Finite Group Databasebeta

← #27278421

Subgroups — #27278421

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283450C211yesmaximal, characteristic
2#20275491C211yesmaximal, characteristic, complemented
3#20273473C211yesmaximal, characteristic, complemented
4#20273141C211yesmaximal, characteristic
5C2×#16810350C211yesmaximal, characteristic
6#20272551C211yesmaximal, characteristic, complemented
7C53×#1301C211yesmaximal, characteristic, complemented
8C2×M16×D53—21no—
9C2×C4×D212C2211yescharacteristic
10#16810700—21no—
11#16810693—22no—
12#16810688—21no—
13C2×#13133153—21no—
14#16810647C2211yescharacteristic
15#16810586—21no—
16C4×#11719917C2211yescharacteristic
17C2×C4×#88322C2211yescharacteristic
18#16810388—21no—
19#16810375—22no—
20#16810352—21no—
21#16810352—21no—
22#16810350C411yescharacteristic
23#16810350C411yescharacteristic
24#16810350C411yescharacteristic
25#16810350C411yescharacteristic
26C53×#431C2211yescharacteristic
27C53×#423C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C22×D212Q811yescharacteristic
30C22×C4×D53—21no—
31C2×#11719917—21no—
32C2×#11719917D411yescharacteristic
33C2×#11719917D411yescharacteristic
34C2×#11719912—22no—
35M16×D53—42no—
36M16×D53—42no—
37C22×#88322—21no—
38C2×C8×D53—22no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212C2×C411yescharacteristic
42C4×D212C2×C411yescharacteristic
43C4×D212D411yescharacteristic
44C4×D212D411yescharacteristic
45C2×C4⋊Dic53Q811yescharacteristic
46C2×#11719921—21no—
47#13133153—42no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50#13133129—22no—
51#13133129—22no—
52#13133128—21no—
53C4×#88322—21no—
54C4×#88322—21no—
55C4×#88322C2×C411yescharacteristic
56C4×#88322C2×C411yescharacteristic
57C2×M16×C53—21no—
58C22×C8×C53—21no—
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53C2×C411yescharacteristic
64C4⋊C8×C53C2×C411yescharacteristic
65C4⋊C8×C53D412yes—
66C22⋊C8×C53—21no—
67C4×C8×C53—21no—
68C23×D53—21no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×Q811yescharacteristic
71C22×C4×C53#4111yescharacteristic
72C22×Dic53—21no—
73C2×D212—22no—
74C2×D212—21no—
75C2×D212—41no—
76C2×D212C4⋊C411yescharacteristic
77C2×D212C4⋊C411yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86M16×C53—41no—
87M16×C53—41no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—22no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C42×C53C22×C411yescharacteristic
95C42×C53C2×D411yescharacteristic
96C42×C53C2×D411yescharacteristic
97C42×C53#4111yescharacteristic
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C4⋊C411yescharacteristic
104C4⋊Dic53C4⋊C411yescharacteristic
105#11719917—21no—
106#11719917—21no—
107#11719917—21no—
108#11719917—41no—
109#11719917—41no—
110#11719917D812yes—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322—42no—
116C2×#88322—41no—
117#11719912—44no—
118C8×D53—44no—
119C23×C53C22⋊Q811yescharacteristic
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—42no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—81no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53C2×C4⋊C411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53C22⋊Q811yescharacteristic
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53C2×D811yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—42no—
147D212#10812yescomplemented
148C4×D53—41no—
149C4×D53—41no—
150C4×D53—41no—
151C4×D53—41no—
152C4×D53—41no—
153C4×D53—41no—
154C4×D53—41no—
155C4×D53—41no—
156C8×C53—41no—
157C8×C53—41no—
158C8×C53—42no—
159C8×C53—81no—
160#88322—22no—
161#88322—22no—
162#88322—41no—
163#88322—41no—
164#88322—81no—
165#8906—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53#38511yescharacteristic
169C22×C53#47411yescharacteristic
170C22×C53#47811yescharacteristic
171C2×D53—22no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—82no—
178C2×D53—82no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53C2×#10811yescharacteristic
183C4×C53#43011yescharacteristic
184C4×C53C4×D811yescharacteristic
185C4×C53#44011yescharacteristic
186C4×C53#44711yescharacteristic
187C4×C53#45211yescharacteristic
188Dic53—41no—
189Dic53—41no—
190D4×M16—531no—
191#1950—531no—
192#1567—531no—
193#1486—531no—
194C2×D4⋊C8—531no—
195#1301D5311yescharacteristic, complemented
196#1278—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53#185511yescharacteristic
200C2×C53#186111yescharacteristic
201C2×C53#186211yescharacteristic
202D53—42no—
203D53—81no—
204C22×M16—1061no—
205C2×C4×D4—531no—
206#436—1061no—
207#435—1062no—
208C8×D4—1061no—
209#431C2×D5311yescharacteristic
210C2×#108—1061no—
211#423—531no—
212#423C2×D5311yescharacteristic
213C2×C4⋊C8C2×D5311yescharacteristic
214#407—1061no—
215C4×M16—531no—
216C2×C4×C8—531no—
217#341—1061no—
218#335—1062no—
219D4⋊C8—531no—
220D4⋊C8—531no—
221D4⋊C8—531no—
222D4⋊C8—531no—
223D4⋊C8—1061no—
224D4⋊C8—1061no—
225C53#890611yescharacteristic, complemented, Sylow
226C22×D4—531no—
227C23×C4—1061no—
228C2×M16—21no—
229C2×M16—531no—
230C2×M16—531no—
231C2×M16—1062no—
232C2×M16—1061no—
233C2×M16—2122no—
234C2×M16—2122no—
235C22×C8—21no—
236C22×C8—1062no—
237C22×C8—1061no—
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—531no—
241C4×D4—531no—
242C4×D4—1061no—
243C4×D4—1061no—
244C2×C4⋊C4—531no—
245C2×C22⋊C4—1061no—
246C2×C42C22×D5311yescharacteristic
247#108—2122no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—1061no—
252C4⋊C8—1061no—
253C4⋊C8C4×D5311yescharacteristic
254C4⋊C8C4×D5311yescharacteristic
255C4⋊C8D21212yes—
256C22⋊C8—21no—
257C22⋊C8—1062no—
258C22⋊C8—1062no—
259#97—1061no—
260C4×C8—21no—
261C4×C8—531no—
262C4×C8—531no—
263C4×C8—1061no—
264C4×C8—1061no—
265C24—1061no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—1062no—
269C2×D4—1061no—
270C2×D4—2121no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4#1171994511yescharacteristic
281C22×C4D4×D5311yescharacteristic
282C22×C4Q8×D5311yescharacteristic
283M16—41no—
284M16—41no—
285M16—532no—
286M16—1061no—
287M16—1061no—
288M16—1061no—
289M16—2124no—
290M16—2121no—
291M16—2121no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—2124no—
303C2×C8—2122no—
304C2×C8—2121no—
305C4⋊C4—531no—
306C4⋊C4—531no—
307C4⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C42C2×C4×D5311yescharacteristic
313C42C2×D21211yescharacteristic
314C42D4×D5311yescharacteristic
315C42#1171994711yescharacteristic
316C23—1061no—
317C23—1061no—
318C23—4241no—
319C23—2122no—
320C23—2121no—
321C23—2121no—
322C23#1313319511yescharacteristic
323D4—532no—
324D4—1061no—
325D4—1061no—
326D4—1061no—
327D4—2122no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4C4⋊C4×D5311yescharacteristic
344C2×C4#1313317711yescharacteristic
345C2×C4C4×D21211yescharacteristic
346C2×C4#1313318711yescharacteristic
347C2×C4#1313319711yescharacteristic
348C2×C4#1313319911yescharacteristic
349C2×C4D8×D5311yescharacteristic
350C2×C4#1313323811yescharacteristic
351C2×C4#1313324411yescharacteristic
352C2×C4#1313324911yescharacteristic
353C8—42no—
354C8—41no—
355C8—41no—
356C8—81no—
357C8—1062no—
358C8—1062no—
359C8—4241no—
360C8—2121no—
361C8—2121no—
362C22—21no—
363C22—21no—
364C22—1062no—
365C22—4242no—
366C22—4242no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22#1681049711yescharacteristic
373C22#1681090211yescharacteristic
374C22#1681092211yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—2121no—
379C4—2121no—
380C4#108×D5311yescharacteristic
381C4#1681067011yescharacteristic
382C4#1681070311yescharacteristic
383C4#1681072611yescharacteristic
384C4#1681075811yescharacteristic
385C4#1681078811yescharacteristic
386C2—21no—
387C2—21no—
388C2—4241no—
389C2—2122no—
390C2#2027492111yescharacteristic
391C2#2027495811yescharacteristic
392C2#2027496311yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1301C2
socleC22×C53#385
prime radicalC22×D212Q8
cosocleC2×C42×C53C23
hypercenter#1301D53
hypocenterC53#8906
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
2. #20275491C2semidirect
3. #20273473C2semidirect
6. #20272551C2semidirect
7. C53×#1301C2semidirect
147. D212#108semidirect
195. #1301D53semidirect
225. C53#8906semidirect

Maximal subgroups

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

subgroupindex
#202834502
#202754912
#202734732
#202731412
C2×#168103502
#202725512
C53×#13012
#890653