Finite Group Databasebeta

← #28477352

Subgroups — #28477352

Order 15104.

400 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 99 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#21432463C211yesmaximal, characteristic
2#21424239C211yesmaximal, characteristic, complemented
3#21421920C212yesmaximal, complemented
4#21421916C212yesmaximal
5C59×#1839C211yesmaximal, characteristic, complemented
6C2×#14273721—21no—
7#16837304C411yescharacteristic
8#16837304C411yescharacteristic
9#16837304C2211yescharacteristic
10#16837300C411yescharacteristic
11#16837300C411yescharacteristic
12#16837300C2211yescharacteristic
13#16837293—21no—
14#16837293—21no—
15#16837293—21no—
16#16837157—21no—
17#16837157—21no—
18#16837157C2212yes—
19C43×C59C2211yescharacteristic
20C59×#336—21no—
21C59×#336—21no—
22C59×#336C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C59⋊C8—22no—
25#14273721—44no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C22×C8×C59—22no—
42C2×C42×C59C2×C411yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59D411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C2×C42×C59D411yescharacteristic
47C2×C42×C59D411yescharacteristic
48C2×C42×C59C2311yescharacteristic
49C23×D59—21no—
50C22×C4×C59C22⋊C411yescharacteristic
51C22×C4×C59C22⋊C411yescharacteristic
52C22×C4×C59C22⋊C411yescharacteristic
53C22×C4×C59C22⋊C411yescharacteristic
54C22×C4×C59C22×C411yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C22×C4×C59C2×D411yescharacteristic
57C22×Dic59—21no—
58C22×Dic59—21no—
59C22×Dic59—21no—
60C2×D236—22no—
61C2×D236—41no—
62C2×D236—41no—
63C2×C8×C59—42no—
64C2×C8×C59—42no—
65C2×C8×C59—42no—
66C42×C59—22no—
67C42×C59—22no—
68C42×C59—22no—
69C42×C59—41no—
70C42×C59—41no—
71C42×C59—41no—
72C42×C59—41no—
73(C2×D59)⋊C4—42no—
74(C2×D59)⋊C4—42no—
75(C2×D59)⋊C4—42no—
76(C2×D59)⋊C4—42no—
77(C2×D59)⋊C4—42no—
78(C2×D59)⋊C4—42no—
79C4⋊Dic59—22no—
80C4⋊Dic59—41no—
81C4⋊Dic59—41no—
82Dic59⋊C4—42no—
83Dic59⋊C4—42no—
84Dic59⋊C4—42no—
85C2×C59⋊C8—42no—
86C2×C59⋊C8—42no—
87C2×C59⋊C8—42no—
88C23×C59C2×C22⋊C411yescharacteristic
89C22×D59—41no—
90C22×D59—41no—
91C22×D59—41no—
92C22×D59—41no—
93C22×D59—41no—
94C22×D59—41no—
95C22×D59—41no—
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—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59#13112yes—
115C2×C4×C59#13512yes—
116C2×C4×C59AGL(1,ℤ/8)12yes—
117C2×Dic59—41no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126D236—82no—
127C8×C59—82no—
128C59⋊C8—82no—
129#7726—591nomaximal, Sylow
130C22×C59#40811yescharacteristic
131C22×C59#41011yescharacteristic
132C22×C59#41311yescharacteristic
133C22×C59#41611yescharacteristic
134C22×C59#41811yescharacteristic
135C22×C59#42111yescharacteristic
136C22×C59#42111yescharacteristic
137C2×D59—81no—
138C2×D59—81no—
139C2×D59—81no—
140C2×D59—81no—
141C2×D59—81no—
142C2×D59—81no—
143C2×D59—81no—
144C4×C59—22no—
145C4×C59—22no—
146C4×C59—22no—
147C4×C59—41no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—41no—
151Dic59—81no—
152Dic59—81no—
153Dic59—81no—
154#2860—591no—
155#1839—591no—
156#1839D5911yescharacteristic, complemented
157#1271—592no—
158#1270—592no—
159C2×C59#147311yescharacteristic
160C2×C59#148211yescharacteristic
161C2×C59#149611yescharacteristic
162C2×C59#151411yescharacteristic
163C2×C59#152111yescharacteristic
164C2×C59#153511yescharacteristic
165C2×C59#155311yescharacteristic
166D59—161no—
167C2×#126—1181no—
168#390—591no—
169#390—591no—
170#390—591no—
171#389—591no—
172#389—591no—
173#389—591no—
174#388—1181no—
175#388—1181no—
176#388—1181no—
177C43C2×D5911yescharacteristic
178#336—21no—
179#336—21no—
180#336—592no—
181#336—1181no—
182#336—1181no—
183#336C2×D5912yes—
184C59#772611yescharacteristic, complemented, Sylow
185C22×D4—591no—
186C22×C8—22no—
187C22×C8—1182no—
188#126—2364no—
189C2×C4⋊C4—591no—
190C2×C4⋊C4—591no—
191C2×C4⋊C4—591no—
192C2×C4⋊C4—1181no—
193C2×C4⋊C4—1181no—
194C2×C4⋊C4—1181no—
195C2×C22⋊C4—1181no—
196C2×C22⋊C4—1181no—
197C2×C22⋊C4—1181no—
198C2×C22⋊C4—1181no—
199C2×C22⋊C4—1181no—
200C2×C22⋊C4—1181no—
201C2×C42C4×D5911yescharacteristic
202C2×C42C4×D5911yescharacteristic
203C2×C42D23611yescharacteristic
204C2×C42D23611yescharacteristic
205C2×C42C22⋊D5911yescharacteristic
206C2×C42C22⋊D5911yescharacteristic
207C2×C42C22×D5911yescharacteristic
208#95—1181no—
209#95—1181no—
210#95—1181no—
211C24—1181no—
212C2×D4—1182no—
213C2×D4—2361no—
214C2×D4—2361no—
215C22×C4—1181no—
216C22×C4—1181no—
217C22×C4—1181no—
218C22×C4(C2×D59)⋊C411yescharacteristic
219C22×C4(C2×D59)⋊C411yescharacteristic
220C22×C4(C2×D59)⋊C411yescharacteristic
221C22×C4(C2×D59)⋊C411yescharacteristic
222C22×C4C2×C4×D5911yescharacteristic
223C22×C4C2×D23611yescharacteristic
224C22×C4C2×C22⋊D5911yescharacteristic
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—2362no—
229C2×C8—2362no—
230C2×C8—2362no—
231C4⋊C4—1182no—
232C4⋊C4—2361no—
233C4⋊C4—2361no—
234C4⋊C4—2362no—
235C4⋊C4—2362no—
236C4⋊C4—2362no—
237C22⋊C4—2362no—
238C22⋊C4—2362no—
239C22⋊C4—2362no—
240C22⋊C4—2362no—
241C22⋊C4—2362no—
242C22⋊C4—2362no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C23—2361no—
251C23—2361no—
252C23—2361no—
253C23—2361no—
254C23—2361no—
255C23—2361no—
256C23—2361no—
257C23C2×(C2×D59)⋊C411yescharacteristic
258D4—4722no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—2361no—
278C2×C4—2361no—
279C2×C4—2361no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4—2361no—
286C2×C4#1427373512yes—
287C2×C4#1427374912yes—
288C2×C4#1427375212yes—
289C8—82no—
290C8—4722no—
291C22—4721no—
292C22—4721no—
293C22—4721no—
294C22—4721no—
295C22—4721no—
296C22—4721no—
297C22—4721no—
298C22#1683737011yescharacteristic
299C22#1683737511yescharacteristic
300C22#1683738411yescharacteristic
301C22#1683739911yescharacteristic
302C22#1683740611yescharacteristic
303C22#1683741811yescharacteristic
304C22#1683742011yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—4721no—
313C4—4721no—
314C4—4721no—
315C2—9441no—
316C2#2142246611yescharacteristic
317C2#2142250511yescharacteristic
318C2#2142256411yescharacteristic
319C2#2142264211yescharacteristic
320C2#2142267411yescharacteristic
321C2#2142272511yescharacteristic
322C2#2142279911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1839C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1839D59
hypocenterC59#7726
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #1839

Complements

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

NH ≅ G/Ntype
2. #21424239C2semidirect
3. #21421920C2semidirect
5. C59×#1839C2semidirect
156. #1839D59semidirect
184. C59#7726semidirect

Maximal subgroups

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

subgroupindex
#214324632
#214242392
#214219202
#214219162
C59×#18392
#772659