Finite Group Databasebeta

← C2×#20276838 #27538653

Subgroups — C2×#20276838 #27538653

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133163C211yesmaximal, characteristic, complemented
2C2×#16811040C211yesmaximal, characteristic
3C2×#16810898C211yesmaximal, characteristic
4#20276838C218yesmaximal, direct product factor
5C2×#16810452C211yesmaximal, characteristic, complemented
6C2×#16810439C211yesmaximal, characteristic, complemented
7C2×#16810438C211yesmaximal, characteristic, complemented
8C2×C53×#473C211yesmaximal, characteristic
9C2×#13133247—21no—
10C22×Dic212C2211yescharacteristic
11C2×#13133163—21no—
12C2×#13133163—21no—
13C2×#13133163—22no—
14C2×#13133163C2214yescomplemented
15#16811040—22no—
16#16811040C2214yes—
17C2×#13133157—21no—
18C2×#13133157C2211yescharacteristic
19C22×#11719908—21no—
20C22×#11719908C2211yescharacteristic
21#16810898—22no—
22#16810898C2214yes—
23C2×#13133153C2211yescharacteristic
24C2×#13133135—21no—
25#16810452C2214yescomplemented
26#16810439C2214yescomplemented
27#16810438C2214yescomplemented
28C2×#137×C53—21no—
29C22×Q16×C53C2211yescharacteristic
30C2×#131×C53C2211yescharacteristic
31C53×#473—22no—
32C53×#473C2214yes—
33C2×#108×C53C2211yescharacteristic
34C2×#101×C53—21no—
35C22×Dic106—21no—
36#13133247—24no—
37#13133247—42no—
38C2×Dic212—21no—
39C2×Dic212—21no—
40C2×Dic212—21no—
41C2×Dic212—21no—
42C2×Dic212—21no—
43C2×Dic212C2312yes—
44C2×#11719925—41no—
45C2×#11719923—41no—
46C2×#11719917—21no—
47#13133163—24no—
48#13133163—24no—
49#13133157—21no—
50#13133157—24no—
51#13133157C2312yes—
52C2×#11719908—21no—
53C2×#11719908—24no—
54C2×#11719908D411yescharacteristic
55C2×#11719908D411yescharacteristic
56C2×#11719908D412yes—
57C2×#11719908C2312yes—
58#13133153—21no—
59#13133153C2312yes—
60#13133135—24no—
61C2×#41×C53D411yescharacteristic
62C22×Q8×C53—21no—
63#137×C53—24no—
64#137×C53—42no—
65C2×Q16×C53—21no—
66C2×Q16×C53—21no—
67C2×Q16×C53—21no—
68C2×Q16×C53—21no—
69C2×Q16×C53—21no—
70C2×Q16×C53—41no—
71C2×Q16×C53C2312yes—
72C2×SD16×C53—41no—
73#131×C53—21no—
74#131×C53—21no—
75#131×C53D412yes—
76#131×C53C2312yes—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53D411yescharacteristic
80C22×C8×C53—21no—
81C22×C8×C53C2311yescharacteristic
82#108×C53—21no—
83#108×C53C2312yes—
84#101×C53—24no—
85#41×C53—21no—
86#41×C53—21no—
87#41×C53D812yes—
88#41×C53C2×D412yes—
89C2×Q8×C53—22no—
90C2×Q8×C53—41no—
91C2×Q8×C53—41no—
92C2×Q8×C53—41no—
93C2×Q8×C53—41no—
94C2×Q8×C53—41no—
95C2×Q8×C53D811yescharacteristic
96C2×D4×C53—21no—
97C2×D4×C53D811yescharacteristic
98C22×C4×C53—21no—
99C22×C4×C53—41no—
100C22×C4×C53C2×D411yescharacteristic
101C22×Dic53—41no—
102C2×Dic106—22no—
103C2×Dic106—41no—
104C2×Dic106—41no—
105C2×Dic106—41no—
106C2×Dic106—41no—
107C2×Dic106—41no—
108Q16×C53—22no—
109Q16×C53—22no—
110Q16×C53—44no—
111Q16×C53—41no—
112Q16×C53—41no—
113SD16×C53—44no—
114M16×C53—21no—
115M16×C53—22no—
116M16×C53—22no—
117M16×C53—22no—
118M16×C53—41no—
119M16×C53C2×D412yes—
120C2×C8×C53—21no—
121C2×C8×C53—22no—
122C2×C8×C53—22no—
123C2×C8×C53—41no—
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53#4111yescharacteristic
128C2×C8×C53#4111yescharacteristic
129C2×C8×C53C2411yescharacteristic
130#11719925—44no—
131#11719923—44no—
132#11719917—22no—
133#11719917—41no—
134C2×#88322—41no—
135Dic212—22no—
136Dic212—22no—
137Dic212—41no—
138Dic212—41no—
139#11719908—22no—
140#11719908C2×D412yes—
141#11719908C2×D412yes—
142C23×C53—21no—
143C23×C53#13511yescharacteristic
144Q8×C53—21no—
145Q8×C53—42no—
146Q8×C53—42no—
147Q8×C53—81no—
148Q8×C53—81no—
149Q8×C53C2×D812yes—
150D4×C53—22no—
151D4×C53—22no—
152D4×C53—21no—
153D4×C53C2×D812yes—
154C2×C4×C53—21no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—42no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53C2×D811yescharacteristic
163C2×C4×C53C2×D811yescharacteristic
164C2×C4×C53#13511yescharacteristic
165C2×C4×C53C22×D411yescharacteristic
166C2×Dic53—42no—
167C2×Dic53—81no—
168C2×Dic53—81no—
169Dic106—42no—
170Dic106—42no—
171Dic106—81no—
172Dic106—81no—
173C8×C53—22no—
174C8×C53—42no—
175C8×C53#12111yescharacteristic
176C8×C53#12111yescharacteristic
177C8×C53C22×D411yescharacteristic
178C8×C53C2×#4111yescharacteristic
179#88322—42no—
180C2×#2222—531nomaximal, Sylow
181C22×C53—22no—
182C22×C53—22no—
183C22×C53—42no—
184C22×C53#46611yescharacteristic
185C22×C53#46611yescharacteristic
186C22×C53C2×#13511yescharacteristic
187C4×C53—22no—
188C4×C53—82no—
189C4×C53#46611yescharacteristic
190C4×C53#46611yescharacteristic
191C4×C53C2×#12111yescharacteristic
192C4×C53C22×D811yescharacteristic
193Dic53—82no—
194C22×Q32—531no—
195C2×#504—531no—
196C2×#473—531no—
197C2×#473D5311yescharacteristic
198#2222—538no—
199C2×#368—531no—
200C2×#362—531no—
201C2×#362—531no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—42no—
205C2×C53#222212yescomplemented
206C2×C53C2×#46611yescharacteristic
207C2×#137—21no—
208C2×#137—1061no—
209C22×Q16—531no—
210C22×Q16C2×D5311yescharacteristic
211C2×#131C2×D5311yescharacteristic
212C2×Q32—534no—
213C2×Q32—1061no—
214C2×Q32—1061no—
215C2×Q32—1062no—
216#504—534no—
217#504—1062no—
218C2×M32—531no—
219C2×M32—1061no—
220C22×C16—531no—
221C22×C16—1061no—
222#473—22no—
223#473—534no—
224#473—1062no—
225#473C2×D5314yes—
226C2×#108—531no—
227C2×#108C2×D5311yescharacteristic
228C2×#101—21no—
229C2×#101—1061no—
230#368—534no—
231#362—534no—
232#362—534no—
233C53C2×#222211yescharacteristic, complemented, Sylow
234C2×#41C22⋊D5311yescharacteristic
235C22×Q8—21no—
236C22×Q8—1061no—
237#137—24no—
238#137—42no—
239#137—1064no—
240#137—2122no—
241C2×Q16—21no—
242C2×Q16—21no—
243C2×Q16—21no—
244C2×Q16—21no—
245C2×Q16—21no—
246C2×Q16—41no—
247C2×Q16—532no—
248C2×Q16—1061no—
249C2×Q16—1061no—
250C2×Q16—1061no—
251C2×Q16—1061no—
252C2×Q16—1061no—
253C2×Q16—2121no—
254C2×Q16C22×D5312yes—
255C2×SD16—41no—
256C2×SD16—2121no—
257#131—21no—
258#131—21no—
259#131C22⋊D5312yes—
260#131C22×D5312yes—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—1061no—
264C2×M16C22⋊D5311yescharacteristic
265C22×C8—21no—
266C22×C8C22×D5311yescharacteristic
267Q32—1064no—
268Q32—1064no—
269M32—532no—
270M32—1064no—
271M32—1061no—
272C2×C16—531no—
273C2×C16—531no—
274C2×C16—532no—
275C2×C16—532no—
276C2×C16—1064no—
277C2×C16—1061no—
278#108—21no—
279#108—532no—
280#108—1061no—
281#108C22×D5312yes—
282#101—24no—
283#101—1064no—
284#41—21no—
285#41—21no—
286#41#1171992212yes—
287#41C2×C22⋊D5312yes—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—1062no—
295C2×Q8—2121no—
296C2×Q8—2121no—
297C2×Q8—2121no—
298C2×Q8—2121no—
299C2×Q8—2121no—
300C2×Q8#1171992211yescharacteristic
301C2×D4—21no—
302C2×D4#1171992211yescharacteristic
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—2121no—
306C22×C4C2×C22⋊D5311yescharacteristic
307Q16—22no—
308Q16—22no—
309Q16—44no—
310Q16—41no—
311Q16—41no—
312Q16—1062no—
313Q16—1062no—
314Q16—2124no—
315Q16—2121no—
316Q16—2121no—
317SD16—44no—
318SD16—2124no—
319M16—21no—
320M16—22no—
321M16—22no—
322M16—22no—
323M16—41no—
324M16—1062no—
325M16—2121no—
326M16C2×C22⋊D5312yes—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—2121no—
332C2×C8D4×D5311yescharacteristic
333C2×C8D4×D5311yescharacteristic
334C2×C8#1171994711yescharacteristic
335C2×C8#1171994711yescharacteristic
336C2×C8C2×C22⋊D5311yescharacteristic
337C2×C8C23×D5311yescharacteristic
338C16—532no—
339C16—532no—
340C16—1062no—
341C23—21no—
342C23#1313323711yescharacteristic
343Q8—21no—
344Q8—42no—
345Q8—42no—
346Q8—81no—
347Q8—81no—
348Q8—4241no—
349Q8—4241no—
350Q8—2122no—
351Q8—2122no—
352Q8C2×#1171992212yes—
353D4—22no—
354D4—22no—
355D4—21no—
356D4C2×#1171992212yes—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—4241no—
364C2×C4—4241no—
365C2×C4—2122no—
366C2×C4#1313318911yescharacteristic
367C2×C4#1313318911yescharacteristic
368C2×C4C2×#1171992211yescharacteristic
369C2×C4C2×#1171992211yescharacteristic
370C2×C4#1313323711yescharacteristic
371C2×C4C22×C22⋊D5311yescharacteristic
372C8—22no—
373C8—42no—
374C8—2122no—
375C8#1313318911yescharacteristic
376C8#1313318911yescharacteristic
377C8C2×D4×D5311yescharacteristic
378C8C2×#1171994711yescharacteristic
379C22—22no—
380C22—22no—
381C22—42no—
382C22#1681085711yescharacteristic
383C22#1681085711yescharacteristic
384C22C2×#1313323711yescharacteristic
385C4—22no—
386C4—82no—
387C4—4242no—
388C4#1681085711yescharacteristic
389C4#1681085711yescharacteristic
390C4C2×#1313318911yescharacteristic
391C4C22×#1171992211yescharacteristic
392C2—21no—
393C2—21no—
394C2—42no—
395C2#2027683812yesdirect product factor
396C2C2×#1681085711yescharacteristic
subgroupG/H
centerC22#16810857
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#473C2
socleC22×C53#466
prime radicalC2×#41×C53D4
hypercenterC2×#473D53
hypocenterC53C2×#2222
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#473

Complements

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

NH ≅ G/Ntype
1. C22×#13133163C2semidirect
4. #20276838C2direct (N × H)
5. C2×#16810452C2semidirect
6. C2×#16810439C2semidirect
7. C2×#16810438C2semidirect
14. C2×#13133163C22semidirect
25. #16810452C22semidirect
26. #16810439C22semidirect
27. #16810438C22semidirect
205. C2×C53#2222semidirect
233. C53C2×#2222semidirect
395. C2#20276838direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331632
C2×#168110402
C2×#168108982
#202768382
C2×#168104522
C2×#168104392
C2×#168104382
C2×C53×#4732
C2×#222253