Finite Group Databasebeta

← C2×#20256365 #27097175

Subgroups — C2×#20256365 #27097175

Order 12032.

2238 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689795C211yesmaximal, characteristic
2#20256365C2116yesmaximal, direct product factor
3C2×#15689497C213yesmaximal, complemented
4C2×#15689323C213yesmaximal, complemented
5C2×#15689325C213yesmaximal, complemented
6C2×Q8×C47⋊C8C211yesmaximal, characteristic, complemented
7C2×C4×#10625336C213yesmaximal, complemented
8C2×C47×#557C211yesmaximal, characteristic, complemented
9C22×#10625354—21no—
10C22×#10625336C2213yes—
11C22×#10625336C2216yes—
12#15689795—24no—
13#15689795C2218yes—
14C2×#13094021C2213yes—
15C2×#13093999—23no—
16C2×C4×D188C2213yes—
17#15689497C22124yescomplemented
18#15689325C22124yescomplemented
19#15689323C22124yescomplemented
20Q8×C47⋊C8C2218yescomplemented
21C4×#10625336C22124yescomplemented
22C2×#13093955C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2213yes—
24C2×#13093948C2213yes—
25C2×#13093944C2213yes—
26C2×C4×C47⋊C8C2213yes—
27C47×#557—24no—
28C47×#557C2218yescomplemented
29C2×C4⋊Q8×C47C2213yes—
30C2×#125×C47—23no—
31C2×C4×Q8×C47—21no—
32C2×C4×Q8×C47C2213yes—
33C2×C4×Q8×C47C2211yescharacteristic
34C2×#10625354—28no—
35C2×#10625354—48no—
36C22×D188—23no—
37C22×D188—23no—
38C22×D188C2313yes—
39C22×C4×D47—23no—
40C2×#10625336—212no—
41C2×#10625336—412no—
42C2×#10625336C23112yes—
43C2×#10625336C23124yes—
44C22×C47⋊C8C2313yes—
45C22×C47⋊C8C2311yescharacteristic
46#13094021—26no—
47#13094021C23112yes—
48#13093999—224no—
49C4×D188—26no—
50C4×D188C23112yes—
51C2×C4⋊Dic47C2311yescharacteristic
52C2×(C2×D47)⋊C4—23no—
53#13093955C2314yes—
54C4⋊(C47⋊C8)C23112yes—
55#13093948C23112yes—
56#13093944C23112yes—
57C4×C47⋊C8C23112yes—
58C22×Q8×C47D411yescharacteristic
59C22×Q8×C47C2313yes—
60C4⋊Q8×C47—26no—
61C4⋊Q8×C47C23112yes—
62#125×C47—224no—
63C4×Q8×C47—28no—
64C4×Q8×C47—26no—
65C4×Q8×C47D414yes—
66C4×Q8×C47C23112yes—
67C4×Q8×C47C2314yes—
68C2×C4⋊C4×C47—23no—
69C2×C4⋊C4×C47—23no—
70C2×C4⋊C4×C47—23no—
71C2×C4⋊C4×C47D413yes—
72C2×C4⋊C4×C47C2311yescharacteristic
73C2×C42×C47—23no—
74C2×C42×C47C2313yes—
75C23×D47—23no—
76C2×Q8×C47—23no—
77C2×Q8×C47—26no—
78C2×Q8×C47—26no—
79C2×Q8×C47SD1618yescomplemented
80C2×Q8×C47C2×D414yes—
81C2×Q8×C47C2416yes—
82C22×C4×C47—21no—
83C22×C4×C47—23no—
84C22×C4×C47C2×D413yes—
85C22×C4×C47C2×D413yes—
86C22×C4×C47C2411yescharacteristic
87C22×Dic47—21no—
88#10625354—432no—
89C2×D188—26no—
90C2×D188—26no—
91C2×D188—23no—
92C2×D188—212no—
93C2×D188—212no—
94C2×D188—412no—
95C2×D188—86no—
96C2×D188C2416yes—
97C2×C4×D47—212no—
98C2×C4×D47—412no—
99C4⋊C4×C47—212no—
100C4⋊C4×C47—212no—
101C4⋊C4×C47—212no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47C2×D4112yes—
104C4⋊C4×C47C2412yes—
105C42×C47—212no—
106C42×C47C2×D416yes—
107C42×C47C2416yes—
108#10625336—224no—
109#10625336—424no—
110(C2×D47)⋊C4—212no—
111C4⋊Dic47—21no—
112C4⋊Dic47C2412yes—
113C2×C47⋊C8—26no—
114C2×C47⋊C8#4114yescomplemented
115C2×C47⋊C8C2416yes—
116C2×C47⋊C8C2412yes—
117C23×C47C22×D411yescharacteristic
118C22×D47—26no—
119C22×D47—46no—
120C22×D47—46no—
121C22×D47—46no—
122Q8×C47—26no—
123Q8×C47—26no—
124Q8×C47—46no—
125Q8×C47C2×SD16116yescomplemented
126C2×C4×C47—26no—
127C2×C4×C47—22no—
128C2×C4×C47—26no—
129C2×C4×C47—46no—
130C2×C4×C47—42no—
131C2×C4×C47C2×SD16112yes—
132C2×C4×C47C22×D416yes—
133C2×C4×C47C22×D411yescharacteristic
134C2×C4×C47C22×D416yes—
135C2×C4×C47C2×#4112yes—
136C2×C4×C47H(5,𝔽2)12yes—
137C2×C4×C47C2511yescharacteristic
138C2×Dic47—22no—
139C2×Dic47—42no—
140D188—26no—
141D188—26no—
142D188—46no—
143D188—424no—
144D188—812no—
145C4×D47—424no—
146C47⋊C8—26no—
147C47⋊C8C2×#4114yescomplemented
148C2×#3392—471nomaximal, Sylow
149C22×C47#55012yes—
150C22×C47C22×SD1612yes—
151C22×C47#57712yes—
152C22×C47C23×D411yescharacteristic
153C2×D47—46no—
154C2×D47—46no—
155C2×D47—46no—
156C2×D47—812no—
157C4×C47—26no—
158C4×C47—46no—
159C4×C47—42no—
160C4×C47#55012yes—
161C4×C47C22×SD16112yes—
162C4×C47C22×#4111yescharacteristic
163C4×C47C2×H(5,𝔽2)11yescharacteristic
164Dic47—42no—
165C2×#557D4711yescharacteristic, complemented
166C2×#550—471no—
167#3392—4716no—
168C2×#492—473no—
169C2×#460—473no—
170C2×#461—473no—
171C2×C8×Q8—471no—
172C2×C4×SD16—473no—
173C2×C47#339214yescomplemented
174C2×C47C2×#55011yescharacteristic
175C2×C47C23×SD1611yescharacteristic
176C2×C47C2×#57711yescharacteristic
177D47—86no—
178C22×#41—941no—
179C22×SD16—473no—
180C22×SD16—476no—
181#557—24no—
182#557C2×D4718yescomplemented
183#550—478no—
184#550—944no—
185C2×C4⋊Q8C2×D4713yes—
186C2×#127—473no—
187C2×#125—23no—
188C2×#121—943no—
189C2×C4×Q8—21no—
190C2×C4×Q8C2×D4713yes—
191C2×C4×Q8C2×D4711yescharacteristic
192C2×C4×D4—473no—
193#492—4724no—
194#461—4724no—
195#460—4724no—
196C8×Q8—478no—
197C4×SD16—4724no—
198C2×#106—471no—
199C2×C4⋊C8—473no—
200C2×#103—473no—
201C2×#102—473no—
202C2×C4×C8—473no—
203C47C2×#339211yescharacteristic, complemented, Sylow
204C2×#41—948no—
205C2×#41—1888no—
206C22×Q8C22⋊D4711yescharacteristic
207C22×Q8C22×D4713yes—
208C22×D4—473no—
209C22×D4—943no—
210C22×D4—943no—
211C23×C4—943no—
212C2×SD16—4712no—
213C2×SD16—4724no—
214C2×SD16—9412no—
215C2×SD16—18812no—
216C22×C8—471no—
217C22×C8—473no—
218C4⋊Q8—26no—
219C4⋊Q8C22×D47112yes—
220#127—4712no—
221#127—946no—
222#125—224no—
223#121—9424no—
224C4×Q8—28no—
225C4×Q8—26no—
226C4×Q8C22⋊D4714yes—
227C4×Q8C22×D47112yes—
228C4×Q8C22×D4714yes—
229C4×D4—4712no—
230C4×D4—946no—
231C2×C4⋊C4—23no—
232C2×C4⋊C4—23no—
233C2×C4⋊C4—23no—
234C2×C4⋊C4—471no—
235C2×C4⋊C4C22⋊D4713yes—
236C2×C4⋊C4C22×D4711yescharacteristic
237C2×C22⋊C4—943no—
238C2×C42—23no—
239C2×C42C22×D4713yes—
240#106—474no—
241C4⋊C8—4712no—
242#103—4712no—
243#102—4712no—
244C4×C8—4712no—
245C24—943no—
246#41—18832no—
247C2×Q8—23no—
248C2×Q8—26no—
249C2×Q8—26no—
250C2×Q8#1062533618yescomplemented
251C2×Q8C2×C22⋊D4714yes—
252C2×Q8C23×D4716yes—
253C2×D4—476no—
254C2×D4—9412no—
255C2×D4—943no—
256C2×D4—9412no—
257C2×D4—946no—
258C2×D4—946no—
259C2×D4—3766no—
260C2×D4—18812no—
261C22×C4—21no—
262C22×C4—23no—
263C22×C4—941no—
264C22×C4—9412no—
265C22×C4—18812no—
266C22×C4C2×C22⋊D4713yes—
267C22×C4C2×C22⋊D4713yes—
268C22×C4C23×D4711yescharacteristic
269SD16—9424no—
270SD16—18824no—
271C2×C8—474no—
272C2×C8—472no—
273C2×C8—476no—
274C2×C8—946no—
275C4⋊C4—212no—
276C4⋊C4—212no—
277C4⋊C4—212no—
278C4⋊C4—21no—
279C4⋊C4—472no—
280C4⋊C4—941no—
281C4⋊C4C2×C22⋊D47112yes—
282C4⋊C4C23×D4712yes—
283C22⋊C4—9412no—
284C42—212no—
285C42C2×C22⋊D4716yes—
286C42C23×D4716yes—
287C23—946no—
288C23—1886no—
289C23—1886no—
290C23—1886no—
291C23C22×C22⋊D4711yescharacteristic
292Q8—26no—
293Q8—26no—
294Q8—46no—
295Q8C2×#10625336116yescomplemented
296D4—946no—
297D4—946no—
298D4—37612no—
299D4—1886no—
300D4—18824no—
301C2×C4—26no—
302C2×C4—22no—
303C2×C4—26no—
304C2×C4—46no—
305C2×C4—42no—
306C2×C4—942no—
307C2×C4—1882no—
308C2×C4—18824no—
309C2×C4C2×#10625336112yes—
310C2×C4C22×C22⋊D4716yes—
311C2×C4C22×C22⋊D4711yescharacteristic
312C2×C4C22×C22⋊D4716yes—
313C2×C4#41×D4712yes—
314C2×C4#1309407012yes—
315C2×C4C24×D4711yescharacteristic
316C8—474no—
317C8—946no—
318C22—37612no—
319C22—1886no—
320C22—1886no—
321C22—1886no—
322C22#1568979212yes—
323C22C22×#1062533612yes—
324C22#1568993112yes—
325C22C23×C22⋊D4711yescharacteristic
326C4—26no—
327C4—42no—
328C4—46no—
329C4—1882no—
330C4#1568979212yes—
331C4C22×#10625336112yes—
332C4C2×#41×D4711yescharacteristic
333C4C2×#1309407011yescharacteristic
334C2—3766no—
335C2#2025636514yesdirect product factor
336C2C2×#1568979211yescharacteristic
337C2C23×#1062533611yescharacteristic
338C2C2×#1568993111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#557C2
socleC23×C47C22×D4
prime radicalC2×#15689795C2
hypercenterC2×#557D47
hypocenterC47C2×#3392
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #20256365C2direct (N × H)
3. C2×#15689497C2semidirect
4. C2×#15689323C2semidirect
5. C2×#15689325C2semidirect
6. C2×Q8×C47⋊C8C2semidirect
7. C2×C4×#10625336C2semidirect
8. C2×C47×#557C2semidirect
17. #15689497C22semidirect
18. #15689325C22semidirect
19. #15689323C22semidirect
20. Q8×C47⋊C8C22semidirect
21. C4×#10625336C22semidirect
28. C47×#557C22semidirect
79. C2×Q8×C47SD16semidirect
114. C2×C47⋊C8#41semidirect
125. Q8×C47C2×SD16semidirect
147. C47⋊C8C2×#41semidirect
165. C2×#557D47semidirect
173. C2×C47#3392semidirect
182. #557C2×D47semidirect
203. C47C2×#3392semidirect
250. C2×Q8#10625336semidirect
295. Q8C2×#10625336semidirect
335. C2#20256365direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897952
#202563652
C2×#156894972
C2×#156893232
C2×#156893252
C2×Q8×C47⋊C82
C2×C4×#106253362
C2×C47×#5572
C2×#339247