Finite Group Databasebeta

← C22×#15689395 #27083662

Subgroups — C22×#15689395 #27083662

Order 12032.

2630 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094055C212yesmaximal
2C22×#13094030C211yesmaximal, characteristic
3C2×#15689395C2124yesmaximal, direct product factor
4C22×#13093960C211yesmaximal, characteristic, complemented
5C22×#13093943C212yesmaximal, complemented
6C23×Q16×C47C211yesmaximal, characteristic, complemented
7C22×#10625350C2211yescharacteristic
8C2×#13094055—212no—
9C2×#13094055C22124yes—
10C22×#10625337—22no—
11C22×#10625336—22no—
12C2×#13094030—26no—
13C2×#13094030C22112yes—
14C22×#10625329C2211yescharacteristic
15C22×#10625329C2212yes—
16C22×#10625324—21no—
17C22×C8×D47—21no—
18#15689395C22164yesdirect product factor
19C2×#13093960C22112yescomplemented
20C2×#13093943C22124yescomplemented
21C23×Q8×C47C2212yes—
22C22×Q16×C47—21no—
23C22×Q16×C47—21no—
24C22×Q16×C47—26no—
25C22×Q16×C47C22112yescomplemented
26C23×C8×C47C2211yescharacteristic
27C2×#10625350—23no—
28C2×#10625350—23no—
29C2×#10625350D416yes—
30C2×#10625350C2316yes—
31C22×Dic94D411yescharacteristic
32C22×D188D411yescharacteristic
33C22×C22⋊D47—21no—
34C22×C4×D47—21no—
35#13094055—248no—
36#13094055C23132yes—
37C2×#10625337—224no—
38C2×#10625336—224no—
39#13094030—224no—
40#13094030C23116yes—
41C2×#10625329—23no—
42C2×#10625329—26no—
43C2×#10625329C2316yes—
44C2×#10625329C23112yes—
45C2×#10625324—212no—
46C22×C47⋊C8—21no—
47C22×C47⋊C8—22no—
48C2×C8×D47—212no—
49#13093960C23116yescomplemented
50#13093943C23132yescomplemented
51C22×Q8×C47—26no—
52C22×Q8×C47—22no—
53C22×Q8×C47—26no—
54C22×Q8×C47—22no—
55C22×Q8×C47—26no—
56C22×Q8×C47C23112yes—
57C23×C4×C47—22no—
58C23×C4×C47C2311yescharacteristic
59C2×Q16×C47—212no—
60C2×Q16×C47—212no—
61C2×Q16×C47—224no—
62C2×Q16×C47C23116yescomplemented
63C22×C8×C47D411yescharacteristic
64C22×C8×C47D411yescharacteristic
65C22×C8×C47D416yes—
66C22×C8×C47C2316yes—
67C24×C47#4111yescharacteristic
68C23×D47—21no—
69C2×Q8×C47—212no—
70C2×Q8×C47—212no—
71C2×Q8×C47—212no—
72C2×Q8×C47—212no—
73C2×Q8×C47—212no—
74C2×Q8×C47—46no—
75C2×Q8×C47—46no—
76C2×Q8×C47—412no—
77C2×Q8×C47C2418yes—
78C22×C4×C47—212no—
79C22×C4×C47—46no—
80C22×C4×C47—42no—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D413yes—
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D413yes—
85C22×C4×C47#4113yes—
86C22×C4×C47C2413yes—
87C2×C22⋊D47—26no—
88C2×C22⋊D47—26no—
89C22×Dic47—21no—
90#10625350—212no—
91#10625350—26no—
92#10625350—26no—
93#10625350C2×D4112yes—
94#10625350C2414yes—
95C2×D188—23no—
96C2×D188C2×D416yes—
97C2×C4×D47—26no—
98C2×C4×D47—26no—
99C2×Dic94—23no—
100C2×Dic94C2×D416yes—
101Q16×C47—216no—
102Q16×C47—216no—
103C2×C8×C47C2×D416yes—
104C2×C8×C47C2×D416yes—
105C2×C8×C47C2×D4112yes—
106C2×C8×C47C2414yes—
107#10625337—232no—
108#10625336—232no—
109#10625329—26no—
110#10625329—212no—
111#10625329C2414yes—
112#10625329C2418yes—
113C2×C47⋊C8—26no—
114C2×C47⋊C8—212no—
115#10625324—216no—
116C8×D47—216no—
117C23×C47—24no—
118C23×C47#12113yes—
119C23×C47#12111yescharacteristic
120C23×C47C2×#4113yes—
121C22×D47—26no—
122C22×D47—44no—
123Q8×C47—28no—
124Q8×C47—28no—
125Q8×C47—412no—
126Q8×C47—412no—
127C2×C4×C47—28no—
128C2×C4×C47—412no—
129C2×C4×C47—412no—
130C2×C4×C47#12113yes—
131C2×C4×C47#12113yes—
132C2×C4×C47#12116yes—
133C2×C4×C47C22×D413yes—
134C2×C4×C47C22×D413yes—
135C2×C4×C47C22×D413yes—
136C2×C4×C47C22×D413yes—
137C2×C4×C47C2×#4113yes—
138C2×C4×C47C2511yescharacteristic
139C22⋊D47—212no—
140C22⋊D47—24no—
141C2×Dic47—26no—
142D188—26no—
143D188C22×D414yes—
144C4×D47—212no—
145C4×D47—24no—
146Dic94—26no—
147Dic94C22×D414yes—
148C8×C47C22×D414yes—
149C8×C47C22×D414yes—
150C47⋊C8—24no—
151C47⋊C8—28no—
152C22×#473—471nomaximal, Sylow
153C22×C47—26no—
154C22×C47—26no—
155C22×C47#47314yescomplemented
156C22×C47C2×#12113yes—
157C22×C47C2×#12113yes—
158C22×C47C22×#4111yescharacteristic
159C2×D47—24no—
160C2×D47—412no—
161C4×C47—48no—
162C4×C47C2×#12113yes—
163C4×C47C2×#12113yes—
164C4×C47C23×D411yescharacteristic
165C4×C47C23×D411yescharacteristic
166Dic47—24no—
167C22×#137—472no—
168C23×Q16D4711yescharacteristic, complemented
169C22×#131—471no—
170C2×#473—4724no—
171C22×#108—471no—
172C22×#101—472no—
173C2×C47—23no—
174C2×C47—21no—
175C2×C47C2×#47316yescomplemented
176C2×C47C22×#12111yescharacteristic
177D47—44no—
178C22×#41—471no—
179C23×Q8C2×D4712yes—
180C2×#137—4724no—
181C2×#137—9412no—
182C22×Q16—21no—
183C22×Q16—21no—
184C22×Q16—26no—
185C22×Q16—942no—
186C22×Q16C2×D47112yescomplemented
187C22×SD16—942no—
188C2×#131—4712no—
189C2×#131—946no—
190C22×M16—471no—
191C22×M16—472no—
192C22×M16—941no—
193C23×C8—941no—
194C23×C8C2×D4711yescharacteristic
195#473—4764no—
196C2×#108—4712no—
197C2×#101—4724no—
198C47C22×#47311yescharacteristic, complemented, Sylow
199C2×#41—476no—
200C2×#41—476no—
201C2×#41—943no—
202C2×#41—943no—
203C22×Q8—26no—
204C22×Q8—22no—
205C22×Q8—26no—
206C22×Q8—26no—
207C22×Q8—22no—
208C22×Q8—471no—
209C22×Q8C22×D47112yes—
210C22×D4—471no—
211C22×D4—941no—
212C23×C4—22no—
213C23×C4—941no—
214C23×C4C22×D4711yescharacteristic
215#137—4732no—
216#137—9448no—
217C2×Q16—212no—
218C2×Q16—212no—
219C2×Q16—224no—
220C2×Q16—9424no—
221C2×Q16C22×D47116yescomplemented
222C2×SD16—9424no—
223#131—4716no—
224#131—9424no—
225C2×M16—476no—
226C2×M16—4712no—
227C2×M16—946no—
228C2×M16—943no—
229C2×M16—9412no—
230C22×C8—942no—
231C22×C8—941no—
232C22×C8—9412no—
233C22×C8C22⋊D4711yescharacteristic
234C22×C8C22⋊D4711yescharacteristic
235C22×C8C22⋊D4716yes—
236C22×C8C22×D4716yes—
237#108—4716no—
238#101—4732no—
239C24—941no—
240C24#1062535211yescharacteristic
241#41—4712no—
242#41—474no—
243#41—9412no—
244#41—946no—
245#41—946no—
246C2×Q8—212no—
247C2×Q8—212no—
248C2×Q8—212no—
249C2×Q8—212no—
250C2×Q8—212no—
251C2×Q8—46no—
252C2×Q8—46no—
253C2×Q8—412no—
254C2×Q8—476no—
255C2×Q8—943no—
256C2×Q8C23×D4718yes—
257C2×D4—476no—
258C2×D4—946no—
259C2×D4—946no—
260C2×D4—943no—
261C22×C4—212no—
262C22×C4—46no—
263C22×C4—42no—
264C22×C4—941no—
265C22×C4—946no—
266C22×C4—946no—
267C22×C4D4×D4713yes—
268C22×C4D4×D4711yescharacteristic
269C22×C4#1062535213yes—
270C22×C4C2×C22⋊D4713yes—
271C22×C4C2×C22⋊D4711yescharacteristic
272C22×C4C23×D4713yes—
273Q16—216no—
274Q16—216no—
275Q16—9432no—
276SD16—9432no—
277M16—474no—
278M16—478no—
279M16—9412no—
280M16—946no—
281M16—9416no—
282C2×C8—9412no—
283C2×C8—946no—
284C2×C8—9416no—
285C2×C8C2×C22⋊D4716yes—
286C2×C8C2×C22⋊D4716yes—
287C2×C8C2×C22⋊D47112yes—
288C2×C8C23×D4714yes—
289C23—24no—
290C23—946no—
291C23—1884no—
292C23#1309399513yes—
293C23#1309399511yescharacteristic
294C23C2×#1062535213yes—
295Q8—28no—
296Q8—28no—
297Q8—412no—
298Q8—412no—
299Q8—474no—
300Q8—946no—
301D4—474no—
302D4—9412no—
303D4—944no—
304D4—946no—
305C2×C4—28no—
306C2×C4—412no—
307C2×C4—412no—
308C2×C4—946no—
309C2×C4—9412no—
310C2×C4—944no—
311C2×C4#1309399513yes—
312C2×C4#1309399513yes—
313C2×C4#1309399516yes—
314C2×C4C2×D4×D4713yes—
315C2×C4C2×D4×D4713yes—
316C2×C4C22×C22⋊D4713yes—
317C2×C4C22×C22⋊D4713yes—
318C2×C4C2×#1062535213yes—
319C2×C4C24×D4711yescharacteristic
320C8—948no—
321C8—944no—
322C8C22×C22⋊D4714yes—
323C8C22×C22⋊D4714yes—
324C22—26no—
325C22—26no—
326C22—944no—
327C22—18812no—
328C22#1568939514yesdirect product factor
329C22C2×#1309399513yes—
330C22C2×#1309399513yes—
331C22C22×#1062535211yescharacteristic
332C4—48no—
333C4—944no—
334C4C2×#1309399513yes—
335C4C2×#1309399513yes—
336C4C22×D4×D4711yescharacteristic
337C4C23×C22⋊D4711yescharacteristic
338C2—23no—
339C2—21no—
340C2—1884no—
341C2C2×#1568939516yesdirect product factor
342C2C22×#1309399511yescharacteristic
subgroupG/H
centerC23#13093995
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC23×Q16×C47C2
socleC23×C47#121
prime radicalC22×#10625350C22
hypercenterC23×Q16D47
hypocenterC47C22×#473
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q16

Complements

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

NH ≅ G/Ntype
3. C2×#15689395C2direct (N × H)
4. C22×#13093960C2semidirect
5. C22×#13093943C2semidirect
6. C23×Q16×C47C2semidirect
18. #15689395C22direct (N × H)
19. C2×#13093960C22semidirect
20. C2×#13093943C22semidirect
25. C22×Q16×C47C22semidirect
49. #13093960C23semidirect
50. #13093943C23semidirect
62. C2×Q16×C47C23semidirect
155. C22×C47#473semidirect
168. C23×Q16D47semidirect
175. C2×C47C2×#473semidirect
186. C22×Q16C2×D47semidirect
198. C47C22×#473semidirect
221. C2×Q16C22×D47semidirect
328. C22#15689395direct (N × H)
341. C2C2×#15689395direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940552
C22×#130940302
C2×#156893952
C22×#130939602
C22×#130939432
C23×Q16×C472
C22×#47347