Finite Group Databasebeta

← #27437962

Subgroups — #27437962

Order 13568.

1082 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811352C211yesmaximal, characteristic
2C2×#16811002C211yesmaximal, characteristic
3C2×#16810830C211yesmaximal, characteristic
4#20275814C214yesmaximal
5#20275652C211yesmaximal, characteristic
6#20275638C211yesmaximal, characteristic
7#20275059C214yesmaximal
8C4×#13133140C211yesmaximal, characteristic
9C53×#2568C211yesmaximal, characteristic
10C22×#11719925C2214yes—
11C22×#11719925C2212yes—
12#16811352—24no—
13#16811352—24no—
14C2×#13133217C2212yes—
15C2×#13133217C2211yescharacteristic
16C2×#13133208—21no—
17C2×#13133208—22no—
18C2×Q8×Dic53—21no—
19C2×C4×Dic106C2212yes—
20C2×C4×Dic106C2211yescharacteristic
21#16811002—24no—
22#16810830—24no—
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133141C2212yes—
25C2×#13133140C2212yes—
26C2×#13133140C2214yes—
27C2×C4×#88322C2211yescharacteristic
28#16810385C2212yes—
29#16810378C2214yes—
30C2×C4⋊Q8×C53C2211yescharacteristic
31C2×C4×Q8×C53C2211yescharacteristic
32C53×#400—22no—
33C53×#391—21no—
34C53×#391C2214yes—
35C53×#382—21no—
36C4×C4⋊C4×C53C2211yescharacteristic
37C22×Dic106C2311yescharacteristic
38C22×Dic106C2312yes—
39C2×#11719925—28no—
40C2×#11719925—24no—
41C2×#11719925—28no—
42C2×#11719925—44no—
43C2×#11719925—44no—
44C22×#88322C2312yes—
45C22×#88322C2312yes—
46#13133217—22no—
47#13133217—24no—
48#13133217—24no—
49#13133217D414yes—
50#13133208—216no—
51#13133208—28no—
52Q8×Dic53—28no—
53C4×Dic106—22no—
54C4×Dic106—24no—
55C4×Dic106—24no—
56C4×Dic106D414yes—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—22no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53C2311yescharacteristic
62C2×#11719919—22no—
63C2×#11719919—21no—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—21no—
66C4⋊#88322—22no—
67#13133141—24no—
68#13133140—24no—
69#13133140—44no—
70C4×#88322—22no—
71C22×Q8×C53D411yescharacteristic
72C22×Q8×C53C2312yes—
73C4⋊Q8×C53—22no—
74C4⋊Q8×C53—22no—
75C4×Q8×C53—22no—
76C4×Q8×C53—22no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53D411yescharacteristic
80C2×C4⋊C4×C53D412yes—
81C2×C4⋊C4×C53C2312yes—
82C2×C42×C53—21no—
83C2×C42×C53C2312yes—
84C2×C42×C53C2311yescharacteristic
85#95×C53—21no—
86#95×C53—24no—
87#95×C53—24no—
88C2×Q8×C53—22no—
89C2×Q8×C53—22no—
90C2×Q8×C53—22no—
91C2×Q8×C53—22no—
92C2×Q8×C53—24no—
93C2×Q8×C53—24no—
94C2×Q8×C53Q1614yescomplemented
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2411yescharacteristic
102C22×Dic53—21no—
103C22×Dic53—22no—
104C22×Dic53—21no—
105C2×Dic106—22no—
106C2×Dic106—24no—
107C2×Dic106—24no—
108C2×Dic106—22no—
109C2×Dic106—22no—
110C2×Dic106—21no—
111C2×Dic106C2×D414yes—
112C2×Dic106C2×D412yes—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—42no—
118C4⋊C4×C53—42no—
119C42×C53—22no—
120C42×C53—22no—
121C42×C53—42no—
122C42×C53C2×D412yes—
123C42×C53C2×D412yes—
124#11719925—416no—
125#11719925—48no—
126C4⋊Dic53—28no—
127C4⋊Dic53—28no—
128C4⋊Dic53—24no—
129C4⋊Dic53—24no—
130C4⋊Dic53—21no—
131C4⋊Dic53C2×D412yes—
132#11719919—28no—
133#11719919—24no—
134C4×Dic53—28no—
135C4×Dic53—24no—
136C2×#88322—22no—
137C2×#88322—22no—
138C2×#88322—22no—
139C2×#88322—24no—
140C2×#88322#4114yes—
141C23×C53C22×D411yescharacteristic
142Q8×C53—28no—
143Q8×C53—42no—
144Q8×C53—42no—
145Q8×C53—44no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—24no—
151C2×C4×C53—42no—
152C2×C4×C53—44no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53#12012yes—
157C2×C4×C53#12012yes—
158C2×C4×C53C2×Q1614yes—
159C2×C4×C53C2×Q1612yes—
160C2×C4×C53#13712yes—
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C22×D411yescharacteristic
163C2×C4×C53C2×#4112yes—
164C2×C4×C53H(5,𝔽2)12yes—
165C2×Dic53—24no—
166C2×Dic53—22no—
167C2×Dic53—22no—
168C2×Dic53—42no—
169C2×Dic53—44no—
170C2×Dic53—42no—
171Dic106—22no—
172Dic106—22no—
173Dic106—28no—
174Dic106—42no—
175Dic106—42no—
176Dic106—42no—
177#88322—24no—
178#88322—42no—
179#25634—531nomaximal, Sylow
180C22×C53C2×#12011yescharacteristic
181C22×C53#55012yes—
182C22×C53C22×Q1611yescharacteristic
183C22×C53C2×#13711yescharacteristic
184C22×C53#57812yes—
185C4×C53—24no—
186C4×C53—22no—
187C4×C53—42no—
188C4×C53—82no—
189C4×C53—82no—
190C4×C53#45114yes—
191C4×C53#53211yescharacteristic
192C4×C53#53511yescharacteristic
193C4×C53#54612yes—
194Dic53—42no—
195Dic53—44no—
196Dic53—42no—
197C2×#557—531no—
198C2×#494—531no—
199C2×#462—531no—
200#2568D5311yescharacteristic
201#2016—534no—
202#1984—531no—
203#1981—531no—
204#1878—534no—
205C4×#103—531no—
206C2×C53#243111yescharacteristic
207C2×C53C2×#45111yescharacteristic
208C2×C53#301111yescharacteristic
209C2×C53#339512yes—
210C2×C53#339812yes—
211C22×Q16—534no—
212C22×Q16—532no—
213#557—1064no—
214#557—1064no—
215C2×C4⋊Q8—531no—
216C2×C4⋊Q8—532no—
217C2×C4⋊Q8C2×D5311yescharacteristic
218C2×#125—1061no—
219C2×#125—1062no—
220C2×C4×Q8—531no—
221C2×C4×Q8—532no—
222C2×C4×Q8—1061no—
223C2×C4×Q8C2×D5311yescharacteristic
224#494—1064no—
225#462—1064no—
226C2×C4⋊C8—531no—
227C2×#103—532no—
228C2×#103—532no—
229C2×#103—534no—
230C2×C4×C8—531no—
231#400—22no—
232#391—21no—
233#391C2×D5314yes—
234#382—21no—
235C4×C4⋊C4C2×D5311yescharacteristic
236#340—532no—
237#336—534no—
238C53#2563411yescharacteristic, complemented, Sylow
239C22×Q8—531no—
240C22×Q8—532no—
241C22×Q8C22⋊D5311yescharacteristic
242C22×Q8C22×D5312yes—
243C2×Q16—1068no—
244C2×Q16—1064no—
245C2×Q16—1068no—
246C2×Q16—2124no—
247C2×Q16—2124no—
248C22×C8—532no—
249C22×C8—532no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—534no—
253C4⋊Q8—1064no—
254C4⋊Q8—1064no—
255C4⋊Q8—1062no—
256#125—10616no—
257#125—1068no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—534no—
261C4×Q8—1064no—
262C4×Q8—1064no—
263C4×Q8—1068no—
264C4×Q8—1062no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—1062no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1062no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1062no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4C22⋊D5311yescharacteristic
275C2×C4⋊C4C22⋊D5312yes—
276C2×C4⋊C4C22×D5312yes—
277C2×C42—21no—
278C2×C42—1062no—
279C2×C42—1061no—
280C2×C42C22×D5312yes—
281C2×C42C22×D5311yescharacteristic
282C4⋊C8—1062no—
283#103—1064no—
284#103—1064no—
285#103—2124no—
286C4×C8—1062no—
287#95—21no—
288#95—24no—
289#95—24no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—532no—
297C2×Q8—534no—
298C2×Q8—1061no—
299C2×Q8—1062no—
300C2×Q8—1062no—
301C2×Q8—1064no—
302C2×Q8—1064no—
303C2×Q8—1062no—
304C2×Q8#1171992514yescomplemented
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—1061no—
308C22×C4—1062no—
309C22×C4—1061no—
310C22×C4C2×C22⋊D5312yes—
311C22×C4C2×C22⋊D5311yescharacteristic
312C22×C4C2×C22⋊D5312yes—
313C22×C4C2×C22⋊D5311yescharacteristic
314C22×C4C23×D5311yescharacteristic
315Q16—21216no—
316Q16—2128no—
317C2×C8—534no—
318C2×C8—1062no—
319C2×C8—1062no—
320C2×C8—1062no—
321C2×C8—1064no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—532no—
329C4⋊C4—1068no—
330C4⋊C4—1068no—
331C4⋊C4—1068no—
332C4⋊C4—1064no—
333C4⋊C4—1064no—
334C4⋊C4—1064no—
335C4⋊C4—1061no—
336C42—22no—
337C42—22no—
338C42—42no—
339C42—1068no—
340C42—1064no—
341C42D4×D5312yes—
342C42D4×D5312yes—
343C23C22×C22⋊D5311yescharacteristic
344Q8—28no—
345Q8—44no—
346Q8—42no—
347Q8—42no—
348Q8—1068no—
349Q8—1062no—
350Q8—1062no—
351Q8—2122no—
352Q8—2122no—
353Q8—2122no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—1062no—
365C2×C4—1064no—
366C2×C4—1062no—
367C2×C4—2122no—
368C2×C4—2124no—
369C2×C4—2122no—
370C2×C4#1313318512yes—
371C2×C4#1313318512yes—
372C2×C4C2×#1171992514yes—
373C2×C4C2×#1171992512yes—
374C2×C4#1313324712yes—
375C2×C4C2×D4×D5311yescharacteristic
376C2×C4C2×D4×D5311yescharacteristic
377C2×C4#41×D5312yes—
378C2×C4#1313326312yes—
379C8—1064no—
380C8—2122no—
381C22C2×#1313318511yescharacteristic
382C22#1681129212yes—
383C22C22×#1171992511yescharacteristic
384C22C2×#1313324711yescharacteristic
385C22#1681143812yes—
386C4—24no—
387C4—22no—
388C4—42no—
389C4—82no—
390C4—82no—
391C4—2124no—
392C4—2122no—
393C4—2122no—
394C4#1681077814yes—
395C4#1681117211yescharacteristic
396C4#1681119111yescharacteristic
397C4#1681126612yes—
398C2#2027833711yescharacteristic
399C2C2×#1681077811yescharacteristic
400C2#2028400811yescharacteristic
401C2#2028771812yes—
402C2#2028774612yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2568C2
socle, prime radicalC23×C53C22×D4
hypercenter#2568D53
hypocenterC53#25634
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
94. C2×Q8×C53Q16semidirect
238. C53#25634semidirect
304. C2×Q8#11719925semidirect

Maximal subgroups

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

subgroupindex
C2×#168113522
C2×#168110022
C2×#168108302
#202758142
#202756522
#202756382
#202750592
C4×#131331402
C53×#25682
#2563453