Finite Group Databasebeta

← C22×#16810899 #28251480

Subgroups — C22×#16810899 #28251480

Order 13568.

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×#13133250C212yesmaximal
2C23×Dic212C211yesmaximal, characteristic, complemented
3C22×#13133223C211yesmaximal, characteristic
4C2×#16810899C2124yesmaximal, direct product factor
5C22×#13133135C212yesmaximal, complemented
6C22×#108×C53C211yesmaximal, characteristic, complemented
7C22×#11719945C2211yescharacteristic
8C23×Dic106C2212yes—
9C2×#13133250—212no—
10C2×#13133250C22124yes—
11C22×Dic212—22no—
12C22×Dic212—21no—
13C22×Dic212—21no—
14C22×Dic212—26no—
15C22×Dic212C22112yescomplemented
16C22×#11719913—22no—
17C2×#13133223—26no—
18C2×#13133223C22112yes—
19C22×#11719917C2211yescharacteristic
20C22×#11719912—21no—
21C22×C8×D53—21no—
22#16810899C22164yesdirect product factor
23C2×#13133135C22124yescomplemented
24C22×M16×C53C2212yes—
25C23×C8×C53C2211yescharacteristic
26C2×#108×C53C22112yescomplemented
27C2×#11719945—23no—
28C2×#11719945—23no—
29C2×#11719945D416yes—
30C2×#11719945C2316yes—
31C22×Dic106—22no—
32C22×Dic106—22no—
33C22×Dic106—26no—
34C22×Dic106—26no—
35C22×Dic106—26no—
36C22×Dic106D411yescharacteristic
37C22×Dic106C23112yes—
38C22×D212D411yescharacteristic
39C22×C22⋊D53—21no—
40C23×Dic53—22no—
41C22×C4×D53—21no—
42#13133250—248no—
43#13133250C23132yes—
44C2×Dic212—212no—
45C2×Dic212—212no—
46C2×Dic212—224no—
47C2×Dic212—224no—
48C2×Dic212C23116yescomplemented
49C2×#11719913—224no—
50#13133223—224no—
51#13133223C23116yes—
52C2×#11719917—23no—
53C2×#11719917C2316yes—
54C2×#11719912—212no—
55C22×#88322—21no—
56C2×C8×D53—212no—
57#13133135C23132yescomplemented
58C23×C4×C53C2311yescharacteristic
59C2×M16×C53—26no—
60C2×M16×C53C23112yes—
61C22×C8×C53—22no—
62C22×C8×C53D411yescharacteristic
63C22×C8×C53D411yescharacteristic
64C22×C8×C53D416yes—
65C22×C8×C53C2316yes—
66#108×C53C23116yescomplemented
67C24×C53#4111yescharacteristic
68C23×D53—21no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D413yes—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D413yes—
73C22×C4×C53#4113yes—
74C22×C4×C53C2413yes—
75C2×C22⋊D53—26no—
76C2×C22⋊D53—26no—
77C22×Dic53—21no—
78C22×Dic53—212no—
79C22×Dic53—46no—
80C22×Dic53—42no—
81#11719945—212no—
82#11719945—26no—
83#11719945—26no—
84#11719945C2×D4112yes—
85#11719945C2414yes—
86C2×D212—23no—
87C2×D212C2×D416yes—
88C2×C4×D53—26no—
89C2×C4×D53—26no—
90C2×Dic106—23no—
91C2×Dic106—212no—
92C2×Dic106—212no—
93C2×Dic106—212no—
94C2×Dic106—212no—
95C2×Dic106—212no—
96C2×Dic106—46no—
97C2×Dic106—46no—
98C2×Dic106—412no—
99C2×Dic106C2×D416yes—
100C2×Dic106C2418yes—
101M16×C53—212no—
102M16×C53C2418yes—
103C2×C8×C53—212no—
104C2×C8×C53C2×D416yes—
105C2×C8×C53C2×D416yes—
106C2×C8×C53C2×D4112yes—
107C2×C8×C53C2414yes—
108#11719917—26no—
109#11719917C2414yes—
110C2×#88322—26no—
111Dic212—232no—
112Dic212—216no—
113Dic212—216no—
114#11719913—232no—
115#11719912—216no—
116C8×D53—216no—
117C23×C53—24no—
118C23×C53#12113yes—
119C23×C53#12111yescharacteristic
120C23×C53C2×#4113yes—
121C22×D53—26no—
122C22×D53—44no—
123C2×C4×C53#12113yes—
124C2×C4×C53#12113yes—
125C2×C4×C53#12116yes—
126C2×C4×C53C22×D413yes—
127C2×C4×C53C22×D413yes—
128C2×C4×C53C22×D413yes—
129C2×C4×C53C22×D413yes—
130C2×C4×C53C2×#4113yes—
131C2×C4×C53C2511yescharacteristic
132C22⋊D53—212no—
133C22⋊D53—24no—
134C2×Dic53—26no—
135C2×Dic53—28no—
136C2×Dic53—412no—
137C2×Dic53—412no—
138D212—26no—
139D212C22×D414yes—
140C4×D53—212no—
141C4×D53—24no—
142Dic106—26no—
143Dic106—28no—
144Dic106—28no—
145Dic106—412no—
146Dic106—412no—
147Dic106C22×D414yes—
148C8×C53—28no—
149C8×C53C22×D414yes—
150C8×C53C22×D414yes—
151#88322—24no—
152C22×#473—531nomaximal, Sylow
153C22×C53—26no—
154C22×C53—26no—
155C22×C53#47314yescomplemented
156C22×C53C2×#12113yes—
157C22×C53C2×#12113yes—
158C22×C53C22×#4111yescharacteristic
159C2×D53—24no—
160C2×D53—412no—
161C4×C53C2×#12113yes—
162C4×C53C2×#12113yes—
163C4×C53C23×D411yescharacteristic
164C4×C53C23×D411yescharacteristic
165Dic53—24no—
166Dic53—48no—
167C22×#137—532no—
168C23×Q16—531no—
169C22×#131—531no—
170C2×#473—5324no—
171C22×#108D5311yescharacteristic, complemented
172C22×#101—532no—
173C2×C53—23no—
174C2×C53—21no—
175C2×C53C2×#47316yescomplemented
176C2×C53C22×#12111yescharacteristic
177D53—44no—
178C22×#41—531no—
179C23×Q8—532no—
180C2×#137—5324no—
181C2×#137—10612no—
182C22×Q16—5312no—
183C22×Q16—1062no—
184C22×Q16—1061no—
185C22×Q16—1061no—
186C22×Q16—1066no—
187C22×SD16—1062no—
188C2×#131—5312no—
189C2×#131—1066no—
190C22×M16—531no—
191C22×M16—1061no—
192C22×M16C2×D5312yes—
193C23×C8—1061no—
194C23×C8C2×D5311yescharacteristic
195#473—5364no—
196C2×#108C2×D53112yescomplemented
197C2×#101—5324no—
198C53C22×#47311yescharacteristic, complemented, Sylow
199C2×#41—536no—
200C2×#41—536no—
201C2×#41—1063no—
202C2×#41—1063no—
203C22×Q8—531no—
204C22×Q8—5312no—
205C22×Q8—1066no—
206C22×Q8—1066no—
207C22×Q8—1062no—
208C22×Q8—1062no—
209C22×Q8—1066no—
210C22×D4—531no—
211C22×D4—1061no—
212C23×C4—1061no—
213C23×C4—1062no—
214C23×C4C22×D5311yescharacteristic
215#137—5332no—
216#137—10648no—
217C2×Q16—5316no—
218C2×Q16—10624no—
219C2×Q16—10612no—
220C2×Q16—10612no—
221C2×Q16—10624no—
222C2×SD16—10624no—
223#131—5316no—
224#131—10624no—
225C2×M16—26no—
226C2×M16—536no—
227C2×M16—1063no—
228C2×M16—10612no—
229C2×M16C22×D53112yes—
230C22×C8—22no—
231C22×C8—1061no—
232C22×C8—10612no—
233C22×C8D21211yescharacteristic
234C22×C8D21211yescharacteristic
235C22×C8D21216yes—
236C22×C8C22×D5316yes—
237#108C22×D53116yescomplemented
238#101—5332no—
239C24—1061no—
240C24#1171994911yescharacteristic
241#41—5312no—
242#41—534no—
243#41—10612no—
244#41—1066no—
245#41—1066no—
246C2×Q8—536no—
247C2×Q8—538no—
248C2×Q8—1063no—
249C2×Q8—10612no—
250C2×Q8—10612no—
251C2×Q8—10612no—
252C2×Q8—10612no—
253C2×Q8—10612no—
254C2×Q8—2126no—
255C2×Q8—2126no—
256C2×Q8—21212no—
257C2×D4—536no—
258C2×D4—1066no—
259C2×D4—1066no—
260C2×D4—1063no—
261C22×C4—1061no—
262C22×C4—1066no—
263C22×C4—1066no—
264C22×C4—10612no—
265C22×C4—2126no—
266C22×C4—2122no—
267C22×C4C2×D21213yes—
268C22×C4C2×D21211yescharacteristic
269C22×C4D4×D5313yes—
270C22×C4D4×D5311yescharacteristic
271C22×C4#1171994913yes—
272C22×C4C23×D5313yes—
273Q16—10632no—
274Q16—10616no—
275Q16—10616no—
276SD16—10632no—
277M16—212no—
278M16—534no—
279M16—1066no—
280M16—10616no—
281M16C23×D5318yes—
282C2×C8—212no—
283C2×C8—1066no—
284C2×C8—10616no—
285C2×C8C2×D21216yes—
286C2×C8C2×D21216yes—
287C2×C8C2×D212112yes—
288C2×C8C23×D5314yes—
289C23—24no—
290C23—1066no—
291C23—2124no—
292C23#1313319213yes—
293C23#1313319211yescharacteristic
294C23C2×#1171994913yes—
295Q8—534no—
296Q8—1066no—
297Q8—1068no—
298Q8—1068no—
299Q8—21212no—
300Q8—21212no—
301D4—534no—
302D4—10612no—
303D4—1064no—
304D4—1066no—
305C2×C4—1066no—
306C2×C4—10612no—
307C2×C4—1064no—
308C2×C4—1068no—
309C2×C4—21212no—
310C2×C4—21212no—
311C2×C4#1313319213yes—
312C2×C4#1313319213yes—
313C2×C4#1313319216yes—
314C2×C4C2×D4×D5313yes—
315C2×C4C2×D4×D5313yes—
316C2×C4C22×D21213yes—
317C2×C4C22×D21213yes—
318C2×C4C2×#1171994913yes—
319C2×C4C24×D5311yescharacteristic
320C8—28no—
321C8—1064no—
322C8C22×D21214yes—
323C8C22×D21214yes—
324C22—26no—
325C22—26no—
326C22—1064no—
327C22—21212no—
328C22#1681089914yesdirect product factor
329C22C2×#1313319213yes—
330C22C2×#1313319213yes—
331C22C22×#1171994911yescharacteristic
332C4—1064no—
333C4—2128no—
334C4C2×#1313319213yes—
335C4C2×#1313319213yes—
336C4C22×D4×D5311yescharacteristic
337C4C23×D21211yescharacteristic
338C2—23no—
339C2—21no—
340C2—2124no—
341C2C2×#1681089916yesdirect product factor
342C2C22×#1313319211yescharacteristic
subgroupG/H
centerC23#13133192
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC22×#108×C53C2
socleC23×C53#121
prime radicalC22×#11719945C22
hypercenterC22×#108D53
hypocenterC53C22×#473
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
2. C23×Dic212C2semidirect
4. C2×#16810899C2direct (N × H)
5. C22×#13133135C2semidirect
6. C22×#108×C53C2semidirect
15. C22×Dic212C22semidirect
22. #16810899C22direct (N × H)
23. C2×#13133135C22semidirect
26. C2×#108×C53C22semidirect
48. C2×Dic212C23semidirect
57. #13133135C23semidirect
66. #108×C53C23semidirect
155. C22×C53#473semidirect
171. C22×#108D53semidirect
175. C2×C53C2×#473semidirect
196. C2×#108C2×D53semidirect
198. C53C22×#473semidirect
237. #108C22×D53semidirect
328. C22#16810899direct (N × H)
341. C2C2×#16810899direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332502
C23×Dic2122
C22×#131332232
C2×#168108992
C22×#131331352
C22×#108×C532
C22×#47353