Finite Group Databasebeta

← #28498998

Subgroups — #28498998

Order 15104.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#21429909C211yesmaximal, characteristic
2C2×#16837128C211yesmaximal, characteristic
3#21422125C212yesmaximal, complemented
4#21422122C212yesmaximal, complemented
5C2×C59×#343C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic
7C2×C22⋊(C59⋊C8)—21no—
8#16837341C411yescharacteristic
9#16837341C2211yescharacteristic
10#16837335—21no—
11#16837335—21no—
12#16837322C411yescharacteristic
13#16837322C2211yescharacteristic
14#16837275—21no—
15#16837265—21no—
16#16837262C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C22×M16×C59—21no—
20C22×C4⋊C4×C59C2211yescharacteristic
21C59×#343—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×C59⋊C8—41no—
25#14273706—41no—
26#14273706—41no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)—41no—
36C59⋊#95—21no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—24no—
48C2×M16×C59—22no—
49C2×M16×C59—41no—
50C2×M16×C59—42no—
51C22×C8×C59—41no—
52C2×C4⋊C4×C59—41no—
53C2×C4⋊C4×C59—41no—
54C2×C4⋊C4×C59C2×C411yescharacteristic
55C2×C4⋊C4×C59D411yescharacteristic
56C2×C4⋊C4×C59D411yescharacteristic
57C2×C4⋊C4×C59C2311yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—81no—
67C22×C4×C59—81no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59SD1612yes—
72C22×C4×C59Q1612yes—
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C22×Dic59—21no—
80C2×Dic118—41no—
81C2×Dic118—81no—
82M16×C59—24no—
83M16×C59—44no—
84M16×C59—42no—
85C2×C8×C59—44no—
86C2×C8×C59—42no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C22⋊Dic59—41no—
92C22⋊Dic59—41no—
93C22⋊Dic59—42no—
94C4⋊Dic59—41no—
95C4⋊Dic59—81no—
96Dic59⋊C4—42no—
97Dic59⋊C4—81no—
98Dic59⋊C4—81no—
99C2×C59⋊C8—42no—
100C2×C59⋊C8—82no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59#9911yescharacteristic
104C23×C59#9911yescharacteristic
105C23×C59#10312yes—
106C23×C59#10412yes—
107C23×C59C2×C22⋊C411yescharacteristic
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—42no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—81no—
126C2×C4×C59—81no—
127C2×C4×C59—81no—
128C2×C4×C59#10312yes—
129C2×C4×C59C2×SD1611yescharacteristic
130C2×C4×C59C2×Q1611yescharacteristic
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—42no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142Dic118—82no—
143C8×C59—44no—
144C59⋊C8—82no—
145#11722—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—42no—
153C22×C59—42no—
154C22×C59#32812yes—
155C22×C59#32812yes—
156C22×C59C2×#9911yescharacteristic
157C22×C59C2×#10311yescharacteristic
158C22×C59C2×#10411yescharacteristic
159C4×C59—21no—
160C4×C59—22no—
161C4×C59—21no—
162C4×C59—41no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59—41no—
166Dic59—81no—
167Dic59—81no—
168Dic59—81no—
169Dic59—81no—
170#2627—591no—
171C2×#343D5911yescharacteristic
172C2×#324—591no—
173#1354—592no—
174#1353—592no—
175C2×C59—21no—
176C2×C59—22no—
177C2×C59—22no—
178C2×C59—21no—
179C2×C59C2×#32811yescharacteristic
180C2×C59#212111yescharacteristic
181C2×C59#213311yescharacteristic
182C22×M16—21no—
183C2×C22⋊Q8—591no—
184C22×C4⋊C4C2×D5911yescharacteristic
185C2×C22⋊C8—1181no—
186#400—591no—
187#400—591no—
188#398—1181no—
189#398—1181no—
190#395—591no—
191#395—591no—
192#385—1181no—
193#381—1181no—
194#380—591no—
195#343—21no—
196#343C2×D5912yes—
197#324—592no—
198#324—1181no—
199C59#1172211yescharacteristic, complemented, Sylow
200C22×Q8—591no—
201C23×C4C4×D5911yescharacteristic
202C23×C4D23611yescharacteristic
203C23×C4C22⋊D5911yescharacteristic
204C2×M16—24no—
205C2×M16—22no—
206C2×M16—41no—
207C2×M16—42no—
208C22×C8—41no—
209C22×C8—2361no—
210C22⋊Q8—2361no—
211C22⋊Q8—2361no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—591no—
215C2×C4⋊C4—1181no—
216C2×C4⋊C4—1181no—
217C2×C4⋊C4—1181no—
218C2×C4⋊C4—1181no—
219C2×C4⋊C4C4×D5911yescharacteristic
220C2×C4⋊C4D23611yescharacteristic
221C2×C4⋊C4C22⋊D5911yescharacteristic
222C2×C4⋊C4C22×D5911yescharacteristic
223C2×C22⋊C4—1181no—
224C2×C22⋊C4—1181no—
225C22⋊C8—1182no—
226C22⋊C8—2361no—
227#95—591no—
228#95—591no—
229#95—1181no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233#95—1181no—
234#95—1181no—
235C24(C2×D59)⋊C411yescharacteristic
236C2×Q8—4721no—
237C2×Q8—2361no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4—1181no—
250C22×C4—1181no—
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4(C2×D59)⋊C411yescharacteristic
254C22×C4#1174567012yes—
255C22×C4#1174567212yes—
256C22×C4C2×C4×D5911yescharacteristic
257C22×C4C2×D23611yescharacteristic
258C22×C4C2×C22⋊D5911yescharacteristic
259M16—24no—
260M16—44no—
261M16—42no—
262C2×C8—44no—
263C2×C8—42no—
264C2×C8—4722no—
265C2×C8—2362no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—4721no—
271C4⋊C4—4721no—
272C4⋊C4—4721no—
273C4⋊C4—2361no—
274C4⋊C4—2362no—
275C22⋊C4—2361no—
276C22⋊C4—2361no—
277C22⋊C4—2362no—
278C23—41no—
279C23—41no—
280C23#1427364111yescharacteristic
281C23#1427364111yescharacteristic
282C23#1427365012yes—
283C23#1427365512yes—
284C23C2×(C2×D59)⋊C411yescharacteristic
285Q8—4722no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2362no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4#1427365012yes—
318C2×C4C2×#1174567011yescharacteristic
319C2×C4C2×#1174567211yescharacteristic
320C8—44no—
321C8—4722no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#1683713812yes—
331C22#1683713812yes—
332C22C2×#1427364111yescharacteristic
333C22C2×#1427365011yescharacteristic
334C22C2×#1427365511yescharacteristic
335C4—21no—
336C4—22no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—4721no—
343C4—4721no—
344C4—4721no—
345C4—4721no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—22no—
350C2C2×#1683713811yescharacteristic
351C2#2142574311yescharacteristic
352C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC2×C59×#343C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenterC2×#343D59
hypocenterC59#11722
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. #21422125C2semidirect
4. #21422122C2semidirect
199. C59#11722semidirect

Maximal subgroups

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

subgroupindex
#214299092
C2×#168371282
#214221252
#214221222
C2×C59×#3432
#1172259