Finite Group Databasebeta

← #27280281

Subgroups — #27280281

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810824C211yesmaximal, characteristic
2C2×#16810747C211yesmaximal, characteristic, complemented
3#20275520C211yesmaximal, characteristic
4#20274717C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272549C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C22×#11719913—21no—
9C2×#13133214C2211yescharacteristic
10C2×C4×Dic106C2211yescharacteristic
11#16810824—22no—
12#16810824C414yescomplemented
13#16810747—22no—
14#16810747—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133146—21no—
17C2×#13133140—21no—
18C2×#13133139—21no—
19C2×#13133137—21no—
20C2×#13133137—21no—
21C2×C4×#88322C2211yescharacteristic
22C2×C8×Dic53—21no—
23#16810385—21no—
24#16810366—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C53×#384C2211yescharacteristic
28C22×Dic106D411yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719913—24no—
33C2×#11719913—42no—
34C2×#11719913—42no—
35C22×#88322—21no—
36C22×#88322D411yescharacteristic
37C22×#88322D411yescharacteristic
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214C2×C412yes—
42C4×Dic106—21no—
43C4×Dic106—21no—
44C4×Dic106C2×C412yes—
45C4×Dic106D412yes—
46C2×C4⋊Dic53D411yescharacteristic
47C2×#11719919—21no—
48C2×C4×Dic53—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51#13133146—24no—
52#13133140—42no—
53#13133139—24no—
54#13133137—42no—
55#13133137—42no—
56C4×#88322—21no—
57C4×#88322—21no—
58C8×Dic53—24no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53C2×C412yes—
68C4⋊C8×C53D412yes—
69C23×D53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4111yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×D212—22no—
77C2×D212—22no—
78C2×D212—22no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—81no—
91C2×D212—81no—
92C2×Dic106—21no—
93C2×Dic106—21no—
94C2×Dic106—21no—
95C2×Dic106C22⋊C412yes—
96C2×Dic106D812yes—
97C2×Dic106SD1612yes—
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—81no—
109C42×C53C22×C411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53#4111yescharacteristic
113C4⋊Dic53—21no—
114C4⋊Dic53C22⋊C412yes—
115#11719919—22no—
116#11719919—22no—
117C4×Dic53—22no—
118C4×Dic53—22no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322SD1612yescomplemented
128#11719913—48no—
129C23×C53#12111yescharacteristic
130C22×D53—42no—
131C22×D53—81no—
132C22×D53—81no—
133C22×D53—81no—
134C22×D53—81no—
135C22×D53—81no—
136C22×D53—81no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12311yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53C2×SD1611yescharacteristic
150C2×C4×C53C2×SD1611yescharacteristic
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162D212—42no—
163D212—42no—
164D212—42no—
165D212—42no—
166D212—82no—
167D212—82no—
168D212—81no—
169D212—81no—
170D212—81no—
171D212—81no—
172Dic106—22no—
173Dic106—22no—
174Dic106—22no—
175Dic106#10214yescomplemented
176C8×C53—42no—
177C8×C53—42no—
178#88322—22no—
179#88322—41no—
180#88322—81no—
181#9222—531nomaximal, Sylow
182C22×C53#38611yescharacteristic
183C22×C53#46011yescharacteristic
184C22×C53#46411yescharacteristic
185C22×C53#46511yescharacteristic
186C22×C53#46611yescharacteristic
187C22×C53#47111yescharacteristic
188C22×C53#47211yescharacteristic
189C2×D53—82no—
190C2×D53—82no—
191C2×D53—82no—
192C2×D53—161no—
193C2×D53—161no—
194C2×D53—161no—
195C2×D53—161no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—161no—
199C4×C53C2×#10211yescharacteristic
200C4×C53#41611yescharacteristic
201C4×C53C4×SD1611yescharacteristic
202C4×C53#44011yescharacteristic
203C4×C53#44811yescharacteristic
204C4×C53#44911yescharacteristic
205C4×C53#46511yescharacteristic
206C4×C53#46611yescharacteristic
207Dic53—24no—
208Dic53—42no—
209C2×#460—531no—
210C2×C8×Q8—531no—
211#1957—531no—
212#1803—531no—
213C2×#331—531no—
214#1296D5311yescharacteristic, complemented
215#1277—531no—
216C2×C53#166611yescharacteristic
217C2×C53#166811yescharacteristic
218C2×C53#167811yescharacteristic
219C2×C53#168211yescharacteristic
220C2×C53#188011yescharacteristic
221C2×C53#189311yescharacteristic
222C2×C53#190711yescharacteristic
223D53—162no—
224C22×SD16—1061no—
225C2×#127—531no—
226C2×C4×Q8—531no—
227#460—534no—
228#460—1062no—
229C8×Q8—1062no—
230C8×Q8—1062no—
231C2×C4⋊C8—531no—
232C2×C4⋊C8—1061no—
233C2×C4⋊C8C2×D5311yescharacteristic
234C2×C4⋊C8C2×D5311yescharacteristic
235C2×#103—1061no—
236C2×#102—1061no—
237C2×#102—1061no—
238C2×#102—1061no—
239C2×C4×C8—531no—
240C2×C4×C8—1061no—
241#384C2×D5311yescharacteristic
242#340—1061no—
243#331—1062no—
244C53#922211yescharacteristic, complemented, Sylow
245C22×Q8—531no—
246C22×D4—1061no—
247C22×D4—1061no—
248C22×D4—1061no—
249C2×SD16—1064no—
250C2×SD16—2122no—
251C2×SD16—2122no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—531no—
255C22×C8—531no—
256C22×C8—1061no—
257#127—532no—
258#127—1061no—
259#127—1061no—
260#127—1061no—
261C4×Q8—532no—
262C4×Q8—532no—
263C4×Q8—1061no—
264C4×Q8—1061no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—1061no—
270C2×C42—1061no—
271C2×C42C22×D5311yescharacteristic
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—1064no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8C4×D5312yes—
278C4⋊C8D21212yes—
279#103—2122no—
280#102—1064no—
281#102—2122no—
282#102—2122no—
283C4×C8—1064no—
284C4×C8—1061no—
285C4×C8—1061no—
286C24—2121no—
287C2×Q8—532no—
288C2×Q8—532no—
289C2×Q8—532no—
290C2×Q8—1061no—
291C2×Q8—1061no—
292C2×Q8—1061no—
293C2×D4—1062no—
294C2×D4—1062no—
295C2×D4—1062no—
296C2×D4—4241no—
297C2×D4—4241no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2121no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4—2121no—
309C22×C4—41no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4#1171994511yescharacteristic
313C22×C4D4×D5311yescharacteristic
314C22×C4D4×D5311yescharacteristic
315SD16—2128no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—532no—
323C2×C8—1061no—
324C2×C8—1061no—
325C2×C8—1061no—
326C2×C8—1061no—
327C2×C8—1061no—
328C2×C8—2121no—
329C2×C8—2121no—
330C2×C8—2121no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—81no—
336C4⋊C4—532no—
337C4⋊C4—1062no—
338C4⋊C4—1061no—
339C4⋊C4—1062no—
340C42—1062no—
341C42—1062no—
342C42C2×C4×D5311yescharacteristic
343C42C2×D21211yescharacteristic
344C42D4×D5311yescharacteristic
345C42#1171994911yescharacteristic
346C23—4241no—
347C23—4241no—
348C23—4241no—
349C23—4241no—
350C23—4241no—
351C23—4241no—
352C23—2122no—
353C23#1313319011yescharacteristic
354Q8—534no—
355Q8—1062no—
356Q8—1062no—
357Q8—1062no—
358D4—4242no—
359D4—4242no—
360D4—4241no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—2122no—
365D4—2122no—
366D4—2122no—
367D4—2122no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—1062no—
374C2×C4—1062no—
375C2×C4—1062no—
376C2×C4—1062no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4C22⋊C4×D5311yescharacteristic
380C2×C4#1313317911yescharacteristic
381C2×C4C4×D21211yescharacteristic
382C2×C4#1313318711yescharacteristic
383C2×C4#1313319211yescharacteristic
384C2×C4#1313320211yescharacteristic
385C2×C4D8×D5311yescharacteristic
386C2×C4SD16×D5311yescharacteristic
387C2×C4SD16×D5311yescharacteristic
388C2×C4#1313323811yescharacteristic
389C2×C4#1313323811yescharacteristic
390C2×C4#1313324011yescharacteristic
391C2×C4#1313324111yescharacteristic
392C2×C4#1313324911yescharacteristic
393C8—42no—
394C8—42no—
395C8—1062no—
396C8—4241no—
397C8—2121no—
398C22—8481no—
399C22—8481no—
400C22—8481no—
401C22—8481no—
402C22—4242no—
403C22—4242no—
404C22—4242no—
405C22#1681050111yescharacteristic
406C22#1681082211yescharacteristic
407C22#1681084811yescharacteristic
408C22#1681085511yescharacteristic
409C22#1681085811yescharacteristic
410C22#1681088911yescharacteristic
411C22#1681089211yescharacteristic
412C4—21no—
413C4—21no—
414C4—161no—
415C4—1064no—
416C4—2122no—
417C4#102×D5311yescharacteristic
418C4#1681061811yescharacteristic
419C4#1681071011yescharacteristic
420C4#1681072211yescharacteristic
421C4#1681076311yescharacteristic
422C4#1681076911yescharacteristic
423C4#1681085311yescharacteristic
424C4#1681086011yescharacteristic
425C2—8482no—
426C2#2027406611yescharacteristic
427C2#2027407411yescharacteristic
428C2#2027413411yescharacteristic
429C2#2027416211yescharacteristic
430C2#2027507111yescharacteristic
431C2#2027514211yescharacteristic
432C2#2027522411yescharacteristic
subgroupG/H
centerC23#13133190
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#121
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#9222
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. C2×#16810747C2semidirect
4. #20274717C2semidirect
6. #20272549C2semidirect
7. C53×#1296C2semidirect
12. #16810824C4semidirect
127. C2×#88322SD16semidirect
175. Dic106#102semidirect
214. #1296D53semidirect
244. C53#9222semidirect

Maximal subgroups

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

subgroupindex
C2×#168108242
C2×#168107472
#202755202
#202747172
C2×#168103662
#202725492
C53×#12962
#922253