Finite Group Databasebeta

← C2×#21432359 #28795099

Subgroups — C2×#21432359 #28795099

Order 15104.

3294 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432359C2116yesmaximal, direct product factor
2C2×#16837339C211yesmaximal, characteristic, complemented
3C2×#16837337C213yesmaximal, complemented
4C2×#16837326C211yesmaximal, characteristic, complemented
5C2×#16837325C213yesmaximal, complemented
6C2×#16837324C213yesmaximal, complemented
7C2×#16837257C213yesmaximal
8C2×C59×#396C211yesmaximal, characteristic, complemented
9C25×D59C2211yescharacteristic
10C22×C4⋊Dic59C2213yescomplemented
11C22×Dic59⋊C4C2213yescomplemented
12C22×Dic59⋊C4C2213yescomplemented
13C22×C22⋊Dic59C2213yes—
14C22×(C2×D59)⋊C4C2213yes—
15C22×(C2×D59)⋊C4C2213yes—
16C22×(C2×D59)⋊C4C2213yes—
17C22×(C2×D59)⋊C4C2213yes—
18#16837339C2218yescomplemented
19#16837337C22124yescomplemented
20#16837326C2218yescomplemented
21#16837325C22124yescomplemented
22#16837324C22124yescomplemented
23#16837257C22124yes—
24C2×C59⋊#95C2213yescomplemented
25C22×C4⋊C4×C59C2213yescomplemented
26C22×C22⋊C4×C59C2213yes—
27C59×#396C2218yescomplemented
28C2×#95×C59C2211yescharacteristic, complemented
29C24×D59—46no—
30C24×D59—46no—
31C24×D59Q818yes—
32C24×D59C2311yescharacteristic
33C24×D59C2311yescharacteristic
34C24×D59C2314yes—
35C23×Dic59C2313yes—
36C23×Dic59C2313yes—
37C2×C4⋊Dic59—46no—
38C2×C4⋊Dic59C23112yescomplemented
39C2×Dic59⋊C4—46no—
40C2×Dic59⋊C4—46no—
41C2×Dic59⋊C4C23112yescomplemented
42C2×Dic59⋊C4C23112yescomplemented
43C2×C22⋊Dic59—46no—
44C2×C22⋊Dic59C23112yes—
45C2×(C2×D59)⋊C4—46no—
46C2×(C2×D59)⋊C4—46no—
47C2×(C2×D59)⋊C4—46no—
48C2×(C2×D59)⋊C4—46no—
49C2×(C2×D59)⋊C4C23112yes—
50C2×(C2×D59)⋊C4C23112yes—
51C2×(C2×D59)⋊C4C23112yes—
52C2×(C2×D59)⋊C4C23112yes—
53C59⋊#95C23112yescomplemented
54C25×C59C2311yescharacteristic
55C23×C4×C59C2313yes—
56C23×C4×C59C2313yes—
57C2×C4⋊C4×C59—46no—
58C2×C4⋊C4×C59C23112yescomplemented
59C2×C22⋊C4×C59—46no—
60C2×C22⋊C4×C59C23112yes—
61#95×C59C2314yescomplemented
62C24×C59—26no—
63C24×C59—26no—
64C24×C59C2×Q814yes—
65C24×C59C2412yes—
66C24×C59C2411yescharacteristic
67C23×D59—26no—
68C23×D59—26no—
69C23×D59—26no—
70C23×D59—26no—
71C23×D59—448no—
72C23×D59—412no—
73C23×D59—412no—
74C23×D59—448no—
75C23×D59—416no—
76C23×D59C2×Q814yes—
77C23×D59C2×Q814yes—
78C23×D59C2×Q8116yes—
79C23×D59C2412yes—
80C23×D59C2412yes—
81C22×C4×C59—26no—
82C22×C4×C59—26no—
83C22×C4×C59—26no—
84C22×C4×C59—26no—
85C22×C4×C59—26no—
86C22×C4×C59—26no—
87C22×C4×C59C2416yes—
88C22×C4×C59C2416yes—
89C22×Dic59—26no—
90C22×Dic59—26no—
91C22×Dic59—26no—
92C22×Dic59—26no—
93C22×Dic59—26no—
94C22×Dic59—26no—
95C22×Dic59C2416yes—
96C22×Dic59C2416yes—
97C4⋊C4×C59—412no—
98C22⋊C4×C59—412no—
99C22⋊Dic59—412no—
100(C2×D59)⋊C4—412no—
101(C2×D59)⋊C4—412no—
102(C2×D59)⋊C4—412no—
103(C2×D59)⋊C4—412no—
104C4⋊Dic59—412no—
105Dic59⋊C4—412no—
106Dic59⋊C4—412no—
107C23×C59—224no—
108C23×C59—26no—
109C23×C59—26no—
110C23×C59—412no—
111C23×C59—44no—
112C23×C59C22×Q814yes—
113C23×C59C22×Q812yes—
114C23×C59H(5,𝔽2)16yes—
115C23×C59H(5,𝔽2)16yes—
116C23×C59C2511yescharacteristic
117C22×D59—224no—
118C22×D59—224no—
119C22×D59—26no—
120C22×D59—26no—
121C22×D59—26no—
122C22×D59—26no—
123C22×D59—448no—
124C22×D59—4128no—
125C22×D59—448no—
126C22×D59—416no—
127C22×D59—412no—
128C22×D59—412no—
129C22×D59—44no—
130C22×D59—44no—
131C22×D59C22×Q814yes—
132C22×D59C22×Q814yes—
133C2×C4×C59—26no—
134C2×C4×C59—26no—
135C2×C4×C59—26no—
136C2×C4×C59—26no—
137C2×C4×C59—26no—
138C2×C4×C59—26no—
139C2×C4×C59—412no—
140C2×C4×C59—412no—
141C2×Dic59—26no—
142C2×Dic59—26no—
143C2×Dic59—26no—
144C2×Dic59—26no—
145C2×Dic59—26no—
146C2×Dic59—26no—
147C2×Dic59—412no—
148C2×Dic59—412no—
149C2×#2842—591nomaximal, Sylow
150C22×C59—212no—
151C22×C59—416no—
152C22×C59—46no—
153C22×C59—42no—
154C22×C59#543112yes—
155C22×C59#560112yes—
156C22×C59#56014yes—
157C22×C59C23×Q811yescharacteristic
158C22×C59C2×H(5,𝔽2)13yes—
159C22×C59C2×H(5,𝔽2)13yes—
160C2×D59—212no—
161C2×D59—212no—
162C2×D59—464no—
163C2×D59—416no—
164C2×D59—416no—
165C2×D59—46no—
166C2×D59—46no—
167C2×D59—42no—
168C2×D59—42no—
169C4×C59—46no—
170C4×C59—46no—
171Dic59—46no—
172Dic59—46no—
173#2842—5916no—
174C2×#399—591no—
175C2×#399—593no—
176C2×#396—591no—
177C2×#396—593no—
178C2×#396—593no—
179C2×#396D5911yescharacteristic, complemented
180C2×#379—593no—
181C2×C59—44no—
182C2×C59#284218yescomplemented
183C2×C59C2×#54313yes—
184C2×C59C2×#56013yes—
185C2×C59C2×#56011yescharacteristic
186D59—44no—
187D59—44no—
188C26—591no—
189C22×C4⋊C4—593no—
190C22×C4⋊C4—593no—
191C22×C4⋊C4—593no—
192C22×C4⋊C4C2×D5913yescomplemented
193C22×C22⋊C4—593no—
194C22×C22⋊C4—593no—
195C22×C22⋊C4—593no—
196C22×C22⋊C4—593no—
197C22×C22⋊C4—593no—
198C22×C22⋊C4C2×D5913yes—
199#399—5924no—
200#399—598no—
201#396—5924no—
202#396—5924no—
203#396—598no—
204#396C2×D5918yescomplemented
205#379—5924no—
206C2×#95—593no—
207C2×#95C2×D5911yescharacteristic, complemented
208C59C2×#284211yescharacteristic, complemented, Sylow
209C25—591no—
210C25—591no—
211C25—598no—
212C25—594no—
213C25—2366no—
214C25—2366no—
215C25C22×D5911yescharacteristic
216C23×C4—593no—
217C23×C4—593no—
218C23×C4C22×D5913yes—
219C23×C4C22×D5913yes—
220C2×C4⋊C4—46no—
221C2×C4⋊C4—5912no—
222C2×C4⋊C4—5912no—
223C2×C4⋊C4—5912no—
224C2×C4⋊C4—2366no—
225C2×C4⋊C4—2366no—
226C2×C4⋊C4—2366no—
227C2×C4⋊C4C22×D59112yescomplemented
228C2×C22⋊C4—46no—
229C2×C22⋊C4—5912no—
230C2×C22⋊C4—5912no—
231C2×C22⋊C4—5912no—
232C2×C22⋊C4—5912no—
233C2×C22⋊C4—5912no—
234C2×C22⋊C4—2366no—
235C2×C22⋊C4—2366no—
236C2×C22⋊C4—2366no—
237C2×C22⋊C4—2366no—
238C2×C22⋊C4—2366no—
239C2×C22⋊C4C22×D59112yes—
240#95—5912no—
241#95C22×D5914yescomplemented
242C24—26no—
243C24—26no—
244C24—594no—
245C24—594no—
246C24—592no—
247C24—592no—
248C24—5916no—
249C24—1186no—
250C24—1186no—
251C24—1186no—
252C24—1186no—
253C24—23612no—
254C24—23612no—
255C24—23648no—
256C24—23648no—
257C24—23616no—
258C24Q8×D5914yes—
259C24C23×D5912yes—
260C24C23×D5911yescharacteristic
261C22×C4—26no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—26no—
265C22×C4—26no—
266C22×C4—26no—
267C22×C4—596no—
268C22×C4—596no—
269C22×C4—1186no—
270C22×C4—1186no—
271C22×C4—1186no—
272C22×C4—1186no—
273C22×C4—1186no—
274C22×C4—1186no—
275C22×C4C23×D5916yes—
276C22×C4C23×D5916yes—
277C4⋊C4—412no—
278C4⋊C4—23612no—
279C4⋊C4—23612no—
280C4⋊C4—23612no—
281C22⋊C4—412no—
282C22⋊C4—23612no—
283C22⋊C4—23612no—
284C22⋊C4—23612no—
285C22⋊C4—23612no—
286C22⋊C4—23612no—
287C23—26no—
288C23—26no—
289C23—224no—
290C23—412no—
291C23—44no—
292C23—594no—
293C23—594no—
294C23—1186no—
295C23—1186no—
296C23—1186no—
297C23—1186no—
298C23—11824no—
299C23—11824no—
300C23—23648no—
301C23—23648no—
302C23—23616no—
303C23—236128no—
304C23—23612no—
305C23—23612no—
306C23—2364no—
307C23—2364no—
308C23C2×Q8×D5914yes—
309C23C2×Q8×D5912yes—
310C23#1427377216yes—
311C23#1427377316yes—
312C23C24×D5911yescharacteristic
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—26no—
319C2×C4—412no—
320C2×C4—412no—
321C2×C4—1186no—
322C2×C4—1186no—
323C2×C4—1186no—
324C2×C4—1186no—
325C2×C4—1186no—
326C2×C4—1186no—
327C2×C4—23612no—
328C2×C4—23612no—
329C22—212no—
330C22—46no—
331C22—42no—
332C22—416no—
333C22—11812no—
334C22—11812no—
335C22—23664no—
336C22—2366no—
337C22—2366no—
338C22—2362no—
339C22—2362no—
340C22—23616no—
341C22—23616no—
342C22#16838027112yes—
343C22#16838140112yes—
344C22#1683814314yes—
345C22C22×Q8×D5911yescharacteristic
346C22C2×#1427377213yes—
347C22C2×#1427377313yes—
348C4—46no—
349C4—46no—
350C4—2366no—
351C4—2366no—
352C2—44no—
353C2—2364no—
354C2—2364no—
355C2#2143235918yesdirect product factor
356C2C2×#1683802713yes—
357C2C2×#1683814013yes—
358C2C2×#1683814311yescharacteristic
subgroupG/H
centerC24C23×D59
commutator, cosocleC23×C59C25
FrattiniC23C24×D59
FittingC2×C59×#396C2
socleC24×C59C24
prime radicalC25×D59C22
hypercenterC2×#396D59
hypocenterC59C2×#2842
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×#396

Complements

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

NH ≅ G/Ntype
1. #21432359C2direct (N × H)
2. C2×#16837339C2semidirect
3. C2×#16837337C2semidirect
4. C2×#16837326C2semidirect
5. C2×#16837325C2semidirect
6. C2×#16837324C2semidirect
8. C2×C59×#396C2semidirect
10. C22×C4⋊Dic59C22semidirect
11. C22×Dic59⋊C4C22semidirect
12. C22×Dic59⋊C4C22semidirect
18. #16837339C22semidirect
19. #16837337C22semidirect
20. #16837326C22semidirect
21. #16837325C22semidirect
22. #16837324C22semidirect
24. C2×C59⋊#95C22semidirect
25. C22×C4⋊C4×C59C22semidirect
27. C59×#396C22semidirect
28. C2×#95×C59C22semidirect
38. C2×C4⋊Dic59C23semidirect
41. C2×Dic59⋊C4C23semidirect
42. C2×Dic59⋊C4C23semidirect
53. C59⋊#95C23semidirect
58. C2×C4⋊C4×C59C23semidirect
61. #95×C59C23semidirect
179. C2×#396D59semidirect
182. C2×C59#2842semidirect
192. C22×C4⋊C4C2×D59semidirect
204. #396C2×D59semidirect
207. C2×#95C2×D59semidirect
208. C59C2×#2842semidirect
227. C2×C4⋊C4C22×D59semidirect
241. #95C22×D59semidirect
355. C2#21432359direct (N × H)

Maximal subgroups

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

subgroupindex
#214323592
C2×#168373392
C2×#168373372
C2×#168373262
C2×#168373252
C2×#168373242
C2×#168372572
C2×C59×#3962
C2×#284259