Finite Group Databasebeta

← #28499220

Subgroups — #28499220

Order 15104.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#16837171C211yesmaximal, characteristic
2#21422426C211yesmaximal, characteristic
3#21422126C212yesmaximal, complemented
4#21422102C212yesmaximal, complemented
5C59×#2583C211yesmaximal, characteristic
6C22×#11745664—21no—
7#16837366—21no—
8#16837171—21no—
9#16837171C2212yes—
10#16837128—21no—
11#16837128C2212yes—
12C2×C22⋊Q8×C59C411yescharacteristic
13C22×C4⋊C4×C59C2211yescharacteristic
14C59×#398C411yescharacteristic
15C59×#398C2211yescharacteristic
16C59×#395C411yescharacteristic
17C59×#395C2211yescharacteristic
18C59×#389—21no—
19C59×#389—21no—
20C59×#385—21no—
21C59×#385—21no—
22C4×C22⋊C4×C59C411yescharacteristic
23C2×#11745664—24no—
24C2×#11745664—22no—
25C2×#11745664—41no—
26C2×#11745664—42no—
27C2×#11745664—41no—
28C22×C59⋊C8—41no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—41no—
31C22×Q8×C59C2×C411yescharacteristic
32C23×C4×C59C2×C411yescharacteristic
33C23×C4×C59D411yescharacteristic
34C23×C4×C59D411yescharacteristic
35C22⋊Q8×C59—41no—
36C22⋊Q8×C59—41no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—41no—
43C2×C4⋊C4×C59—41no—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59C2311yescharacteristic
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C42×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59C2×C411yescharacteristic
57#95×C59C2×C411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C2×Q8×C59—41no—
60C2×Q8×C59—81no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—81no—
75C22×C4×C59—81no—
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22×C411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—42no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—42no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—81no—
92C4⋊C4×C59—81no—
93C22⋊C4×C59—24no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C42×C59—81no—
97#11745664—24no—
98#11745664—44no—
99#11745664—42no—
100#11745664—82no—
101C2×C59⋊C8—44no—
102C2×C59⋊C8—42no—
103C2×C59⋊C8—42no—
104C23×C59—22no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—41no—
108C23×C59#9911yescharacteristic
109C23×C59#9911yescharacteristic
110C23×C59C2×C22⋊C411yescharacteristic
111Q8×C59—82no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—42no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—81no—
142C2×C4×C59—81no—
143C2×C4×C59—81no—
144C2×C4×C59#10412yes—
145C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C59#13711yescharacteristic
147C59⋊C8—44no—
148C59⋊C8—82no—
149#11768—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—42no—
158C22×C59—42no—
159C22×C59#35612yes—
160C22×C59C2×#9911yescharacteristic
161C22×C59#41711yescharacteristic
162C22×C59C2×#10411yescharacteristic
163C4×C59—21no—
164C4×C59—22no—
165C4×C59—21no—
166C4×C59—42no—
167C4×C59—42no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59—41no—
171C4×C59—41no—
172C4×C59—81no—
173C4×C59—81no—
174#2583D5911yescharacteristic
175C2×#343—591no—
176#1463—591no—
177#1354—592no—
178#1346—592no—
179C2×C59—21no—
180C2×C59—21no—
181C2×C59—22no—
182C2×C59—41no—
183C2×C59#150411yescharacteristic
184C2×C59#212111yescharacteristic
185C2×C59C2×#35611yescharacteristic
186C22×M16—1181no—
187C2×C22⋊Q8Dic5911yescharacteristic
188C22×C4⋊C4C2×D5911yescharacteristic
189#407—1181no—
190#398Dic5911yescharacteristic
191#398C2×D5911yescharacteristic
192#395Dic5911yescharacteristic
193#395C2×D5911yescharacteristic
194#389—21no—
195#389—21no—
196#385—21no—
197#385—21no—
198C4×C22⋊C4Dic5911yescharacteristic
199#343—592no—
200#343—1181no—
201#324—592no—
202#324—1181no—
203C59#1176811yescharacteristic, complemented, Sylow
204C22×Q8C2×Dic5911yescharacteristic
205C23×C4C2×Dic5911yescharacteristic
206C23×C4C22⋊D5911yescharacteristic
207C23×C4C22⋊D5911yescharacteristic
208C2×M16—1184no—
209C2×M16—1182no—
210C2×M16—2361no—
211C2×M16—2361no—
212C2×M16—2362no—
213C22×C8—2361no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×Dic5911yescharacteristic
224C2×C4⋊C4C2×Dic5911yescharacteristic
225C2×C4⋊C4C22⋊D5911yescharacteristic
226C2×C4⋊C4C22⋊D5911yescharacteristic
227C2×C4⋊C4C22×D5911yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C22⋊C8—1182no—
232C22⋊C8—2361no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95C2×Dic5911yescharacteristic
238#95C2×Dic5911yescharacteristic
239C24C22⋊Dic5911yescharacteristic
240C2×Q8—41no—
241C2×Q8—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4C22⋊Dic5911yescharacteristic
258C22×C4C22⋊Dic5911yescharacteristic
259C22×C4C22⋊Dic5911yescharacteristic
260C22×C4C22×Dic5911yescharacteristic
261C22×C4C2×C22⋊D5911yescharacteristic
262C22×C4C2×C22⋊D5911yescharacteristic
263M16—1184no—
264M16—4722no—
265M16—2364no—
266M16—2362no—
267C2×C8—2362no—
268C2×C8—2364no—
269C2×C8—2362no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C22⋊C4—24no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—81no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—41no—
289C23#1427364211yescharacteristic
290C23#1427364211yescharacteristic
291C23C2×C22⋊Dic5911yescharacteristic
292Q8—82no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4#1427365412yes—
326C2×C4#1427375211yescharacteristic
327C2×C4#1427375711yescharacteristic
328C8—4722no—
329C8—2364no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#1683720112yes—
340C22C2×#1427364211yescharacteristic
341C22#1683740311yescharacteristic
342C22C2×#1427365411yescharacteristic
343C4—21no—
344C4—22no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2#2142260011yescharacteristic
359C2#2142574211yescharacteristic
360C2C2×#1683720111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2583C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2583D59
hypocenterC59#11768
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
3. #21422126C2semidirect
4. #21422102C2semidirect
203. C59#11768semidirect

Maximal subgroups

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

subgroupindex
C2×#168371712
#214224262
#214221262
#214221022
C59×#25832
#1176859