Finite Group Databasebeta

← #28477254

Subgroups — #28477254

Order 15104.

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

normal subgroups only

This group has 500 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#21424164C211yesmaximal, characteristic
2#21423858C211yesmaximal, characteristic
3#21421919C214yesmaximal, complemented
4C59×#2857C211yesmaximal, characteristic
5C2×C22⋊(C59⋊C8)—21no—
6C2×C22⋊(C59⋊C8)—21no—
7C2×#14273638—21no—
8#16837157—21no—
9#16837157C2212yes—
10#16837157C2212yes—
11C2×C4⋊Q8×C59C411yescharacteristic
12C2×C22⋊Q8×C59—21no—
13C22×C42×C59C2211yescharacteristic
14C59×#391—21no—
15C59×#389—21no—
16C59×#389C2212yes—
17C59×#384C411yescharacteristic
18C59×#380—21no—
19C59×#380C412yes—
20C22×C59⋊C8—22no—
21C22×C59⋊C8—22no—
22C22⋊(C59⋊C8)—24no—
23C22⋊(C59⋊C8)—42no—
24#14273638—42no—
25C22×Q8×C59—21no—
26C23×C4×C59C2×C411yescharacteristic
27C23×C4×C59D411yescharacteristic
28C23×C4×C59D411yescharacteristic
29C4⋊Q8×C59—42no—
30C22⋊Q8×C59—44no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59C2×C412yes—
36C2×C4⋊C4×C59C2×C412yes—
37C2×C22⋊C4×C59—22no—
38C2×C42×C59—42no—
39C2×C42×C59C2×C411yescharacteristic
40C2×C42×C59D411yescharacteristic
41C2×C42×C59D411yescharacteristic
42C2×C42×C59C2311yescharacteristic
43#95×C59—22no—
44#95×C59—22no—
45C24×C59C22⋊C411yescharacteristic
46C2×Q8×C59—42no—
47C2×Q8×C59—42no—
48C2×Q8×C59—42no—
49C22×C4×C59—22no—
50C22×C4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59—22no—
53C22×C4×C59—22no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59—41no—
58C22×C4×C59—41no—
59C22×C4×C59—41no—
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C4⋊C4×C59—24no—
67C4⋊C4×C59—42no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—42no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—44no—
72C22⋊C4×C59—44no—
73C42×C59—22no—
74C42×C59—22no—
75C42×C59—41no—
76C42×C59—41no—
77C2×C59⋊C8—24no—
78C2×C59⋊C8—42no—
79C2×C59⋊C8—42no—
80C2×C59⋊C8—42no—
81C2×C59⋊C8—42no—
82C2×C59⋊C8—42no—
83C23×C59—21no—
84C23×C59—21no—
85C23×C59—21no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59#10112yes—
90C23×C59C2×C22⋊C411yescharacteristic
91Q8×C59—84no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—44no—
106C2×C4×C59—44no—
107C2×C4×C59—44no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59#9912yes—
115C2×C4×C59#10412yes—
116C2×C4×C59#13112yes—
117C2×C4×C59AGL(1,ℤ/8)12yes—
118C2×C4×C59#13712yes—
119C59⋊C8—44no—
120C59⋊C8—82no—
121#7705—591nomaximal, Sylow
122C22×C59—22no—
123C22×C59—22no—
124C22×C59—22no—
125C22×C59—41no—
126C22×C59—41no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59#40811yescharacteristic
130C22×C59C2×#9911yescharacteristic
131C22×C59C2×#10111yescharacteristic
132C22×C59#41811yescharacteristic
133C22×C59#41911yescharacteristic
134C22×C59C2×#10411yescharacteristic
135C22×C59#42111yescharacteristic
136C4×C59—22no—
137C4×C59—22no—
138C4×C59—22no—
139C4×C59—22no—
140C4×C59—22no—
141C4×C59—41no—
142C4×C59—82no—
143C4×C59—82no—
144#2857D5911yescharacteristic
145#1817—591no—
146#1749—591no—
147#1271—594no—
148C2×C59—42no—
149C2×C59#147211yescharacteristic
150C2×C59#148511yescharacteristic
151C2×C59#149611yescharacteristic
152C2×C59#150711yescharacteristic
153C2×C59#151111yescharacteristic
154C2×C59#153111yescharacteristic
155C2×C59#155611yescharacteristic
156C2×C4⋊Q8Dic5911yescharacteristic
157C2×C22⋊Q8—21no—
158C22×C42C2×D5911yescharacteristic
159C2×C22⋊C8—1181no—
160C2×C22⋊C8—1181no—
161C2×#97—1181no—
162#391—21no—
163#389—21no—
164#389C2×D5912yes—
165#384Dic5911yescharacteristic
166#380—21no—
167#380Dic5912yes—
168#336—592no—
169#336—592no—
170#336—1181no—
171C59#770511yescharacteristic, complemented, Sylow
172C22×Q8—21no—
173C23×C4C2×Dic5911yescharacteristic
174C23×C4C22⋊D5911yescharacteristic
175C23×C4C22⋊D5911yescharacteristic
176C22×C8—1182no—
177C22×C8—1182no—
178C4⋊Q8—42no—
179C22⋊Q8—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C2×Dic5912yes—
185C2×C4⋊C4C2×Dic5912yes—
186C2×C22⋊C4—22no—
187C2×C42—42no—
188C2×C42C2×Dic5911yescharacteristic
189C2×C42C22⋊D5911yescharacteristic
190C2×C42C22⋊D5911yescharacteristic
191C2×C42C22×D5911yescharacteristic
192C22⋊C8—1184no—
193C22⋊C8—2362no—
194#97—2362no—
195#95—22no—
196#95—22no—
197C24C22⋊Dic5911yescharacteristic
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4C22⋊Dic5911yescharacteristic
213C22×C4C22⋊Dic5911yescharacteristic
214C22×C4C22⋊Dic5911yescharacteristic
215C22×C4C22×Dic5911yescharacteristic
216C22×C4C2×C22⋊D5911yescharacteristic
217C22×C4C2×C22⋊D5911yescharacteristic
218C2×C8—1184no—
219C2×C8—2362no—
220C2×C8—2362no—
221C2×C8—2362no—
222C2×C8—2362no—
223C2×C8—2362no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C22⋊C4—44no—
231C42—22no—
232C42—22no—
233C42—41no—
234C42—41no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23#1427364612yes—
242C23C2×C22⋊Dic5911yescharacteristic
243Q8—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4#1427364212yes—
267C2×C4#1427365412yes—
268C2×C4#1427373412yes—
269C2×C4#1427375312yes—
270C2×C4#1427375812yes—
271C8—4722no—
272C8—2364no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#1683736911yescharacteristic
281C22C2×#1427364211yescharacteristic
282C22C2×#1427364611yescharacteristic
283C22#1683740711yescharacteristic
284C22#1683741111yescharacteristic
285C22C2×#1427365411yescharacteristic
286C22#1683741911yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—82no—
294C4—82no—
295C2—42no—
296C2#2142246011yescharacteristic
297C2#2142251711yescharacteristic
298C2#2142256511yescharacteristic
299C2#2142261211yescharacteristic
300C2#2142262911yescharacteristic
301C2#2142271111yescharacteristic
302C2#2142281311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2857C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2857D59
hypocenterC59#7705
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. #21421919C2semidirect
171. C59#7705semidirect

Maximal subgroups

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

subgroupindex
#214241642
#214238582
#214219192
C59×#28572
#770559