Finite Group Databasebeta

← #27301277

Subgroups — #27301277

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283599C211yesmaximal, characteristic, complemented
2#20276451C211yesmaximal, characteristic, complemented
3C2×#16810433C211yesmaximal, characteristic, complemented
4#20275614C211yesmaximal, characteristic, complemented
5#20272821C211yesmaximal, characteristic, complemented
6#20272802C211yesmaximal, characteristic, complemented
7C53×#3414C211yesmaximal, characteristic
8#16811106—21no—
9C2×#13133157C2211yescharacteristic
10#16810735—21no—
11#16810730—21no—
12SD16×Dic53—22no—
13#16810661C2211yescharacteristic
14C2×#13133149C2211yescharacteristic
15#16810640—21no—
16#16810624—21no—
17#16810597—21no—
18#16810578C2211yescharacteristic
19#16810433—22no—
20#16810433—22no—
21C2×#135×C53C2211yescharacteristic
22C2×#135×C53C2211yescharacteristic
23C53×#510—21no—
24C53×#510C412yescomplemented
25C53×#509—21no—
26C53×#509C412yescomplemented
27C53×#508—22no—
28C2×M32×C53C2211yescharacteristic
29C2×#11719917D411yescharacteristic
30Q8×Dic53—22no—
31D4×Dic53—22no—
32D4×Dic53—42no—
33#13133176—21no—
34#13133176—41no—
35#13133176D411yescharacteristic
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—41no—
38#13133157—21no—
39#13133157—21no—
40C2×#11719908—21no—
41#13133151—21no—
42#13133149—21no—
43#13133149—21no—
44#13133149D412yes—
45#13133144—42no—
46#13133141—22no—
47#13133138—22no—
48#13133132—42no—
49#13133127—21no—
50C8×Dic53—21no—
51C2×#41×C53—21no—
52C2×#41×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53—21no—
56#135×C53—21no—
57#135×C53—21no—
58#135×C53—21no—
59#135×C53C2×C412yes—
60#135×C53D412yes—
61C2×Q16×C53C2×C411yescharacteristic, complemented
62C2×Q16×C53D411yescharacteristic
63C2×SD16×C53—21no—
64C2×SD16×C53—21no—
65C2×D8×C53C2×C411yescharacteristic, complemented
66C2×D8×C53D411yescharacteristic
67C22×C8×C53C2311yescharacteristic
68Q32×C53—22no—
69SD32×C53—22no—
70SD32×C53—22no—
71D16×C53—22no—
72M32×C53—21no—
73M32×C53C2×C412yescomplemented
74C2×C16×C53—21no—
75#41×C53—22no—
76#41×C53—22no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82#41×C53—41no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C22×Dic53—42no—
93C22×Dic53—41no—
94Q16×C53—21no—
95Q16×C53—21no—
96Q16×C53—21no—
97Q16×C53C22⋊C412yes—
98SD16×C53—22no—
99SD16×C53—22no—
100SD16×C53—41no—
101SD16×C53—41no—
102D8×C53—21no—
103D8×C53—21no—
104D8×C53—21no—
105D8×C53C22⋊C412yes—
106C2×C8×C53C22×C411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C22⋊Dic53—42no—
113C22⋊Dic53—41no—
114C22⋊Dic53D812yes—
115C4⋊Dic53—21no—
116C4⋊Dic53—22no—
117C4⋊Dic53—42no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C4×Dic53—21no—
121C4×Dic53—42no—
122C4×Dic53—41no—
123#11719917—21no—
124#11719917—21no—
125C2×#88322—21no—
126C16×C53—22no—
127#11719908—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53C2×D811yescharacteristic
131Q8×C53—22no—
132Q8×C53—41no—
133Q8×C53—41no—
134Q8×C53—41no—
135D4×C53—22no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—41no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—81no—
142D4×C53—81no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×Dic53—22no—
156C2×Dic53—42no—
157C2×Dic53—81no—
158C2×Dic53—162no—
159C2×Dic53—161no—
160C8×C53C2×C22⋊C411yescharacteristic
161C8×C53C4×D411yescharacteristic
162C8×C53#12111yescharacteristic
163C8×C53#12311yescharacteristic
164#88322—41no—
165#13030—531nomaximal, Sylow
166C22×C53—41no—
167C22×C53—42no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—161no—
172C22×C53C4×D811yescharacteristic
173C22×C53#44711yescharacteristic
174C22×C53#45911yescharacteristic
175C4×C53—42no—
176C4×C53—81no—
177C4×C53#38611yescharacteristic
178C4×C53#44211yescharacteristic
179C4×C53#44911yescharacteristic
180C4×C53#46311yescharacteristic
181Dic53—42no—
182Dic53—82no—
183Dic53—161no—
184#3414D5311yescharacteristic
185#2952—531no—
186#2145—531no—
187C2×#360—531no—
188#1976—531no—
189#1386—531no—
190#1378—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—82no—
195C2×C53—161no—
196C2×C53#187911yescharacteristic
197C2×#135C2×D5311yescharacteristic
198C2×#135C2×D5311yescharacteristic
199#520—1061no—
200#510—21no—
201#510Dic5312yescomplemented
202#509—21no—
203#509Dic5312yescomplemented
204#508—22no—
205C2×M32—531no—
206C2×M32C2×D5311yescharacteristic
207#442—1061no—
208#441—1061no—
209C4×SD16—1062no—
210#427—531no—
211C2×#106—531no—
212#421—1061no—
213#417—1061no—
214#410—1061no—
215#405—531no—
216#360—1062no—
217#360—1062no—
218C53#1303011yescharacteristic, complemented, Sylow
219C2×#41—21no—
220C2×#41—21no—
221#135—21no—
222#135—21no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135C2×Dic5312yes—
228#135C22⋊D5312yes—
229C2×Q16C2×Dic5311yescharacteristic, complemented
230C2×Q16C22⋊D5311yescharacteristic
231C2×SD16—21no—
232C2×SD16—21no—
233C2×D8C2×Dic5311yescharacteristic, complemented
234C2×D8C22⋊D5311yescharacteristic
235C2×M16—531no—
236C22×C8C22×D5311yescharacteristic
237C4×Q8—1062no—
238C4×D4—1062no—
239C4×D4—2122no—
240#117—531no—
241#117—1061no—
242#117—2121no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—2121no—
245Q32—22no—
246SD32—22no—
247SD32—22no—
248D16—22no—
249M32—21no—
250M32—1061no—
251M32—1061no—
252M32C2×Dic5312yescomplemented
253C2×C16—21no—
254C2×C16—1061no—
255#107—1061no—
256#106—532no—
257#106—1061no—
258#106—1061no—
259#104—2122no—
260#103—1062no—
261#102—1062no—
262#99—2122no—
263#97—1061no—
264C4×C8—1061no—
265#41—22no—
266#41—22no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—2122no—
282C22×C4—2121no—
283C22×C4D4×D5311yescharacteristic
284Q16—21no—
285Q16—21no—
286Q16—21no—
287Q16C22⋊Dic5312yes—
288SD16—22no—
289SD16—22no—
290SD16—41no—
291SD16—41no—
292D8—21no—
293D8—21no—
294D8—21no—
295D8C22⋊Dic5312yes—
296M16—1061no—
297M16—1061no—
298C2×C8—1061no—
299C2×C8D4×D5311yescharacteristic
300C2×C8#1171994711yescharacteristic
301C2×C8#1171994711yescharacteristic
302C2×C8C22×Dic5311yescharacteristic
303C2×C8C2×C22⋊D5311yescharacteristic
304C2×C8C2×C22⋊D5311yescharacteristic
305C4⋊C4—1061no—
306C4⋊C4—1062no—
307C4⋊C4—2122no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C22⋊C4—532no—
311C22⋊C4—2122no—
312C22⋊C4—2121no—
313C42—1061no—
314C42—2122no—
315C42—2121no—
316C16—22no—
317C16—2121no—
318C23—41no—
319C23—41no—
320C23D8×D5311yescharacteristic
321Q8—22no—
322Q8—41no—
323Q8—41no—
324Q8—41no—
325D4—22no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—81no—
332D4—81no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—8481no—
340C2×C4—8482no—
341C2×C4—1062no—
342C2×C4—4241no—
343C2×C4—2122no—
344C2×C4D4×Dic5311yescharacteristic
345C2×C4#1313318511yescharacteristic
346C2×C4#1313318911yescharacteristic
347C2×C4#1313323611yescharacteristic
348C2×C4#1313324111yescharacteristic
349C2×C4#1313324111yescharacteristic
350C8—2121no—
351C8C2×C22⋊Dic5311yescharacteristic
352C8D4×Dic5311yescharacteristic
353C8#1313318911yescharacteristic
354C8#1313319811yescharacteristic
355C22—41no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—161no—
361C22D8×Dic5311yescharacteristic
362C22#1681075511yescharacteristic
363C22#1681081611yescharacteristic
364C4—42no—
365C4—81no—
366C4—8481no—
367C4—4242no—
368C4—2122no—
369C4#1681050311yescharacteristic
370C4#1681073511yescharacteristic
371C4#1681076511yescharacteristic
372C4#1681083611yescharacteristic
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2—161no—
378C2#2027506511yescharacteristic
subgroupG/H
centerC2#20275065
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3414C2
socleC2×C53#1879
cosocleC22×C8×C53C23
hypercenter#3414D53
hypocenterC53#13030
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #20283599C2semidirect
2. #20276451C2semidirect
3. C2×#16810433C2semidirect
4. #20275614C2semidirect
5. #20272821C2semidirect
6. #20272802C2semidirect
24. C53×#510C4semidirect
26. C53×#509C4semidirect
61. C2×Q16×C53C2×C4semidirect
65. C2×D8×C53C2×C4semidirect
73. M32×C53C2×C4semidirect
201. #510Dic53semidirect
203. #509Dic53semidirect
218. C53#13030semidirect
229. C2×Q16C2×Dic53semidirect
233. C2×D8C2×Dic53semidirect
252. M32C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202835992
#202764512
C2×#168104332
#202756142
#202728212
#202728022
C53×#34142
#1303053