Finite Group Databasebeta

← #28482701

Subgroups — #28482701

Order 15104.

460 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 111 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#21433125C211yesmaximal, characteristic
2#21424394C211yesmaximal, characteristic
3#21424303C212yesmaximal, complemented
4#21421984C212yesmaximal, complemented
5C59×#1459C211yesmaximal, characteristic
6C22×#11745664C2211yescharacteristic
7#16837474—22no—
8#16837474—22no—
9#16837459—21no—
10C2×#14273663—22no—
11#16837366C2212yes—
12C2×C22⋊(C59⋊C8)—21no—
13C2×C22⋊(C59⋊C8)—21no—
14C2×C22⋊(C59⋊C8)C2211yescharacteristic
15#16837167—22no—
16#16837157—21no—
17C2×C4×D4×C59C2211yescharacteristic
18C2×C22⋊C8×C59C2212yes—
19C59×#324C412yes—
20C59×#323C412yes—
21C2×#11745664—22no—
22C2×#11745664—42no—
23C2×#11745664D412yes—
24C2×#11745664D412yes—
25C22×C59⋊C8—21no—
26C22×C59⋊C8—21no—
27C22×C59⋊C8—21no—
28C22×C59⋊C8—21no—
29#14273663—44no—
30C4⋊(C59⋊C8)—21no—
31C4⋊(C59⋊C8)—21no—
32C22⋊(C59⋊C8)—22no—
33C22⋊(C59⋊C8)—22no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—42no—
37#14273638—22no—
38C22×D4×C59Q811yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41C22×C8×C59—22no—
42C4×D4×C59—22no—
43C4×D4×C59—22no—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59Q811yescharacteristic
47C2×C42×C59D411yescharacteristic
48C22⋊C8×C59—22no—
49C22⋊C8×C59C2×C414yes—
50C24×C59C4⋊C411yescharacteristic
51C24×C59C2×Q811yescharacteristic
52C2×D4×C59—22no—
53C2×D4×C59—22no—
54C2×D4×C59—41no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—41no—
58C22×C4×C59—41no—
59C22×C4×C59—41no—
60C22×C4×C59—41no—
61C22×C4×C59C4⋊C411yescharacteristic
62C22×C4×C59C4⋊C411yescharacteristic
63C22×C4×C59C4⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D412yes—
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59#4112yes—
68C2×C8×C59—24no—
69C2×C8×C59—42no—
70C2×C8×C59—42no—
71C4⋊C4×C59—22no—
72C22⋊C4×C59—22no—
73C22⋊C4×C59—22no—
74C42×C59—21no—
75C42×C59—21no—
76#11745664—24no—
77#11745664—42no—
78#11745664—42no—
79#11745664—42no—
80C2×C59⋊C8—22no—
81C2×C59⋊C8—22no—
82C2×C59⋊C8—22no—
83C2×C59⋊C8—41no—
84C2×C59⋊C8—41no—
85C2×C59⋊C8—41no—
86C2×C59⋊C8—41no—
87C2×C59⋊C8—41no—
88C2×C59⋊C8—81no—
89C2×C59⋊C8—81no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59#10812yescomplemented
99C23×C59C2×C4⋊C411yescharacteristic
100C23×C59C22⋊Q812yes—
101D4×C59—44no—
102C2×C4×C59—21no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—41no—
112C2×C4×C59—42no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—81no—
118C2×C4×C59C4×D412yes—
119C2×C4×C59#12011yescharacteristic
120C2×C4×C59#12311yescharacteristic
121C8×C59—44no—
122C59⋊C8—42no—
123C59⋊C8—42no—
124C59⋊C8—42no—
125C59⋊C8—81no—
126#8859—591nomaximal, Sylow
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—22no—
130C22×C59—22no—
131C22×C59—22no—
132C22×C59—22no—
133C22×C59—42no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—81no—
137C22×C59—81no—
138C22×C59—81no—
139C22×C59#38511yescharacteristic
140C22×C59C2×#10811yescharacteristic
141C22×C59#43011yescharacteristic
142C4×C59—21no—
143C4×C59—22no—
144C4×C59—41no—
145C4×C59—41no—
146C4×C59—41no—
147C4×C59B(4,𝔽2)11yescharacteristic
148C4×C59#45811yescharacteristic
149#2973—591no—
150#1870—591no—
151#1855—592no—
152#1459D5911yescharacteristic
153#1298—592no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59—42no—
157C2×C59—42no—
158C2×C59#185511yescharacteristic
159C2×C59#185811yescharacteristic
160C2×C59#186711yescharacteristic
161C22×M16—591no—
162C2×C4×D4C2×D5911yescharacteristic
163#435—1182no—
164#435—1182no—
165#432—1181no—
166C2×#108—1182no—
167#407—592no—
168C2×C22⋊C8—591no—
169C2×C22⋊C8—1181no—
170C2×C22⋊C8—1181no—
171C2×C22⋊C8C2×D5912yes—
172#341—1182no—
173#336—1181no—
174#324Dic5912yes—
175#323Dic5912yes—
176C59#885911yescharacteristic, complemented, Sylow
177C22×D4Dic11811yescharacteristic
178C23×C4D23611yescharacteristic
179C23×C4C22×D5911yescharacteristic
180C2×M16—592no—
181C2×M16—592no—
182C2×M16—1182no—
183C2×M16—2362no—
184C22×C8—22no—
185C22×C8—1181no—
186C22×C8—1181no—
187C22×C8—1181no—
188C22×C8—1181no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4C2×Dic5911yescharacteristic
192C2×C22⋊C4Dic11811yescharacteristic
193C2×C22⋊C4C2×Dic5911yescharacteristic
194C2×C42D23611yescharacteristic
195#108—2364no—
196C4⋊C8—1181no—
197C4⋊C8—1181no—
198C22⋊C8—22no—
199C22⋊C8—1182no—
200C22⋊C8—1182no—
201C22⋊C8—1182no—
202C22⋊C8—1182no—
203C22⋊C8—2362no—
204C22⋊C8C2×Dic5914yes—
205#97—1182no—
206C24C4⋊Dic5911yescharacteristic
207C24C2×Dic11811yescharacteristic
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4C4⋊Dic5911yescharacteristic
218C22×C4C4⋊Dic5911yescharacteristic
219C22×C4C4⋊Dic5911yescharacteristic
220C22×C4C2×D23611yescharacteristic
221C22×C4D4×D5912yes—
222C22×C4#1174568712yes—
223C22×C4C22×Dic5911yescharacteristic
224M16—1184no—
225M16—2362no—
226M16—2362no—
227M16—2362no—
228C2×C8—24no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—1182no—
232C2×C8—1182no—
233C2×C8—1182no—
234C2×C8—4721no—
235C2×C8—4721no—
236C2×C8—2361no—
237C2×C8—2361no—
238C2×C8—2361no—
239C2×C8—2361no—
240C2×C8—2361no—
241C4⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C42—21no—
245C42—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1427366312yescomplemented
255C23C2×C4⋊Dic5911yescharacteristic
256C23#1427370412yes—
257D4—44no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4D4×Dic5912yes—
275C2×C4#1427369711yescharacteristic
276C2×C4#1427371111yescharacteristic
277C8—44no—
278C8—4721no—
279C8—2362no—
280C8—2362no—
281C8—2362no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#1683727711yescharacteristic
295C22C2×#1427366311yescharacteristic
296C22#1683745111yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4#1683759011yescharacteristic
303C4#1683759311yescharacteristic
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2142430311yescharacteristic
309C2#2142431811yescharacteristic
310C2#2142437911yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1459C2
socleC22×C59#385
prime radicalC22×D4×C59Q8
cosocleC23×C4×C59C23
hypercenter#1459D59
hypocenterC59#8859
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #21424303C2semidirect
4. #21421984C2semidirect
98. C23×C59#108semidirect
176. C59#8859semidirect
254. C23#14273663semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214243942
#214243032
#214219842
C59×#14592
#885959