Finite Group Databasebeta

← C2×#21427309 #28754883

Subgroups — C2×#21427309 #28754883

Order 15104.

6238 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D59C213yesmaximal, complemented
2#21427309C2116yesmaximal, direct product factor
3C2×#16837321C211yesmaximal, characteristic, complemented
4C2×#16837316C213yesmaximal, complemented
5C2×#16837313C211yesmaximal, characteristic, complemented
6C2×#16837310C213yesmaximal, complemented
7C2×C4×C22⋊Dic59C213yesmaximal, complemented
8C23×#41×C59C211yesmaximal, characteristic, complemented
9C23×C22⋊D59C2213yes—
10C23×C4×D59C2211yescharacteristic
11C2×C4×C22⋊D59—224no—
12C2×C4×C22⋊D59C22124yescomplemented
13C22×Dic59⋊C4C2213yes—
14C22×C22⋊Dic59C2213yes—
15C22×C22⋊Dic59C2216yes—
16C22×(C2×D59)⋊C4C2213yes—
17C22×C4×Dic59C2213yes—
18#16837321C2218yescomplemented
19#16837316C22124yescomplemented
20#16837313C2218yescomplemented
21#16837310C22124yescomplemented
22C4×C22⋊Dic59C22124yescomplemented
23C2×C59⋊#95C2213yes—
24C2×C59⋊#95C2213yes—
25C22×#41×C59—224no—
26C22×#41×C59C2218yescomplemented
27C23×Q8×C59C2211yescharacteristic
28C23×D4×C59C2213yes—
29C24×C4×C59C2213yes—
30C24×D59C2311yescharacteristic
31C22×C22⋊D59—212no—
32C22×C22⋊D59—212no—
33C22×C22⋊D59—212no—
34C22×C22⋊D59C23112yes—
35C23×Dic59C2311yescharacteristic
36C23×Dic59C2313yes—
37C23×Dic59C2313yes—
38C22×C4×D59—212no—
39C22×C4×D59C2314yes—
40C4×C22⋊D59—296no—
41C2×Dic59⋊C4—212no—
42C2×Dic59⋊C4C23112yes—
43C2×C22⋊Dic59—212no—
44C2×C22⋊Dic59—224no—
45C2×C22⋊Dic59C23124yes—
46C2×C22⋊Dic59C23112yes—
47C2×(C2×D59)⋊C4—212no—
48C2×(C2×D59)⋊C4C23112yes—
49C2×C4×Dic59—212no—
50C2×C4×Dic59C23112yes—
51C59⋊#95C23112yes—
52C59⋊#95C23112yes—
53C25×C59C2313yes—
54C2×#41×C59—2128no—
55C2×#41×C59—296no—
56C22×Q8×C59—212no—
57C22×Q8×C59C2314yes—
58C22×D4×C59—224no—
59C22×D4×C59—212no—
60C22×D4×C59C23112yes—
61C23×C4×C59—224no—
62C23×C4×C59D4124yescomplemented
63C23×C4×C59C2313yes—
64C23×C4×C59C23112yes—
65C23×C4×C59C2311yescharacteristic
66C24×C59—212no—
67C24×C59—212no—
68C24×C59—26no—
69C24×C59—26no—
70C24×C59C2×D4112yes—
71C24×C59C2416yes—
72C24×C59C2411yescharacteristic
73C23×D59—26no—
74C23×D59—26no—
75C23×D59#4114yescomplemented
76C23×D59C2412yes—
77#41×C59—2256no—
78C2×Q8×C59—232no—
79C2×Q8×C59—224no—
80C2×D4×C59—296no—
81C2×D4×C59—248no—
82C2×D4×C59—224no—
83C22×C4×C59—212no—
84C22×C4×C59—296no—
85C22×C4×C59—248no—
86C22×C4×C59C2×D4112yes—
87C22×C4×C59C2×D4112yes—
88C22×C4×C59C2×D4148yescomplemented
89C22×C4×C59C2416yes—
90C22×C4×C59C2412yes—
91C2×C22⋊D59—296no—
92C2×C22⋊D59—224no—
93C2×C22⋊D59—224no—
94C2×C22⋊D59—224no—
95C22×Dic59—26no—
96C22×Dic59—26no—
97C22×Dic59—26no—
98C22×Dic59—26no—
99C22×Dic59—212no—
100C22×Dic59#41112yescomplemented
101C22×Dic59C2416yes—
102C22×Dic59C2412yes—
103C22×Dic59C2416yes—
104C2×C4×D59—224no—
105C2×C4×D59—416no—
106C22⋊Dic59—224no—
107C22⋊Dic59—248no—
108(C2×D59)⋊C4—224no—
109Dic59⋊C4—224no—
110C4×Dic59—224no—
111C23×C59—224no—
112C23×C59—224no—
113C23×C59—224no—
114C23×C59—212no—
115C23×C59—212no—
116C23×C59—26no—
117C23×C59—26no—
118C23×C59—424no—
119C23×C59—424no—
120C23×C59C22×D4112yes—
121C23×C59C22×D416yes—
122C23×C59C2×#4112yes—
123C23×C59C2×#4116yes—
124C23×C59C2511yescharacteristic
125C22×D59—224no—
126C22×D59—26no—
127C22×D59—26no—
128C22×D59—44no—
129C22×D59—412no—
130C22×D59C2×#4114yescomplemented
131Q8×C59—232no—
132D4×C59—296no—
133C2×C4×C59—224no—
134C2×C4×C59—212no—
135C2×C4×C59—296no—
136C2×C4×C59#120116yes—
137C2×C4×C59C22×D4112yes—
138C2×C4×C59C22×D4112yes—
139C22⋊D59—296no—
140C2×Dic59—224no—
141C2×Dic59—26no—
142C2×Dic59—26no—
143C2×Dic59—26no—
144C2×Dic59—26no—
145C2×Dic59—212no—
146C2×Dic59—44no—
147C2×Dic59—412no—
148C2×Dic59C2×#41112yescomplemented
149C4×D59—416no—
150C2×#2388—591nomaximal, Sylow
151C22×C59—212no—
152C22×C59—212no—
153C22×C59—212no—
154C22×C59—448no—
155C22×C59—412no—
156C22×C59—412no—
157C22×C59C2×#12014yes—
158C22×C59#525112yes—
159C22×C59#532112yes—
160C22×C59C23×D413yes—
161C22×C59C22×#4111yescharacteristic
162C22×C59C22×#4113yes—
163C2×D59—212no—
164C2×D59—416no—
165C2×D59—42no—
166C2×D59—46no—
167C4×C59—212no—
168C4×C59C2×#12018yes—
169Dic59—212no—
170Dic59—42no—
171Dic59—46no—
172C23×#41D5911yescharacteristic, complemented
173C22×C4×D4—593no—
174#2388—5916no—
175C2×#395—591no—
176C2×#394—593no—
177C2×#393—591no—
178C2×#392—593no—
179C2×C4×C22⋊C4—593no—
180C2×C59—412no—
181C2×C59#238818yescomplemented
182C2×C59C22×#12011yescharacteristic
183C2×C59C2×#52513yes—
184C2×C59C2×#53213yes—
185D59—44no—
186C22×#41—224no—
187C22×#41C2×D5918yescomplemented
188C23×Q8C2×D5911yescharacteristic
189C23×D4—593no—
190C23×D4C2×D5913yes—
191C24×C4—591no—
192C24×C4C2×D5913yes—
193C2×C4×D4—5924no—
194C2×C4×D4—11824no—
195C22×C4⋊C4—593no—
196C22×C22⋊C4—596no—
197C22×C22⋊C4—593no—
198C22×C22⋊C4—593no—
199C22×C42—593no—
200#395—598no—
201#394—5924no—
202#393—598no—
203#392—5924no—
204C4×C22⋊C4—5924no—
205C2×#95—593no—
206C2×#95—593no—
207C59C2×#238811yescharacteristic, complemented, Sylow
208C25—591no—
209C25C22×D5913yes—
210C2×#41—2128no—
211C2×#41—296no—
212C22×Q8—212no—
213C22×Q8C22×D5914yes—
214C22×D4—224no—
215C22×D4—212no—
216C22×D4—5912no—
217C22×D4—11812no—
218C22×D4—11812no—
219C22×D4—11812no—
220C22×D4C22×D59112yes—
221C23×C4—224no—
222C23×C4—593no—
223C23×C4—593no—
224C23×C4—591no—
225C23×C4—594no—
226C23×C4—11812no—
227C23×C4C22⋊D59124yescomplemented
228C23×C4C22×D5913yes—
229C23×C4C22×D59112yes—
230C23×C4C22×D5911yescharacteristic
231C4×D4—11896no—
232C2×C4⋊C4—5912no—
233C2×C4⋊C4—11812no—
234C2×C22⋊C4—5912no—
235C2×C22⋊C4—5924no—
236C2×C22⋊C4—5912no—
237C2×C22⋊C4—11824no—
238C2×C22⋊C4—11812no—
239C2×C22⋊C4—11812no—
240C2×C42—5912no—
241C2×C42—11812no—
242#95—5912no—
243#95—5912no—
244C24—212no—
245C24—212no—
246C24—26no—
247C24—26no—
248C24—594no—
249C24—592no—
250C24—1186no—
251C24—1186no—
252C24C2×C22⋊D59112yes—
253C24C23×D5916yes—
254C24C23×D5911yescharacteristic
255#41—2256no—
256C2×Q8—232no—
257C2×Q8—224no—
258C2×D4—296no—
259C2×D4—248no—
260C2×D4—224no—
261C2×D4—11824no—
262C2×D4—11824no—
263C2×D4—11896no—
264C2×D4—11824no—
265C22×C4—212no—
266C22×C4—296no—
267C22×C4—248no—
268C22×C4—5912no—
269C22×C4—596no—
270C22×C4—596no—
271C22×C4—592no—
272C22×C4—1186no—
273C22×C4—1186no—
274C22×C4—1186no—
275C22×C4—1186no—
276C22×C4—11812no—
277C22×C4—11824no—
278C22×C4—23616no—
279C22×C4C2×C22⋊D59112yes—
280C22×C4C2×C22⋊D59112yes—
281C22×C4C2×C22⋊D59148yescomplemented
282C22×C4C23×D5916yes—
283C22×C4C23×D5912yes—
284C4⋊C4—11824no—
285C22⋊C4—11848no—
286C22⋊C4—11824no—
287C22⋊C4—11824no—
288C42—11824no—
289C23—224no—
290C23—224no—
291C23—224no—
292C23—212no—
293C23—212no—
294C23—26no—
295C23—26no—
296C23—424no—
297C23—424no—
298C23—594no—
299C23—11824no—
300C23—1186no—
301C23—1186no—
302C23—2364no—
303C23—23612no—
304C23C22×C22⋊D59112yes—
305C23C22×C22⋊D5916yes—
306C23#41×D5912yes—
307C23#41×D5916yes—
308C23C24×D5911yescharacteristic
309Q8—232no—
310D4—296no—
311D4—11896no—
312C2×C4—224no—
313C2×C4—212no—
314C2×C4—296no—
315C2×C4—5912no—
316C2×C4—11824no—
317C2×C4—1186no—
318C2×C4—1186no—
319C2×C4—1186no—
320C2×C4—1186no—
321C2×C4—11812no—
322C2×C4—2364no—
323C2×C4—23612no—
324C2×C4—23616no—
325C2×C4#14273696116yes—
326C2×C4C22×C22⋊D59112yes—
327C2×C4C22×C22⋊D59112yes—
328C22—212no—
329C22—212no—
330C22—212no—
331C22—448no—
332C22—412no—
333C22—412no—
334C22—11812no—
335C22—23616no—
336C22—2362no—
337C22—2366no—
338C22C2×#1427369614yes—
339C22#16837909112yes—
340C22#16837948112yes—
341C22C23×C22⋊D5913yes—
342C22C2×#41×D5911yescharacteristic
343C22C2×#41×D5913yes—
344C4—212no—
345C4—11812no—
346C4—2362no—
347C4—2366no—
348C4C2×#1427369618yes—
349C2—412no—
350C2—2364no—
351C2#2142730918yesdirect product factor
352C2C22×#1427369611yescharacteristic
353C2C2×#1683790913yes—
354C2C2×#1683794813yes—
subgroupG/H
centerC23×C4C22×D59
commutator, cosocleC23×C59C25
FrattiniC23C24×D59
FittingC23×#41×C59C2
socleC24×C59C24
hypercenterC23×#41D59
hypocenterC59C2×#2388
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C22×C4×C22⋊D59C2semidirect
2. #21427309C2direct (N × H)
3. C2×#16837321C2semidirect
4. C2×#16837316C2semidirect
5. C2×#16837313C2semidirect
6. C2×#16837310C2semidirect
7. C2×C4×C22⋊Dic59C2semidirect
8. C23×#41×C59C2semidirect
12. C2×C4×C22⋊D59C22semidirect
18. #16837321C22semidirect
19. #16837316C22semidirect
20. #16837313C22semidirect
21. #16837310C22semidirect
22. C4×C22⋊Dic59C22semidirect
26. C22×#41×C59C22semidirect
62. C23×C4×C59D4semidirect
75. C23×D59#41semidirect
88. C22×C4×C59C2×D4semidirect
100. C22×Dic59#41semidirect
130. C22×D59C2×#41semidirect
148. C2×Dic59C2×#41semidirect
172. C23×#41D59semidirect
181. C2×C59#2388semidirect
187. C22×#41C2×D59semidirect
207. C59C2×#2388semidirect
227. C23×C4C22⋊D59semidirect
281. C22×C4C2×C22⋊D59semidirect
351. C2#21427309direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D592
#214273092
C2×#168373212
C2×#168373162
C2×#168373132
C2×#168373102
C2×C4×C22⋊Dic592
C23×#41×C592
C2×#238859