Finite Group Databasebeta

← #28486351

Subgroups — #28486351

Order 15104.

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

normal subgroups only

This group has 486 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#21433181C211yesmaximal, characteristic, complemented
2#21429072C211yesmaximal, characteristic
3#21422719C212yesmaximal
4#21421953C212yesmaximal, complemented
5C59×#1467C211yesmaximal, characteristic, complemented
6C2×#14273725C411yescharacteristic
7C2×#14273716C411yescharacteristic
8C2×#14273715C2211yescharacteristic
9#16837533—24no—
10#16837459—21no—
11#16837459C2212yes—
12#16837455—21no—
13#16837324—21no—
14#16837324—21no—
15#16837307C411yescharacteristic
16#16837303C411yescharacteristic
17#16837293—21no—
18#16837293—21no—
19C4×C4⋊Dic59C2211yescharacteristic
20#16837141—24no—
21C2×C4×Q8×C59C2211yescharacteristic
22C59×#432C2212yes—
23C59×#324—21no—
24C59×#324—21no—
25C22×Dic118C2×C411yescharacteristic
26C22×D236C2×C411yescharacteristic
27#14273725—42no—
28#14273716—42no—
29#14273715—22no—
30#14273715—22no—
31C2×C4⋊Dic59—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×C4×Dic59C2×C411yescharacteristic
41C2×C4×Dic59C2×C411yescharacteristic
42C4⋊(C59⋊C8)—22no—
43C4⋊(C59⋊C8)—22no—
44C4⋊(C59⋊C8)—22no—
45C22⋊(C59⋊C8)—22no—
46#14273638—22no—
47#14273638D414yes—
48C4×C59⋊C8—22no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C22×Q8×C59C2×C411yescharacteristic
53C4×Q8×C59—22no—
54C4×Q8×C59—22no—
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C4⋊C4×C59D411yescharacteristic
57C2×C4⋊C4×C59D411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59—22no—
62C22⋊C8×C59—22no—
63#97×C59—22no—
64C23×D59—21no—
65C2×Q8×C59—22no—
66C2×Q8×C59—22no—
67C2×Q8×C59—41no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C2×D236—22no—
79C2×D236—41no—
80C2×D236—41no—
81C2×Dic118—22no—
82C2×Dic118—41no—
83C2×Dic118—41no—
84C2×C8×C59—42no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—22no—
87C4⋊C4×C59—22no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59—21no—
92C42×C59C2×D412yes—
93C42×C59#4112yes—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—42no—
96(C2×D59)⋊C4—42no—
97(C2×D59)⋊C4—42no—
98(C2×D59)⋊C4—41no—
99(C2×D59)⋊C4—42no—
100(C2×D59)⋊C4—41no—
101C4⋊Dic59—42no—
102Dic59⋊C4—42no—
103Dic59⋊C4—42no—
104C4×Dic59—41no—
105C4×Dic59—41no—
106C2×C59⋊C8—24no—
107C23×C59C2×C22⋊C411yescharacteristic
108C22×D59—41no—
109C22×D59—41no—
110C22×D59—41no—
111C22×D59—41no—
112C22×D59—41no—
113C22×D59—41no—
114C22×D59—41no—
115Q8×C59—44no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59#10012yes—
130C2×C4×C59C4×D412yes—
131C2×C4×C59#12011yescharacteristic
132C2×C4×C59#12112yes—
133C2×C4×C59#12311yescharacteristic
134C2×Dic59—22no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143D236—82no—
144Dic118—82no—
145C8×C59—82no—
146C59⋊C8—44no—
147#9448—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59#38611yescharacteristic
151C22×C59C2×#10011yescharacteristic
152C22×C59#41311yescharacteristic
153C2×D59—81no—
154C2×D59—81no—
155C2×D59—81no—
156C2×D59—81no—
157C2×D59—81no—
158C2×D59—81no—
159C2×D59—81no—
160C4×C59—22no—
161C4×C59—22no—
162C4×C59—22no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168Dic59—42no—
169Dic59—81no—
170Dic59—81no—
171#2983—591no—
172#2545—591no—
173#1533—592no—
174#1467D5911yescharacteristic, complemented
175#1285—592no—
176C2×C59—21no—
177C2×C59—21no—
178C2×C59#188411yescharacteristic
179C2×C59#190011yescharacteristic
180C2×C59#191211yescharacteristic
181D59—161no—
182C2×C4⋊Q8—591no—
183C2×#124—591no—
184C2×#124—591no—
185C2×C4×Q8C2×D5911yescharacteristic
186#446—1184no—
187#432—592no—
188#432—1181no—
189#432C2×D5912yes—
190#431—1181no—
191#396—1181no—
192#396—1181no—
193#391—591no—
194#390—591no—
195#388—1181no—
196#388—1181no—
197C4×C4⋊C4—591no—
198#329—1184no—
199#324—21no—
200#324—21no—
201C59#944811yescharacteristic, complemented, Sylow
202C22×Q8—591no—
203C22×Q8C4×D5911yescharacteristic
204C22×D4—591no—
205C4⋊Q8—2362no—
206#124—1182no—
207#124—1182no—
208#124—2362no—
209C4×Q8—22no—
210C4×Q8—22no—
211C2×C4⋊C4—1181no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4C4×D5911yescharacteristic
215C2×C4⋊C4C22⋊D5911yescharacteristic
216C2×C4⋊C4C22⋊D5911yescharacteristic
217C2×C22⋊C4—1181no—
218C2×C22⋊C4—1181no—
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1181no—
221C2×C22⋊C4—1181no—
222C2×C22⋊C4—1181no—
223C2×C42—591no—
224C2×C42—591no—
225C2×C42D23611yescharacteristic
226C2×C42D23611yescharacteristic
227C2×C42C22×D5911yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—1182no—
230C4⋊C8—1182no—
231C4⋊C8—1182no—
232C22⋊C8—22no—
233C22⋊C8—1182no—
234#97—22no—
235#97—594no—
236#97—1182no—
237C4×C8—1182no—
238#95—1181no—
239#95—1181no—
240#95—1181no—
241C24—1181no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—1182no—
246C2×Q8—2361no—
247C2×Q8—2361no—
248C2×D4—1182no—
249C2×D4—2361no—
250C2×D4—2361no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4(C2×D59)⋊C411yescharacteristic
255C22×C4(C2×D59)⋊C411yescharacteristic
256C22×C4(C2×D59)⋊C411yescharacteristic
257C22×C4(C2×D59)⋊C411yescharacteristic
258C22×C4C2×C4×D5911yescharacteristic
259C22×C4C2×D23611yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261C2×C8—42no—
262C2×C8—1184no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—2362no—
267C4⋊C4—2362no—
268C4⋊C4—2362no—
269C22⋊C4—2362no—
270C22⋊C4—2362no—
271C22⋊C4—2362no—
272C22⋊C4—2362no—
273C22⋊C4—2361no—
274C22⋊C4—2362no—
275C22⋊C4—2361no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—2361no—
281C42—2361no—
282C42D4×D5912yes—
283C42#1174568712yes—
284C23—2361no—
285C23—2361no—
286C23—2361no—
287C23—2361no—
288C23—2361no—
289C23—2361no—
290C23—2361no—
291C23C2×(C2×D59)⋊C411yescharacteristic
292Q8—44no—
293Q8—4722no—
294D4—4722no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—1182no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4#1427364312yes—
318C2×C4(C22⋊D59)⋊C412yes—
319C2×C4#1427369711yescharacteristic
320C2×C4#1427369912yes—
321C2×C4#1427371111yescharacteristic
322C8—82no—
323C8—2364no—
324C22—21no—
325C22—21no—
326C22—4721no—
327C22—4721no—
328C22—4721no—
329C22—4721no—
330C22—4721no—
331C22—4721no—
332C22—4721no—
333C22#1683727911yescharacteristic
334C22C2×#1427364311yescharacteristic
335C22#1683738411yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—4721no—
345C4—4721no—
346C4—2362no—
347C2—21no—
348C2—21no—
349C2—9441no—
350C2#2142447711yescharacteristic
351C2#2142455711yescharacteristic
352C2#2142463111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1467C2
socleC22×C59#386
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1467D59
hypocenterC59#9448
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
1. #21433181C2semidirect
4. #21421953C2semidirect
5. C59×#1467C2semidirect
174. #1467D59semidirect
201. C59#9448semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214290722
#214227192
#214219532
C59×#14672
#944859