Finite Group Databasebeta

← #27300614

Subgroups — #27300614

Order 13568.

424 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
1#20283303C211yesmaximal, characteristic
2#20276504C211yesmaximal, characteristic, complemented
3#20276503C211yesmaximal, characteristic, complemented
4C2×#16810404C211yesmaximal, characteristic
5#20274745C211yesmaximal, characteristic
6#20272822C211yesmaximal, characteristic, complemented
7C53×#1386C211yesmaximal, characteristic, complemented
8C22×#11719912C2211yescharacteristic
9#117×D53—21no—
10C2×#13133157C2211yescharacteristic
11#16810664—21no—
12#16810664—21no—
13#16810662—21no—
14#16810662—21no—
15#16810661C2211yescharacteristic
16#16810659—21no—
17C2×#13133151C2211yescharacteristic
18#16810480—21no—
19#16810451—21no—
20#16810450—21no—
21#16810449—21no—
22#16810448—21no—
23#16810404C414yescomplemented
24#16810385—21no—
25C2×M32×C53C2211yescharacteristic
26C53×#427C2211yescharacteristic
27C2×#106×C53C2211yescharacteristic
28C22×C4×D53D411yescharacteristic
29C2×#11719912—21no—
30C2×#11719912—21no—
31C2×#11719912—21no—
32C2×#11719912—21no—
33C2×#11719912C2×C412yes—
34C2×#11719912Q812yes—
35C22×#88322D411yescharacteristic
36#13133176—21no—
37#13133175—21no—
38#13133175—21no—
39#13133174—41no—
40#13133173—41no—
41C2×C4⋊Dic53—21no—
42C4⋊C4×D53—21no—
43C4⋊C4×D53—21no—
44C2×#11719921—41no—
45C22⋊C4×D53—41no—
46C42×D53—41no—
47#13133157—21no—
48#13133157C2×C412yescomplemented
49C2×#11719908—21no—
50#13133151—21no—
51#13133151—21no—
52#13133151D411yescharacteristic
53#13133151D411yescharacteristic
54#13133150—21no—
55#13133150—21no—
56#13133149—21no—
57#13133148—21no—
58#13133148—21no—
59#13133124—41no—
60C22×C8×C53C2311yescharacteristic
61#117×C53—21no—
62C2×C4⋊C4×C53—21no—
63M32×C53—21no—
64M32×C53C2×C412yescomplemented
65C2×C16×C53—21no—
66#107×C53D411yescharacteristic
67#107×C53D411yescharacteristic
68#106×C53—21no—
69#106×C53—21no—
70#106×C53—21no—
71C23×D53D811yescharacteristic
72C22×C4×C53—41no—
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—41no—
75C22×Dic53D811yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53C4⋊C412yes—
83C2×C4×D53Q1612yes—
84C2×C8×C53C22×C411yescharacteristic
85C2×C8×C53C2×D411yescharacteristic
86C2×C8×C53C2×D411yescharacteristic
87C2×C8×C53C2×Q811yescharacteristic
88C2×C8×C53#4111yescharacteristic
89C2×C8×C53#4111yescharacteristic
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C42×C53—41no—
96C22⋊Dic53—41no—
97#11719921—41no—
98#11719921—41no—
99#11719921—82no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104#11719919—41no—
105#11719919—41no—
106C4×Dic53—41no—
107C4×Dic53—41no—
108C4×Dic53—41no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322C4⋊C412yes—
112#11719912—22no—
113#11719912—22no—
114#11719912—22no—
115#11719912—22no—
116C16×C53—22no—
117#11719908—22no—
118C23×C53C2×D811yescharacteristic
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53#10712yes—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53C2×C4⋊C411yescharacteristic
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12311yescharacteristic
132C2×C4×C53C2×Q1611yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C53#13711yescharacteristic
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142C2×Dic53#10712yes—
143C4×D53—22no—
144C4×D53—22no—
145C4×D53—22no—
146C4×D53—22no—
147C4×D53—82no—
148C4×D53—82no—
149C8×C53C4×D411yescharacteristic
150C8×C53C4×D411yescharacteristic
151C8×C53C22⋊Q811yescharacteristic
152C8×C53C22⋊Q811yescharacteristic
153#88322—22no—
154#12923—531nomaximal, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53C2×#10711yescharacteristic
158C22×C53#44711yescharacteristic
159C22×C53#48011yescharacteristic
160C2×D53—22no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—42no—
164C2×D53—81no—
165C4×C53—81no—
166C4×C53—81no—
167C4×C53—161no—
168C4×C53#38511yescharacteristic
169C4×C53#42811yescharacteristic
170C4×C53#45111yescharacteristic
171C4×C53#48411yescharacteristic
172Dic53—22no—
173Dic53—81no—
174Dic53—81no—
175Dic53—161no—
176#2911—531no—
177#2159—531no—
178#2159—531no—
179C2×#349—531no—
180#1809—531no—
181#1386—531no—
182#1386D5311yescharacteristic, complemented
183C2×C53—21no—
184C2×C53—21no—
185C2×C53#185211yescharacteristic
186C2×C53#219011yescharacteristic
187C2×C53#219111yescharacteristic
188D53—42no—
189C22×M16—531no—
190C2×#117—1061no—
191C2×M32—531no—
192C2×M32C2×D5311yescharacteristic
193#428—1061no—
194#428—1061no—
195#428—1061no—
196#428—1061no—
197#427—531no—
198#427—1061no—
199#427C2×D5311yescharacteristic
200C2×#107—531no—
201C2×#106C2×D5311yescharacteristic
202#380—1061no—
203#367—1061no—
204#367—1061no—
205#366—1061no—
206#366—1061no—
207#349—534no—
208#340—1061no—
209C53#1292311yescharacteristic, complemented, Sylow
210C23×C4—531no—
211C2×M16—532no—
212C2×M16—532no—
213C2×M16—1061no—
214C2×M16—1061no—
215C2×M16—1061no—
216C2×M16—1061no—
217C22×C8—531no—
218C22×C8C22×D5311yescharacteristic
219#117—21no—
220#117—1061no—
221#117—1061no—
222#117—1061no—
223#117—2121no—
224#117—2121no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C22⋊C4—2121no—
230C2×C22⋊C4—2121no—
231C2×C42—2121no—
232M32—21no—
233M32—532no—
234M32—1061no—
235M32C4×D5312yescomplemented
236C2×C16—21no—
237C2×C16—1061no—
238#107—531no—
239#107—531no—
240#107—1061no—
241#107—1061no—
242#107—1061no—
243#107—1061no—
244#107D21211yescharacteristic
245#107D21211yescharacteristic
246#106—21no—
247#106—21no—
248#106—21no—
249#106—1061no—
250#106—1061no—
251#106—1061no—
252#95—2121no—
253C24—531no—
254C22×C4—41no—
255C22×C4—531no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4D4×D5311yescharacteristic
266M16—1062no—
267M16—1062no—
268M16—1062no—
269M16—1062no—
270C2×C8—532no—
271C2×C8—1061no—
272C2×C8—1061no—
273C2×C8C2×C4×D5311yescharacteristic
274C2×C8C2×D21211yescharacteristic
275C2×C8#1171994511yescharacteristic
276C2×C8D4×D5311yescharacteristic
277C2×C8#1171994711yescharacteristic
278C2×C8Q8×D5311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4—2121no—
289C22⋊C4—41no—
290C22⋊C4—4242no—
291C22⋊C4—2121no—
292C22⋊C4—2121no—
293C22⋊C4—2121no—
294C42—41no—
295C42—2121no—
296C42—2121no—
297C42—2121no—
298C16—22no—
299C16—1062no—
300C23—532no—
301C23—1061no—
302C23—1061no—
303C23—2121no—
304C23—2121no—
305C23D8×D5311yescharacteristic
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—532no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1062no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—4242no—
319C2×C4—4242no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4C4⋊C4×D5311yescharacteristic
326C2×C4#1313318711yescharacteristic
327C2×C4#1313319911yescharacteristic
328C2×C4Q16×D5311yescharacteristic
329C2×C4#1313324411yescharacteristic
330C2×C4#1313324611yescharacteristic
331C8—1062no—
332C8#1313317711yescharacteristic
333C8C4×D21211yescharacteristic
334C8#1313319511yescharacteristic
335C8#1313319711yescharacteristic
336C22—21no—
337C22—21no—
338C22—1062no—
339C22—4241no—
340C22—2122no—
341C22—2121no—
342C22—2121no—
343C22#107×D5311yescharacteristic
344C22#1681075811yescharacteristic
345C22#1681093511yescharacteristic
346C4—81no—
347C4—81no—
348C4—161no—
349C4—8481no—
350C4—1062no—
351C4—4241no—
352C4—4241no—
353C4#1681049711yescharacteristic
354C4#1681066211yescharacteristic
355C4#1681078011yescharacteristic
356C4#1681095711yescharacteristic
357C2—21no—
358C2—21no—
359C2—2122no—
360C2#2027490411yescharacteristic
361C2#2027666211yescharacteristic
362C2#2027666711yescharacteristic
subgroupG/H
centerC22#16810758
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1386C2
socleC22×C53#447
prime radicalC23×D53D8
cosocleC22×C8×C53C23
hypercenter#1386D53
hypocenterC53#12923
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
2. #20276504C2semidirect
3. #20276503C2semidirect
6. #20272822C2semidirect
7. C53×#1386C2semidirect
23. #16810404C4semidirect
48. #13133157C2×C4semidirect
64. M32×C53C2×C4semidirect
182. #1386D53semidirect
209. C53#12923semidirect
235. M32C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202833032
#202765042
#202765032
C2×#168104042
#202747452
#202728222
C53×#13862
#1292353