Finite Group Databasebeta

← #27301006

Subgroups — #27301006

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811040C211yesmaximal, characteristic, complemented
2#20283616C211yesmaximal, characteristic
3C2×#16810408C211yesmaximal, characteristic, complemented
4#20276335C211yesmaximal, characteristic, complemented
5#20272805C211yesmaximal, characteristic, complemented
6#20272800C211yesmaximal, characteristic, complemented
7C53×#2117C211yesmaximal, characteristic, complemented
8#16811421C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10#16811040—22no—
11#16811040—22no—
12C2×#13133157—21no—
13C2×#13133157C2211yescharacteristic
14C22×#11719908—21no—
15C22×#11719908C2211yescharacteristic
16#16810743—22no—
17#16810738—22no—
18#16810671C411yescharacteristic
19#16810640C411yescharacteristic
20#16810639C411yescharacteristic
21#16810578C2211yescharacteristic
22#16810408—21no—
23#16810408—21no—
24#16810404—22no—
25C2×#131×C53C2211yescharacteristic
26C2×M32×C53C2211yescharacteristic
27C2×M32×C53C2211yescharacteristic
28C53×#350—22no—
29C2×#11719945C2×C411yescharacteristic
30#13133248—21no—
31#13133247C812yes—
32#13133243—21no—
33#13133242C812yes—
34#13133237—22no—
35#13133223—21no—
36#13133223—21no—
37#13133223—21no—
38#13133223—21no—
39C2×#11719917C2×C411yescharacteristic
40C2×#11719917C2×C411yescharacteristic
41C2×#11719912—21no—
42C2×C8×D53—21no—
43#13133176C2×C411yescharacteristic
44#13133157—24no—
45#13133157—21no—
46#13133157D412yes—
47C2×#11719908—24no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50C2×#11719908—21no—
51C2×#11719908D411yescharacteristic
52C2×#11719908D411yescharacteristic
53#13133152—22no—
54#13133144—22no—
55#13133143—22no—
56#13133127—21no—
57C8×Dic53—21no—
58C2×#41×C53C2×C411yescharacteristic
59#131×C53—21no—
60#131×C53—21no—
61#131×C53D412yes—
62#131×C53D412yes—
63C2×M16×C53—21no—
64C2×M16×C53C2×C411yescharacteristic
65C22×C8×C53—21no—
66C22×C8×C53C2311yescharacteristic
67M32×C53—21no—
68M32×C53—21no—
69M32×C53—21no—
70M32×C53—21no—
71C2×C16×C53—21no—
72C2×C16×C53—21no—
73#41×C53—21no—
74#41×C53—21no—
75#41×C53C22⋊C412yes—
76#41×C53C2×C812yes—
77C2×Q8×C53M1611yescharacteristic
78C2×D4×C53—21no—
79C2×D4×C53M1611yescharacteristic
80C22×C4×C53—21no—
81C22×C4×C53C22×C411yescharacteristic
82C2×C22⋊D53—21no—
83#11719945—21no—
84#11719945—21no—
85#11719945—21no—
86#11719945—21no—
87C2×D212C2×C811yescharacteristic
88C2×C4×D53—21no—
89C2×Dic106C2×C811yescharacteristic
90M16×C53—21no—
91M16×C53—22no—
92M16×C53—22no—
93M16×C53C22⋊C412yes—
94C2×C8×C53—21no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C2×C8×C53#4111yescharacteristic
103C22⋊Dic53—21no—
104#11719925—22no—
105#11719924—22no—
106#11719923—22no—
107#11719922—22no—
108C4⋊Dic53—21no—
109C4×Dic53—21no—
110#11719917—21no—
111#11719917—21no—
112#11719917—21no—
113#11719917C2×C812yes—
114C2×#88322—21no—
115C2×#88322—21no—
116#11719912—41no—
117#11719912—41no—
118C8×D53—41no—
119C8×D53—41no—
120C16×C53—41no—
121C16×C53—41no—
122#11719908—21no—
123#11719908—22no—
124#11719908—21no—
125C23×C53—21no—
126C23×C53#13111yescharacteristic
127C22×D53—21no—
128Q8×C53—21no—
129Q8×C53C22⋊C812yes—
130D4×C53—22no—
131D4×C53—22no—
132D4×C53—21no—
133D4×C53C22⋊C812yes—
134C2×C4×C53—21no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C22×C811yescharacteristic
141C2×C4×C53C2×M1611yescharacteristic
142C2×C4×C53#13111yescharacteristic
143C22⋊D53—41no—
144C22⋊D53—41no—
145C2×Dic53—21no—
146C2×Dic53—41no—
147D212—21no—
148D212—21no—
149C4×D53—41no—
150C4×D53—41no—
151Dic106—21no—
152Dic106—21no—
153C8×C53—22no—
154C8×C53#12011yescharacteristic
155C8×C53#12111yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12311yescharacteristic
158#88322—22no—
159#88322—41no—
160#12987—531nomaximal, Sylow
161C22×C53—22no—
162C22×C53—41no—
163C22×C53—42no—
164C22×C53#40811yescharacteristic
165C22×C53C8×D411yescharacteristic
166C22×C53#43511yescharacteristic
167C2×D53—41no—
168C2×D53—81no—
169C4×C53—22no—
170C4×C53#38611yescharacteristic
171C4×C53C2×C22⋊C811yescharacteristic
172C4×C53C8×D411yescharacteristic
173C4×C53#43511yescharacteristic
174Dic53—41no—
175Dic53—81no—
176C2×#504—531no—
177#2955—531no—
178#2117D5311yescharacteristic, complemented
179C2×#350—531no—
180#2115—531no—
181#1379—531no—
182#1378—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—42no—
187C2×C53#186911yescharacteristic
188D53—81no—
189#575—531no—
190C2×#131—531no—
191C2×#131C2×D5311yescharacteristic
192#504—1062no—
193#504—1062no—
194C2×M32—531no—
195C2×M32—1061no—
196C2×M32C2×D5311yescharacteristic
197C2×M32C2×D5311yescharacteristic
198C22×C16—531no—
199C22×C16—1061no—
200#444—1062no—
201#443—1062no—
202#430—531no—
203#421—531no—
204#421—531no—
205#405—531no—
206#350—22no—
207#350—1061no—
208#350—1061no—
209#349—1062no—
210C53#1298711yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41C4×D5311yescharacteristic
213#137—532no—
214#137—1061no—
215AGL(1,ℤ/8)—532no—
216AGL(1,ℤ/8)—1061no—
217#135—1062no—
218#131—21no—
219#131—21no—
220#131—1061no—
221#131—1061no—
222#131—1061no—
223#131—1061no—
224#131D21212yes—
225#131C22⋊D5312yes—
226C2×M16—21no—
227C2×M16—531no—
228C2×M16—531no—
229C2×M16—1061no—
230C2×M16C4×D5311yescharacteristic
231C22×C8—21no—
232C22×C8—1061no—
233C22×C8C22×D5311yescharacteristic
234#117—531no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—532no—
240M32—1064no—
241M32—1061no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—531no—
245C2×C16—531no—
246C2×C16—1064no—
247C2×C16—1061no—
248C2×C16—1061no—
249C2×C16—1061no—
250#108—1062no—
251#104—1062no—
252#104—1062no—
253#97—1061no—
254C4×C8—1061no—
255#41—21no—
256#41—21no—
257#41—1061no—
258#41—1061no—
259#41—1061no—
260#41—1061no—
261#41C8×D5312yes—
262#41#1171992112yes—
263C2×Q8—531no—
264C2×Q8#1171991211yescharacteristic
265C2×D4—21no—
266C2×D4—531no—
267C2×D4—1061no—
268C2×D4#1171991211yescharacteristic
269C22×C4—21no—
270C22×C4—1061no—
271C22×C4C2×C4×D5311yescharacteristic
272Q16—1062no—
273SD16—1062no—
274SD16—1062no—
275D8—1062no—
276M16—21no—
277M16—22no—
278M16—22no—
279M16—532no—
280M16—1061no—
281M16—1061no—
282M16—1061no—
283M16—2121no—
284M16—2121no—
285M16#1171992112yes—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8C2×D21211yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C4⋊C4—1061no—
300C22⋊C4—1061no—
301C42—1061no—
302C16—41no—
303C16—41no—
304C16—1061no—
305C16—1061no—
306C16—1062no—
307C23—21no—
308C23—1061no—
309C23#1313322511yescharacteristic
310Q8—21no—
311Q8—1061no—
312Q8—1061no—
313Q8#1313312912yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—1061no—
318D4—1061no—
319D4—2121no—
320D4—2121no—
321D4#1313312912yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—1061no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4C2×#1171992111yescharacteristic
330C2×C4#1313317711yescharacteristic
331C2×C4#1313317711yescharacteristic
332C2×C4C2×C8×D5311yescharacteristic
333C2×C4C2×#1171991211yescharacteristic
334C2×C4#1313322511yescharacteristic
335C8—22no—
336C8—1062no—
337C8—2121no—
338C8#1313318711yescharacteristic
339C8#1313318911yescharacteristic
340C8#1313318911yescharacteristic
341C8#1313320111yescharacteristic
342C22—22no—
343C22—41no—
344C22—42no—
345C22—4241no—
346C22—2121no—
347C22#1681059111yescharacteristic
348C22#1681068611yescharacteristic
349C22#1681069111yescharacteristic
350C4—22no—
351C4—4241no—
352C4—2121no—
353C4#1681050011yescharacteristic
354C4C2×#1313312911yescharacteristic
355C4#1681068611yescharacteristic
356C4#1681069111yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—4241no—
362C2#2027500511yescharacteristic
subgroupG/H
centerC8#13133189
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#2117C2
socleC2×C53#1869
prime radical#16811421C4
cosocleC22×C8×C53C23
hypercenter#2117D53
hypocenterC53#12987
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
1. C2×#16811040C2semidirect
3. C2×#16810408C2semidirect
4. #20276335C2semidirect
5. #20272805C2semidirect
6. #20272800C2semidirect
7. C53×#2117C2semidirect
178. #2117D53semidirect
210. C53#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#168110402
#202836162
C2×#168104082
#202763352
#202728052
#202728002
C53×#21172
#1298753