Finite Group Databasebeta

← #27301532

Subgroups — #27301532

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811040C211yesmaximal, characteristic, complemented
2C2×#16810893C211yesmaximal, characteristic
3C2×#16810435C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic, complemented
5#20272823C211yesmaximal, characteristic, complemented
6#20272808C211yesmaximal, characteristic, complemented
7C53×#1816C211yesmaximal, characteristic, complemented
8C2×#13133242—21no—
9C22×D424C2211yescharacteristic
10#16811040—22no—
11#16811040—22no—
12C2×#13133157—21no—
13C2×#13133157C2211yescharacteristic
14C22×#11719908—21no—
15C22×#11719908C2211yescharacteristic
16#16810893—22no—
17#16810893C414yescomplemented
18C2×#13133153C2211yescharacteristic
19C2×#13133133—21no—
20#16810435—22no—
21#16810423—21no—
22#16810423—21no—
23C2×#131×C53C2211yescharacteristic
24C2×#108×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C53×#417—21no—
27C53×#343—21no—
28C22×D212—21no—
29#13133242—24no—
30#13133242—42no—
31C2×#11719924—41no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424—21no—
35C2×D424—21no—
36C2×D424—21no—
37C2×D424C2×C412yes—
38C2×#11719922—41no—
39C2×#11719917—21no—
40#13133157—24no—
41#13133157—21no—
42#13133157D412yes—
43C2×#11719908—24no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47C2×#11719908D411yescharacteristic
48C2×#11719908D411yescharacteristic
49#13133153—21no—
50#13133153C2×C412yes—
51#13133133—24no—
52C2×#41×C53D411yescharacteristic
53#131×C53—21no—
54#131×C53—21no—
55#131×C53C2×C412yes—
56#131×C53D412yes—
57C2×M16×C53—21no—
58C2×M16×C53—21no—
59C2×M16×C53D411yescharacteristic
60C22×C8×C53—21no—
61C22×C8×C53C2311yescharacteristic
62C2×C4⋊C4×C53—21no—
63#108×C53—21no—
64#108×C53—21no—
65#107×C53—21no—
66#107×C53—21no—
67#103×C53—41no—
68#102×C53—41no—
69C23×D53—41no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53C22⋊C412yes—
73#41×C53SD1612yes—
74C2×Q8×C53D811yescharacteristic
75C2×D4×C53—21no—
76C2×D4×C53D811yescharacteristic
77C22×C4×C53—21no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C2×D212—22no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86M16×C53—21no—
87M16×C53—22no—
88M16×C53—22no—
89M16×C53—41no—
90M16×C53—41no—
91M16×C53C22⋊C412yes—
92C2×C8×C53—21no—
93C2×C8×C53—22no—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53C22×C411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104#11719924—44no—
105#11719922—44no—
106#11719917—22no—
107#11719917—41no—
108C2×#88322—41no—
109D424—22no—
110D424—22no—
111D424—41no—
112D424—41no—
113#11719908—21no—
114#11719908—22no—
115#11719908—21no—
116C23×C53—21no—
117C23×C53#13511yescharacteristic
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—161no—
122C22×D53—161no—
123Q8×C53—21no—
124Q8×C53#10212yes—
125D4×C53—22no—
126D4×C53—22no—
127D4×C53—21no—
128D4×C53#10212yes—
129C2×C4×C53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—81no—
133C2×C4×C53—161no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53C2×D811yescharacteristic
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53#13511yescharacteristic
140D212—42no—
141D212—42no—
142D212—81no—
143D212—81no—
144C8×C53—22no—
145C8×C53—81no—
146C8×C53C4×D411yescharacteristic
147C8×C53C4×D411yescharacteristic
148C8×C53#12011yescharacteristic
149C8×C53#12311yescharacteristic
150#88322—42no—
151#13073—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53#41611yescharacteristic
158C22×C53#46511yescharacteristic
159C22×C53#46611yescharacteristic
160C2×D53—82no—
161C2×D53—162no—
162C2×D53—161no—
163C2×D53—161no—
164C2×D53—162no—
165C4×C53—22no—
166C4×C53—161no—
167C4×C53#38611yescharacteristic
168C4×C53C2×#10211yescharacteristic
169C4×C53#46511yescharacteristic
170C4×C53#46611yescharacteristic
171C2×#504—531no—
172C2×#472—531no—
173C2×#361—531no—
174C2×#357—531no—
175#1816D5311yescharacteristic, complemented
176#1387—531no—
177#1381—531no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53—42no—
181C2×C53#189311yescharacteristic
182C2×C53#222111yescharacteristic
183C2×C53#222311yescharacteristic
184D53—162no—
185C2×AGL(1,ℤ/8)—1061no—
186C22×D8—531no—
187C2×#131C2×D5311yescharacteristic
188#504—1062no—
189#504—1062no—
190C2×M32—531no—
191C2×M32—1061no—
192C22×C16—531no—
193C22×C16—1061no—
194#472—534no—
195#472—1062no—
196C2×#108—531no—
197C2×#108C2×D5311yescharacteristic
198C2×#107C2×D5311yescharacteristic
199#417—21no—
200C2×#100—1061no—
201#361—1062no—
202#357—1061no—
203#357—1061no—
204#343—21no—
205C53#1307311yescharacteristic, complemented, Sylow
206C2×#41C22⋊D5311yescharacteristic
207C22×D4—1061no—
208AGL(1,ℤ/8)—1064no—
209AGL(1,ℤ/8)—2122no—
210C2×SD16—2121no—
211C2×D8—532no—
212C2×D8—1061no—
213C2×D8—1061no—
214C2×D8—1061no—
215C2×D8—1061no—
216C2×D8—1061no—
217C2×D8—2121no—
218#131—21no—
219#131—21no—
220#131C4×D5312yes—
221#131D21212yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—1061no—
225C2×M16C22⋊D5311yescharacteristic
226C22×C8—21no—
227C22×C8C22×D5311yescharacteristic
228C2×C4⋊C4—21no—
229M32—532no—
230M32—1064no—
231M32—1061no—
232C2×C16—531no—
233C2×C16—531no—
234C2×C16—1064no—
235C2×C16—1061no—
236C2×C16—1061no—
237C2×C16—1061no—
238#108—21no—
239#108—21no—
240#108—532no—
241#108—1061no—
242#107—21no—
243#107—21no—
244#103—41no—
245#102—41no—
246#100—1064no—
247C24—2121no—
248#41—21no—
249#41—21no—
250#41#1171992112yes—
251#41#1171992412yes—
252C2×Q8#1171992211yescharacteristic
253C2×D4—21no—
254C2×D4—1062no—
255C2×D4—2121no—
256C2×D4—2121no—
257C2×D4—2121no—
258C2×D4—2121no—
259C2×D4—2121no—
260C2×D4#1171992211yescharacteristic
261C22×C4—21no—
262C22×C4—41no—
263C22×C4C2×C22⋊D5311yescharacteristic
264SD16—2124no—
265D8—1062no—
266D8—1062no—
267D8—2124no—
268D8—2121no—
269D8—2121no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—41no—
274M16—41no—
275M16—1062no—
276M16—2121no—
277M16#1171992112yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—2121no—
283C2×C8C2×C4×D5311yescharacteristic
284C2×C8C2×D21211yescharacteristic
285C2×C8D4×D5311yescharacteristic
286C2×C8D4×D5311yescharacteristic
287C2×C8#1171994711yescharacteristic
288C2×C8#1171994711yescharacteristic
289C4⋊C4—41no—
290C4⋊C4—41no—
291C16—1061no—
292C16—1061no—
293C16—1062no—
294C23—21no—
295C23—8481no—
296C23—8481no—
297C23—4241no—
298C23—4241no—
299C23—2122no—
300C23#1313323711yescharacteristic
301Q8—21no—
302Q8#1313313712yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—4241no—
307D4—4241no—
308D4—2122no—
309D4—2122no—
310D4#1313313712yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—81no—
315C2×C4—161no—
316C2×C4C2×#1171992111yescharacteristic
317C2×C4#1313318911yescharacteristic
318C2×C4#1313318911yescharacteristic
319C2×C4C2×#1171992211yescharacteristic
320C2×C4C2×#1171992411yescharacteristic
321C2×C4#1313323711yescharacteristic
322C8—22no—
323C8—81no—
324C8—2122no—
325C8#1313317711yescharacteristic
326C8#1313317711yescharacteristic
327C8#1313318711yescharacteristic
328C8#1313320111yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—8482no—
335C22—8482no—
336C22—8481no—
337C22—8481no—
338C22—4242no—
339C22#1681061711yescharacteristic
340C22#1681085411yescharacteristic
341C22#1681085711yescharacteristic
342C4—22no—
343C4—161no—
344C4#1681050011yescharacteristic
345C4C2×#1313313711yescharacteristic
346C4#1681085411yescharacteristic
347C4#1681085711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—8482no—
352C2#2027514111yescharacteristic
353C2#2027683311yescharacteristic
354C2#2027684711yescharacteristic
subgroupG/H
centerC22#16810857
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1816C2
socleC22×C53#466
prime radicalC2×#16810893C2
cosocleC22×C8×C53C23
hypercenter#1816D53
hypocenterC53#13073
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ #1816

Complements

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

NH ≅ G/Ntype
1. C2×#16811040C2semidirect
3. C2×#16810435C2semidirect
4. C2×#16810423C2semidirect
5. #20272823C2semidirect
6. #20272808C2semidirect
7. C53×#1816C2semidirect
17. #16810893C4semidirect
175. #1816D53semidirect
205. C53#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#168110402
C2×#168108932
C2×#168104352
C2×#168104232
#202728232
#202728082
C53×#18162
#1307353