Finite Group Databasebeta

← #27278406

Subgroups — #27278406

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274735C211yesmaximal, characteristic
2#20273485C211yesmaximal, characteristic, complemented
3#20273470C211yesmaximal, characteristic, complemented
4#20273097C211yesmaximal, characteristic
5#20272616C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C53×#2906C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9#16810647C2211yescharacteristic
10C2×#13133144—21no—
11C2×C4×#88322C2211yescharacteristic
12C4×C4⋊Dic53C2211yescharacteristic
13#16810382—21no—
14#16810373—22no—
15#16810373—21no—
16#16810373—21no—
17Q8⋊#88322C411yescharacteristic, complemented
18Q8⋊#88322C411yescharacteristic, complemented
19D4⋊#88322C411yescharacteristic, complemented
20D4⋊#88322C411yescharacteristic, complemented
21C2×#131×C53—21no—
22C4×#41×C53C2211yescharacteristic
23C53×#424—21no—
24C53×#424—22no—
25C53×#423—21no—
26C53×#423C2211yescharacteristic
27C2×C4⋊C8×C53—21no—
28C2×C4⋊C8×C53C2211yescharacteristic
29C2×#11719917—21no—
30C22×#88322—21no—
31C2×C4⋊Dic53—21no—
32C2×C4×Dic53—21no—
33C4⋊#88322—21no—
34C4⋊#88322—21no—
35C4⋊#88322C2×C411yescharacteristic
36C4⋊#88322C2×C411yescharacteristic
37#13133144—42no—
38C22⋊#88322—21no—
39C4×#88322—21no—
40C4×#88322—21no—
41C4×#88322C2×C411yescharacteristic
42C4×#88322C2×C411yescharacteristic
43#13133124—21no—
44C2×#41×C53D411yescharacteristic
45#131×C53—42no—
46#131×C53—42no—
47C2×M16×C53—22no—
48C2×M16×C53—21no—
49C22×C8×C53—22no—
50C22×C8×C53—21no—
51C4×Q8×C53C2×C411yescharacteristic
52C4×Q8×C53Q811yescharacteristic
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55C4×D4×C53C2×C411yescharacteristic
56C4×D4×C53Q811yescharacteristic
57#117×C53—21no—
58#117×C53D411yescharacteristic
59C2×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—22no—
62C4⋊C8×C53—22no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53D411yescharacteristic
66C4⋊C8×C53D411yescharacteristic
67C4⋊C8×C53D412yes—
68#41×C53—21no—
69#41×C53—21no—
70#41×C53SD1612yes—
71#41×C53SD1612yes—
72C2×Q8×C53C4⋊C411yescharacteristic
73C2×D4×C53—21no—
74C2×D4×C53C4⋊C411yescharacteristic
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—41no—
82M16×C53—44no—
83M16×C53—41no—
84M16×C53—41no—
85C2×C8×C53—21no—
86C2×C8×C53—22no—
87C2×C8×C53—21no—
88C2×C8×C53—44no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53C4⋊C411yescharacteristic
94C4⋊C4×C53C4⋊C411yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C42×C53—21no—
99C42×C53—21no—
100C42×C53—21no—
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×Q811yescharacteristic
105C4⋊Dic53—22no—
106C4⋊Dic53—41no—
107C4×Dic53—41no—
108C4×Dic53—41no—
109#11719917—41no—
110#11719917—41no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322—42no—
116C2×#88322—41no—
117C23×C53—21no—
118C23×C53#12311yescharacteristic
119Q8×C53—21no—
120Q8×C53#10612yes—
121D4×C53—22no—
122D4×C53—22no—
123D4×C53—21no—
124D4×C53#10612yes—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—81no—
136C2×C4×C53C2×C4⋊C411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53C22⋊Q811yescharacteristic
141C2×C4×C53C22⋊Q811yescharacteristic
142C2×C4×C53C2×SD1611yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53#13711yescharacteristic
146C2×Dic53—42no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C8×C53—41no—
150C8×C53—42no—
151C8×C53—41no—
152#88322—22no—
153#88322—41no—
154#88322—41no—
155#88322—41no—
156#88322—81no—
157#8904—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—22no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53#38511yescharacteristic
164C22×C53#48111yescharacteristic
165C22×C53#48311yescharacteristic
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#10611yescharacteristic
173C4×C53#42811yescharacteristic
174C4×C53#44311yescharacteristic
175C4×C53#44411yescharacteristic
176C4×C53#44811yescharacteristic
177C4×C53#45011yescharacteristic
178Dic53—82no—
179#2906D5311yescharacteristic
180#1806—531no—
181#1570—531no—
182#1566—531no—
183#1476—531no—
184#1303—531no—
185#1279—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—42no—
189C2×C53#185311yescharacteristic
190C2×C53#186111yescharacteristic
191C2×C53#186611yescharacteristic
192C2×#131—21no—
193C4×#41C2×D5311yescharacteristic
194#431—531no—
195#424—21no—
196#424—22no—
197#423—21no—
198#423—531no—
199#423C2×D5311yescharacteristic
200C2×C4⋊C8—21no—
201C2×C4⋊C8C2×D5311yescharacteristic
202C2×#104—1061no—
203C2×C4×C8—531no—
204C4×C4⋊C4—531no—
205#339—1061no—
206#334—1062no—
207#334—1061no—
208#334—1061no—
209Q8⋊C8—531no—
210Q8⋊C8—531no—
211D4⋊C8—531no—
212D4⋊C8—531no—
213C53#890411yescharacteristic, complemented, Sylow
214C2×#41C22⋊D5311yescharacteristic
215#131—42no—
216#131—42no—
217C2×M16—22no—
218C2×M16—21no—
219C2×M16—1061no—
220C22×C8—22no—
221C22×C8—21no—
222C22×C8—1061no—
223C4×Q8Dic10611yescharacteristic
224C4×Q8C4×D5311yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic10611yescharacteristic
228C4×D4C4×D5311yescharacteristic
229#117—21no—
230#117C22⋊D5311yescharacteristic
231C2×C4⋊C4—1061no—
232C2×C42—21no—
233C2×C42—1061no—
234C2×C42C22×D5311yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—531no—
240C4⋊C8—531no—
241C4⋊C8—1061no—
242C4⋊C8—1061no—
243C4⋊C8C22⋊D5311yescharacteristic
244C4⋊C8C22⋊D5311yescharacteristic
245C4⋊C8C22⋊D5312yes—
246#104—2122no—
247C22⋊C8—1061no—
248C4×C8—531no—
249C4×C8—531no—
250C4×C8—1061no—
251C4×C8—1061no—
252#95—1061no—
253#41—21no—
254#41—21no—
255#41#1171992312yes—
256#41#1171992412yes—
257C2×Q8#1171991911yescharacteristic
258C2×D4—21no—
259C2×D4#1171991911yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—2121no—
264C22×C4#1171994511yescharacteristic
265C22×C4#1171994511yescharacteristic
266C22×C4C2×C22⋊D5311yescharacteristic
267M16—44no—
268M16—41no—
269M16—41no—
270M16—2121no—
271M16—2121no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—21no—
275C2×C8—44no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—1061no—
279C2×C8—1061no—
280C2×C8—1061no—
281C2×C8—1061no—
282C2×C8—2122no—
283C2×C8—2121no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—1062no—
287C4⋊C4—2121no—
288C4⋊C4#1171991911yescharacteristic
289C4⋊C4#1171991911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—2121no—
297C42—2121no—
298C42C2×Dic10611yescharacteristic
299C42C2×C4×D5311yescharacteristic
300C42C2×C22⋊D5311yescharacteristic
301C42C2×C22⋊D5311yescharacteristic
302C23—21no—
303C23#1313320011yescharacteristic
304Q8—21no—
305Q8#1313314812yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1313314812yes—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—2122no—
324C2×C4C2×#1171991911yescharacteristic
325C2×C4C4×C22⋊D5311yescharacteristic
326C2×C4C4×C22⋊D5311yescharacteristic
327C2×C4#1313318611yescharacteristic
328C2×C4#1313319611yescharacteristic
329C2×C4#1313319611yescharacteristic
330C2×C4C2×#1171992411yescharacteristic
331C2×C4C2×#1171992311yescharacteristic
332C2×C4#1313324311yescharacteristic
333C2×C4#1313324811yescharacteristic
334C8—41no—
335C8—42no—
336C8—41no—
337C8—1062no—
338C8—4241no—
339C8—2121no—
340C8—2121no—
341C8—2121no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1681049611yescharacteristic
348C22#1681094111yescharacteristic
349C22#1681095311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—4242no—
357C4C2×#1313314811yescharacteristic
358C4#1681066311yescharacteristic
359C4#1681073811yescharacteristic
360C4#1681074311yescharacteristic
361C4#1681076211yescharacteristic
362C4#1681077411yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2027490811yescharacteristic
367C2#2027495511yescharacteristic
368C2#2027499011yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2906C2
socleC22×C53#385
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenter#2906D53
hypocenterC53#8904
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #20273485C2semidirect
3. #20273470C2semidirect
5. #20272616C2semidirect
6. #20272554C2semidirect
17. Q8⋊#88322C4semidirect
18. Q8⋊#88322C4semidirect
19. D4⋊#88322C4semidirect
20. D4⋊#88322C4semidirect
213. C53#8904semidirect

Maximal subgroups

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

subgroupindex
#202747352
#202734852
#202734702
#202730972
#202726162
#202725542
C53×#29062
#890453