Finite Group Databasebeta

← #28484698

Subgroups — #28484698

Order 15104.

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

normal subgroups only

This group has 458 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#21422777C212yesmaximal, complemented
3#21422410C211yesmaximal, characteristic
4#21421954C212yesmaximal, complemented
5C59×#2628C211yesmaximal, characteristic
6#16837463—24no—
7#16837455C2212yes—
8#16837426C2212yes—
9#16837366—21no—
10C4×#11745664—21no—
11#16837151—24no—
12#16837128—21no—
13#16837126—21no—
14C2×C4⋊Q8×C59C2211yescharacteristic
15C2×#117×C59C2211yescharacteristic
16C59×#400—21no—
17C59×#399—21no—
18C59×#395C412yes—
19C59×#393C412yes—
20C59×#389—21no—
21C59×#382C2211yescharacteristic
22C59×#380—21no—
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×C59C2×C412yes—
33C23×C4×C59D411yescharacteristic
34C4⋊Q8×C59—22no—
35C4⋊Q8×C59—22no—
36#117×C59—22no—
37#117×C59—22no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—22no—
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C42×C59D411yescharacteristic
48C2×C42×C59C2311yescharacteristic
49#95×C59—22no—
50#95×C59—22no—
51#95×C59—21no—
52#95×C59C2×C412yes—
53C24×C59C22⋊C411yescharacteristic
54C2×Q8×C59—24no—
55C2×Q8×C59—82no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—22no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C4⋊C4×C59—22no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—41no—
73C4⋊C4×C59—42no—
74C4⋊C4×C59—82no—
75C22⋊C4×C59—22no—
76C22⋊C4×C59—22no—
77C22⋊C4×C59—82no—
78C42×C59—21no—
79C42×C59—21no—
80C42×C59C2×D412yes—
81C42×C59#4112yes—
82#11745664—42no—
83#11745664—42no—
84C2×C59⋊C8—24no—
85C2×C59⋊C8—42no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59C2×C22⋊C411yescharacteristic
92Q8×C59—84no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59C4×D412yes—
111C2×C4×C59#12011yescharacteristic
112C2×C4×C59#12112yes—
113C2×C4×C59#12311yescharacteristic
114C2×C4×C59#13512yes—
115C2×C4×C59#13712yes—
116C59⋊C8—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—82no—
135C4×C59—82no—
136C4×C59#45212yes—
137C4×C59#48512yes—
138#2921—591no—
139#2628D5911yescharacteristic
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⋊Q8C2×D5911yescharacteristic
150C2×#117C2×D5911yescharacteristic
151#433—1184no—
152#431—592no—
153#423—592no—
154#407—1181no—
155C4×M16—1181no—
156#400—21no—
157#399—21no—
158#395Dic5912yes—
159#393Dic5912yes—
160#389—21no—
161#382C2×D5911yescharacteristic
162#380—21no—
163#333—1184no—
164#324—1181no—
165#323—1181no—
166C59#919211yescharacteristic, complemented, Sylow
167C22×Q8C2×Dic5912yes—
168C23×C4C22⋊D5911yescharacteristic
169C2×M16—1182no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C2×Dic5911yescharacteristic
179C2×C4⋊C4C22⋊D5911yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×Dic5911yescharacteristic
182C2×C22⋊C4C22⋊D5911yescharacteristic
183C2×C42C22⋊D5911yescharacteristic
184C2×C42C22×D5911yescharacteristic
185C4⋊C8—594no—
186C4⋊C8—1182no—
187C4⋊C8—1182no—
188C22⋊C8—1184no—
189C22⋊C8—1182no—
190#97—1182no—
191C4×C8—1182no—
192C4×C8—1182no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95C2×Dic5912yes—
197C24C22⋊Dic5911yescharacteristic
198C2×Q8—24no—
199C2×Q8—82no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4C22⋊Dic5911yescharacteristic
207C22×C4C22⋊Dic5911yescharacteristic
208C22×C4C22⋊Dic5911yescharacteristic
209C22×C4C22×Dic5911yescharacteristic
210C22×C4C2×C22⋊D5911yescharacteristic
211C22×C4C2×C22⋊D5911yescharacteristic
212M16—2362no—
213M16—2362no—
214C2×C8—1184no—
215C2×C8—2362no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42D4×D5912yes—
229C42#1174568712yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊Dic5911yescharacteristic
236Q8—84no—
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—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4D4×Dic5912yes—
255C2×C4#1427369511yescharacteristic
256C2×C4#1427369912yes—
257C2×C4#1427370811yescharacteristic
258C2×C4#1427374712yes—
259C2×C4#1427375812yes—
260C8—4722no—
261C8—2364no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1683728211yescharacteristic
270C22#1683739811yescharacteristic
271C22#1683741111yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—82no—
279C4#1683756212yes—
280C4#1683774012yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2142440911yescharacteristic
285C2#2142461911yescharacteristic
286C2#2142464411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2628C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2628D59
hypocenterC59#9192
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #21422777C2semidirect
4. #21421954C2semidirect
166. C59#9192semidirect

Maximal subgroups

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

subgroupindex
#214327342
#214227772
#214224102
#214219542
C59×#26282
#919259