Finite Group Databasebeta

← #27278030

Subgroups — #27278030

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810702C211yesmaximal, characteristic
2C4×#13133137C211yesmaximal, characteristic
3C2×#16810375C211yesmaximal, characteristic, complemented
4C2×#16810350C211yesmaximal, characteristic
5#20272556C211yesmaximal, characteristic, complemented
6#20272542C211yesmaximal, characteristic, complemented
7C53×#1939C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C2×C4×D212C2211yescharacteristic
10#16810702—22no—
11#16810702—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133137C411yescharacteristic
14C2×#13133137C411yescharacteristic
15C2×#13133129—21no—
16C2×C4×#88322C2211yescharacteristic
17C2×C4×#88322C2211yescharacteristic
18#16810385C411yescharacteristic
19#16810385C411yescharacteristic
20#16810375—22no—
21#16810350—22no—
22C2×C4⋊C8×C53—21no—
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4×C8×C53C2211yescharacteristic
25C4×C4⋊C4×C53C2211yescharacteristic
26C53×#336—21no—
27C22×D212C2×C411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719912—24no—
30C2×#11719912—42no—
31C2×#11719912—42no—
32C22×#88322—21no—
33C22×#88322—21no—
34C22×#88322C2×C411yescharacteristic
35C22×#88322C2×C411yescharacteristic
36C4×D212—21no—
37C4×D212—21no—
38C4×D212D412yes—
39C4×D212Q812yes—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719921—21no—
42C4⋊#88322—21no—
43C4⋊#88322D412yes—
44#13133137—22no—
45#13133137—22no—
46#13133129—24no—
47C4×#88322—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50C4×#88322—21no—
51C22×C8×C53—21no—
52C22×C8×C53D411yescharacteristic
53C22×C8×C53D411yescharacteristic
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C42×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53—42no—
61C4×C8×C53—21no—
62C4×C8×C53—21no—
63#95×C53—21no—
64C23×D53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4111yescharacteristic
70C22×Dic53—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212C4⋊C412yes—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C8×C53—21no—
84C2×C8×C53—21no—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53D812yes—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53M1612yescomplemented
96C42×C53—41no—
97C42×C53—41no—
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×Q811yescharacteristic
101C42×C53#4111yescharacteristic
102#11719921—22no—
103#11719921—22no—
104C4⋊Dic53—21no—
105C4⋊Dic53C4⋊C412yes—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322—22no—
109C2×#88322—21no—
110C2×#88322—22no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322M1612yescomplemented
118#11719912—48no—
119C23×C53C4×D411yescharacteristic
120C22×D53—22no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—41no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C2×C4⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53C22⋊Q811yescharacteristic
139C2×C4×C53C22⋊Q811yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53C2×M1611yescharacteristic
142C2×C4×C53C2×M1611yescharacteristic
143C2×C4×C53#13111yescharacteristic
144C2×C4×C53#13111yescharacteristic
145C2×C4×C53C2×D811yescharacteristic
146C2×C4×C53#13511yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—22no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152D212—22no—
153D212—22no—
154D212—41no—
155D212—41no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—42no—
159C4×D53—42no—
160C8×C53—22no—
161C8×C53—41no—
162C8×C53—81no—
163#88322—22no—
164#88322—42no—
165#88322—41no—
166#88322—42no—
167#8847—531nomaximal, Sylow
168C22×C53#38511yescharacteristic
169C22×C53#43511yescharacteristic
170C22×C53#43611yescharacteristic
171C22×C53C4×D811yescharacteristic
172C22×C53#44011yescharacteristic
173C22×C53#44311yescharacteristic
174C22×C53#44411yescharacteristic
175C2×D53—42no—
176C2×D53—42no—
177C2×D53—42no—
178C2×D53—81no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—42no—
185C4×C53—81no—
186C4×C53#42311yescharacteristic
187C4×C53#42411yescharacteristic
188C4×C53#43511yescharacteristic
189C4×C53#43611yescharacteristic
190C4×C53#44711yescharacteristic
191C4×C53#45211yescharacteristic
192C4×C53#47411yescharacteristic
193C4×C53#47811yescharacteristic
194Dic53—42no—
195C2×#436—531no—
196#1939D5311yescharacteristic, complemented
197C4×#102—531no—
198C2×#335—531no—
199C2×D4⋊C8—531no—
200#1279—531no—
201#1273—531no—
202C2×C53#158111yescharacteristic
203C2×C53#158811yescharacteristic
204C2×C53#158911yescharacteristic
205C2×C53#159211yescharacteristic
206C2×C53#185111yescharacteristic
207C2×C53#186211yescharacteristic
208C2×C53#186611yescharacteristic
209D53—82no—
210C22×M16—1061no—
211C2×C4×D4—531no—
212#436—1062no—
213#436—1062no—
214C2×C4⋊C8—21no—
215C2×C4⋊C8—531no—
216C2×C4⋊C8C2×D5311yescharacteristic
217C2×#102—531no—
218C2×#102—531no—
219C2×C22⋊C8—1061no—
220C2×C4×C8—531no—
221C2×C4×C8—531no—
222C2×C4×C8C2×D5311yescharacteristic
223C4×C4⋊C4C2×D5311yescharacteristic
224#340—531no—
225#340—531no—
226#336—21no—
227#335—1062no—
228D4⋊C8—1062no—
229C53#884711yescharacteristic, complemented, Sylow
230C22×D4—531no—
231C23×C4—1061no—
232C2×M16—1064no—
233C2×M16—2122no—
234C2×M16—2122no—
235C22×C8—21no—
236C22×C8—531no—
237C22×C8—531no—
238C22×C8—1061no—
239C22×C8—1061no—
240C22×C8C22⋊D5311yescharacteristic
241C22×C8C22⋊D5311yescharacteristic
242C4×D4—532no—
243C4×D4—532no—
244C4×D4—1061no—
245C4×D4—1061no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4C4×D5311yescharacteristic
248C2×C4⋊C4C4×D5311yescharacteristic
249C2×C22⋊C4—1061no—
250C2×C42—21no—
251C2×C42C22×D5311yescharacteristic
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—42no—
255C4⋊C8—532no—
256C4⋊C8—1061no—
257#102—1062no—
258#102—1062no—
259C22⋊C8—1064no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—1061no—
263C4×C8—1061no—
264C4×C8—1061no—
265C4×C8—1061no—
266#95—21no—
267C24—1061no—
268C2×D4—532no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—1061no—
277C22×C4—1062no—
278C22×C4—1062no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4—2121no—
283C22×C4C2×C4×D5311yescharacteristic
284C22×C4#1171994511yescharacteristic
285C22×C4C2×C22⋊D5311yescharacteristic
286M16—2128no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—532no—
296C2×C8—1061no—
297C2×C8—1061no—
298C2×C8—1062no—
299C2×C8—1061no—
300C2×C8—1062no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—2121no—
304C2×C8—2121no—
305C2×C8—2121no—
306C2×C8—2121no—
307C2×C8#1171992212yes—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—532no—
312C4⋊C4—1061no—
313C4⋊C4#1171991212yescomplemented
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C42—41no—
317C42—41no—
318C42D4×D5311yescharacteristic
319C42D4×D5311yescharacteristic
320C42Q8×D5311yescharacteristic
321C42#1171994911yescharacteristic
322C23—1062no—
323C23—2121no—
324C23—2121no—
325C23—2121no—
326C23—2121no—
327C23—2121no—
328C23—2121no—
329C23C4×C22⋊D5311yescharacteristic
330D4—1062no—
331D4—1062no—
332D4—2121no—
333D4—2121no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—1062no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4C4⋊C4×D5311yescharacteristic
350C2×C4#1313317911yescharacteristic
351C2×C4#1313318511yescharacteristic
352C2×C4#1313319311yescharacteristic
353C2×C4#1313319511yescharacteristic
354C2×C4#1313320211yescharacteristic
355C2×C4M16×D5311yescharacteristic
356C2×C4C2×#1171991211yescharacteristic
357C2×C4#1313322311yescharacteristic
358C2×C4#1313322511yescharacteristic
359C2×C4C2×#1171992211yescharacteristic
360C2×C4#1313323711yescharacteristic
361C2×C4#1313324211yescharacteristic
362C2×C4#1313324811yescharacteristic
363C8—22no—
364C8—41no—
365C8—81no—
366C8—1062no—
367C8—2122no—
368C8—2122no—
369C8—2121no—
370C22—4241no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22—2122no—
375C22—2122no—
376C22—2122no—
377C22#1681049411yescharacteristic
378C22#1681069411yescharacteristic
379C22#1681069911yescharacteristic
380C22C4×#1171992211yescharacteristic
381C22#1681072111yescharacteristic
382C22#1681073811yescharacteristic
383C22#1681074311yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—81no—
388C4—2122no—
389C4#1681064511yescharacteristic
390C4#1681064811yescharacteristic
391C4#1681069311yescharacteristic
392C4#1681070011yescharacteristic
393C4#1681075611yescharacteristic
394C4#1681078311yescharacteristic
395C4#1681090011yescharacteristic
396C4#1681092011yescharacteristic
397C2—4242no—
398C2#2027354411yescharacteristic
399C2#2027359311yescharacteristic
400C2#2027359811yescharacteristic
401C2#2027361511yescharacteristic
402C2#2027489611yescharacteristic
403C2#2027496111yescharacteristic
404C2#2027498911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1939C2
socleC23×C53C4×D4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1939D53
hypocenterC53#8847
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1939

Complements

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

NH ≅ G/Ntype
3. C2×#16810375C2semidirect
5. #20272556C2semidirect
6. #20272542C2semidirect
7. C53×#1939C2semidirect
95. C4⋊C4×C53M16semidirect
117. C2×#88322M16semidirect
196. #1939D53semidirect
229. C53#8847semidirect
313. C4⋊C4#11719912semidirect

Maximal subgroups

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

subgroupindex
C2×#168107022
C4×#131331372
C2×#168103752
C2×#168103502
#202725562
#202725422
C53×#19392
#884753