Finite Group Databasebeta

← #27301456

Subgroups — #27301456

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C16×D53C211yesmaximal, characteristic
2C2×#16810861C211yesmaximal, characteristic
3C2×#16810427C211yesmaximal, characteristic, complemented
4#20274741C211yesmaximal, characteristic
5#20272807C211yesmaximal, characteristic, complemented
6#20272806C211yesmaximal, characteristic, complemented
7C2×C53×#358C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×#13133193—21no—
10C22×#11719908C2211yescharacteristic
11C2×C16×D53—22no—
12C2×C16×D53—22no—
13C2×C16×D53—22no—
14#16810861—22no—
15#16810861C414yescomplemented
16C2×#13133151C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133141—21no—
19#16810480—21no—
20#16810427—22no—
21#16810385—21no—
22C22×Q16×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C53×#358—22no—
26C22×C4×D53D411yescharacteristic
27C22×#88322D411yescharacteristic
28C2×C8×D53—21no—
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53C2×C412yes—
33C2×C8×D53D412yes—
34#13133193—24no—
35#13133193—42no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×#11719919—41no—
39C2×#11719921—41no—
40C2×#11719921—41no—
41C2×#11719908—21no—
42C2×#11719908—21no—
43C2×#11719908D412yes—
44C16×D53—28no—
45#13133151—21no—
46#13133151—21no—
47#13133151—21no—
48#13133151C2×C412yes—
49#13133141—24no—
50#13133124—41no—
51C22×Q8×C53—21no—
52C2×Q16×C53—21no—
53C2×Q16×C53—21no—
54C2×Q16×C53—21no—
55C2×Q16×C53—21no—
56C2×Q16×C53—21no—
57C2×Q16×C53C2×C412yes—
58C22×C8×C53C2311yescharacteristic
59C2×C4⋊C4×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53D412yes—
63#107×C53—21no—
64#107×C53—21no—
65C23×D53D811yescharacteristic
66C2×Q8×C53—22no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53D811yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53C22⋊C412yes—
83C2×C4×D53SD1612yes—
84Q16×C53—22no—
85Q16×C53—22no—
86Q16×C53—41no—
87Q16×C53—41no—
88C2×C8×C53C22×C411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96#11719921—44no—
97#11719921—82no—
98C4⋊Dic53—22no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102#11719919—44no—
103C2×#88322—21no—
104C2×#88322—21no—
105C2×#88322C22⋊C412yes—
106C8×D53—22no—
107C8×D53—22no—
108C8×D53—22no—
109C8×D53—22no—
110C16×C53—22no—
111#11719908—22no—
112C23×C53C2×D811yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53#10212yes—
118C22×D53SD3212yes—
119C22×D53Q3212yes—
120Q8×C53—42no—
121Q8×C53—42no—
122Q8×C53—81no—
123Q8×C53—81no—
124C2×C4×C53—42no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53C2×SD1611yescharacteristic
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—42no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144C2×Dic53#10212yes—
145C4×D53—22no—
146C4×D53—22no—
147C4×D53—22no—
148C4×D53—22no—
149C8×C53C4×D411yescharacteristic
150C8×C53C4×D411yescharacteristic
151C8×C53#12011yescharacteristic
152C8×C53#12311yescharacteristic
153#88322—22no—
154#13060—531nomaximal, Sylow
155C22×C53C2×#10211yescharacteristic
156C22×C53#46611yescharacteristic
157C22×C53#46611yescharacteristic
158C22×C53C2×SD3211yescharacteristic
159C22×C53C2×Q3211yescharacteristic
160C22×C53#50811yescharacteristic
161C22×C53#50811yescharacteristic
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—22no—
168C2×D53#35814yescomplemented
169C4×C53—82no—
170C4×C53—161no—
171C4×C53#38611yescharacteristic
172C4×C53#41611yescharacteristic
173C4×C53#46511yescharacteristic
174C4×C53#46511yescharacteristic
175Dic53—22no—
176Dic53—82no—
177Dic53—161no—
178C23×C16—531no—
179C2×#467—531no—
180C2×#358—531no—
181C2×#358D5311yescharacteristic, complemented
182#1808—531no—
183#1380—531no—
184#1380—531no—
185C2×C53#189311yescharacteristic
186C2×C53C2×#35811yescharacteristic
187C2×C53#213811yescharacteristic
188C2×C53#221611yescharacteristic
189C2×C53#221611yescharacteristic
190C2×C53#221711yescharacteristic
191C2×C53#221711yescharacteristic
192D53—24no—
193C22×Q16C2×D5311yescharacteristic
194C23×C8—531no—
195C2×C22⋊Q8—1061no—
196C22×C16—531no—
197C22×C16—1062no—
198C22×C16—1062no—
199C22×C16—1062no—
200C22×C16C2×D5311yescharacteristic
201#467—534no—
202#467—1062no—
203C2×#107—531no—
204C2×#107—531no—
205C2×#107C2×D5311yescharacteristic
206C2×#103—1061no—
207#380—1061no—
208#358—22no—
209#358—1062no—
210#340—1061no—
211C53#1306011yescharacteristic, complemented, Sylow
212C22×Q8—21no—
213C23×C4—531no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D5312yes—
220C22×C8—531no—
221C22×C8—532no—
222C22×C8—532no—
223C22×C8—1061no—
224C22×C8—1061no—
225C22×C8—1061no—
226C22×C8—1061no—
227C22×C8C22×D5311yescharacteristic
228C22⋊Q8—1064no—
229C22⋊Q8—2122no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—2121no—
234C2×C22⋊C4—2121no—
235C2×C22⋊C4—2121no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—532no—
239C2×C16—1068no—
240C2×C16—1061no—
241C2×C16—1061no—
242C2×C16C22⋊D5312yes—
243#107—21no—
244#107—21no—
245#107—532no—
246#107—1061no—
247#107—1061no—
248#107—1061no—
249#103—1064no—
250#95—2121no—
251C24—531no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—531no—
261C22×C4—532no—
262C22×C4—532no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4D4×D5311yescharacteristic
270Q16—22no—
271Q16—22no—
272Q16—41no—
273Q16—41no—
274C2×C8—532no—
275C2×C8—1061no—
276C2×C8—1061no—
277C2×C8—1062no—
278C2×C8—1062no—
279C2×C8—1062no—
280C2×C8—1062no—
281C2×C8C2×C4×D5311yescharacteristic
282C2×C8#1171994511yescharacteristic
283C2×C8D4×D5311yescharacteristic
284C2×C8D4×D5311yescharacteristic
285C2×C8#1171994711yescharacteristic
286C2×C8C2×C22⋊D5311yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—1062no—
290C4⋊C4—2124no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C22⋊C4—4242no—
295C22⋊C4—2124no—
296C16—22no—
297C16—1062no—
298C23—532no—
299C23—532no—
300C23—532no—
301C23—1061no—
302C23—1061no—
303C23—1061no—
304C23—1061no—
305C23D8×D5311yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—81no—
309Q8—81no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—532no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—4241no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—2122no—
329C2×C4C22⋊C4×D5311yescharacteristic
330C2×C4#1313318811yescharacteristic
331C2×C4#1313319011yescharacteristic
332C2×C4SD16×D5311yescharacteristic
333C2×C4#1313323611yescharacteristic
334C2×C4#1313323811yescharacteristic
335C8—1062no—
336C8#1313317711yescharacteristic
337C8C4×C22⋊D5311yescharacteristic
338C8#1313318511yescharacteristic
339C8#1313319911yescharacteristic
340C22—534no—
341C22—1062no—
342C22—1062no—
343C22—1062no—
344C22—1062no—
345C22—1062no—
346C22—1062no—
347C22#102×D5311yescharacteristic
348C22#1681085611yescharacteristic
349C22#1681085811yescharacteristic
350C22SD32×D5311yescharacteristic
351C22Q32×D5311yescharacteristic
352C22#1681105411yescharacteristic
353C22#1681105611yescharacteristic
354C4—82no—
355C4—161no—
356C4—8481no—
357C4—1062no—
358C4—4242no—
359C4#1681049911yescharacteristic
360C4#1681061611yescharacteristic
361C4#1681084911yescharacteristic
362C4#1681085111yescharacteristic
363C2—1064no—
364C2#2027514011yescharacteristic
365C2#358×D5311yescharacteristic
366C2#2027642111yescharacteristic
367C2#2027680311yescharacteristic
368C2#2027680511yescharacteristic
369C2#2027680811yescharacteristic
370C2#2027681411yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#358C2
socleC23×C53C2×D8
prime radicalC23×D53D8
cosocleC22×C8×C53C23
hypercenterC2×#358D53
hypocenterC53#13060
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
3. C2×#16810427C2semidirect
5. #20272807C2semidirect
6. #20272806C2semidirect
7. C2×C53×#358C2semidirect
15. #16810861C4semidirect
168. C2×D53#358semidirect
181. C2×#358D53semidirect
211. C53#13060semidirect

Maximal subgroups

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

subgroupindex
C22×C16×D532
C2×#168108612
C2×#168104272
#202747412
#202728072
#202728062
C2×C53×#3582
#1306053