Finite Group Databasebeta

← #28477077

Subgroups — #28477077

Order 15104.

568 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432445C211yesmaximal, characteristic
2#21424075C211yesmaximal, characteristic
3#336×D59C211yesmaximal, characteristic
4#21421922C211yesmaximal, characteristic, complemented
5#21421921C211yesmaximal, characteristic, complemented
6#21421921C211yesmaximal, characteristic, complemented
7C59×#1272C211yesmaximal, characteristic, complemented
8C22×C8×D59—21no—
9C2×#14273725C411yescharacteristic
10C2×#14273707C411yescharacteristic
11C2×#14273703C411yescharacteristic
12C2×C42×D59C2211yescharacteristic
13C2×Dic59⋊C8—21no—
14C2×(C2×D59)⋊C8—21no—
15#16837308C2211yescharacteristic
16#16837299—21no—
17#16837299—21no—
18#16837271C411yescharacteristic
19#16837261—21no—
20#16837261—21no—
21#16837157C2211yescharacteristic
22#16837157C2211yescharacteristic
23#16837156—21no—
24C59×#391C2211yescharacteristic
25C59×#336C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×Dic118C2×C411yescharacteristic
28C22×C4×D59C2×C411yescharacteristic
29C22×C4×D59D411yescharacteristic
30C22×C4×D59D411yescharacteristic
31C22×C59⋊C8—21no—
32C22×C59⋊C8—21no—
33C2×C8×D59—24no—
34C2×C8×D59—42no—
35C2×C8×D59—42no—
36#14273725—42no—
37#14273707—42no—
38#14273703—42no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59C2×C411yescharacteristic
41C2×Dic59⋊C4—21no—
42C2×Dic59⋊C4—21no—
43C2×Dic59⋊C4—21no—
44C2×Dic59⋊C4—21no—
45C2×Dic59⋊C4—21no—
46C2×(C2×D59)⋊C4—21no—
47C2×(C2×D59)⋊C4—21no—
48C2×C4×Dic59C2×C411yescharacteristic
49C2×C4×Dic59D411yescharacteristic
50C2×C4×Dic59D411yescharacteristic
51C42×D59—42no—
52Dic59⋊C8—42no—
53(C2×D59)⋊C8—42no—
54C59⋊#95—21no—
55C59⋊#95—21no—
56C22×Q8×C59C2×C411yescharacteristic
57C22×C8×C59—21no—
58C22×C8×C59—21no—
59C2×C4⋊C4×C59C2×C411yescharacteristic
60C2×C42×C59C2311yescharacteristic
61#95×C59—21no—
62#95×C59—21no—
63C23×D59C22⋊C411yescharacteristic
64C2×Q8×C59—21no—
65C2×Q8×C59—21no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59C22⋊C411yescharacteristic
76C22×Dic59C22⋊C411yescharacteristic
77C22×Dic59C22⋊C411yescharacteristic
78C2×C4×D59—22no—
79C2×C4×D59—22no—
80C2×C4×D59—22no—
81C2×C4×D59—21no—
82C2×C4×D59—21no—
83C2×C4×D59—41no—
84C2×C4×D59—41no—
85C2×C4×D59—41no—
86C2×C4×D59SD1612yes—
87C2×C4×D59SD1612yes—
88C2×C4×D59Q1612yes—
89C2×C4×D59Q1612yes—
90C2×Dic118—21no—
91C2×Dic118—21no—
92C2×Dic118—41no—
93C2×Dic118—41no—
94C2×C8×C59—22no—
95C2×C8×C59—41no—
96C2×C8×C59—41no—
97C2×C8×C59—41no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C4⋊C4×C59—41no—
101C42×C59—41no—
102(C2×D59)⋊C4—42no—
103(C2×D59)⋊C4—42no—
104C4⋊Dic59—42no—
105C4⋊Dic59—41no—
106Dic59⋊C4—42no—
107Dic59⋊C4—42no—
108Dic59⋊C4—42no—
109Dic59⋊C4—42no—
110Dic59⋊C4—42no—
111C4×Dic59—22no—
112C4×Dic59—22no—
113C4×Dic59—41no—
114C2×C59⋊C8—22no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—41no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—41no—
120C8×D59—48no—
121C23×C59C2×C22⋊C411yescharacteristic
122C22×D59—21no—
123C22×D59—21no—
124C22×D59—21no—
125C22×D59—21no—
126C22×D59#10012yes—
127C22×D59#10312yes—
128C22×D59#10312yes—
129Q8×C59—81no—
130Q8×C59—81no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59#13111yescharacteristic
144C2×C4×C59#13111yescharacteristic
145C2×C4×C59C2×SD1611yescharacteristic
146C2×C4×C59C2×SD1611yescharacteristic
147C2×C4×C59C2×Q1611yescharacteristic
148C2×C4×C59C2×Q1611yescharacteristic
149C2×Dic59—21no—
150C2×Dic59—21no—
151C2×Dic59—21no—
152C2×Dic59—21no—
153C2×Dic59—21no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59#9912yes—
161C2×Dic59#10312yes—
162C2×Dic59#10312yes—
163C2×Dic59#10412yes—
164C4×D59—24no—
165C4×D59—24no—
166C4×D59—42no—
167C4×D59—42no—
168C4×D59—42no—
169C4×D59—42no—
170Dic118—81no—
171Dic118—81no—
172C8×C59—42no—
173C8×C59—81no—
174C59⋊C8—42no—
175C59⋊C8—81no—
176#7667—591nomaximal, Sylow
177C22×C59#40811yescharacteristic
178C22×C59C2×#9911yescharacteristic
179C22×C59C2×#10011yescharacteristic
180C22×C59C2×#10311yescharacteristic
181C22×C59C2×#10311yescharacteristic
182C22×C59C2×#10411yescharacteristic
183C22×C59#42111yescharacteristic
184C2×D59—22no—
185C2×D59—22no—
186C2×D59—22no—
187C2×D59—22no—
188C2×D59—22no—
189C2×D59—22no—
190C2×D59#33314yes—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—81no—
198C4×C59—81no—
199Dic59—22no—
200Dic59—41no—
201Dic59—81no—
202Dic59—81no—
203Dic59#32814yes—
204#2857—591no—
205#1797—591no—
206C2×#336—591no—
207#1272—591no—
208#1272—591no—
209#1272—591no—
210#1272D5911yescharacteristic, complemented
211C2×C59#147111yescharacteristic
212C2×C59#148111yescharacteristic
213C2×C59#149311yescharacteristic
214C2×C59C2×#32811yescharacteristic
215C2×C59#152011yescharacteristic
216C2×C59#153011yescharacteristic
217C2×C59C2×#33311yescharacteristic
218D59—24no—
219C23×C8—1181no—
220C2×C4⋊Q8—591no—
221C2×C22⋊Q8—591no—
222C2×C22⋊Q8—591no—
223C22×C42—591no—
224C2×C4⋊C8—1181no—
225C2×C22⋊C8—1181no—
226#391—591no—
227#391C2×D5911yescharacteristic
228#389—1181no—
229#389—1181no—
230#384—591no—
231#380—1181no—
232#380—1181no—
233#336—591no—
234#336—591no—
235#336—1181no—
236#336C2×D5911yescharacteristic
237#336C2×D5911yescharacteristic
238C59#766711yescharacteristic, complemented, Sylow
239C22×Q8—591no—
240C22×Q8C4×D5911yescharacteristic
241C23×C4—591no—
242C23×C4—591no—
243C23×C4—591no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—1184no—
247C22×C8—1181no—
248C22×C8—1181no—
249C22×C8—2362no—
250C22×C8—2362no—
251C4⋊Q8—2362no—
252C22⋊Q8—2362no—
253C22⋊Q8—2362no—
254C2×C4⋊C4—591no—
255C2×C4⋊C4—1181no—
256C2×C4⋊C4—1181no—
257C2×C4⋊C4—1181no—
258C2×C4⋊C4—1181no—
259C2×C4⋊C4—1181no—
260C2×C4⋊C4—1181no—
261C2×C4⋊C4C4×D5911yescharacteristic
262C2×C22⋊C4—1181no—
263C2×C22⋊C4—1181no—
264C2×C42—591no—
265C2×C42—591no—
266C2×C42—591no—
267C2×C42—2362no—
268C2×C42C22×D5911yescharacteristic
269C4⋊C8—2362no—
270C22⋊C8—2362no—
271#95—21no—
272#95—21no—
273#95—1181no—
274#95—1181no—
275C24—591no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—1181no—
281C2×Q8—1181no—
282C2×Q8—2361no—
283C2×Q8—2361no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—591no—
287C22×C4—591no—
288C22×C4—592no—
289C22×C4—592no—
290C22×C4—592no—
291C22×C4—592no—
292C22×C4—591no—
293C22×C4—1182no—
294C22×C4—1182no—
295C22×C4—1182no—
296C22×C4—1181no—
297C22×C4—1181no—
298C22×C4—1181no—
299C22×C4—1181no—
300C22×C4—2361no—
301C22×C4—2361no—
302C22×C4—2361no—
303C22×C4C2×C4×D5911yescharacteristic
304C22×C4D4×D5911yescharacteristic
305C22×C4D4×D5911yescharacteristic
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—1182no—
313C2×C8—2368no—
314C2×C8—2361no—
315C2×C8—2361no—
316C2×C8—2361no—
317C2×C8—2361no—
318C2×C8—2361no—
319C4⋊C4—41no—
320C4⋊C4—2361no—
321C4⋊C4—2362no—
322C4⋊C4—2362no—
323C4⋊C4—2362no—
324C4⋊C4—2362no—
325C4⋊C4—2362no—
326C4⋊C4—2362no—
327C22⋊C4—2362no—
328C22⋊C4—2362no—
329C42—41no—
330C42—1182no—
331C42—1182no—
332C42—2361no—
333C23—592no—
334C23—592no—
335C23—592no—
336C23—1181no—
337C23—1181no—
338C23—1181no—
339C23—1181no—
340C23C22⋊C4×D5911yescharacteristic
341Q8—81no—
342Q8—81no—
343Q8—4721no—
344Q8—4721no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—592no—
358C2×C4—592no—
359C2×C4—592no—
360C2×C4—592no—
361C2×C4—1184no—
362C2×C4—1184no—
363C2×C4—1181no—
364C2×C4—1181no—
365C2×C4—1181no—
366C2×C4—1181no—
367C2×C4—1181no—
368C2×C4—2362no—
369C2×C4—2362no—
370C2×C4—2362no—
371C2×C4—2362no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4—2361no—
377C2×C4—2361no—
378C2×C4#1427373311yescharacteristic
379C2×C4#1427373511yescharacteristic
380C2×C4SD16×D5911yescharacteristic
381C2×C4SD16×D5911yescharacteristic
382C2×C4Q16×D5911yescharacteristic
383C2×C4Q16×D5911yescharacteristic
384C8—42no—
385C8—81no—
386C8—4721no—
387C8—2362no—
388C22—594no—
389C22—1182no—
390C22—1182no—
391C22—1182no—
392C22—1182no—
393C22—1182no—
394C22—1182no—
395C22#1683736711yescharacteristic
396C22#99×D5911yescharacteristic
397C22#100×D5911yescharacteristic
398C22#103×D5911yescharacteristic
399C22#103×D5911yescharacteristic
400C22#104×D5911yescharacteristic
401C22#1683741711yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—594no—
411C4—1182no—
412C4—4721no—
413C4—4721no—
414C4—2361no—
415C2—1184no—
416C2#2142245411yescharacteristic
417C2#2142249711yescharacteristic
418C2#2142254911yescharacteristic
419C2#328×D5911yescharacteristic
420C2#2142266611yescharacteristic
421C2#2142270511yescharacteristic
422C2#333×D5911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socleC23×C59C2×C22⋊C4
prime radicalC23×D59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1272D59
hypocenterC59#7667
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #21421922C2semidirect
5. #21421921C2semidirect
6. #21421921C2semidirect
7. C59×#1272C2semidirect
210. #1272D59semidirect
238. C59#7667semidirect

Maximal subgroups

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

subgroupindex
#214324452
#214240752
#336×D592
#214219222
#214219212
C59×#12722
#766759