—1181no—
169C59#919611yescharacteristic, complemented, Sylow
170C22×Q8—21no—
171C23×C4C22⋊D5911yescharacteristic
172C2×M16—1182no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C22⋊Q8—44no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C2×Dic5912yes—
182C2×C4⋊C4C2×Dic5911yescharacteristic
183C2×C4⋊C4C22⋊D5911yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×Dic5911yescharacteristic
186C2×C22⋊C4C22⋊D5911yescharacteristic
187C2×C42C22⋊D5911yescharacteristic
188C2×C42C22×D5911yescharacteristic
189C4⋊C8—594no—
190C4⋊C8—1182no—
191C4⋊C8—1182no—
192C22⋊C8—1184no—
193C22⋊C8—1182no—
194#97—1182no—
195C4×C8—1182no—
196C4×C8—1182no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95C2×Dic5912yes—
201C24C22⋊Dic5911yescharacteristic
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—42no—
205C2×Q8—42no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4C22⋊Dic5911yescharacteristic
213C22×C4C22⋊Dic5911yescharacteristic
214C22×C4C22⋊Dic5911yescharacteristic
215C22×C4C22×Dic5911yescharacteristic
216C22×C4C2×C22⋊D5911yescharacteristic
217C22×C4C2×C22⋊D5911yescharacteristic
218M16—2362no—
219M16—2362no—
220C2×C8—1184no—
221C2×C8—2362no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—44no—
231C42—21no—
232C42—21no—
233C42D4×D5912yes—
234C42#1174568712yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×C22⋊Dic5911yescharacteristic
241Q8—82no—
242Q8—82no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4D4×Dic5912yes—
261C2×C4#1427369511yescharacteristic
262C2×C4#1427369912yes—
263C2×C4#1427370811yescharacteristic
264C2×C4#1427374712yes—
265C2×C4#1427375312yes—
266C8—4722no—
267C8—2364no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1683728211yescharacteristic
276C22#1683739811yescharacteristic
277C22#1683740711yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4#1683754512yes—
286C4#1683772312yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#2142441911yescharacteristic
291C2#2142461911yescharacteristic
292C2#2142464911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2621C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2621D59
hypocenterC59#9196
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2621

Complements

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

NH ≅ G/Ntype
2. #21422768C2semidirect
4. #21421954C2semidirect
169. C59#9196semidirect

Maximal subgroups

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

subgroupindex
#214327342
#214227682
#214224102
#214219542
C59×#26212
#919659