Finite Group Databasebeta

← C2×#21429927 #28755757

Subgroups — C2×#21429927 #28755757

Order 15104.

2614 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273684C211yesmaximal, characteristic
2#21429927C2116yesmaximal, direct product factor
3C2×#16837335C214yesmaximal, complemented
4C2×#16837324C214yesmaximal
5C2×#16837298C212yesmaximal, complemented
6C2×#16837272C211yesmaximal, characteristic, complemented
7C2×#16837259C212yesmaximal
8C22×#125×C59C211yesmaximal, characteristic, complemented
9C23×C4×D59C2211yescharacteristic
10C2×#14273684—28no—
11C2×#14273684C2218yes—
12C22×C4⋊Dic59C2212yes—
13C22×C4⋊Dic59C2214yes—
14C22×Dic59⋊C4C2212yes—
15C22×Dic59⋊C4C2214yes—
16C22×(C2×D59)⋊C4C2212yes—
17C22×(C2×D59)⋊C4C2214yes—
18C22×C4×Dic59C2211yescharacteristic
19#16837335C22132yescomplemented
20#16837324C22132yes—
21#16837298C22116yescomplemented
22#16837272C2218yescomplemented
23#16837259C22116yes—
24C2×C59⋊#95C2214yes—
25C2×C59⋊#95C2214yes—
26C2×#125×C59—28no—
27C2×#125×C59C2218yescomplemented
28C22×C4⋊C4×C59C2212yes—
29C22×C4⋊C4×C59C2214yes—
30C22×C42×C59C2211yescharacteristic
31C24×D59C2311yescharacteristic
32C23×Dic59C2312yes—
33C23×Dic59C2314yes—
34C23×Dic59C2311yescharacteristic
35C22×C4×D59—24no—
36C22×C4×D59—24no—
37C22×C4×D59Q818yes—
38C22×C4×D59C2314yes—
39#14273684—232no—
40C2×C4⋊Dic59—28no—
41C2×C4⋊Dic59—48no—
42C2×C4⋊Dic59C2318yes—
43C2×C4⋊Dic59C23116yes—
44C2×Dic59⋊C4—28no—
45C2×Dic59⋊C4—48no—
46C2×Dic59⋊C4C2318yes—
47C2×Dic59⋊C4C23116yes—
48C2×(C2×D59)⋊C4—28no—
49C2×(C2×D59)⋊C4—48no—
50C2×(C2×D59)⋊C4C2318yes—
51C2×(C2×D59)⋊C4C23116yes—
52C2×C4×Dic59—24no—
53C2×C4×Dic59C2314yes—
54C59⋊#95C23116yes—
55C59⋊#95C23116yes—
56C23×C4×C59C2314yes—
57C23×C4×C59C2312yes—
58C23×C4×C59C2311yescharacteristic
59#125×C59—232no—
60C2×C4⋊C4×C59—216no—
61C2×C4⋊C4×C59—28no—
62C2×C4⋊C4×C59C2318yes—
63C2×C4⋊C4×C59C23116yes—
64C2×C42×C59D418yes—
65C2×C42×C59C2314yes—
66C24×C59C2411yescharacteristic
67C23×D59—22no—
68C23×D59—22no—
69C23×D59—24no—
70C23×D59—24no—
71C23×D59C2×Q814yes—
72C23×D59C2412yes—
73C22×C4×C59—24no—
74C22×C4×C59—24no—
75C22×C4×C59—28no—
76C22×C4×C59—28no—
77C22×C4×C59—28no—
78C22×C4×C59C2×D418yes—
79C22×C4×C59C2×D414yes—
80C22×C4×C59C2×Q814yes—
81C22×C4×C59#4114yes—
82C22×C4×C59C2418yes—
83C22×C4×C59C2414yes—
84C22×C4×C59C2412yes—
85C22×Dic59—22no—
86C22×Dic59—22no—
87C22×Dic59—24no—
88C22×Dic59—28no—
89C22×Dic59—28no—
90C22×Dic59—24no—
91C22×Dic59—28no—
92C22×Dic59—24no—
93C22×Dic59C2×Q814yes—
94C22×Dic59C2414yes—
95C22×Dic59C2418yes—
96C22×Dic59C2412yes—
97C2×C4×D59—232no—
98C2×C4×D59—28no—
99C2×C4×D59—28no—
100C2×C4×D59C2×Q8116yes—
101C4⋊C4×C59—232no—
102C4⋊C4×C59—216no—
103C42×C59C2×D4116yes—
104(C2×D59)⋊C4—216no—
105(C2×D59)⋊C4—416no—
106C4⋊Dic59—216no—
107C4⋊Dic59—416no—
108Dic59⋊C4—216no—
109Dic59⋊C4—416no—
110C4×Dic59—28no—
111C23×C59C22×D412yes—
112C23×C59C22×Q812yes—
113C23×C59C2×#4112yes—
114C23×C59H(5,𝔽2)14yes—
115C23×C592−1+414yes—
116C23×C59C2511yescharacteristic
117C22×D59—28no—
118C22×D59—216no—
119C22×D59—22no—
120C22×D59—22no—
121C22×D59—24no—
122C22×D59—24no—
123C22×D59—48no—
124C22×D59—44no—
125C22×D59—44no—
126C22×D59C22×Q814yes—
127C2×C4×C59—216no—
128C2×C4×C59—24no—
129C2×C4×C59—24no—
130C2×C4×C59—28no—
131C2×C4×C59—28no—
132C2×C4×C59—28no—
133C2×C4×C59—416no—
134C2×C4×C59C22⋊Q8116yes—
135C2×C4×C59C22×D418yes—
136C2×C4×C59C22×D414yes—
137C2×C4×C59C22×Q814yes—
138C2×C4×C59C2×#4114yes—
139C2×Dic59—28no—
140C2×Dic59—22no—
141C2×Dic59—22no—
142C2×Dic59—24no—
143C2×Dic59—28no—
144C2×Dic59—24no—
145C2×Dic59—28no—
146C2×Dic59—28no—
147C2×Dic59—24no—
148C2×Dic59—48no—
149C2×Dic59—416no—
150C2×Dic59C22×Q814yes—
151C4×D59—232no—
152C2×#2629—591nomaximal, Sylow
153C22×C59C2×C22⋊Q814yes—
154C22×C59#53718yes—
155C22×C59#54014yes—
156C22×C59#54114yes—
157C22×C59#54418yes—
158C22×C59C23×D411yescharacteristic
159C22×C59C23×Q811yescharacteristic
160C22×C59C22×#4111yescharacteristic
161C22×C59C2×H(5,𝔽2)12yes—
162C22×C59C2×2−1+412yes—
163C2×D59—24no—
164C2×D59—28no—
165C2×D59—416no—
166C2×D59—44no—
167C2×D59—42no—
168C2×D59—42no—
169C4×C59—28no—
170C4×C59—48no—
171C4×C59C2×C22⋊Q818yes—
172Dic59—24no—
173Dic59—44no—
174Dic59—48no—
175C22×#125D5911yescharacteristic, complemented
176C22×#117—591no—
177#2629—5916no—
178C2×#398—594no—
179C2×#396—594no—
180C2×#389—592no—
181C2×#384—591no—
182C2×#380—592no—
183C2×C59#262918yescomplemented
184C2×C59C22×C22⋊Q811yescharacteristic
185C2×C59C2×#53712yes—
186C2×C59C2×#54011yescharacteristic
187C2×C59C2×#54111yescharacteristic
188C2×C59C2×#54412yes—
189D59—44no—
190C24×C4—591no—
191C2×#125—28no—
192C2×#125C2×D5918yescomplemented
193C2×#117—598no—
194C2×#117—1188no—
195C22×C4⋊C4—592no—
196C22×C4⋊C4—592no—
197C22×C4⋊C4—594no—
198C22×C4⋊C4—594no—
199C22×C4⋊C4C2×D5914yes—
200C22×C4⋊C4C2×D5912yes—
201C22×C22⋊C4—592no—
202C22×C22⋊C4—594no—
203C22×C42—591no—
204C22×C42C2×D5911yescharacteristic
205#398—5932no—
206#396—5932no—
207#389—5916no—
208#384—598no—
209#380—5916no—
210C2×#95—594no—
211C2×#95—594no—
212C59C2×#262911yescharacteristic, complemented, Sylow
213C25—591no—
214C23×C4—591no—
215C23×C4—598no—
216C23×C4—594no—
217C23×C4—594no—
218C23×C4—592no—
219C23×C4—1184no—
220C23×C4—1184no—
221C23×C4C22×D5914yes—
222C23×C4C22×D5912yes—
223C23×C4C22×D5911yescharacteristic
224#125—232no—
225#117—11832no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—598no—
229C2×C4⋊C4—598no—
230C2×C4⋊C4—5916no—
231C2×C4⋊C4—5916no—
232C2×C4⋊C4—1188no—
233C2×C4⋊C4—1188no—
234C2×C4⋊C4—2368no—
235C2×C4⋊C4—2368no—
236C2×C4⋊C4C22×D5918yes—
237C2×C4⋊C4C22×D59116yes—
238C2×C22⋊C4—598no—
239C2×C22⋊C4—5916no—
240C2×C22⋊C4—1188no—
241C2×C22⋊C4—2368no—
242C2×C42—594no—
243C2×C42—1184no—
244C2×C42C22⋊D5918yes—
245C2×C42C22×D5914yes—
246#95—5916no—
247#95—5916no—
248C24—594no—
249C24—592no—
250C24—1182no—
251C24—1182no—
252C24—1184no—
253C24—1184no—
254C24C23×D5911yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—594no—
261C22×C4—592no—
262C22×C4—5916no—
263C22×C4—594no—
264C22×C4—598no—
265C22×C4—1182no—
266C22×C4—1182no—
267C22×C4—11832no—
268C22×C4—1188no—
269C22×C4—1188no—
270C22×C4—1184no—
271C22×C4—1188no—
272C22×C4—1188no—
273C22×C4—1184no—
274C22×C4—1188no—
275C22×C4—1184no—
276C22×C4Q8×D5914yes—
277C22×C4#1174568914yes—
278C22×C4C2×C22⋊D5918yes—
279C22×C4C2×C22⋊D5914yes—
280C22×C4C23×D5918yes—
281C22×C4C23×D5914yes—
282C22×C4C23×D5912yes—
283C4⋊C4—232no—
284C4⋊C4—216no—
285C4⋊C4—11816no—
286C4⋊C4—11816no—
287C4⋊C4—23616no—
288C4⋊C4—23616no—
289C22⋊C4—11816no—
290C22⋊C4—23616no—
291C42—1188no—
292C42C2×C22⋊D59116yes—
293C23—594no—
294C23—1188no—
295C23—1182no—
296C23—1182no—
297C23—1184no—
298C23—1184no—
299C23—11816no—
300C23—2368no—
301C23—2364no—
302C23—2364no—
303C23C22×C22⋊D5912yes—
304C23C2×Q8×D5912yes—
305C23C2×#1174568912yes—
306C23#1427377314yes—
307C23#1427377414yes—
308C23C24×D5911yescharacteristic
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—216no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—416no—
316C2×C4—594no—
317C2×C4—1188no—
318C2×C4—1182no—
319C2×C4—1182no—
320C2×C4—11832no—
321C2×C4—1184no—
322C2×C4—1188no—
323C2×C4—1188no—
324C2×C4—1188no—
325C2×C4—1184no—
326C2×C4—1184no—
327C2×C4—2368no—
328C2×C4—23616no—
329C2×C4#14273703116yes—
330C2×C4C22×C22⋊D5918yes—
331C2×C4C22×C22⋊D5914yes—
332C2×C4C2×Q8×D5914yes—
333C2×C4C2×#1174568914yes—
334C22—1188no—
335C22—1184no—
336C22—2362no—
337C22—2362no—
338C22—23616no—
339C22—2364no—
340C22C2×#1427370314yes—
341C22#1683798018yes—
342C22#1683800814yes—
343C22#1683801514yes—
344C22#1683803218yes—
345C22C23×C22⋊D5911yescharacteristic
346C22C22×Q8×D5911yescharacteristic
347C22C22×#1174568911yescharacteristic
348C22C2×#1427377312yes—
349C22C2×#1427377412yes—
350C4—28no—
351C4—48no—
352C4—1184no—
353C4—2368no—
354C4—2364no—
355C4C2×#1427370318yes—
356C2—2364no—
357C2#2142992718yesdirect product factor
358C2C22×#1427370311yescharacteristic
359C2C2×#1683798012yes—
360C2C2×#1683800811yescharacteristic
361C2C2×#1683801511yescharacteristic
362C2C2×#1683803212yes—
subgroupG/H
centerC24C23×D59
commutator, cosocleC23×C59C25
FrattiniC23C24×D59
FittingC22×#125×C59C2
socleC24×C59C24
prime radicalC24×D59C23
hypercenterC22×#125D59
hypocenterC59C2×#2629
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #21429927C2direct (N × H)
3. C2×#16837335C2semidirect
5. C2×#16837298C2semidirect
6. C2×#16837272C2semidirect
8. C22×#125×C59C2semidirect
19. #16837335C22semidirect
21. #16837298C22semidirect
22. #16837272C22semidirect
27. C2×#125×C59C22semidirect
175. C22×#125D59semidirect
183. C2×C59#2629semidirect
192. C2×#125C2×D59semidirect
212. C59C2×#2629semidirect
357. C2#21429927direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142736842
#214299272
C2×#168373352
C2×#168373242
C2×#168372982
C2×#168372722
C2×#168372592
C22×#125×C592
C2×#262959