Finite Group Databasebeta

← #28476751

Subgroups — #28476751

Order 15104.

578 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426634C211yesmaximal, characteristic
2#21424075C211yesmaximal, characteristic
3#336×D59C211yesmaximal, characteristic
4#21421922C211yesmaximal, characteristic, complemented
5#21421920C211yesmaximal, characteristic, complemented
6#21421918C211yesmaximal, characteristic, complemented
7C59×#1272C211yesmaximal, characteristic, complemented
8C22×C8×D59—21no—
9C2×#14273703C411yescharacteristic
10C2×Q8×Dic59C411yescharacteristic
11C2×C42×D59C2211yescharacteristic
12C2×Dic59⋊C8—21no—
13C2×(C2×D59)⋊C8—21no—
14#16837300C2211yescharacteristic
15#16837289—21no—
16#16837276—21no—
17#16837259C411yescharacteristic
18C4⋊C4×Dic59C411yescharacteristic
19#16837253—21no—
20#16837249—21no—
21#16837157C2211yescharacteristic
22#16837157C2211yescharacteristic
23#16837156—21no—
24C59×#391C2211yescharacteristic
25C59×#336C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×C4×D59C2×C411yescharacteristic
28C22×C4×D59D411yescharacteristic
29C22×C4×D59D411yescharacteristic
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32C2×C8×D59—24no—
33C2×C8×D59—42no—
34C2×C8×D59—42no—
35#14273703—22no—
36#14273703C814yescomplemented
37Q8×Dic59—42no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59C2×C411yescharacteristic
40C2×C4⋊Dic59C2×C411yescharacteristic
41C2×Dic59⋊C4—21no—
42C2×Dic59⋊C4—21no—
43C2×(C2×D59)⋊C4—21no—
44C2×(C2×D59)⋊C4—21no—
45C2×C4×Dic59—21no—
46C2×C4×Dic59—21no—
47C2×C4×Dic59C2×C411yescharacteristic
48C2×C4×Dic59D411yescharacteristic
49C2×C4×Dic59D411yescharacteristic
50C42×D59—42no—
51Dic59⋊C8—42no—
52(C2×D59)⋊C8—42no—
53C59⋊#95—21no—
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—41no—
66C2×Q8×C59—41no—
67C2×Q8×C59C2×C812yescomplemented
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—21no—
80C2×C4×D59—21no—
81C2×C4×D59—22no—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84C2×C4×D59—41no—
85C2×C4×D59—41no—
86C2×C4×D59C2×C812yes—
87C2×C4×D59M1612yes—
88C2×C4×D59SD1612yes—
89C2×C4×D59Q1612yes—
90C2×C8×C59—22no—
91C2×C8×C59—41no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C2×C8×C59—41no—
95C2×C8×C59—41no—
96C4⋊C4×C59—41no—
97C42×C59—41no—
98(C2×D59)⋊C4—24no—
99(C2×D59)⋊C4—42no—
100C4⋊Dic59—21no—
101C4⋊Dic59—42no—
102C4⋊Dic59—41no—
103C4⋊Dic59C2×C812yescomplemented
104Dic59⋊C4—24no—
105Dic59⋊C4—41no—
106Dic59⋊C4—41no—
107C4×Dic59—22no—
108C4×Dic59—22no—
109C4×Dic59—41no—
110C4×Dic59—42no—
111C4×Dic59—42no—
112C2×C59⋊C8—22no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—41no—
118C8×D59—48no—
119C23×C59C2×C22⋊C411yescharacteristic
120C22×D59—21no—
121C22×D59—21no—
122C22×D59—21no—
123C22×D59—21no—
124C22×D59C22⋊C812yes—
125C22×D59#10312yes—
126C22×D59#10412yes—
127Q8×C59—42no—
128Q8×C59—42no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—22no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59C22×C811yescharacteristic
142C2×C4×C59C2×M1611yescharacteristic
143C2×C4×C59C2×SD1611yescharacteristic
144C2×C4×C59C2×Q1611yescharacteristic
145C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C59#13711yescharacteristic
147C2×Dic59—21no—
148C2×Dic59—21no—
149C2×Dic59—21no—
150C2×Dic59—21no—
151C2×Dic59—21no—
152C2×Dic59—21no—
153C2×Dic59—22no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59C22⋊C812yes—
160C2×Dic59#9912yes—
161C2×Dic59#10312yes—
162C4×D59—24no—
163C4×D59—24no—
164C4×D59—42no—
165C4×D59—42no—
166C4×D59—42no—
167C4×D59—42no—
168C8×C59—42no—
169C8×C59—81no—
170C59⋊C8—42no—
171C59⋊C8—81no—
172#7589—591nomaximal, Sylow
173C22×C59C2×C22⋊C811yescharacteristic
174C22×C59C2×#9911yescharacteristic
175C22×C59#41311yescharacteristic
176C22×C59C2×#10311yescharacteristic
177C22×C59#41711yescharacteristic
178C22×C59C2×#10411yescharacteristic
179C22×C59#42111yescharacteristic
180C2×D59—22no—
181C2×D59—22no—
182C2×D59—22no—
183C2×D59—22no—
184C2×D59—22no—
185C2×D59—22no—
186C2×D59Q8⋊C814yes—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—42no—
194C4×C59—81no—
195Dic59—22no—
196Dic59—22no—
197Dic59—22no—
198Dic59—42no—
199Dic59—81no—
200C4×C22⋊Q8—591no—
201#1797—591no—
202C2×#336—591no—
203#1272—591no—
204#1272D5911yescharacteristic, complemented
205#1271—591no—
206#1271—591no—
207C2×C59#145911yescharacteristic
208C2×C59C2×Q8⋊C811yescharacteristic
209C2×C59#147711yescharacteristic
210C2×C59#150411yescharacteristic
211C2×C59#152011yescharacteristic
212C2×C59#153211yescharacteristic
213C2×C59#155011yescharacteristic
214D59—24no—
215C23×C8—1181no—
216C2×C22⋊Q8—591no—
217C2×C4×Q8—591no—
218C22×C42—591no—
219C2×C4⋊C8—1181no—
220C2×C22⋊C8—1181no—
221#391C2×D5911yescharacteristic
222#389—591no—
223#387—1181no—
224#385—1181no—
225#380—591no—
226C4×C4⋊C4—591no—
227C4×C4⋊C4—1181no—
228C4×C22⋊C4—1181no—
229#336—591no—
230#336—591no—
231#336—1181no—
232#336C2×D5911yescharacteristic
233#336C2×D5911yescharacteristic
234C59#758911yescharacteristic, complemented, Sylow
235C22×Q8C4×D5911yescharacteristic
236C23×C4—591no—
237C23×C4—591no—
238C23×C4—591no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—1184no—
242C22×C8—1181no—
243C22×C8—1181no—
244C22×C8—2362no—
245C22×C8—2362no—
246C22⋊Q8—594no—
247C22⋊Q8—1182no—
248C4×Q8—2362no—
249C2×C4⋊C4—591no—
250C2×C4⋊C4—591no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—1181no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4C4×D5911yescharacteristic
255C2×C22⋊C4—1181no—
256C2×C22⋊C4—1181no—
257C2×C42—591no—
258C2×C42—591no—
259C2×C42—591no—
260C2×C42—1181no—
261C2×C42—1181no—
262C2×C42—2362no—
263C2×C42C22×D5911yescharacteristic
264C4⋊C8—2362no—
265C22⋊C8—2362no—
266#95—21no—
267#95—21no—
268#95—1181no—
269#95—1181no—
270#95—1181no—
271C24—591no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8C8×D5912yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—591no—
279C22×C4—591no—
280C22×C4—591no—
281C22×C4—592no—
282C22×C4—592no—
283C22×C4—592no—
284C22×C4—592no—
285C22×C4—1182no—
286C22×C4—1182no—
287C22×C4—1181no—
288C22×C4—1181no—
289C22×C4—1181no—
290C22×C4—1181no—
291C22×C4—2361no—
292C22×C4—2361no—
293C22×C4—2361no—
294C22×C4—2361no—
295C22×C4C2×C4×D5911yescharacteristic
296C22×C4D4×D5911yescharacteristic
297C22×C4D4×D5911yescharacteristic
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—1182no—
305C2×C8—2368no—
306C2×C8—2361no—
307C2×C8—2361no—
308C2×C8—2361no—
309C2×C8—2361no—
310C2×C8—2361no—
311C4⋊C4—41no—
312C4⋊C4—592no—
313C4⋊C4—1184no—
314C4⋊C4—1181no—
315C4⋊C4—2361no—
316C4⋊C4—2362no—
317C4⋊C4—2361no—
318C4⋊C4—2361no—
319C22⋊C4—1184no—
320C22⋊C4—2362no—
321C42—41no—
322C42—1182no—
323C42—1182no—
324C42—2361no—
325C42—2362no—
326C42—2362no—
327C23—592no—
328C23—592no—
329C23—592no—
330C23—1181no—
331C23—1181no—
332C23—1181no—
333C23—1181no—
334C23C22⋊C4×D5911yescharacteristic
335Q8—42no—
336Q8—42no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—592no—
350C2×C4—592no—
351C2×C4—592no—
352C2×C4—1184no—
353C2×C4—1184no—
354C2×C4—1181no—
355C2×C4—1181no—
356C2×C4—1181no—
357C2×C4—1181no—
358C2×C4—1181no—
359C2×C4—1181no—
360C2×C4—1182no—
361C2×C4—2362no—
362C2×C4—2362no—
363C2×C4—2362no—
364C2×C4—2362no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4—2361no—
368C2×C4—2361no—
369C2×C4—2361no—
370C2×C4C2×C8×D5911yescharacteristic
371C2×C4M16×D5911yescharacteristic
372C2×C4SD16×D5911yescharacteristic
373C2×C4Q16×D5911yescharacteristic
374C2×C4#1427375411yescharacteristic
375C2×C4#1427375911yescharacteristic
376C8—42no—
377C8—81no—
378C8—4721no—
379C8—2362no—
380C22—594no—
381C22—1182no—
382C22—1182no—
383C22—1182no—
384C22—1182no—
385C22—1182no—
386C22—1182no—
387C22C22⋊C8×D5911yescharacteristic
388C22#99×D5911yescharacteristic
389C22#1683738311yescharacteristic
390C22#103×D5911yescharacteristic
391C22#1683740211yescharacteristic
392C22#104×D5911yescharacteristic
393C22#1683741711yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—1182no—
403C4—1182no—
404C4—1182no—
405C4—4721no—
406C4—2362no—
407C2—1184no—
408C2#2142241111yescharacteristic
409C2Q8⋊C8×D5911yescharacteristic
410C2#2142248211yescharacteristic
411C2#2142259911yescharacteristic
412C2#2142266611yescharacteristic
413C2#2142271311yescharacteristic
414C2#2142278311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socleC23×C59C2×C22⋊C4
prime radicalC23×D59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1272D59
hypocenterC59#7589
LCSG ▷ C2×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. #21421920C2semidirect
6. #21421918C2semidirect
7. C59×#1272C2semidirect
36. #14273703C8semidirect
67. C2×Q8×C59C2×C8semidirect
103. C4⋊Dic59C2×C8semidirect
204. #1272D59semidirect
234. C59#7589semidirect
275. C2×Q8C8×D59semidirect

Maximal subgroups

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

subgroupindex
#214266342
#214240752
#336×D592
#214219222
#214219202
#214219182
C59×#12722
#758959