Finite Group Databasebeta

← #27285329

Subgroups — #27285329

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810975C211yesmaximal, characteristic, complemented
2C2×#16810900C211yesmaximal, characteristic
3#20279884C211yesmaximal, characteristic
4C2×#16810371C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1981C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810975—22no—
11#16810975—22no—
12#16810900—22no—
13#16810900—22no—
14C2×#13133150—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133140—22no—
17C2×#13133137—21no—
18C2×C4×#88322C2211yescharacteristic
19#16810536—22no—
20#16810525—21no—
21#16810489—21no—
22#16810480—22no—
23#16810371—22no—
24#16810350—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×#103×C53—22no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×Dic106D412yes—
30C22×D212D411yescharacteristic
31C22×C4×D53—21no—
32C22×#88322—21no—
33C22×#88322—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53D411yescharacteristic
45C2×#11719921—24no—
46C2×#11719921—21no—
47#13133150—24no—
48C4⋊#88322—21no—
49C4⋊#88322D412yes—
50#13133140—28no—
51#13133137—24no—
52C4×#88322—21no—
53C4×#88322D412yes—
54#13133124—24no—
55C22×Q8×C53—21no—
56C22×C8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53D412yes—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53D412yes—
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68#103×C53—44no—
69#95×C53—21no—
70C23×D53—21no—
71C2×Q8×C53—22no—
72C2×Q8×C53—41no—
73C2×Q8×C53—42no—
74C2×Q8×C53—42no—
75C22×C4×C53—22no—
76C22×C4×C53—41no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—22no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212Q1612yes—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—42no—
91C2×C4×D53—42no—
92C2×Dic106—22no—
93C2×Dic106—22no—
94C2×Dic106—44no—
95C2×C8×C53—41no—
96C2×C8×C53—42no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C42×C53—41no—
104C42×C53—41no—
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109#11719921—41no—
110#11719921—41no—
111#11719921—84no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118C2×#88322—22no—
119C2×#88322—22no—
120C2×#88322—42no—
121C2×#88322—41no—
122C2×#88322—41no—
123C23×C53#12011yescharacteristic
124C22×D53—22no—
125C22×D53—41no—
126C22×D53—42no—
127C22×D53—41no—
128C22×D53—42no—
129Q8×C53—44no—
130Q8×C53—81no—
131Q8×C53—81no—
132C2×C4×C53—22no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—82no—
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12711yescharacteristic
144C2×C4×C53C2×Q1611yescharacteristic
145C2×C4×C53#13512yes—
146C2×C4×C53#13511yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)12yes—
148C2×C4×C53#13711yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C2×Dic53—42no—
151C2×Dic53—41no—
152C2×Dic53—42no—
153C2×Dic53—44no—
154D212—22no—
155D212—22no—
156D212—41no—
157D212—41no—
158C4×D53—82no—
159C4×D53—81no—
160C4×D53—81no—
161Dic106—44no—
162Dic106—82no—
163C8×C53—81no—
164#88322—42no—
165#88322—42no—
166#10072—531nomaximal, Sylow
167C22×C53#39211yescharacteristic
168C22×C53#44912yes—
169C22×C53#45111yescharacteristic
170C22×C53#45211yescharacteristic
171C22×C53#45512yes—
172C2×D53—44no—
173C2×D53—42no—
174C2×D53—82no—
175C2×D53—81no—
176C2×D53—81no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—82no—
180C4×C53—82no—
181C4×C53#45111yescharacteristic
182C4×C53#45211yescharacteristic
183C4×C53#46312yes—
184C4×C53#46711yescharacteristic
185C4×C53#47011yescharacteristic
186C4×C53#49511yescharacteristic
187C4×C53#49611yescharacteristic
188Dic53—82no—
189Dic53—81no—
190C2×#487—531no—
191C2×#474—531no—
192#2563—531no—
193#1981D5311yescharacteristic, complemented
194C2×#333—531no—
195C2×D4⋊C8—531no—
196#1279—531no—
197C2×C53#164412yes—
198C2×C53#164612yes—
199C2×C53#200111yescharacteristic
200C2×C53#200411yescharacteristic
201C2×C53#201211yescharacteristic
202D53—82no—
203C2×C4⋊Q8—531no—
204C2×C4⋊Q8C2×D5311yescharacteristic
205C2×C4×D4—531no—
206#487—1062no—
207#487—1062no—
208#474—1062no—
209#474—1062no—
210C2×#107—1061no—
211C2×C4⋊C8—531no—
212C2×C4⋊C8C2×D5311yescharacteristic
213C2×#103—22no—
214C2×#103—1062no—
215C2×#102—1061no—
216C2×C4×C8—531no—
217#393—1062no—
218#390—1061no—
219#382—1061no—
220#380—1062no—
221C4×C4⋊C4C2×D5311yescharacteristic
222#333—1062no—
223D4⋊C8—1062no—
224C53#1007211yescharacteristic, complemented, Sylow
225C22×Q8—21no—
226C22×Q8—532no—
227C22×D4—531no—
228C23×C4—1061no—
229C22×C8—21no—
230C22×C8—1061no—
231C22×C8—1061no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—1061no—
236C4⋊Q8—1061no—
237C4⋊Q8—1061no—
238C4⋊Q8—1061no—
239C4⋊Q8D21212yes—
240C4×D4—1061no—
241C4×D4—1061no—
242C4×D4—1061no—
243C4×D4—1061no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4C22⋊D5312yes—
250C2×C22⋊C4—1064no—
251C2×C22⋊C4—1061no—
252C2×C42—21no—
253C2×C42C22×D5311yescharacteristic
254#107—1064no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—532no—
258C4⋊C8—1061no—
259#103—44no—
260#103—1068no—
261#102—1064no—
262C4×C8—532no—
263C4×C8—1061no—
264#95—21no—
265#95—1064no—
266C24—1061no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—1062no—
272C2×Q8—1062no—
273C2×Q8—2124no—
274C2×D4—532no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C2×D4—1061no—
280C22×C4—22no—
281C22×C4—41no—
282C22×C4—1061no—
283C22×C4—1062no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4—2122no—
287C22×C4—2122no—
288C22×C4D4×D5311yescharacteristic
289C22×C4D4×D5311yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—1062no—
294C2×C8—1062no—
295C2×C8—2122no—
296C2×C8—2121no—
297C2×C8—2121no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C22⋊C4—4244no—
311C22⋊C4—2121no—
312C22⋊C4—2121no—
313C42—41no—
314C42—41no—
315C42C2×D21211yescharacteristic
316C42#1171994511yescharacteristic
317C42D4×D5311yescharacteristic
318C42D4×D5311yescharacteristic
319C23—1062no—
320C23—2122no—
321C23—2121no—
322C23—2121no—
323C23—2122no—
324C23#1313318511yescharacteristic
325Q8—44no—
326Q8—81no—
327Q8—81no—
328Q8—4242no—
329Q8—2124no—
330D4—1062no—
331D4—1062no—
332D4—2121no—
333D4—2121no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—4241no—
341C2×C4—4242no—
342C2×C4—4241no—
343C2×C4—2122no—
344C2×C4—2121no—
345C2×C4—2122no—
346C2×C4—2124no—
347C2×C4#1313318711yescharacteristic
348C2×C4#1313318711yescharacteristic
349C2×C4#1313319011yescharacteristic
350C2×C4#1313319011yescharacteristic
351C2×C4#1313319111yescharacteristic
352C2×C4#1313321311yescharacteristic
353C2×C4Q16×D5311yescharacteristic
354C2×C4#1313323712yes—
355C2×C4#1313323811yescharacteristic
356C2×C4#1313324312yes—
357C2×C4#1313324611yescharacteristic
358C2×C4#1313324911yescharacteristic
359C8—81no—
360C8—2122no—
361C8—2122no—
362C22—4242no—
363C22—4241no—
364C22—4241no—
365C22—2124no—
366C22—2122no—
367C22#1681053211yescharacteristic
368C22#1681076612yes—
369C22#1681077711yescharacteristic
370C22#1681078211yescharacteristic
371C22#1681079912yes—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—82no—
376C4—4242no—
377C4—4241no—
378C4#1681078011yescharacteristic
379C4#1681078811yescharacteristic
380C4#1681083812yes—
381C4#1681086311yescharacteristic
382C4#1681088011yescharacteristic
383C4#1681100511yescharacteristic
384C4#1681101011yescharacteristic
385C2—4242no—
386C2#2027392712yes—
387C2#2027394112yes—
388C2#2027572411yescharacteristic
389C2#2027574511yescharacteristic
390C2#2027579111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1981C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1981D53
hypocenterC53#10072
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
1. C2×#16810975C2semidirect
4. C2×#16810371C2semidirect
6. #20272556C2semidirect
7. C53×#1981C2semidirect
193. #1981D53semidirect
224. C53#10072semidirect

Maximal subgroups

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

subgroupindex
C2×#168109752
C2×#168109002
#202798842
C2×#168103712
C2×#168103502
#202725562
C53×#19812
#1007253