Finite Group Databasebeta

← #27301145

Subgroups — #27301145

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811053C211yesmaximal, characteristic
2C2×#16810741C211yesmaximal, characteristic, complemented
3C2×#16810430C211yesmaximal, characteristic, complemented
4C2×#16810429C211yesmaximal, characteristic, complemented
5#20272808C211yesmaximal, characteristic, complemented
6#20272793C211yesmaximal, characteristic, complemented
7C53×#1974C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9C2×#13133223—21no—
10#16811053—22no—
11#16811053—22no—
12C2×#13133163C411yescharacteristic, complemented
13C2×#13133161C411yescharacteristic, complemented
14C2×#13133160C411yescharacteristic, complemented
15C2×#13133158C411yescharacteristic, complemented
16C22×#11719908C2211yescharacteristic
17C22×#11719908C2211yescharacteristic
18#16810741—22no—
19#16810741—22no—
20C2×#13133153C2211yescharacteristic
21C2×#13133145—21no—
22#16810430—22no—
23#16810429—22no—
24C2×#135×C53C2211yescharacteristic
25C2×#108×C53C2211yescharacteristic
26C2×#104×C53—21no—
27C53×#408—21no—
28C2×C4×C8×C53C2211yescharacteristic
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239D412yes—
33#13133239D412yes—
34C2×Dic212C2×C411yescharacteristic
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic
37#13133223—24no—
38#13133223—42no—
39C2×#11719917—21no—
40C2×#11719912—41no—
41C2×C8×D53—41no—
42#13133163—22no—
43#13133161—22no—
44#13133160—22no—
45#13133158—22no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50C2×#11719908—21no—
51C2×#11719908C2×C411yescharacteristic
52C2×#11719908C2×C411yescharacteristic
53#13133153—21no—
54#13133153D412yes—
55#13133145—24no—
56C2×#41×C53—21no—
57#135×C53—21no—
58#135×C53—21no—
59#135×C53—21no—
60#135×C53—21no—
61C2×Q16×C53C2×C411yescharacteristic
62C2×SD16×C53—21no—
63C2×D8×C53C2×C411yescharacteristic
64C2×M16×C53—21no—
65C22×C8×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67C2×C42×C53D411yescharacteristic
68#108×C53—21no—
69#108×C53—21no—
70#104×C53—42no—
71C22⋊C8×C53—41no—
72C22⋊C8×C53—41no—
73C4×C8×C53—21no—
74C4×C8×C53—21no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79C2×Q8×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—41no—
82C22×C4×C53—21no—
83C22×C4×C53—41no—
84C22×C4×C53C2×D411yescharacteristic
85C2×C22⋊D53—41no—
86#11719945—22no—
87#11719945—22no—
88#11719945—41no—
89#11719945—41no—
90C2×D212—21no—
91C2×C4×D53—41no—
92C2×Dic106—21no—
93Q16×C53—21no—
94Q16×C53—21no—
95SD16×C53—41no—
96SD16×C53—41no—
97D8×C53—21no—
98D8×C53—21no—
99M16×C53—41no—
100M16×C53—41no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—41no—
104C2×C8×C53D811yescharacteristic
105C2×C8×C53D811yescharacteristic
106C2×C8×C53C22×C411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C42×C53—21no—
113C42×C53—21no—
114#11719917—22no—
115#11719917—41no—
116C2×#88322—41no—
117Dic212—21no—
118Dic212C22⋊C412yes—
119D424—21no—
120D424C22⋊C412yes—
121#11719913—22no—
122#11719913—22no—
123#11719912—44no—
124C8×D53—44no—
125#11719908—21no—
126#11719908—21no—
127#11719908—41no—
128C23×C53—41no—
129C23×C53#13511yescharacteristic
130C22×D53—41no—
131Q8×C53—41no—
132Q8×C53—41no—
133D4×C53—41no—
134D4×C53—41no—
135D4×C53—81no—
136D4×C53—81no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C22⋊D53—42no—
150C22⋊D53—42no—
151C2×Dic53—41no—
152D212—22no—
153D212—41no—
154C4×D53—42no—
155C4×D53—42no—
156Dic106—22no—
157Dic106—41no—
158C8×C53—21no—
159C8×C53—21no—
160C8×C53—81no—
161C8×C53C2×C22⋊C411yescharacteristic
162C8×C53C4×D411yescharacteristic
163C8×C53#12111yescharacteristic
164C8×C53#12311yescharacteristic
165#88322—42no—
166#13011—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53C4×D811yescharacteristic
173C22×C53#44911yescharacteristic
174C22×C53#46311yescharacteristic
175C2×D53—42no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—42no—
179C4×C53—81no—
180C4×C53#38611yescharacteristic
181C4×C53#44211yescharacteristic
182C4×C53#44711yescharacteristic
183C4×C53#45911yescharacteristic
184Dic53—42no—
185C2×#508—531no—
186C2×#443—531no—
187C2×#359—531no—
188C2×#359—531no—
189#1974D5311yescharacteristic, complemented
190#1381—531no—
191#1374—531no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—161no—
195C2×C53#187911yescharacteristic
196C2×C53#217811yescharacteristic
197C2×C53#217911yescharacteristic
198D53—82no—
199C2×#135—531no—
200C2×#135C2×D5311yescharacteristic
201C2×#131—1061no—
202#508—1062no—
203#508—1062no—
204C2×Q32—531no—
205C2×SD32—531no—
206C2×SD32—531no—
207C2×D16—531no—
208C22×C16—531no—
209C22×C16—531no—
210#443—1062no—
211#443—1062no—
212C2×#108—531no—
213C2×#108C2×D5311yescharacteristic
214C2×#104—21no—
215C2×#104—1061no—
216#408—21no—
217C2×C4×C8C2×D5311yescharacteristic
218#359—1062no—
219#359—1062no—
220C53#1301111yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—1061no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—532no—
228#135—532no—
229#135—1061no—
230#135—1061no—
231C2×Q16—531no—
232C2×Q16C4×D5311yescharacteristic
233C2×SD16—21no—
234C2×SD16—1061no—
235C2×D8—531no—
236C2×D8C4×D5311yescharacteristic
237#131—1064no—
238#131—2122no—
239C2×M16—21no—
240C2×M16—1061no—
241C2×M16—2121no—
242C22×C8—2121no—
243C22×C8C22⋊D5311yescharacteristic
244C22×C8C22×D5311yescharacteristic
245C2×C42C22⋊D5311yescharacteristic
246Q32—1062no—
247SD32—1062no—
248SD32—1062no—
249D16—1062no—
250C2×C16—531no—
251C2×C16—531no—
252C2×C16—1061no—
253C2×C16—1061no—
254C2×C16—1061no—
255C2×C16—1061no—
256C2×C16—1061no—
257#108—21no—
258#108—21no—
259#108—532no—
260#108—1061no—
261#104—42no—
262#104—1064no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C4×C8—21no—
266C4×C8—21no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—1062no—
272#41—1062no—
273#41—2121no—
274#41—2121no—
275C2×Q8—21no—
276C2×Q8—1061no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—1061no—
280C2×D4—2121no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—2121no—
284C22×C4C2×C22⋊D5311yescharacteristic
285Q16—21no—
286Q16—21no—
287Q16—532no—
288Q16—1061no—
289SD16—41no—
290SD16—41no—
291SD16—1062no—
292SD16—1062no—
293D8—21no—
294D8—21no—
295D8—532no—
296D8—1061no—
297M16—41no—
298M16—41no—
299M16—1062no—
300M16—2124no—
301M16—2121no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—2124no—
306C2×C8—2121no—
307C2×C8#1171992211yescharacteristic
308C2×C8#1171992211yescharacteristic
309C2×C8C2×C4×D5311yescharacteristic
310C2×C8#1171994511yescharacteristic
311C2×C8D4×D5311yescharacteristic
312C2×C8D4×D5311yescharacteristic
313C2×C8D4×D5311yescharacteristic
314C2×C8#1171994711yescharacteristic
315C42—21no—
316C42—21no—
317C16—1061no—
318C16—1061no—
319C16—2121no—
320C23—41no—
321C23—2121no—
322C23#1313323711yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—1062no—
326Q8—2121no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331D4—1062no—
332D4—2122no—
333D4—2122no—
334D4—2121no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—2121no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4C4×C22⋊D5311yescharacteristic
345C2×C4#1313318511yescharacteristic
346C2×C4#1313318811yescharacteristic
347C2×C4C2×#1171992211yescharacteristic
348C2×C4#1313324211yescharacteristic
349C2×C4#1313324311yescharacteristic
350C8—21no—
351C8—21no—
352C8—81no—
353C8—2122no—
354C8C22⋊C4×D5311yescharacteristic
355C8#1313317711yescharacteristic
356C8#1313319011yescharacteristic
357C8#1313319911yescharacteristic
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—4241no—
364C22—4241no—
365C22—2122no—
366C22C4×#1171992211yescharacteristic
367C22#1681076611yescharacteristic
368C22#1681083511yescharacteristic
369C4—42no—
370C4—81no—
371C4—2122no—
372C4#1681049911yescharacteristic
373C4#1681073311yescharacteristic
374C4#1681075611yescharacteristic
375C4#1681081511yescharacteristic
376C2—21no—
377C2—21no—
378C2—161no—
379C2—4242no—
380C2#2027506611yescharacteristic
381C2#2027659411yescharacteristic
382C2#2027659911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1974C2
socleC22×C53C4×D8
prime radicalC2×#16811053C2
cosocleC22×C8×C53C23
hypercenter#1974D53
hypocenterC53#13011
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ #1974

Complements

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

NH ≅ G/Ntype
2. C2×#16810741C2semidirect
3. C2×#16810430C2semidirect
4. C2×#16810429C2semidirect
5. #20272808C2semidirect
6. #20272793C2semidirect
7. C53×#1974C2semidirect
12. C2×#13133163C4semidirect
13. C2×#13133161C4semidirect
14. C2×#13133160C4semidirect
15. C2×#13133158C4semidirect
189. #1974D53semidirect
220. C53#13011semidirect

Maximal subgroups

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

subgroupindex
C2×#168110532
C2×#168107412
C2×#168104302
C2×#168104292
#202728082
#202727932
C53×#19742
#1301153