Finite Group Databasebeta

← #27286654

Subgroups — #27286654

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810970C211yesmaximal, characteristic
2C2×#16810918C211yesmaximal, characteristic, complemented
3C2×#16810823C211yesmaximal, characteristic
4C2×#16810371C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272550C211yesmaximal, characteristic, complemented
7C53×#1984C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133217C2211yescharacteristic
10C2×#13133214C2211yescharacteristic
11#16810970—22no—
12#16810970—22no—
13#16810918—22no—
14#16810918—22no—
15#16810823—22no—
16#16810823—22no—
17C2×#13133150—21no—
18C2×C4⋊#88322C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133141—21no—
21C2×#13133139—22no—
22C2×#13133137—21no—
23#16810371—22no—
24#16810366—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C53×#336—22no—
29C22×Dic106—21no—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212—21no—
33C2×#11719924—24no—
34C2×#11719924—42no—
35C2×#11719924—42no—
36C22×#88322—21no—
37C22×#88322—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217—21no—
41#13133217D412yes—
42#13133214—21no—
43#13133214—21no—
44#13133214—21no—
45#13133214D412yes—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48#13133150—24no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322D412yes—
52C4⋊#88322D412yes—
53#13133141—24no—
54#13133139—28no—
55#13133137—24no—
56C22×Q8×C53D411yescharacteristic
57C22×C8×C53—21no—
58C22×C8×C53D412yes—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4×C8×C53—21no—
70C4×C8×C53—21no—
71C23×D53—41no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53SD1612yes—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×Dic53—41no—
84C2×D212—22no—
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—42no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—42no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—82no—
97C2×Dic106—22no—
98C2×Dic106—41no—
99C2×Dic106—82no—
100C2×C8×C53—22no—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53—41no—
104C2×C8×C53—42no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C42×C53—41no—
110C42×C53—41no—
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115#11719924—48no—
116C4⋊Dic53—22no—
117C4⋊Dic53—22no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C2×#88322—22no—
121C2×#88322—22no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C23×C53#12111yescharacteristic
127C22×D53—42no—
128C22×D53—82no—
129C22×D53—81no—
130C22×D53—81no—
131C22×D53—82no—
132Q8×C53—22no—
133Q8×C53—22no—
134Q8×C53—41no—
135Q8×C53—41no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12711yescharacteristic
148C2×C4×C53C2×SD1611yescharacteristic
149C2×C4×C53#13512yes—
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C53#13712yes—
153C2×C4×C53#13711yescharacteristic
154C2×Dic53—42no—
155C2×Dic53—82no—
156D212—42no—
157D212—44no—
158D212—42no—
159D212—82no—
160D212—82no—
161D212—81no—
162D212—81no—
163D212—81no—
164D212—81no—
165Dic106—82no—
166Dic106—82no—
167C8×C53—42no—
168C8×C53—81no—
169#88322—42no—
170#88322—42no—
171#10269—531nomaximal, Sylow
172C22×C53#39211yescharacteristic
173C22×C53#46412yes—
174C22×C53#46511yescharacteristic
175C22×C53#46811yescharacteristic
176C22×C53#47212yes—
177C2×D53—82no—
178C2×D53—82no—
179C2×D53—82no—
180C2×D53—162no—
181C2×D53—162no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—42no—
185C4×C53—81no—
186C4×C53#44811yescharacteristic
187C4×C53#44911yescharacteristic
188C4×C53#45011yescharacteristic
189C4×C53#45212yes—
190C4×C53#45211yescharacteristic
191C4×C53#49511yescharacteristic
192C4×C53#49711yescharacteristic
193Dic53—82no—
194C2×#486—531no—
195C2×#477—531no—
196C2×#460—531no—
197#1984D5311yescharacteristic, complemented
198C2×#333—531no—
199C2×#331—531no—
200#1277—531no—
201C2×C53#167812yes—
202C2×C53#169812yes—
203C2×C53#200011yescharacteristic
204C2×C53#201511yescharacteristic
205C2×C53#201811yescharacteristic
206D53—162no—
207C22×SD16—1061no—
208C2×C4⋊Q8—531no—
209C2×#127—531no—
210C2×C4×Q8C2×D5311yescharacteristic
211#486—1062no—
212#486—1062no—
213#477—1062no—
214#477—1062no—
215#460—1062no—
216#460—1062no—
217C2×#107—1061no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8—531no—
220C2×C4⋊C8C2×D5311yescharacteristic
221C2×#103—1061no—
222C2×#102—1062no—
223C2×#102—1061no—
224C2×C4×C8C2×D5311yescharacteristic
225#336—22no—
226#333—1062no—
227#331—1062no—
228C53#1026911yescharacteristic, complemented, Sylow
229C22×Q8—1061no—
230C22×Q8D21211yescharacteristic
231C22×D4—1061no—
232C22×D4—1061no—
233C22×D4—1061no—
234C2×SD16—1064no—
235C2×SD16—2122no—
236C2×SD16—2122no—
237C22×C8—21no—
238C22×C8—1061no—
239C22×C8—1061no—
240C22×C8C22⋊D5312yes—
241C4⋊Q8—532no—
242C4⋊Q8—1061no—
243C4⋊Q8—1061no—
244C4⋊Q8—1061no—
245#127—532no—
246#127—1061no—
247#127—1061no—
248#127—1061no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4D21211yescharacteristic
257C2×C42—21no—
258C2×C42C22×D5311yescharacteristic
259#107—1064no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—532no—
263C4⋊C8—532no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266#103—1064no—
267#102—1064no—
268#102—1068no—
269C4×C8—21no—
270C4×C8—21no—
271C24—2121no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—1062no—
278C2×Q8—4242no—
279C2×Q8—2121no—
280C2×Q8#1171991312yes—
281C2×D4—1062no—
282C2×D4—1062no—
283C2×D4—1062no—
284C2×D4—4242no—
285C2×D4—2121no—
286C2×D4—2121no—
287C2×D4—2121no—
288C2×D4—2121no—
289C2×D4—2122no—
290C2×D4—2121no—
291C2×D4—2121no—
292C2×D4—2122no—
293C2×D4—2121no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—2121no—
297C22×C4C2×D21211yescharacteristic
298C22×C4#1171994511yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300SD16—2128no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—1062no—
307C2×C8—1062no—
308C2×C8—2121no—
309C2×C8—2121no—
310C2×C8—2121no—
311C2×C8—2121no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C42—41no—
321C42—41no—
322C42D4×D5311yescharacteristic
323C42D4×D5311yescharacteristic
324C42D4×D5311yescharacteristic
325C42D4×D5311yescharacteristic
326C23—4242no—
327C23—4241no—
328C23—4241no—
329C23—4242no—
330C23—2122no—
331C23#1313319111yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—4242no—
337Q8—4242no—
338D4—4242no—
339D4—4242no—
340D4—4241no—
341D4—4241no—
342D4—4241no—
343D4—4241no—
344D4—2122no—
345D4—2124no—
346D4—2122no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—4242no—
354C2×C4—2122no—
355C2×C4#1313318511yescharacteristic
356C2×C4#1313318711yescharacteristic
357C2×C4#1313318711yescharacteristic
358C2×C4#1313319011yescharacteristic
359C2×C4#1313319011yescharacteristic
360C2×C4#1313321311yescharacteristic
361C2×C4C2×#1171991311yescharacteristic
362C2×C4#1313323712yes—
363C2×C4#1313323911yescharacteristic
364C2×C4#1313324511yescharacteristic
365C2×C4#1313324812yes—
366C2×C4#1313325011yescharacteristic
367C8—42no—
368C8—81no—
369C8—2122no—
370C8—2122no—
371C22—8482no—
372C22—8482no—
373C22—4242no—
374C22—4242no—
375C22—4242no—
376C22#1681053211yescharacteristic
377C22#1681084512yes—
378C22#1681085211yescharacteristic
379C22#1681086911yescharacteristic
380C22#1681089312yes—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—4242no—
386C4#1681076411yescharacteristic
387C4#1681077011yescharacteristic
388C4#1681077611yescharacteristic
389C4#1681078312yes—
390C4#1681078711yescharacteristic
391C4#1681100511yescharacteristic
392C4#1681101511yescharacteristic
393C2—8482no—
394C2#2027413112yes—
395C2#2027425512yes—
396C2#2027572011yescharacteristic
397C2#2027580811yescharacteristic
398C2#2027582411yescharacteristic
subgroupG/H
centerC23#13133191
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1984C2
socleC23×C53#121
prime radicalC2×#13133214C22
hypercenter#1984D53
hypocenterC53#10269
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. C2×#16810918C2semidirect
4. C2×#16810371C2semidirect
6. #20272550C2semidirect
7. C53×#1984C2semidirect
197. #1984D53semidirect
228. C53#10269semidirect

Maximal subgroups

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

subgroupindex
C2×#168109702
C2×#168109182
C2×#168108232
C2×#168103712
C2×#168103662
#202725502
C53×#19842
#1026953