Finite Group Databasebeta

← C4×#108×D59 #28586783

Subgroups — C4×#108×D59 #28586783

Order 15104.

1086 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#108×D59C212yesmaximal
2C4×M16×D59C212yesmaximal
3C2×C4×C8×D59C211yesmaximal, characteristic
4#108×Dic59C212yesmaximal, complemented
5C4×#14273662C212yesmaximal, complemented
6C4×#14273663C211yesmaximal, characteristic, complemented
7#21423997C212yesmaximal, complemented
8#341×D59C212yesmaximal
9C4×#108×C59C211yesmaximal, characteristic, complemented
10C2×M16×D59C2214yes—
11C22×C8×D59C2211yescharacteristic
12C22×C8×D59C2211yescharacteristic
13C2×C42×D59C2211yescharacteristic
14C2×#14273663C2212yes—
15C2×#14273662C2214yes—
16#108×D59C4116yesdirect product factor
17C4×#11745664C2212yes—
18#16837354—22no—
19M16×Dic59C2214yes—
20C4×#11745659—22no—
21#97×D59—22no—
22C2×C4×C59⋊C8C2211yescharacteristic
23C2×C8×Dic59C2211yescharacteristic
24C2×C8×Dic59C2211yescharacteristic
25C4×C8×D59—22no—
26C4×C8×D59C414yes—
27C4×C8×D59C414yes—
28#16837167C2212yes—
29#16837166C2214yes—
30C2×#108×C59C2212yes—
31C4×M16×C59C2212yes—
32C2×C4×C8×C59C2211yescharacteristic
33C59×#341C2212yes—
34C22×C4×D59Q812yes—
35C22×C4×D59C2311yescharacteristic
36C2×#11745664C2314yes—
37C2×#11745659—24no—
38M16×D59—28no—
39M16×D59C2×C4116yes—
40C22×C59⋊C8C2311yescharacteristic
41C22×C59⋊C8C2311yescharacteristic
42C2×C8×D59—24no—
43C2×C8×D59—24no—
44C2×C8×D59C2×C412yes—
45C2×C8×D59C2×C412yes—
46C2×C8×D59C2×C418yes—
47C2×C8×D59C2×C412yes—
48C2×C8×D59C2×C412yes—
49C2×C4×Dic59Q812yes—
50C2×C4×Dic59C2311yescharacteristic
51C42×D59C2×C412yes—
52C42×D59C2×C412yes—
53C42×D59D411yescharacteristic
54C42×D59D411yescharacteristic
55C42×D59D411yescharacteristic
56C42×D59D411yescharacteristic
57#14273663C2×C418yescomplemented
58#14273662C2×C4116yescomplemented
59#14273638—22no—
60#14273637—24no—
61C4×C59⋊C8—22no—
62C4×C59⋊C8C2×C414yes—
63C8×Dic59—24no—
64C8×Dic59C2×C414yes—
65C8×Dic59C2×C414yes—
66C2×M16×C59C2314yes—
67C22×C8×C59C2311yescharacteristic
68C22×C8×C59C2311yescharacteristic
69C2×C42×C59C2311yescharacteristic
70#108×C59C2×C418yescomplemented
71#97×C59—22no—
72C4×C8×C59—22no—
73C4×C8×C59C2×C414yes—
74C23×D59C2×Q811yescharacteristic
75C22×C4×C59C2×Q812yes—
76C22×C4×C59C2411yescharacteristic
77C22×Dic59C2×Q812yes—
78C22×Dic59C2×Q811yescharacteristic
79C2×C4×D59—28no—
80C2×C4×D59C4⋊C412yes—
81C2×C4×D59C4⋊C412yes—
82C2×C4×D59C4⋊C412yes—
83C2×C4×D59C4⋊C412yes—
84C2×C4×D59C22×C411yescharacteristic
85C2×C4×D59C22×C411yescharacteristic
86C2×C4×D59C22×C414yes—
87C2×C4×D59C2×D411yescharacteristic
88C2×C4×D59C2×D411yescharacteristic
89C2×C4×D59#4114yes—
90M16×C59—24no—
91M16×C59C22×C418yes—
92C2×C8×C59—24no—
93C2×C8×C59—22no—
94C2×C8×C59C22×C412yes—
95C2×C8×C59C22×C412yes—
96C2×C8×C59C22×C414yes—
97C42×C59C22×C412yes—
98C42×C59C2×D411yescharacteristic
99C42×C59C2×D411yescharacteristic
100C4×Dic59C4⋊C412yes—
101C4×Dic59C4⋊C412yes—
102C4×Dic59C4⋊C412yes—
103C4×Dic59C4⋊C412yes—
104C4×Dic59C22×C412yes—
105C4×Dic59C2×D411yescharacteristic
106C4×Dic59C2×D411yescharacteristic
107#11745664—24no—
108#11745664C22×C418yes—
109C2×C59⋊C8—24no—
110C2×C59⋊C8—22no—
111C2×C59⋊C8C22×C412yes—
112C2×C59⋊C8C22×C412yes—
113C2×C59⋊C8C22×C414yes—
114#11745659—216no—
115C8×D59—24no—
116C8×D59—24no—
117C8×D59—216no—
118C8×D59C4218yes—
119C8×D59C4218yes—
120C23×C59C22×Q811yescharacteristic
121C22×D59—24no—
122C22×D59C2×C4⋊C411yescharacteristic
123C22×D59C2×C4⋊C411yescharacteristic
124C22×D59C4×Q814yes—
125C2×C4×C59—24no—
126C2×C4×C59C2×C4⋊C412yes—
127C2×C4×C59C2×C4⋊C412yes—
128C2×C4×C59C23×C411yescharacteristic
129C2×C4×C59C23×C412yes—
130C2×C4×C59C22×D411yescharacteristic
131C2×C4×C59C2×#4112yes—
132C2×Dic59—24no—
133C2×Dic59C2×C4⋊C412yes—
134C2×Dic59C2×C4⋊C411yescharacteristic
135C2×Dic59C2×C4⋊C411yescharacteristic
136C2×Dic59C2×C4⋊C412yes—
137C2×Dic59C4×Q814yes—
138C4×D59—24no—
139C4×D59—24no—
140C4×D59—24no—
141C4×D59#10818yesdirect product factor
142C4×D59C2×C4212yes—
143C4×D59C2×C4212yes—
144C4×D59#11712yes—
145C4×D59#11712yes—
146C4×D59C4×D414yes—
147C4×D59C4×D414yes—
148C8×C59—22no—
149C8×C59—28no—
150C8×C59C2×C4214yes—
151C59⋊C8—22no—
152C59⋊C8—28no—
153C59⋊C8C2×C4214yes—
154C2×C4×#108—591nomaximal, Sylow
155C22×C59—22no—
156C22×C59C22×C4⋊C411yescharacteristic
157C22×C59C2×C4×Q812yes—
158C2×D59—22no—
159C2×D59—28no—
160C2×D59C4×C4⋊C412yes—
161C2×D59C4×C4⋊C412yes—
162C2×D59C2×#10814yes—
163C4×C59—22no—
164C4×C59C2×#10814yescomplemented
165C4×C59C22×C4211yescharacteristic
166C4×C59C2×#11711yescharacteristic
167C4×C59C2×C4×D412yes—
168Dic59—22no—
169Dic59C4×C4⋊C412yes—
170Dic59C4×C4⋊C412yes—
171Dic59C2×#10814yescomplemented
172C22×#108—592no—
173C2×C4×M16—592no—
174C22×C4×C8—591no—
175C4×#108—592no—
176C4×#108—592no—
177C4×#108—591no—
178C4×#108—592no—
179C4×#108D5911yescharacteristic, direct product factor
180C2×#341—592no—
181C2×C59—22no—
182C2×C59C2×C4×C4⋊C411yescharacteristic
183C2×C59C22×#10812yes—
184D59—22no—
185D59C4×#10814yesdirect product factor
186C22×M16—594no—
187C23×C8—591no—
188C23×C8—591no—
189C22×C42—591no—
190C2×#108—594no—
191C2×#108—592no—
192C2×#108—5916no—
193C2×#108C2×D5912yes—
194C4×M16—594no—
195C4×M16—592no—
196C4×M16—1182no—
197C4×M16—1182no—
198C4×M16C2×D5912yes—
199C2×#97—1182no—
200C2×C4×C8—591no—
201C2×C4×C8—591no—
202C2×C4×C8—591no—
203C2×C4×C8—594no—
204C2×C4×C8—594no—
205C2×C4×C8—1182no—
206C2×C4×C8C2×D5911yescharacteristic
207#341—594no—
208#341—592no—
209#341C2×D5912yes—
210C59C2×C4×#10811yescharacteristic, complemented, Sylow
211C23×C4—591no—
212C23×C4—592no—
213C2×M16—594no—
214C2×M16—5916no—
215C2×M16—1184no—
216C2×M16—1188no—
217C2×M16C22×D5914yes—
218C22×C8—591no—
219C22×C8—592no—
220C22×C8—592no—
221C22×C8—598no—
222C22×C8—591no—
223C22×C8—592no—
224C22×C8—592no—
225C22×C8—1184no—
226C22×C8—1184no—
227C22×C8C22×D5911yescharacteristic
228C22×C8C22×D5911yescharacteristic
229C2×C42—592no—
230C2×C42—591no—
231C2×C42—592no—
232C2×C42—592no—
233C2×C42—591no—
234C2×C42—591no—
235C2×C42—591no—
236C2×C42—591no—
237C2×C42C22×D5911yescharacteristic
238#108—5916no—
239#108—598no—
240#108C4×D5918yesdirect product factor
241#97—22no—
242#97—1184no—
243#97—1182no—
244C4×C8—22no—
245C4×C8—594no—
246C4×C8—594no—
247C4×C8—594no—
248C4×C8—1184no—
249C4×C8—1182no—
250C4×C8C4×D5914yes—
251C24—591no—
252C22×C4—591no—
253C22×C4—592no—
254C22×C4—591no—
255C22×C4—591no—
256C22×C4—594no—
257C22×C4—591no—
258C22×C4—591no—
259C22×C4—594no—
260C22×C4—592no—
261C22×C4—592no—
262C22×C4—592no—
263C22×C4—592no—
264C22×C4—1188no—
265C22×C4Q8×D5912yes—
266C22×C4C23×D5911yescharacteristic
267M16—24no—
268M16—598no—
269M16—11816no—
270M16—1184no—
271M16C2×C4×D5918yes—
272C2×C8—24no—
273C2×C8—22no—
274C2×C8—592no—
275C2×C8—594no—
276C2×C8—598no—
277C2×C8—598no—
278C2×C8—592no—
279C2×C8—1184no—
280C2×C8—1184no—
281C2×C8—11816no—
282C2×C8—1184no—
283C2×C8—1182no—
284C2×C8C2×C4×D5912yes—
285C2×C8C2×C4×D5912yes—
286C2×C8C2×C4×D5914yes—
287C42—592no—
288C42—592no—
289C42—592no—
290C42—592no—
291C42—592no—
292C42—591no—
293C42—591no—
294C42C2×C4×D5912yes—
295C42D4×D5911yescharacteristic
296C42D4×D5911yescharacteristic
297C23—591no—
298C23—591no—
299C23—594no—
300C23—1184no—
301C23C2×Q8×D5911yescharacteristic
302C2×C4—24no—
303C2×C4—592no—
304C2×C4—591no—
305C2×C4—591no—
306C2×C4—594no—
307C2×C4—598no—
308C2×C4—592no—
309C2×C4—592no—
310C2×C4—592no—
311C2×C4—592no—
312C2×C4—594no—
313C2×C4—594no—
314C2×C4—592no—
315C2×C4—1184no—
316C2×C4—1184no—
317C2×C4—1184no—
318C2×C4—1184no—
319C2×C4C4⋊C4×D5912yes—
320C2×C4C4⋊C4×D5912yes—
321C2×C4C22×C4×D5911yescharacteristic
322C2×C4C22×C4×D5912yes—
323C2×C4C2×D4×D5911yescharacteristic
324C2×C4#41×D5912yes—
325C8—22no—
326C8—28no—
327C8—594no—
328C8—1182no—
329C8—1188no—
330C8C42×D5914yes—
331C22—22no—
332C22—594no—
333C22—592no—
334C22—592no—
335C22—1188no—
336C22—1182no—
337C22C2×C4⋊C4×D5911yescharacteristic
338C22C4×Q8×D5912yes—
339C4—22no—
340C4—594no—
341C4—592no—
342C4—592no—
343C4—1182no—
344C4#108×D5914yesdirect product factor
345C4C2×C42×D5911yescharacteristic
346C4#117×D5911yescharacteristic
347C4C4×D4×D5912yes—
348C2—22no—
349C2—594no—
350C2—1182no—
351C2C4×C4⋊C4×D5911yescharacteristic
352C2C2×#108×D5912yes—
subgroupG/H
centerC42D4×D59
commutatorC4×C59C22×C42
FrattiniC22×C4C23×D59
FittingC4×#108×C59C2
socleC22×C59C22×C4⋊C4
prime radicalC23×D59C2×Q8
cosocleC22×C4×C59C24
hypercenterC4×#108D59
hypocenterC59C2×C4×#108
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#108

Complements

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

NH ≅ G/Ntype
4. #108×Dic59C2semidirect
5. C4×#14273662C2semidirect
6. C4×#14273663C2semidirect
7. #21423997C2semidirect
9. C4×#108×C59C2semidirect
16. #108×D59C4direct (N × H)
57. #14273663C2×C4semidirect
58. #14273662C2×C4semidirect
70. #108×C59C2×C4semidirect
141. C4×D59#108direct (N × H)
164. C4×C59C2×#108semidirect
171. Dic59C2×#108semidirect
179. C4×#108D59direct (N × H)
185. D59C4×#108direct (N × H)
210. C59C2×C4×#108semidirect
240. #108C4×D59direct (N × H)
344. C4#108×D59direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#108×D592
C4×M16×D592
C2×C4×C8×D592
#108×Dic592
C4×#142736622
C4×#142736632
#214239972
#341×D592
C4×#108×C592
C2×C4×#10859