Finite Group Databasebeta

← #28558368

Subgroups — #28558368

Order 15104.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433176C211yesmaximal, characteristic, complemented
2#21433136C211yesmaximal, characteristic
3C2×C4×#11745660C211yesmaximal, characteristic
4#21422973C214yesmaximal, complemented
5#21422938C214yesmaximal, complemented
6#21422490C211yesmaximal, characteristic, complemented
7#21422486C211yesmaximal, characteristic
8#21422388C211yesmaximal, characteristic, complemented
9C59×#1564C211yesmaximal, characteristic, complemented
10C22×#11745660C411yescharacteristic
11C2×C4×Dic118C2211yescharacteristic
12C2×C4×D236C2211yescharacteristic
13C8×Dic118—22no—
14#16837526C2214yescomplemented
15C4×#11745660C412yes—
16C4×#11745660C412yes—
17C4×#11745660C2212yes—
18C4×#11745660C2212yes—
19#16837481—22no—
20#16837479C2214yes—
21#16837455C2211yescharacteristic
22#16837455C2211yescharacteristic
23#16837452—22no—
24C2×#14273659C411yescharacteristic
25C2×#14273652C411yescharacteristic
26C2×#14273649C411yescharacteristic
27#16837364—22no—
28#16837132C2214yescomplemented
29#16837129C2214yes—
30#16837124C2212yescomplemented
31#16837124C2212yescomplemented
32C59×#431C2211yescharacteristic
33C59×#431C2211yescharacteristic
34C2×C4×C8×C59C2211yescharacteristic
35C59×#335C2212yescomplemented
36C59×#334C2212yescomplemented
37C22×Dic118C2×C411yescharacteristic
38C22×D236C2×C411yescharacteristic
39C22×C4×D59—21no—
40C2×#11745660—24no—
41C2×#11745660C2×C412yes—
42C2×#11745660C2×C412yes—
43C2×#11745659—24no—
44C4×Dic118—21no—
45C4×Dic118—21no—
46C4×Dic118C2×C412yes—
47C4×Dic118C2312yes—
48C4×D236—21no—
49C4×D236—21no—
50C4×D236C2×C412yes—
51C4×D236C2312yes—
52C2×C4⋊Dic59C2×C411yescharacteristic
53C2×C4⋊Dic59C2×C411yescharacteristic
54C2×Dic59⋊C4—21no—
55C2×(C2×D59)⋊C4—21no—
56C2×C4×Dic59—21no—
57#14273659C2×C412yes—
58#14273659C2×C412yes—
59C4⋊(C59⋊C8)—21no—
60C4⋊(C59⋊C8)D412yes—
61Dic59⋊C8—24no—
62#14273652C2×C412yes—
63#14273652C2×C412yes—
64#14273649C2×C412yes—
65#14273649C2×C412yes—
66C22⋊(C59⋊C8)—21no—
67C22⋊(C59⋊C8)D412yes—
68(C2×D59)⋊C8—24no—
69C4×C59⋊C8C2312yes—
70C4×C59⋊C8C2312yes—
71C8×Dic59—24no—
72C22×C8×C59C2×C411yescharacteristic
73C22×C8×C59C2×C411yescharacteristic
74C2×C42×C59C2311yescharacteristic
75C4⋊C8×C59C2312yes—
76C4⋊C8×C59C2312yes—
77C22⋊C8×C59—21no—
78C22⋊C8×C59—21no—
79C4×C8×C59—21no—
80C4×C8×C59—21no—
81C4×C8×C59C2×C411yescharacteristic
82C4×C8×C59C2×C411yescharacteristic
83C4×C8×C59C2311yescharacteristic
84C4×C8×C59C2311yescharacteristic
85C23×D59—21no—
86C22×C4×C59C22×C411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×Dic59—21no—
90C22×Dic59—21no—
91C22×Dic59—21no—
92C2×D236—22no—
93C2×D236—22no—
94C2×D236—21no—
95C2×D236C22×C412yes—
96C2×C4×D59—22no—
97C2×C4×D59—22no—
98C2×C4×D59—82no—
99C2×Dic118—21no—
100C2×Dic118—22no—
101C2×Dic118M1614yescomplemented
102C2×Dic118C22×C412yes—
103C2×C8×C59—22no—
104C2×C8×C59—22no—
105C2×C8×C59—22no—
106C2×C8×C59—22no—
107C2×C8×C59C22×C411yescharacteristic
108C2×C8×C59C22×C411yescharacteristic
109C2×C8×C59C22×C411yescharacteristic
110C2×C8×C59C22×C411yescharacteristic
111C42×C59C22×C411yescharacteristic
112C42×C59C2×D411yescharacteristic
113C42×C59#4111yescharacteristic
114C42×C59C2411yescharacteristic
115(C2×D59)⋊C4—22no—
116(C2×D59)⋊C4—22no—
117C4⋊Dic59—21no—
118C4⋊Dic59—21no—
119C4⋊Dic59C22×C412yes—
120C4⋊Dic59C22×C412yes—
121Dic59⋊C4—22no—
122Dic59⋊C4—22no—
123C4×Dic59—22no—
124C4×Dic59—22no—
125C2×C59⋊C8—22no—
126C2×C59⋊C8C2×D414yes—
127#11745660—24no—
128#11745660—24no—
129#11745659—84no—
130C23×C59C4×D411yescharacteristic
131C22×D59—22no—
132C22×D59—42no—
133C22×D59—81no—
134C22×D59—81no—
135C2×C4×C59—22no—
136C2×C4×C59—41no—
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59C4×D411yescharacteristic
139C2×C4×C59C4×D411yescharacteristic
140C2×C4×C59C2×M1612yes—
141C2×C4×C59#13112yes—
142C2×C4×C59C23×C411yescharacteristic
143C2×C4×C59C22×D411yescharacteristic
144C2×C4×C59C2×#4111yescharacteristic
145C2×Dic59—22no—
146C2×Dic59—24no—
147C2×Dic59—22no—
148C2×Dic59—22no—
149C2×Dic59—81no—
150C2×Dic59—81no—
151D236—22no—
152D236—22no—
153D236—42no—
154C4×D59—44no—
155C4×D59—82no—
156Dic118—22no—
157Dic118—24no—
158Dic118C2×M1614yes—
159C8×C59—21no—
160C8×C59—21no—
161C8×C59—21no—
162C8×C59—21no—
163C8×C59—42no—
164C8×C59—42no—
165C59⋊C8—24no—
166C59⋊C8—42no—
167#17878—591nomaximal, Sylow
168C22×C59—21no—
169C22×C59—21no—
170C22×C59#43512yes—
171C22×C59C2×C4×D411yescharacteristic
172C2×D59—42no—
173C2×D59—42no—
174C2×D59—42no—
175C2×D59—84no—
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59#43512yes—
180C4×C59C22×M1611yescharacteristic
181C4×C59C2×#13111yescharacteristic
182C4×C59#57712yes—
183Dic59—24no—
184Dic59—42no—
185Dic59—42no—
186#2982—591no—
187#2975—591no—
188C2×C4×SD16—591no—
189#1588—594no—
190#1583—594no—
191#1564D5911yescharacteristic, complemented
192#1479—591no—
193#1478—591no—
194#1452—591no—
195C2×C59—22no—
196C2×C59C2×#43511yescharacteristic
197C2×C59#294911yescharacteristic
198C2×C59#295711yescharacteristic
199D59—82no—
200C22×SD16—591no—
201C2×C4×Q8—591no—
202C2×C4×D4—591no—
203C8×Q8—594no—
204C8×Q8—1182no—
205C4×SD16—592no—
206C4×SD16—592no—
207C4×SD16—592no—
208C4×SD16—592no—
209#436—594no—
210#436—1182no—
211#431—591no—
212#431—591no—
213#431—1182no—
214#431C2×D5911yescharacteristic
215#431C2×D5911yescharacteristic
216C2×#106—591no—
217C2×#103—591no—
218C2×#102—591no—
219#407—1182no—
220C2×C4×C8C2×D5911yescharacteristic
221#335C2×D5912yescomplemented
222#334C2×D5912yescomplemented
223Q8⋊C8—594no—
224D4⋊C8—594no—
225#322—592no—
226#322—592no—
227C59#1787811yescharacteristic, complemented, Sylow
228C22×Q8—591no—
229C22×D4—591no—
230C23×C4—1181no—
231C2×SD16—592no—
232C2×SD16—592no—
233C2×SD16—1184no—
234C2×M16—1184no—
235C22×C8C4×D5911yescharacteristic
236C22×C8C4×D5911yescharacteristic
237C4×Q8—592no—
238C4×Q8—592no—
239C4×Q8—1181no—
240C4×Q8—1181no—
241C4×D4—592no—
242C4×D4—592no—
243C4×D4—1181no—
244C4×D4—1181no—
245C2×C4⋊C4—591no—
246C2×C4⋊C4—591no—
247C2×C4⋊C4—1181no—
248C2×C22⋊C4—1181no—
249C2×C42—1181no—
250C2×C42C22×D5911yescharacteristic
251#106—592no—
252#106—592no—
253C4⋊C8—592no—
254C4⋊C8—1184no—
255C4⋊C8—1181no—
256C4⋊C8C22×D5912yes—
257C4⋊C8C22×D5912yes—
258#103—592no—
259#103—592no—
260#102—592no—
261#102—592no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—592no—
265C22⋊C8—1184no—
266C22⋊C8—1181no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—592no—
270C4×C8—592no—
271C4×C8—1184no—
272C4×C8C4×D5911yescharacteristic
273C4×C8C4×D5911yescharacteristic
274C4×C8C22×D5911yescharacteristic
275C4×C8C22×D5911yescharacteristic
276C24—1181no—
277C2×Q8—594no—
278C2×Q8—592no—
279C2×Q8—1181no—
280C2×Q8—1182no—
281C2×D4—592no—
282C2×D4—1181no—
283C2×D4—1182no—
284C2×D4—1182no—
285C22×C4—1181no—
286C22×C4—1181no—
287C22×C4—1181no—
288C22×C4—1182no—
289C22×C4—1182no—
290C22×C4—4722no—
291C22×C4C2×C4×D5911yescharacteristic
292C22×C4D4×D5911yescharacteristic
293C22×C4#1174568911yescharacteristic
294SD16—1184no—
295SD16—1184no—
296M16—4724no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—594no—
302C2×C8—1182no—
303C2×C8C2×C4×D5911yescharacteristic
304C2×C8C2×C4×D5911yescharacteristic
305C2×C8C2×C4×D5911yescharacteristic
306C2×C8C2×C4×D5911yescharacteristic
307C4⋊C4—592no—
308C4⋊C4—592no—
309C4⋊C4—1181no—
310C4⋊C4—1181no—
311C4⋊C4—1182no—
312C4⋊C4—1182no—
313C22⋊C4—1182no—
314C22⋊C4—1182no—
315C42—1182no—
316C42—1182no—
317C42C2×C4×D5911yescharacteristic
318C42D4×D5911yescharacteristic
319C42#1174568911yescharacteristic
320C42C23×D5911yescharacteristic
321C23—1182no—
322C23—4721no—
323C23—4721no—
324C23—2362no—
325C23#1427368911yescharacteristic
326Q8—594no—
327Q8—1182no—
328Q8—1184no—
329D4—1182no—
330D4—1182no—
331D4—2362no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—1184no—
335C2×C4—1182no—
336C2×C4—1182no—
337C2×C4—1182no—
338C2×C4—4722no—
339C2×C4—4721no—
340C2×C4—4721no—
341C2×C4—2364no—
342C2×C4#1427368911yescharacteristic
343C2×C4#1427368911yescharacteristic
344C2×C4#1427368911yescharacteristic
345C2×C4M16×D5912yes—
346C2×C4#1427373312yes—
347C2×C4C22×C4×D5911yescharacteristic
348C2×C4C2×D4×D5911yescharacteristic
349C2×C4C2×#1174568911yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—42no—
355C8—42no—
356C8—1184no—
357C8—2362no—
358C22—21no—
359C22—21no—
360C22—4724no—
361C22—2362no—
362C22—2362no—
363C22—2362no—
364C22#1683747212yes—
365C22C2×#1427368911yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—1184no—
370C4—2362no—
371C4—2362no—
372C4#1683747212yes—
373C4C2×M16×D5911yescharacteristic
374C4C2×#1427373311yescharacteristic
375C4#1683820912yes—
376C2—22no—
377C2—4722no—
378C2C2×#1683747211yescharacteristic
379C2#2143295411yescharacteristic
380C2#2143301211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1564C2
socleC22×C59C2×C4×D4
prime radicalC22×D236C2×C4
cosocleC42×C59C24
hypercenter#1564D59
hypocenterC59#17878
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #21433176C2semidirect
4. #21422973C2semidirect
5. #21422938C2semidirect
6. #21422490C2semidirect
8. #21422388C2semidirect
9. C59×#1564C2semidirect
14. #16837526C22semidirect
28. #16837132C22semidirect
30. #16837124C22semidirect
31. #16837124C22semidirect
35. C59×#335C22semidirect
36. C59×#334C22semidirect
101. C2×Dic118M16semidirect
191. #1564D59semidirect
221. #335C2×D59semidirect
222. #334C2×D59semidirect
227. C59#17878semidirect

Maximal subgroups

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

subgroupindex
#214331762
#214331362
C2×C4×#117456602
#214229732
#214229382
#214224902
#214224862
#214223882
C59×#15642
#1787859