Finite Group Databasebeta

← #27301518

Subgroups — #27301518

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811040C211yesmaximal, characteristic, complemented
2#20284435C211yesmaximal, characteristic
3#20276450C211yesmaximal, characteristic, complemented
4C2×#16810429C211yesmaximal, characteristic, complemented
5#20272826C211yesmaximal, characteristic, complemented
6#20272810C211yesmaximal, characteristic, complemented
7C53×#1814C211yesmaximal, characteristic, complemented
8#16811421—21no—
9C2×#13133239C2211yescharacteristic
10#16811040—22no—
11#16811040—22no—
12C2×#13133157—21no—
13C2×#13133157C2211yescharacteristic
14C22×#11719908—21no—
15C22×#11719908C2211yescharacteristic
16#16810898C412yescomplemented
17#16810893C412yescomplemented
18#16810886—22no—
19C2×#13133153C2211yescharacteristic
20#16810605—21no—
21#16810432—22no—
22#16810429—21no—
23#16810429—21no—
24C2×#131×C53C2211yescharacteristic
25C2×#108×C53C2211yescharacteristic
26C53×#427C2211yescharacteristic
27C53×#421—21no—
28C53×#344—21no—
29C2×#11719945—21no—
30#13133248—41no—
31#13133247—22no—
32#13133243—41no—
33#13133242—22no—
34#13133239—21no—
35#13133239—21no—
36#13133239—21no—
37#13133239—21no—
38#13133237—42no—
39C2×Dic212C2×C411yescharacteristic
40C2×#11719913—21no—
41C2×D424C2×C411yescharacteristic
42C2×#11719917—21no—
43#13133157—24no—
44#13133157—21no—
45#13133157D412yes—
46C2×#11719908—24no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50C2×#11719908D411yescharacteristic
51C2×#11719908D411yescharacteristic
52#13133153—21no—
53#13133153C2×C412yes—
54#13133135—22no—
55#13133133—22no—
56C2×#41×C53D411yescharacteristic
57#131×C53—21no—
58#131×C53—21no—
59#131×C53C2×C412yes—
60#131×C53D412yes—
61C2×M16×C53—21no—
62C2×M16×C53—21no—
63C2×M16×C53D411yescharacteristic
64C22×C8×C53—21no—
65C22×C8×C53C2311yescharacteristic
66#117×C53—21no—
67#108×C53—21no—
68#108×C53—21no—
69#107×C53—21no—
70#106×C53—21no—
71#104×C53—42no—
72#41×C53—21no—
73#41×C53—21no—
74#41×C53C22⋊C412yes—
75#41×C53D812yes—
76C2×Q8×C53SD1611yescharacteristic
77C2×D4×C53—21no—
78C2×D4×C53SD1611yescharacteristic
79C22×C4×C53—21no—
80C22×C4×C53C2×D411yescharacteristic
81C2×C22⋊D53—41no—
82#11719945—41no—
83#11719945—41no—
84#11719945—41no—
85#11719945—41no—
86C2×D212—21no—
87C2×C4×D53—41no—
88C2×Dic106—21no—
89M16×C53—21no—
90M16×C53—22no—
91M16×C53—22no—
92M16×C53—41no—
93M16×C53—41no—
94M16×C53C22⋊C412yes—
95C2×C8×C53—21no—
96C2×C8×C53—22no—
97C2×C8×C53—22no—
98C2×C8×C53—41no—
99C2×C8×C53C22×C411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53#4111yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C4⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53—41no—
108#11719925—42no—
109#11719924—42no—
110#11719923—42no—
111#11719922—42no—
112#11719917—22no—
113#11719917—41no—
114C2×#88322—41no—
115Dic212—21no—
116Dic212—21no—
117D424—21no—
118D424—21no—
119#11719913—41no—
120#11719913—41no—
121#11719908—21no—
122#11719908—22no—
123#11719908—21no—
124C23×C53—21no—
125C23×C53#13511yescharacteristic
126C22×D53—41no—
127Q8×C53—21no—
128Q8×C53#10212yes—
129D4×C53—22no—
130D4×C53—22no—
131D4×C53—21no—
132D4×C53#10212yes—
133C2×C4×C53—21no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—81no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53C2×D811yescharacteristic
141C2×C4×C53C2×SD1611yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C22⋊D53—81no—
144C22⋊D53—81no—
145C2×Dic53—41no—
146D212—41no—
147D212—41no—
148C4×D53—81no—
149C4×D53—81no—
150Dic106—41no—
151Dic106—41no—
152C8×C53—22no—
153C8×C53—81no—
154C8×C53C4×D411yescharacteristic
155C8×C53C4×D411yescharacteristic
156C8×C53#12011yescharacteristic
157C8×C53#12311yescharacteristic
158#88322—42no—
159#13071—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—41no—
162C22×C53—42no—
163C22×C53#41611yescharacteristic
164C22×C53#46511yescharacteristic
165C22×C53#46611yescharacteristic
166C2×D53—81no—
167C2×D53—161no—
168C4×C53—22no—
169C4×C53—161no—
170C4×C53#38611yescharacteristic
171C4×C53C2×#10211yescharacteristic
172C4×C53#46511yescharacteristic
173C4×C53#46611yescharacteristic
174Dic53—81no—
175C2×#504—531no—
176#3067—531no—
177#2145—531no—
178C2×#359—531no—
179#1814D5311yescharacteristic, complemented
180#1388—531no—
181#1382—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—42no—
186C2×C53#189311yescharacteristic
187D53—161no—
188#575—1061no—
189C2×#135—531no—
190C2×#131C2×D5311yescharacteristic
191#504—1062no—
192#504—1062no—
193C2×M32—531no—
194C2×M32—1061no—
195C22×C16—531no—
196C22×C16—1061no—
197#473—532no—
198#472—532no—
199#471—1062no—
200C2×#108—531no—
201C2×#108C2×D5311yescharacteristic
202#427C2×D5311yescharacteristic
203#421—21no—
204#413—1061no—
205#360—1062no—
206#359—1061no—
207#359—1061no—
208#344—21no—
209C53#1307111yescharacteristic, complemented, Sylow
210C2×#41—1061no—
211C2×#41C22⋊D5311yescharacteristic
212#137—1062no—
213#137—2121no—
214AGL(1,ℤ/8)—1062no—
215AGL(1,ℤ/8)—2121no—
216#135—1061no—
217#135—1061no—
218#135—1061no—
219#135—1061no—
220#135—2122no—
221C2×Q16—531no—
222C2×SD16—1061no—
223C2×D8—531no—
224#131—21no—
225#131—21no—
226#131C4×D5312yes—
227#131D21212yes—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—1061no—
231C2×M16C22⋊D5311yescharacteristic
232C22×C8—21no—
233C22×C8C22×D5311yescharacteristic
234#117—21no—
235M32—532no—
236M32—1064no—
237M32—1061no—
238C2×C16—531no—
239C2×C16—531no—
240C2×C16—1064no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16—1061no—
244#108—21no—
245#108—21no—
246#108—532no—
247#108—1061no—
248#107—21no—
249#106—21no—
250#104—42no—
251#101—1062no—
252#100—1062no—
253#41—21no—
254#41—21no—
255#41—2121no—
256#41—2121no—
257#41—2121no—
258#41—2121no—
259#41#1171992112yes—
260#41#1171992212yes—
261C2×Q8—1061no—
262C2×Q8#1171992411yescharacteristic
263C2×D4—21no—
264C2×D4—1061no—
265C2×D4—2121no—
266C2×D4#1171992411yescharacteristic
267C22×C4—21no—
268C22×C4—2121no—
269C22×C4C2×C22⋊D5311yescharacteristic
270Q16—1061no—
271Q16—1061no—
272Q16—2122no—
273SD16—2122no—
274SD16—2122no—
275SD16—2121no—
276SD16—2121no—
277D8—1061no—
278D8—1061no—
279D8—2122no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—41no—
284M16—41no—
285M16—1062no—
286M16—2121no—
287M16#1171992112yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—2121no—
293C2×C8C2×C4×D5311yescharacteristic
294C2×C8C2×D21211yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8#1171994711yescharacteristic
299C4⋊C4—41no—
300C22⋊C4—41no—
301C42—41no—
302C16—1061no—
303C16—1061no—
304C16—1062no—
305C23—21no—
306C23—2121no—
307C23#1313323711yescharacteristic
308Q8—21no—
309Q8—2121no—
310Q8—2121no—
311Q8#1313313712yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—4241no—
316D4—4241no—
317D4—2121no—
318D4—2121no—
319D4#1313313712yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—81no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—2121no—
327C2×C4C2×#1171992111yescharacteristic
328C2×C4#1313318911yescharacteristic
329C2×C4#1313318911yescharacteristic
330C2×C4C2×#1171992211yescharacteristic
331C2×C4C2×#1171992411yescharacteristic
332C2×C4#1313323711yescharacteristic
333C8—22no—
334C8—81no—
335C8—2122no—
336C8#1313317711yescharacteristic
337C8#1313317711yescharacteristic
338C8#1313318711yescharacteristic
339C8#1313320111yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—8481no—
344C22—4241no—
345C22#1681061711yescharacteristic
346C22#1681085411yescharacteristic
347C22#1681085711yescharacteristic
348C4—22no—
349C4—161no—
350C4—4241no—
351C4#1681050011yescharacteristic
352C4C2×#1313313711yescharacteristic
353C4#1681085411yescharacteristic
354C4#1681085711yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—8481no—
360C2#2027514111yescharacteristic
subgroupG/H
centerC4#16810857
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1814C2
socleC2×C53#1893
prime radical#20284435C2
cosocleC22×C8×C53C23
hypercenter#1814D53
hypocenterC53#13071
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #1814

Complements

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

NH ≅ G/Ntype
1. C2×#16811040C2semidirect
3. #20276450C2semidirect
4. C2×#16810429C2semidirect
5. #20272826C2semidirect
6. #20272810C2semidirect
7. C53×#1814C2semidirect
16. #16810898C4semidirect
17. #16810893C4semidirect
179. #1814D53semidirect
209. C53#13071semidirect

Maximal subgroups

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

subgroupindex
C2×#168110402
#202844352
#202764502
C2×#168104292
#202728262
#202728102
C53×#18142
#1307153